1.6. w-local rings
Definition1.6.1
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 9
Associated Lean declarations
-
IsWLocalRing[complete]
A ring A is w-local if \Spec(A) is w-local.
(Bhatt–Scholze, Definition 2.2.1(i))
Lean code for Definition1.6.1●1 definition
Associated Lean declarations
-
IsWLocalRing[complete]
Associated Lean declarations
-
IsWLocalRing[complete]
-
classdefined in Proetale/Algebra/WLocal.leancomplete
class IsWLocalRing.{u_1} (R : Type u_1) [CommSemiring R] : Prop
class IsWLocalRing.{u_1} (R : Type u_1) [CommSemiring R] : Prop
A ring is w-local if it has a w-local prime spectrum.
Methods
wLocalSpace_primeSepectrum : WLocalSpace (PrimeSpectrum R)
Definition1.6.2
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 3
Used by 3
Lean status
- No associated Lean code or declarations.
A ring map f \colon A \to B between w-local rings is w-local if the induced
map of w-local spaces \Spec(f) : \Spec(B) \to \Spec(A) is w-local.
(Bhatt–Scholze, Definition 2.2.1(iii))
Lemma1.6.3
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
-
IsWLocalRing.of_surjective[complete]
If A \to B is a surjective ring map and A is w-local, then B is
w-local.
Lean code for Lemma1.6.3●1 theorem
Associated Lean declarations
-
IsWLocalRing.of_surjective[complete]
Associated Lean declarations
-
IsWLocalRing.of_surjective[complete]
-
theoremdefined in Proetale/Algebra/WLocal.leancomplete
theorem IsWLocalRing.of_surjective.{u_1, u_2} {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (hf : Function.Surjective ⇑f) [IsWLocalRing R] : IsWLocalRing S
theorem IsWLocalRing.of_surjective.{u_1, u_2} {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (hf : Function.Surjective ⇑f) [IsWLocalRing R] : IsWLocalRing S
Proof for Lemma 1.6.3
This is Lemma 1.5.7.
Lemma1.6.4
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Lean status
- No associated Lean code or declarations.
Let A be a w-local ring. Then \Spec (A) is set theoretically the disjoint
union of the spectra of the local rings at closed points.
Proof for Lemma 1.6.4
uses 0
This follows from the fact that every point specializes to a unique closed point and the spectrum of the local ring at a point is the set of all points specializing to it.
Lemma1.6.5
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 2
Associated Lean declarations
Let A \to B be a ring map such that
-
A \to Bidentifies local rings, and -
\Spec (B) \to \Spec (A)is bijective.
Then A \to B is an isomorphism.
Lean code for Lemma1.6.5●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/StalkIso.leancomplete
theorem RingHom.BijectiveOnStalks.bijective_of_bijective.{u, v} {R : Type u} {S : Type v} [CommRing R] [CommRing S] {f : R →+* S} (hf : f.BijectiveOnStalks) (hb : Function.Bijective (PrimeSpectrum.comap f)) : Function.Bijective ⇑f
theorem RingHom.BijectiveOnStalks.bijective_of_bijective.{u, v} {R : Type u} {S : Type v} [CommRing R] [CommRing S] {f : R →+* S} (hf : f.BijectiveOnStalks) (hb : Function.Bijective (PrimeSpectrum.comap f)) : Function.Bijective ⇑f
A ring homomorphism that is bijective on stalks and induces a bijection on prime spectra is itself bijective.
Proof for Lemma 1.6.5
Since A \to B is flat (can be checked on stalks of B), it has going down.
Thus Lemma 1.1.27 shows for any prime
\q \subset B lying over \p \subset A we have B_{\q} = B_{\p B}. Since
B_{\q} = A_{\p} by assumption, we see that A_{\p} = B_{\p B} for all
primes \p of A. Thus A = B since being an isomorphism can be checked
locally on A.
Lemma1.6.6
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 3
Associated Lean declarations
-
RingHom.IsWLocal.bijective_of_bijective[complete]
Let A \to B be a w-local ring map between w-local rings such that
-
A \to Bidentifies local rings, and -
\pi_0(\Spec (B)) \to \pi_0(\Spec (A))is bijective.
Then A \to B is an isomorphism.
(Stacks Project, Tag 097E)
Lean code for Lemma1.6.6●1 theorem
Associated Lean declarations
-
RingHom.IsWLocal.bijective_of_bijective[complete]
Associated Lean declarations
-
RingHom.IsWLocal.bijective_of_bijective[complete]
-
theoremdefined in Proetale/Algebra/WLocal.leancomplete
theorem RingHom.IsWLocal.bijective_of_bijective.{u_1, u_2} {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsWLocalRing R] [IsWLocalRing S] {f : R →+* S} (hw : f.IsWLocal) (hs : f.BijectiveOnStalks) (hb : Function.Bijective ⋯.connectedComponentsMap) : Function.Bijective ⇑f
theorem RingHom.IsWLocal.bijective_of_bijective.{u_1, u_2} {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsWLocalRing R] [IsWLocalRing S] {f : R →+* S} (hw : f.IsWLocal) (hs : f.BijectiveOnStalks) (hb : Function.Bijective ⋯.connectedComponentsMap) : Function.Bijective ⇑f
A w-local ring map between w-local rings that is bijective on stalks and bijective on connected components is bijective.
Proof for Lemma 1.6.6
Proof uses 3
By Lemma 1.6.5, it suffices to show
that Y = \Spec (B) \to X = \Spec (A) is bijective. Let X^c \subset X and
Y^c \subset Y be the sets of closed points. By assumption Y^c maps into
X^c and the induced map Y^c \to X^c is a bijection since
Lemma 1.5.16 holds. By
Lemma 1.6.4, we see that both X and Y, as
sets, are disjoint unions of the spectra of the local rings at closed points. As
A \to B identifies local rings, Y \to X is a bijection.
The goal of the remaining part of this section is to show that for every ring
A, there exists a faithfully flat, ind-Zariski A-algebra A_{w} with
A_w w-local. Let A be a ring.
Definition1.6.7
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 4
Associated Lean declarations
-
WLocalization.Generalization.submonoid[complete] -
WLocalization.Generalization[complete] -
WLocalization.Generalization.ideal[complete]
Let f \in A be an element and I \subseteq A an ideal. Let
S_{f, I} \subseteq A be the multiplicative subset of elements which map to
invertible elements of (A/I)_{f}. We define \tildering{A}{I}{f} to be the
A-algebra S_{f, I}^{-1}A and
\tildeideal{I}{f} \subseteq \tildering{A}{I}{f} to be the kernel of the
induced map \tildering{A}{I}{f} \to (A / I)_{f}.
Lean code for Definition1.6.7●3 definitions
Associated Lean declarations
-
WLocalization.Generalization.submonoid[complete]
-
WLocalization.Generalization[complete]
-
WLocalization.Generalization.ideal[complete]
Associated Lean declarations
-
WLocalization.Generalization.submonoid[complete] -
WLocalization.Generalization[complete] -
WLocalization.Generalization.ideal[complete]
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.Generalization.submonoid.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Submonoid A
def WLocalization.Generalization.submonoid.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Submonoid A
The submonoid of `A` consisting of elements that become invertible in `(A / I)_f`.
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.Generalization.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Type u_1
def WLocalization.Generalization.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Type u_1
The localization of `A` at all elements invertible in `(A / I)_f`.
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.Generalization.ideal.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Ideal (WLocalization.Generalization f I)
def WLocalization.Generalization.ideal.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Ideal (WLocalization.Generalization f I)
The kernel of the canonical map from the generalization at `(f, I)` to `(A ⧸ I)_f`. This ideal defines a closed subset of the prime spectrum of the generalization at `(f, I)` that maps homeomorphically to `D(f) ∩ V(I)`.
