Dependencies

Let \(R\) be a commutative ring.

• The forgetful functors from commutative \(R\)-algebras to \(R\)-algebras and from \(R\)-algebras to \(R\)-modules preserve filtered colimits.

• The forgetful functor from commutative \(R\)-algebras to commutative rings preserves colimits.

• Tensoring on the left with an \(R\)-algebra preserves colimits on the category of commutative \(R\)-algebras.

A category \(\mathcal{C}\) is precoherent if given a finite effective epimorphism family \(f\colon \colon X_i \