If Z \subseteq \spec{A} is a locally closed subscheme of the form
D(f) \cap V(I), we also write A_{\widetilde{Z}} instead of
\tildering{A}{I}{f}. In the following, when we use the notation
A_{\widetilde{Z}} there is always a canonical choice of I and f for
Z. In general this notation is justified by the fact that, up to isomorphism,
the ring A_{\widetilde{Z}} does not depend on the choice of f and I
(Stacks Project, Tag 096V).
Lemma1.6.8
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Used by 3
Associated Lean declarations
-
WLocalization.Generalization.range_algebraMap_generalization[complete] -
WLocalization.Generalization.bijOn_algebraMap_generalization_specComap_zeroLocus_ideal[complete] -
WLocalization.Generalization.submonoid_le[complete] -
WLocalization.Generalization.map[complete] -
WLocalization.Generalization.exists_specializes_zeroLocus_ideal[complete]
Let f \in A be an element and I \subseteq A an ideal.
-
The map
\spec{\tildering{A}{I}{f}} \to \spec{A}induces a homeomorphism onto the generalization ofD(f) \cap V(I). -
The closed subscheme
V(\tildeideal{I}{f})induces an isomorphism ontoD(f) \cap V(I). -
If
A'is a ring with an elementf' \in A'and idealI' \subseteq A'andA \to A'is a ring map that mapsD(f') \cap V(I')intoD(f) \cap V(I), then there is a uniqueA-algebra map\tildering{A}{I}{f} \to \tildering{A'}{I'}{f'}making the obvious diagram withA \to A'commute.
In particular, every point in \spec{\tildering{A}{I}{f}} specializes to a
point in V(\tildeideal{I}{f}).
(Stacks Project, Tag 096V)
Lean code for Lemma1.6.8●5 declarations
Associated Lean declarations
-
WLocalization.Generalization.range_algebraMap_generalization[complete]
-
WLocalization.Generalization.bijOn_algebraMap_generalization_specComap_zeroLocus_ideal[complete]
-
WLocalization.Generalization.submonoid_le[complete]
-
WLocalization.Generalization.map[complete]
-
WLocalization.Generalization.exists_specializes_zeroLocus_ideal[complete]
Associated Lean declarations
-
WLocalization.Generalization.range_algebraMap_generalization[complete] -
WLocalization.Generalization.bijOn_algebraMap_generalization_specComap_zeroLocus_ideal[complete] -
WLocalization.Generalization.submonoid_le[complete] -
WLocalization.Generalization.map[complete] -
WLocalization.Generalization.exists_specializes_zeroLocus_ideal[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.Generalization.range_algebraMap_generalization.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Set.range (PrimeSpectrum.comap (algebraMap A (WLocalization.Generalization f I))) = generalizationHull (WLocalization.Generalization.locClosedSubset f I)
theorem WLocalization.Generalization.range_algebraMap_generalization.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Set.range (PrimeSpectrum.comap (algebraMap A (WLocalization.Generalization f I))) = generalizationHull (WLocalization.Generalization.locClosedSubset f I)
The image of `Spec (Generalization f I)` in `Spec A` is equal to the generalization hull of `D(f) ∩ V(I)`.
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.Generalization.bijOn_algebraMap_generalization_specComap_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Set.BijOn (PrimeSpectrum.comap (algebraMap A (WLocalization.Generalization f I))) (PrimeSpectrum.zeroLocus ↑(WLocalization.Generalization.ideal f I)) (WLocalization.Generalization.locClosedSubset f I)
theorem WLocalization.Generalization.bijOn_algebraMap_generalization_specComap_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Set.BijOn (PrimeSpectrum.comap (algebraMap A (WLocalization.Generalization f I))) (PrimeSpectrum.zeroLocus ↑(WLocalization.Generalization.ideal f I)) (WLocalization.Generalization.locClosedSubset f I)
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.Generalization.submonoid_le.{u_1} {A : Type u_1} [CommRing A] {f f' : A} {I I' : Ideal A} (h : WLocalization.Generalization.locClosedSubset f' I' ⊆ WLocalization.Generalization.locClosedSubset f I) : WLocalization.Generalization.submonoid f I ≤ WLocalization.Generalization.submonoid f' I'
theorem WLocalization.Generalization.submonoid_le.{u_1} {A : Type u_1} [CommRing A] {f f' : A} {I I' : Ideal A} (h : WLocalization.Generalization.locClosedSubset f' I' ⊆ WLocalization.Generalization.locClosedSubset f I) : WLocalization.Generalization.submonoid f I ≤ WLocalization.Generalization.submonoid f' I'
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.Generalization.map.{u_1} {A : Type u_1} [CommRing A] {f f' : A} {I I' : Ideal A} (h : WLocalization.Generalization.locClosedSubset f' I' ⊆ WLocalization.Generalization.locClosedSubset f I) : WLocalization.Generalization f I →ₐ[A] WLocalization.Generalization f' I'
def WLocalization.Generalization.map.{u_1} {A : Type u_1} [CommRing A] {f f' : A} {I I' : Ideal A} (h : WLocalization.Generalization.locClosedSubset f' I' ⊆ WLocalization.Generalization.locClosedSubset f I) : WLocalization.Generalization f I →ₐ[A] WLocalization.Generalization f' I'
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.Generalization.exists_specializes_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] {f : A} (I : Ideal A) (x : PrimeSpectrum (WLocalization.Generalization f I)) : ∃ y ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.Generalization.ideal f I), x ⤳ y
theorem WLocalization.Generalization.exists_specializes_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] {f : A} (I : Ideal A) (x : PrimeSpectrum (WLocalization.Generalization f I)) : ∃ y ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.Generalization.ideal f I), x ⤳ y
Proof for Lemma 1.6.8
uses 0
Let S = S_{f, I}, Z = D(f) \cap V(I) and J = \tildeideal{I}{f}.
Note that Z is homeomorphic to \spec{(A/I)_f}. Since the map
\spec{S^{-1}A} \to \spec{A} is a homeomorphism onto its image, to show (1) it
suffices to show that the image is the set of points of \spec{A} specializing
to D(f) \cap V(I). Let \mathfrak{p} be a prime of A specializing to a
\mathfrak{q} in Z, so I \subseteq \mathfrak{q} and
f \not\in \mathfrak{q}. In particular, the image of every element of
\mathfrak{p} in (A / I)_f is contained in \mathfrak{q} and is therefore
not invertible. Conversely, let \mathfrak{p} be a prime of A which does
not specialize to a point in Z, hence the image of \mathfrak{p} in
(A/I)_f contains the unit element. So there exists g \in \mathfrak{p} that
becomes invertible in (A / I)_f, so g \in S and \mathfrak{p} is not in
the image of \spec{S^{-1} A}.
By construction, the induced map S^{-1}A \to (A / I)_f is surjective and
hence induces an isomorphism S^{-1}A / J \to (A / I)_f. This shows (2).
Let g \in A. Suppose the image of g in (A' / I')_{f'} is not
invertible. Then there exists a prime ideal \mathfrak{p} of
(A' / I')_{f'} that contains the image of g. Hence g is in the preimage
of \mathfrak{p} under A \to A', which lies in Z by assumption. Thus
g is not invertible in (A / I)_f, which shows g \not\in S.
In view of Lemma 1.6.8, given
Z = D(f) \cap V(I) we may view Z as a closed subscheme of
\spec{A_{\widetilde{Z}}} defined by the ideal V(\tildeideal{I}{f}).
Lemma1.6.9
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 3
Associated Lean declarations
-
WLocalization.Generalization.indZariski[complete]
Let f \in A be an element and I \subseteq A an ideal. Then
\tildering{A}{I}{f} is ind-Zariski over A.
Lean code for Lemma1.6.9●1 theorem
Associated Lean declarations
-
WLocalization.Generalization.indZariski[complete]
Associated Lean declarations
-
WLocalization.Generalization.indZariski[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.Generalization.indZariski.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Algebra.IndZariski A (WLocalization.Generalization f I)
theorem WLocalization.Generalization.indZariski.{u_1} {A : Type u_1} [CommRing A] (f : A) (I : Ideal A) : Algebra.IndZariski A (WLocalization.Generalization f I)
Proof for Lemma 1.6.9
\tildering{A}{I}{f} = S^{-1} A for some multiplicative subset S of A.
Definition1.6.10
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 3
Associated Lean declarations
-
WLocalization.stratum[complete]
Let E, F \subseteq A be subsets. We define
Z(E, F) = \bigcap_{f \in E} D(f) \cap \bigcap_{f \in F} V(f).
Lean code for Definition1.6.10●1 definition
Associated Lean declarations
-
WLocalization.stratum[complete]
Associated Lean declarations
-
WLocalization.stratum[complete]
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.stratum.{u_1} {A : Type u_1} [CommRing A] (E F : Finset A) : Set (PrimeSpectrum A)
def WLocalization.stratum.{u_1} {A : Type u_1} [CommRing A] (E F : Finset A) : Set (PrimeSpectrum A)
The single stratum `Z(E, F)` in `Spec A`.
Lemma1.6.11
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
Let A be a ring, E, F \subseteq A finite subsets. Then
Z(E, F) = D\left( \prod_{f \in E} f \right) \cap V\left( \sum_{f \in F} fA \right).
Lean code for Lemma1.6.11●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.stratum_eq_basicOpen_inter_zeroLocus.{u_1} {A : Type u_1} [CommRing A] (E F : Finset A) : WLocalization.stratum E F = ↑(PrimeSpectrum.basicOpen (∏ f ∈ E, f)) ∩ PrimeSpectrum.zeroLocus ↑(Ideal.span ↑F)
theorem WLocalization.stratum_eq_basicOpen_inter_zeroLocus.{u_1} {A : Type u_1} [CommRing A] (E F : Finset A) : WLocalization.stratum E F = ↑(PrimeSpectrum.basicOpen (∏ f ∈ E, f)) ∩ PrimeSpectrum.zeroLocus ↑(Ideal.span ↑F)
Proof for Lemma 1.6.11
uses 0
This is immediate from the definitions of D and V.
Lemma1.6.12
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 2
Associated Lean declarations
-
WLocalization.stratum_anti[complete]
Let E_1, F_1, E_2, F_2 \subseteq A be subsets such that E_1 \subseteq E_2
and F_1 \subseteq F_2. Then Z(E_2, F_2) \subseteq Z(E_1, F_1).
Lean code for Lemma1.6.12●1 theorem
Associated Lean declarations
-
WLocalization.stratum_anti[complete]
Associated Lean declarations
-
WLocalization.stratum_anti[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.stratum_anti.{u_1} {A : Type u_1} [CommRing A] {E F E' F' : Finset A} (hEE' : E ⊆ E') (hFF' : F ⊆ F') : WLocalization.stratum E' F' ⊆ WLocalization.stratum E F
theorem WLocalization.stratum_anti.{u_1} {A : Type u_1} [CommRing A] {E F E' F' : Finset A} (hEE' : E ⊆ E') (hFF' : F ⊆ F') : WLocalization.stratum E' F' ⊆ WLocalization.stratum E F
Proof for Lemma 1.6.12
uses 0
This is immediate from the definition of Z(-, -).
Lemma1.6.13
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
Let A be a ring, E \subseteq A a finite subset. Then
\spec{A} = \coprod_{E = E' \coprod E''} Z(E', E'')
set theoretically.
Lean code for Lemma1.6.13●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.Stratification.Index.iUnion_stratum.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : ⋃ i, WLocalization.stratum i.left i.right = Set.univ
theorem WLocalization.Stratification.Index.iUnion_stratum.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : ⋃ i, WLocalization.stratum i.left i.right = Set.univ
Proof for Lemma 1.6.13
uses 0
Let x \in \spec{A} and E = E' \coprod E'' a disjoint union decomposition.
Then x \in Z(E', E'') if and only if f(x) \neq 0 for all f \in E' and
f(x) = 0 for all f \in E''. Hence x is contained in
Z(\{ f \in E \mid f(x) \neq 0\}, \{f \in E \mid f(x) = 0\}),
and this is the only set of this form in which x is contained.
Definition1.6.14
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 3
Used by 4
Associated Lean declarations
-
WLocalization.ProdStrata[complete] -
WLocalization.ProdStrata.ideal[complete]
Let A be a ring and E \subseteq A a finite subset. Let
A_{E} = \prod_{E = E' \coprod E''} A_{\widetilde{Z(E', E'')}}.
Further, we set I_E \subseteq A_E to be the product of the ideals defining
Z(E', E'') in \spec{A_{\widetilde{Z(E', E'')}}}.
Lean code for Definition1.6.14●2 definitions
Associated Lean declarations
-
WLocalization.ProdStrata[complete]
-
WLocalization.ProdStrata.ideal[complete]
Associated Lean declarations
-
WLocalization.ProdStrata[complete] -
WLocalization.ProdStrata.ideal[complete]
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.ProdStrata.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : Type u_1
def WLocalization.ProdStrata.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : Type u_1
The product of the generalizations of `Z(E', E'')` indexed by all disjoint union decompositions `E = E' ⨿ E''`.
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.ProdStrata.ideal.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : Ideal (WLocalization.ProdStrata E)
def WLocalization.ProdStrata.ideal.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : Ideal (WLocalization.ProdStrata E)
The ideal of the stratification product, given by the product of the ideals defining `Z(E', E'')` in its generalization.
Note that Definition 1.6.14 is well-defined, because
for finite subsets E', E'' of A, the locally closed subset Z(E', E'')
is always of the right form by Lemma 1.6.11.
Lemma1.6.15
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
Let E \subseteq A be a finite subset. Then
V(I_E) = \coprod_{E = E' \coprod E''} Z(E', E''). In particular, the map
\spec{A_E} \to \spec{A} induces a bijection V(I_E) \to \spec{A}.
Lean code for Lemma1.6.15●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.ProdStrata.bijOn_algebraMap_specComap_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : Set.BijOn (PrimeSpectrum.comap (algebraMap A (WLocalization.ProdStrata E))) (PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E)) Set.univ
theorem WLocalization.ProdStrata.bijOn_algebraMap_specComap_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] (E : Finset A) : Set.BijOn (PrimeSpectrum.comap (algebraMap A (WLocalization.ProdStrata E))) (PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E)) Set.univ
Proof for Lemma 1.6.15
By definition I_E is the product of the ideals defining Z(E', E'') in
\spec{A_{\widetilde{Z(E', E'')}}}. Hence the first claim. The second follows
from the first, because
\spec{A} = \coprod_{E = E' \coprod E''} Z(E', E'') by
Lemma 1.6.13.
Lemma1.6.16
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
Let E \subseteq A be a finite subset. Then every point of \spec{A_E}
specializes to a point in V(I_E). In particular, V(I_E) contains all
closed points of \spec{A_E}.
Lean code for Lemma1.6.16●2 theorems
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.ProdStrata.exists_specializes_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] {E : Finset A} (x : PrimeSpectrum (WLocalization.ProdStrata E)) : ∃ y ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E), x ⤳ y
theorem WLocalization.ProdStrata.exists_specializes_zeroLocus_ideal.{u_1} {A : Type u_1} [CommRing A] {E : Finset A} (x : PrimeSpectrum (WLocalization.ProdStrata E)) : ∃ y ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E), x ⤳ y
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.ProdStrata.mem_zeroLocus_ideal_of_isClosed.{u_1} {A : Type u_1} [CommRing A] {E : Finset A} {x : PrimeSpectrum (WLocalization.ProdStrata E)} (hx : IsClosed {x}) : x ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E)
theorem WLocalization.ProdStrata.mem_zeroLocus_ideal_of_isClosed.{u_1} {A : Type u_1} [CommRing A] {E : Finset A} {x : PrimeSpectrum (WLocalization.ProdStrata E)} (hx : IsClosed {x}) : x ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E)
Proof for Lemma 1.6.16
Let x \in \spec{A_E}. Then x is in \spec{A_{\widetilde{Z(E', E'')}}}
for some E = E' \coprod E''. By
Lemma 1.6.8, x specializes to a point in
Z(E', E'') \subseteq V(I_E). If x is closed, then x specializes to
itself, so x \in V(I_E).
Let A be a ring. Given finite subsets E_1, E_2 \subseteq A with
E_1 \subseteq E_2, there is a canonical transition map A_{E_1} \to A_{E_2}
of A-algebras: For a disjoint union decomposition
E_2 = E_2' \coprod E_2'', we set E_1' = E_1 \cap E_2' and
E_1'' = E_1 \cap E_2''. By Lemma 1.6.12, we obtain
Z(E_2', E_2'') \subseteq Z(E_1', E_1'') and hence by
Lemma 1.6.8 a unique A-algebra map
A_{\widetilde{Z(E_1', E_1'')}} \to A_{\widetilde{Z(E_2', E_2'')}}.
Lemma1.6.17
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Associated Lean declarations
Let E_1, E_2 \subseteq A be finite subsets with E_1 \subseteq E_2. The
induced map \spec{A_{E_2}} \to \spec{A_{E_1}} maps V(I_{E_2}) into
V(I_{E_1}).
Lean code for Lemma1.6.17●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.ProdStrata.mapsTo_map_specComap.{u_1} {A : Type u_1} [CommRing A] {E F : Finset A} (h : E ⊆ F) : Set.MapsTo (PrimeSpectrum.comap (WLocalization.ProdStrata.map h).toRingHom) (PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal F)) (PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E))
theorem WLocalization.ProdStrata.mapsTo_map_specComap.{u_1} {A : Type u_1} [CommRing A] {E F : Finset A} (h : E ⊆ F) : Set.MapsTo (PrimeSpectrum.comap (WLocalization.ProdStrata.map h).toRingHom) (PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal F)) (PrimeSpectrum.zeroLocus ↑(WLocalization.ProdStrata.ideal E))
Proof for Lemma 1.6.17
uses 0
This holds because for every decomposition E_2 = E_2' \coprod E_2'' and
E_1' = E_1 \cap E_2' and E_1'' = E_1 \cap E_2'', the induced map
\spec{A_{\widetilde{Z(E_2', E_2'')}}} \to \spec{A_{\widetilde{Z(E_1', E_1'')}}}
maps Z(E_2', E_2'') into Z(E_1', E_1'').
Definition1.6.18
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 3
Used by 6
Associated Lean declarations
-
WLocalization.commAlgCat[complete] -
WLocalization[complete] -
WLocalization.ideal[complete]
Let I(A) be the partially ordered set of all finite subsets of A. We
define the A-algebra A_{w} as
A_w = \colim_{E \in I(A)} A_E.
We denote by I_w \subseteq A_w the colimit of the ideals I_E, i.e., the
union of the images of the I_E in A_w.
Lean code for Definition1.6.18●3 definitions
Associated Lean declarations
-
WLocalization.commAlgCat[complete]
-
WLocalization[complete]
-
WLocalization.ideal[complete]
Associated Lean declarations
-
WLocalization.commAlgCat[complete] -
WLocalization[complete] -
WLocalization.ideal[complete]
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.commAlgCat.{u_1} (A : Type u_1) [CommRing A] : CommAlgCat A
def WLocalization.commAlgCat.{u_1} (A : Type u_1) [CommRing A] : CommAlgCat A
The w-localization of a ring as an object of `CommAlgCat A` is the colimit over the rings `A_E`.
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.{u} (A : Type u) [CommRing A] : Type u
def WLocalization.{u} (A : Type u) [CommRing A] : Type u
The w-localization of a ring.
-
defdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
def WLocalization.ideal.{u} (A : Type u) [CommRing A] : Ideal (WLocalization A)
def WLocalization.ideal.{u} (A : Type u) [CommRing A] : Ideal (WLocalization A)
By the following lemma, we see that V(I_w) is the inverse limit of the closed
subsets V(I_E).
Lemma1.6.19
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
-
zeroLocus_eq_iInter_specComap_preimage[complete] -
specComap_preimage_zeroLocus_subset[complete]
Let (A_i, I_i)_{i \in I} be a filtered system of rings with ideals. Set
A = \colim_{i \in I} A_i and
I = \colim_{i \in I} I_i = \colim_{i \in I} I_i A. Then
V(I) = \lim_{i \in I} V(I_i) = \bigcap_{i \in I} \text{preimage of } V(I_i) \text{ in } \spec{A}.
Lean code for Lemma1.6.19●2 theorems
Associated Lean declarations
-
zeroLocus_eq_iInter_specComap_preimage[complete]
-
specComap_preimage_zeroLocus_subset[complete]
Associated Lean declarations
-
zeroLocus_eq_iInter_specComap_preimage[complete] -
specComap_preimage_zeroLocus_subset[complete]
-
theoremdefined in Proetale/Algebra/Preliminaries/Ideal.leancomplete
theorem zeroLocus_eq_iInter_specComap_preimage.{u_1, u_2, u_3} {ι : Type u_1} [CategoryTheory.Category.{u_2, u_1} ι] {F : CategoryTheory.Functor ι CommRingCat} {C : CategoryTheory.Limits.Cocone F} {Iι : (i : ι) → Ideal ↑(F.obj i)} (I : Ideal ↑C.pt) (h : I = ⨆ i, Ideal.map (CommRingCat.Hom.hom (C.ι.app i)) (Iι i)) : PrimeSpectrum.zeroLocus ↑I = ⋂ i, PrimeSpectrum.comap (CommRingCat.Hom.hom (C.ι.app i)) ⁻¹' PrimeSpectrum.zeroLocus ↑(Iι i)
theorem zeroLocus_eq_iInter_specComap_preimage.{u_1, u_2, u_3} {ι : Type u_1} [CategoryTheory.Category.{u_2, u_1} ι] {F : CategoryTheory.Functor ι CommRingCat} {C : CategoryTheory.Limits.Cocone F} {Iι : (i : ι) → Ideal ↑(F.obj i)} (I : Ideal ↑C.pt) (h : I = ⨆ i, Ideal.map (CommRingCat.Hom.hom (C.ι.app i)) (Iι i)) : PrimeSpectrum.zeroLocus ↑I = ⋂ i, PrimeSpectrum.comap (CommRingCat.Hom.hom (C.ι.app i)) ⁻¹' PrimeSpectrum.zeroLocus ↑(Iι i)
-
theoremdefined in Proetale/Algebra/Preliminaries/Ideal.leancomplete
theorem specComap_preimage_zeroLocus_subset.{u_1, u_2, u_3} {ι : Type u_1} [CategoryTheory.Category.{u_2, u_1} ι] {F : CategoryTheory.Functor ι CommRingCat} (C : CategoryTheory.Limits.Cocone F) {i j : ι} (f : i ⟶ j) {Iι : (i : ι) → Ideal ↑(F.obj i)} (h : Iι j ≤ Ideal.map (CommRingCat.Hom.hom (F.map f)) (Iι i)) : PrimeSpectrum.comap (CommRingCat.Hom.hom (C.ι.app i)) ⁻¹' PrimeSpectrum.zeroLocus ↑(Iι i) ⊆ PrimeSpectrum.comap (CommRingCat.Hom.hom (C.ι.app j)) ⁻¹' PrimeSpectrum.zeroLocus ↑(Iι j)
theorem specComap_preimage_zeroLocus_subset.{u_1, u_2, u_3} {ι : Type u_1} [CategoryTheory.Category.{u_2, u_1} ι] {F : CategoryTheory.Functor ι CommRingCat} (C : CategoryTheory.Limits.Cocone F) {i j : ι} (f : i ⟶ j) {Iι : (i : ι) → Ideal ↑(F.obj i)} (h : Iι j ≤ Ideal.map (CommRingCat.Hom.hom (F.map f)) (Iι i)) : PrimeSpectrum.comap (CommRingCat.Hom.hom (C.ι.app i)) ⁻¹' PrimeSpectrum.zeroLocus ↑(Iι i) ⊆ PrimeSpectrum.comap (CommRingCat.Hom.hom (C.ι.app j)) ⁻¹' PrimeSpectrum.zeroLocus ↑(Iι j)
Lemma1.6.20
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 3
Associated Lean declarations
-
WLocalization.indZariski[complete]
The A-algebra A_{w} is ind-Zariski.
Lean code for Lemma1.6.20●1 theorem
Associated Lean declarations
-
WLocalization.indZariski[complete]
Associated Lean declarations
-
WLocalization.indZariski[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.indZariski.{u} (A : Type u) [CommRing A] : Algebra.IndZariski A (WLocalization A)
theorem WLocalization.indZariski.{u} (A : Type u) [CommRing A] : Algebra.IndZariski A (WLocalization A)
Proof for Lemma 1.6.20
Proof uses 2
For E \subseteq A a finite subset, the A-algebra A_E is ind-Zariski as
a finite product of ind-Zariski algebras by
Lemma 1.2.5. Thus A_w is a filtered colimit of
ind-Zariski algebras, hence ind-Zariski by Lemma 1.2.4.
Lemma1.6.21
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 5
Associated Lean declarations
-
WLocalization.bijOn_algebraMap_specComap_zeroLocus_ideal[complete] -
WLocalization.exists_mem_zeroLocus_ideal_specializes[complete] -
WLocalization.zeroLocus_ideal_eq[sorry in proof]
Let A be a ring.
-
The composition
V(I_w) \to \spec{A_w} \to \spec{A}is a bijection. -
Every point of
\spec{A_w}specializes to a unique point ofV(I_w). -
V(I_w)is the set of closed points of\spec{A_w}.
Lean code for Lemma1.6.21●3 theorems, 1 incomplete
Associated Lean declarations
-
WLocalization.bijOn_algebraMap_specComap_zeroLocus_ideal[complete]
-
WLocalization.exists_mem_zeroLocus_ideal_specializes[complete]
-
WLocalization.zeroLocus_ideal_eq[sorry in proof]
Associated Lean declarations
-
WLocalization.bijOn_algebraMap_specComap_zeroLocus_ideal[complete] -
WLocalization.exists_mem_zeroLocus_ideal_specializes[complete] -
WLocalization.zeroLocus_ideal_eq[sorry in proof]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.bijOn_algebraMap_specComap_zeroLocus_ideal.{u} (A : Type u) [CommRing A] : Set.BijOn (PrimeSpectrum.comap (algebraMap A (WLocalization A))) (PrimeSpectrum.zeroLocus ↑(WLocalization.ideal A)) Set.univ
theorem WLocalization.bijOn_algebraMap_specComap_zeroLocus_ideal.{u} (A : Type u) [CommRing A] : Set.BijOn (PrimeSpectrum.comap (algebraMap A (WLocalization A))) (PrimeSpectrum.zeroLocus ↑(WLocalization.ideal A)) Set.univ
The composition `V(ideal A) ↪ Spec (WLocalization A) → Spec A` is a bijection.
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.exists_mem_zeroLocus_ideal_specializes.{u} (A : Type u) [CommRing A] (x : PrimeSpectrum (WLocalization A)) : ∃ y ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.ideal A), x ⤳ y
theorem WLocalization.exists_mem_zeroLocus_ideal_specializes.{u} (A : Type u) [CommRing A] (x : PrimeSpectrum (WLocalization A)) : ∃ y ∈ PrimeSpectrum.zeroLocus ↑(WLocalization.ideal A), x ⤳ y
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancontains sorry
theorem WLocalization.zeroLocus_ideal_eq.{u} (A : Type u) [CommRing A] : PrimeSpectrum.zeroLocus ↑(WLocalization.ideal A) = closedPoints (PrimeSpectrum (WLocalization A))
theorem WLocalization.zeroLocus_ideal_eq.{u} (A : Type u) [CommRing A] : PrimeSpectrum.zeroLocus ↑(WLocalization.ideal A) = closedPoints (PrimeSpectrum (WLocalization A))
Proof for Lemma 1.6.21
Proof uses 5
Since V(I_w) = \lim_{E} V(I_E) (by
Lemma 1.6.19) and
V(I_E) \to \spec{A_E} \to \spec{A} is bijective for all E by
Lemma 1.6.15, the first claim holds.
We first show that every point of \spec{A_w} specializes to a point in
V(I_w). For this let y \in \spec{A_w}. For all E \subseteq A finite,
denote by T_E the closure of the image of y in \spec{A_E}. By
Lemma 1.6.16, the intersection
T_E \cap V(I_E) is non-empty. Since both T_E and V(I_E) are closed, the
intersection is closed and hence quasi-compact. Thus
(\lim_E T_E) \cap V(I_w) = \lim_E (T_E \cap V(I_E)) is non-empty. Applying
Lemma 1.1.8 and
Lemma 1.1.9 to the singleton \{y\}, the subset
\lim_E T_E is the closure of y in \spec{A_w}. (This should be a
cofiltered version of
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed.) Hence
y specializes to a point in V(I_w).
For uniqueness, suppose y \in \spec{A_w} specializes to z, z' \in V(I_w)
with z \neq z'. Denote the images of z, z' in \spec{A} by x, x'.
Because V(I_w) \to \spec{A_w} \to \spec{A} is injective, x and x' are
distinct. Hence we may assume there exists f \in A such that x \in D(f)
and x' \in V(f). Set E = \{f\}. Then
\spec{A_E} = \spec{\tildering{A}{0}{f}} \coprod \spec{\tildering{A}{(f)}{1}}.
Denote the images of y, z, z' in \spec{A_E} by y_E, x_E, x'_E. Since
x_E maps to x \in D(f) and x_E' maps to x' \in V(f), they lie in
different components of the disjoint union decomposition above. But since
\spec{A_w} \to \spec{A_E} is continuous, y_E specializes to both x_E
and x'_E, which is not possible.
Finally, every closed point specializes to itself and is hence contained in
V(I_{w}). Conversely, every point in V(I_{w}) specializes only to itself,
so it is closed.
Corollary1.6.22
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Associated Lean declarations
-
WLocalization.isWLocalRing[sorry in proof]
Let A be a ring. The A-algebra A_w is w-local.
Lean code for Corollary1.6.22●1 theorem, incomplete
Associated Lean declarations
-
WLocalization.isWLocalRing[sorry in proof]
Associated Lean declarations
-
WLocalization.isWLocalRing[sorry in proof]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancontains sorry
theorem WLocalization.isWLocalRing.{u} (A : Type u) [CommRing A] : IsWLocalRing (WLocalization A)
theorem WLocalization.isWLocalRing.{u} (A : Type u) [CommRing A] : IsWLocalRing (WLocalization A)
Proof for Corollary 1.6.22
Immediate from parts (2) and (3) of Lemma 1.6.21.
Lemma1.6.23
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
-
WLocalization.faithfullyFlat[complete]
The A-algebra A_w is faithfully flat.
Lean code for Lemma1.6.23●1 theorem
Associated Lean declarations
-
WLocalization.faithfullyFlat[complete]
Associated Lean declarations
-
WLocalization.faithfullyFlat[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Basic.leancomplete
theorem WLocalization.faithfullyFlat.{u} (A : Type u) [CommRing A] : Module.FaithfullyFlat A (WLocalization A)
theorem WLocalization.faithfullyFlat.{u} (A : Type u) [CommRing A] : Module.FaithfullyFlat A (WLocalization A)
Proof for Lemma 1.6.23
Proof uses 2
By Lemma 1.6.20, A \to A_w is flat. Because
V(I_w) surjects onto \spec{A} by
Lemma 1.6.21, in particular
\spec{A_w} \to \spec{A} is surjective, hence A \to A_w is faithfully flat.
Lemma1.6.24
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
-
Ideal.IsMaximal.of_isAlgebraic[complete]
Let B be an A-algebra and \mathfrak{m} a maximal ideal of A. Let
\mathfrak{q} be a prime ideal of B lying over \mathfrak{m} such that
\kappa(\mathfrak{q}) is algebraic over \kappa(\mathfrak{m}). Then
\mathfrak{q} is maximal.
(Stacks Project, Tag 00GA)
Lean code for Lemma1.6.24●1 theorem
Associated Lean declarations
-
Ideal.IsMaximal.of_isAlgebraic[complete]
Associated Lean declarations
-
Ideal.IsMaximal.of_isAlgebraic[complete]
-
theoremdefined in Proetale/Algebra/StalkAlgebraic.leancomplete
theorem Ideal.IsMaximal.of_isAlgebraic.{u_1, u_2} {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (m : Ideal A) [m.IsMaximal] (q : Ideal B) [q.IsPrime] [q.LiesOver m] [Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [Localization.AtPrime.IsLiesOverAlgebra m q] [Algebra.IsAlgebraic m.ResidueField q.ResidueField] : q.IsMaximal
theorem Ideal.IsMaximal.of_isAlgebraic.{u_1, u_2} {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (m : Ideal A) [m.IsMaximal] (q : Ideal B) [q.IsPrime] [q.LiesOver m] [Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [Localization.AtPrime.IsLiesOverAlgebra m q] [Algebra.IsAlgebraic m.ResidueField q.ResidueField] : q.IsMaximal
If `q` lies over a maximal ideal `m` and the residue field extension is algebraic, `q` is maximal.
Proof for Lemma 1.6.24
uses 0
We have a factorization
\kappa(\mathfrak{m}) = A / \mathfrak{m} \to B / \mathfrak{q} \to \kappa(\mathfrak{q}).
Hence B / \mathfrak{q} is an intermediate subring of an algebraic field
extension and therefore a field.
Lemma1.6.25
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 2
Associated Lean declarations
Let A be w-local and I \subseteq A an ideal. Then
A_{\widetilde{V(I)}} = A_{\widetilde{(I,1)}} is w-local. If V(I) only
contains closed points, then the set of closed points is
V(IA_{\widetilde{V(I)}}).
Lean code for Lemma1.6.25●2 theorems
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem isWLocalRing_generalization_one.{u} {A : Type u} [CommRing A] (I : Ideal A) [IsWLocalRing A] : IsWLocalRing (WLocalization.Generalization 1 I)
theorem isWLocalRing_generalization_one.{u} {A : Type u} [CommRing A] (I : Ideal A) [IsWLocalRing A] : IsWLocalRing (WLocalization.Generalization 1 I)
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem IsWLocalRing.zeroLocus_map_algebraMap_generalization_one_eq.{u} {A : Type u} [CommRing A] {I : Ideal A} [IsWLocalRing A] (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A (WLocalization.Generalization 1 I)) I) = closedPoints (PrimeSpectrum (WLocalization.Generalization 1 I))
theorem IsWLocalRing.zeroLocus_map_algebraMap_generalization_one_eq.{u} {A : Type u} [CommRing A] {I : Ideal A} [IsWLocalRing A] (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A (WLocalization.Generalization 1 I)) I) = closedPoints (PrimeSpectrum (WLocalization.Generalization 1 I))
Proof for Lemma 1.6.25
Proof uses 3
By Lemma 1.6.8,
\spec{A_{\widetilde{V(I)}}} is homeomorphic to the generalization V(I)^g.
Since \spec{A} is w-local and V(I) is closed under specialization, also
V(I)^g is closed under specialization by
Lemma 1.5.5. Hence
\spec{A_{\widetilde{V(I)}}} \cong V(I)^g is w-local by
Lemma 1.5.6.
By part (2) of Lemma 1.6.8, we may identify
V(IA_{\widetilde{V(I)}}) with V(I). Since every point in
\spec{A_{\widetilde{V(I)}}} \cong V(I)^g specializes to a unique point in
V(I), the closed subset V(I) contains all closed points of
\spec{A_{\widetilde{V(I)}}}. Hence, if V(I) only contains closed points,
the closed points are exactly V(I).
Proposition1.6.26
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Lean status
- No associated Lean code or declarations.
Let A \to B be a ring map with B w-local. There exists a unique
factorization A \to A_w \to B such that A_w \to B is w-local.
(Stacks Project, Tag 0977)
Definition1.6.27
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Lean status
- No associated Lean code or declarations.
Let f \colon A \to B be a ring map. We denote by
f_w \colon A_w \to B_w the unique w-local ring map compatible with f and
the maps A \to A_w and B \to B_w.
Lemma1.6.28
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 2
Associated Lean declarations
Let A \to B be a ring map that induces algebraic extensions on residue
fields and let I \subseteq A be an ideal, such that V(I) only contains
closed points. Then V(IB) only contains closed points.
Lean code for Lemma1.6.28●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem zeroLocus_map_algebraMap_subset_closedPoints.{u} {A B : Type u} [CommRing A] [CommRing B] {I : Ideal A} [Algebra A B] (h : ∀ (m : Ideal A) (q : Ideal B) [inst : q.LiesOver m] [inst_1 : m.IsMaximal] [inst_2 : q.IsPrime] [inst_3 : Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [inst_4 : Localization.AtPrime.IsLiesOverAlgebra m q], Algebra.IsAlgebraic m.ResidueField q.ResidueField) (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A B) I) ⊆ closedPoints (PrimeSpectrum B)
theorem zeroLocus_map_algebraMap_subset_closedPoints.{u} {A B : Type u} [CommRing A] [CommRing B] {I : Ideal A} [Algebra A B] (h : ∀ (m : Ideal A) (q : Ideal B) [inst : q.LiesOver m] [inst_1 : m.IsMaximal] [inst_2 : q.IsPrime] [inst_3 : Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [inst_4 : Localization.AtPrime.IsLiesOverAlgebra m q], Algebra.IsAlgebraic m.ResidueField q.ResidueField) (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A B) I) ⊆ closedPoints (PrimeSpectrum B)
Proof for Lemma 1.6.28
By Lemma 1.6.24, every prime of B lying
above a maximal ideal of A is maximal. In particular, the closed subset
V(I B) = (\spec{B} \to \spec{A})^{-1}(V(I)) only contains closed points.
Lemma1.6.29
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Associated Lean declarations
Let B be an A-algebra satisfying going down. If every closed point of
\spec{A} has a preimage in \spec{B}, the map \spec{B} \to \spec{A} is
surjective.
Lean code for Lemma1.6.29●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Mathlib/RingTheory/Ideal/GoingDown.leancomplete
theorem Algebra.HasGoingDown.specComap_surjective_of_closedPoints_subset_preimage.{u_1, u_2} {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.HasGoingDown R S] (h : closedPoints (PrimeSpectrum R) ⊆ Set.range (PrimeSpectrum.comap (algebraMap R S))) : Function.Surjective (PrimeSpectrum.comap (algebraMap R S))
theorem Algebra.HasGoingDown.specComap_surjective_of_closedPoints_subset_preimage.{u_1, u_2} {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.HasGoingDown R S] (h : closedPoints (PrimeSpectrum R) ⊆ Set.range (PrimeSpectrum.comap (algebraMap R S))) : Function.Surjective (PrimeSpectrum.comap (algebraMap R S))
Proof for Lemma 1.6.29
uses 0
Let \mathfrak{p} be a prime of A. Then it is contained in a maximal ideal
\mathfrak{m} that is the preimage of a prime \mathfrak{q} of B. By
going-down, there exists therefore a prime \mathfrak{q}' \subseteq \mathfrak{q}
lying above \mathfrak{p}.
Definition1.6.30
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Used by 7
Associated Lean declarations
-
Ideal.WLocalization[complete]
Let A be a ring and I \subseteq A an ideal. We define the
w-localization of A with respect to I to be the ring
(A_w)_{\widetilde{V(IA_{w})}}, denoted by A_{w,I}.
Lean code for Definition1.6.30●1 definition
Associated Lean declarations
-
Ideal.WLocalization[complete]
Associated Lean declarations
-
Ideal.WLocalization[complete]
-
defdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
def Ideal.WLocalization.{u} {A : Type u} [CommRing A] (I : Ideal A) : Type u
def Ideal.WLocalization.{u} {A : Type u} [CommRing A] (I : Ideal A) : Type u
The w-localization with respect to an ideal `I`.
Lemma1.6.31
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Associated Lean declarations
-
isWLocalRing_generalization_one[complete] -
Ideal.WLocalization.isWLocalRing[complete]
Let A be a ring and I \subseteq A an ideal. The ring A_{w,I} is
w-local.
Lean code for Lemma1.6.31●2 theorems
Associated Lean declarations
-
isWLocalRing_generalization_one[complete]
-
Ideal.WLocalization.isWLocalRing[complete]
Associated Lean declarations
-
isWLocalRing_generalization_one[complete] -
Ideal.WLocalization.isWLocalRing[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem isWLocalRing_generalization_one.{u} {A : Type u} [CommRing A] (I : Ideal A) [IsWLocalRing A] : IsWLocalRing (WLocalization.Generalization 1 I)
theorem isWLocalRing_generalization_one.{u} {A : Type u} [CommRing A] (I : Ideal A) [IsWLocalRing A] : IsWLocalRing (WLocalization.Generalization 1 I)
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.isWLocalRing.{u} {A : Type u} [CommRing A] {I : Ideal A} : IsWLocalRing I.WLocalization
theorem Ideal.WLocalization.isWLocalRing.{u} {A : Type u} [CommRing A] {I : Ideal A} : IsWLocalRing I.WLocalization
Proof for Lemma 1.6.31
Proof uses 2
Follows immediately from Lemma 1.6.25 and Corollary 1.6.22.
Lemma1.6.32
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.33
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 2
Used by 2
Associated Lean declarations
-
Ideal.WLocalization.indZariski[complete]
Let A be a ring and I \subseteq A an ideal. The ring map
A \to A_{w, I} is ind-Zariski.
Lean code for Lemma1.6.32●1 theorem
Associated Lean declarations
-
Ideal.WLocalization.indZariski[complete]
Associated Lean declarations
-
Ideal.WLocalization.indZariski[complete]
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.indZariski.{u} {A : Type u} [CommRing A] {I : Ideal A} : Algebra.IndZariski A I.WLocalization
theorem Ideal.WLocalization.indZariski.{u} {A : Type u} [CommRing A] {I : Ideal A} : Algebra.IndZariski A I.WLocalization
Proof for Lemma 1.6.32
Proof uses 3
The map A \to A_w is ind-Zariski by
Lemma 1.6.20 and the map A_w \to A_{w, I} is
ind-Zariski by Lemma 1.6.9. Hence the
composition A \to A_{w, I} is ind-Zariski by
Lemma 1.2.6.
Lemma1.6.33
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.34
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 2
Associated Lean declarations
Let I \subseteq A be an ideal such that V(I) \subseteq \spec{A} only
contains closed points. Then
-
the set of closed points of
\spec{A_{w, I}}isV(IA_{w, I}), and -
the map
A / I \to A_{w, I} / I A_{w, I}is an isomorphism.
Lean code for Lemma1.6.33●3 theorems
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.zeroLocus_map_algebraMap_eq_closedPoints.{u} {A : Type u} [CommRing A] {I : Ideal A} (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A I.WLocalization) I) = closedPoints (PrimeSpectrum I.WLocalization)
theorem Ideal.WLocalization.zeroLocus_map_algebraMap_eq_closedPoints.{u} {A : Type u} [CommRing A] {I : Ideal A} (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A I.WLocalization) I) = closedPoints (PrimeSpectrum I.WLocalization)
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.quotientMap_algebraMap_bijective.{u} {A : Type u} [CommRing A] (I : Ideal A) (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : Function.Bijective ⇑(Ideal.quotientMap (Ideal.map (algebraMap A I.WLocalization) I) (algebraMap A I.WLocalization) ⋯)
theorem Ideal.WLocalization.quotientMap_algebraMap_bijective.{u} {A : Type u} [CommRing A] (I : Ideal A) (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : Function.Bijective ⇑(Ideal.quotientMap (Ideal.map (algebraMap A I.WLocalization) I) (algebraMap A I.WLocalization) ⋯)
[Stacks Tag 097A](https://stacks.math.columbia.edu/tag/097A) ((2)(a))
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.bijOn_zeroLocus_map.{u} {A : Type u} [CommRing A] (I : Ideal A) (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : Set.BijOn (PrimeSpectrum.comap (algebraMap A I.WLocalization)) (PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A I.WLocalization) I)) (PrimeSpectrum.zeroLocus ↑I)
theorem Ideal.WLocalization.bijOn_zeroLocus_map.{u} {A : Type u} [CommRing A] (I : Ideal A) (hI : PrimeSpectrum.zeroLocus ↑I ⊆ closedPoints (PrimeSpectrum A)) : Set.BijOn (PrimeSpectrum.comap (algebraMap A I.WLocalization)) (PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A I.WLocalization) I)) (PrimeSpectrum.zeroLocus ↑I)
Proof for Lemma 1.6.33
Proof uses 8
A \to A_w is ind-Zariski by Lemma 1.6.20 and
therefore induces algebraic extensions of residue fields by
Lemma 1.2.2. In particular, the closed
subset V(I A_w) only contains closed points by
Lemma 1.6.28 and the first claim follows
from Lemma 1.6.25.
The map A \to A_{w, I} is ind-Zariski as a composition of ind-Zariski maps by
Lemma 1.6.20 and
Lemma 1.6.9. Since ind-Zariski is stable under
base change, the same holds for A/I \to A_{w, I}/IA_{w, I}. Hence it
identifies local rings. By part (2) of
Lemma 1.6.8,
V(IA_{w, I}) \to V(I A_w) is an isomorphism. Moreover
V(I A_{w}) \to V(I) is a bijection by part (1) of
Lemma 1.6.21 because
V(I A_{w}) \subseteq V(I_w). Hence, the composition
\spec{A_{w, I} / I A_{w, I}} = V(I A_{w, I}) \to V(I A_w) \to V(I) is
bijective. Thus A / I \to A_{w, I} / I A_{w, I} is an isomorphism by
Lemma 1.6.5.
Lemma1.6.34
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.35
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Statement uses 4
Used by 2
Associated Lean declarations
Let A be a w-local ring. Let I \subset A be an ideal cutting out the set
X^c of closed points in X = \Spec (A). Let A \to B be a ring map
inducing algebraic extensions on residue fields at primes. Let C be the ring
B_{w, IB} constructed in Definition 1.6.30. Then
-
B/IB \to C/ICis an isomorphism, -
Cis w-local, -
the map
p \colon Y = \Spec (C) \to Xinduced by the ring mapA \to B \to Cis w-local, and -
p^{-1}(X^c)is the set of closed points ofY, i.e.V(IC)is the set of closed points ofY.
(Stacks Project, Tag 097A, (2))
Lean code for Lemma1.6.34●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.algebraMap_specComap_preimage_closedPoints_eq.{u} {A B : Type u} [CommRing A] [CommRing B] {I : Ideal A} [IsWLocalRing A] [Algebra A B] (hI : PrimeSpectrum.zeroLocus ↑I = closedPoints (PrimeSpectrum A)) (h : ∀ (m : Ideal A) (q : Ideal B) [inst : q.LiesOver m] [inst_1 : m.IsMaximal] [inst_2 : q.IsPrime] [inst_3 : Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [inst_4 : Localization.AtPrime.IsLiesOverAlgebra m q], Algebra.IsAlgebraic m.ResidueField q.ResidueField) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A (Ideal.map (algebraMap A B) I).WLocalization) I) = closedPoints (PrimeSpectrum (Ideal.map (algebraMap A B) I).WLocalization)
theorem Ideal.WLocalization.algebraMap_specComap_preimage_closedPoints_eq.{u} {A B : Type u} [CommRing A] [CommRing B] {I : Ideal A} [IsWLocalRing A] [Algebra A B] (hI : PrimeSpectrum.zeroLocus ↑I = closedPoints (PrimeSpectrum A)) (h : ∀ (m : Ideal A) (q : Ideal B) [inst : q.LiesOver m] [inst_1 : m.IsMaximal] [inst_2 : q.IsPrime] [inst_3 : Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [inst_4 : Localization.AtPrime.IsLiesOverAlgebra m q], Algebra.IsAlgebraic m.ResidueField q.ResidueField) : PrimeSpectrum.zeroLocus ↑(Ideal.map (algebraMap A (Ideal.map (algebraMap A B) I).WLocalization) I) = closedPoints (PrimeSpectrum (Ideal.map (algebraMap A B) I).WLocalization)
[Stacks Tag 097A](https://stacks.math.columbia.edu/tag/097A) ((2)(d))
Proof for Lemma 1.6.34
Proof uses 6
Every point of V(IB) is a closed point of \spec{B} by
Lemma 1.6.28. Hence we may choose
C = (B_w)_{\widetilde{V(IB_w)}} as in
Definition 1.6.30. Part (1) holds by
Lemma 1.6.33. Part (2) holds by
Lemma 1.6.31. Then the set of closed points of
\spec{C} is V(IC), which is the preimage of V(I) under
\spec{C} \to \spec{B} \to \spec{A}.
Lemma1.6.35
Group: Topological and ring-theoretic preliminaries on w-local spaces and w-local, w-strictly local and w-contractible rings, following Section 2 of Bhatt–Scholze and the Stacks Project. (136)
- Definition 1.1.1
- Theorem 1.1.2
- Definition 1.1.3
- Theorem 1.1.4
- Proposition 1.1.5
- Lemma 1.1.6
- Lemma 1.1.7
- Lemma 1.1.8
- Lemma 1.1.9
- Lemma 1.1.10
- Definition 1.1.11
- Lemma 1.1.12
- Definition 1.1.13
- Lemma 1.1.14
- Lemma 1.1.15
- Lemma 1.1.16
- Lemma 1.1.17
- Lemma 1.1.18
- Lemma 1.1.19
- Lemma 1.1.20
- Lemma 1.1.21
- Lemma 1.1.22
- Lemma 1.1.23
- Lemma 1.1.24
- Lemma 1.1.25
- Lemma 1.1.26
- Lemma 1.1.27
- Definition 1.1.28
- Definition 1.1.29
- Lemma 1.1.30
- Lemma 1.1.31
- Lemma 1.1.32
- Definition 1.1.33
- Lemma 1.1.34
- Definition 1.1.35
- Lemma 1.1.36
- Lemma 1.1.37
- Definition 1.2.1
- Lemma 1.2.2
- Lemma 1.2.3
- Lemma 1.2.4
- Lemma 1.2.5
- Lemma 1.2.6
- Lemma 1.2.7
- Definition 1.3.1
- Lemma 1.3.2
- Definition 1.3.3
- Lemma 1.3.4
- Proposition 1.3.5
- Proposition 1.3.6
- Definition 1.4.1
- Lemma 1.4.2
- Lemma 1.4.3
- Lemma 1.4.4
- Lemma 1.4.5
- Definition 1.5.1
- Definition 1.5.2
- Definition 1.5.3
- Lemma 1.5.4
- Lemma 1.5.5
- Lemma 1.5.6
- Lemma 1.5.7
- Lemma 1.5.8
- Lemma 1.5.9
- Lemma 1.5.10
- Lemma 1.5.11
- Definition 1.5.12
- Lemma 1.5.13
- Lemma 1.5.14
- Lemma 1.5.15
- Lemma 1.5.16
- Lemma 1.5.17
- Definition 1.6.1
- Definition 1.6.2
- Lemma 1.6.3
- Lemma 1.6.4
- Lemma 1.6.5
- Lemma 1.6.6
- Definition 1.6.7
- Lemma 1.6.8
- Lemma 1.6.9
- Definition 1.6.10
- Lemma 1.6.11
- Lemma 1.6.12
- Lemma 1.6.13
- Definition 1.6.14
- Lemma 1.6.15
- Lemma 1.6.16
- Lemma 1.6.17
- Definition 1.6.18
- Lemma 1.6.19
- Lemma 1.6.20
- Lemma 1.6.21
- Corollary 1.6.22
- Lemma 1.6.23
- Lemma 1.6.24
- Lemma 1.6.25
- Proposition 1.6.26
- Definition 1.6.27
- Lemma 1.6.28
- Lemma 1.6.29
- Definition 1.6.30
- Lemma 1.6.31
- Lemma 1.6.32
- Lemma 1.6.33
- Lemma 1.6.34
- Definition 1.7.1
- Lemma 1.7.2
- Lemma 1.7.3
- Definition 1.7.4
- Lemma 1.7.5
- Lemma 1.7.6
- Lemma 1.7.7
- Lemma 1.7.8
- Definition 1.7.9
- Lemma 1.7.10
- Proposition 1.7.11
- Corollary 1.7.12
- Lemma 1.7.13
- Proposition 1.7.14
- Proposition 1.7.15
- Definition 1.7.16
- Lemma 1.7.17
- Definition 1.7.18
- Lemma 1.7.19
- Definition 1.7.20
- Lemma 1.7.21
- Definition 1.7.22
- Lemma 1.7.23
- Lemma 1.7.24
- Definition 1.8.1
- Lemma 1.8.2
- Proposition 1.8.3
- Lemma 1.8.4
- Theorem 1.8.5
- Corollary 1.8.6
Used by 2
Associated Lean declarations
Let A be a w-local ring. Let I \subset A be an ideal cutting out the set
X^c of closed points in X = \Spec (A). Let B be a faithfully flat
A-algebra. Let C be the ring B_{w, IB} constructed in
Definition 1.6.30. Then the composition A \to B \to C is
also faithfully flat.
Lean code for Lemma1.6.35●1 theorem
Associated Lean declarations
Associated Lean declarations
-
theoremdefined in Proetale/Algebra/WLocalization/Ideal.leancomplete
theorem Ideal.WLocalization.faithfullyFlat_map_algebraMap.{u} {A B : Type u} [CommRing A] [CommRing B] {I : Ideal A} [IsWLocalRing A] [Algebra A B] [Module.FaithfullyFlat A B] (hI : PrimeSpectrum.zeroLocus ↑I = closedPoints (PrimeSpectrum A)) (h : ∀ (m : Ideal A) (q : Ideal B) [inst : q.LiesOver m] [inst_1 : m.IsMaximal] [inst_2 : q.IsPrime] [inst_3 : Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [inst_4 : Localization.AtPrime.IsLiesOverAlgebra m q], Algebra.IsAlgebraic m.ResidueField q.ResidueField) : Module.FaithfullyFlat A (Ideal.map (algebraMap A B) I).WLocalization
theorem Ideal.WLocalization.faithfullyFlat_map_algebraMap.{u} {A B : Type u} [CommRing A] [CommRing B] {I : Ideal A} [IsWLocalRing A] [Algebra A B] [Module.FaithfullyFlat A B] (hI : PrimeSpectrum.zeroLocus ↑I = closedPoints (PrimeSpectrum A)) (h : ∀ (m : Ideal A) (q : Ideal B) [inst : q.LiesOver m] [inst_1 : m.IsMaximal] [inst_2 : q.IsPrime] [inst_3 : Algebra (Localization.AtPrime m) (Localization.AtPrime q)] [inst_4 : Localization.AtPrime.IsLiesOverAlgebra m q], Algebra.IsAlgebraic m.ResidueField q.ResidueField) : Module.FaithfullyFlat A (Ideal.map (algebraMap A B) I).WLocalization
Proof for Lemma 1.6.35
Proof uses 4
By Lemma 1.2.3 and
Lemma 1.6.32, A \to B \to C is flat, so
by Lemma 1.6.29 it suffices to show every
closed point of \spec{A} is in the image. But since
\spec{B} \to \spec{A} is surjective, also V(IC) \cong V(IB) \to V(I) is
surjective (the first map is an isomorphism by part (2) of
Lemma 1.6.33).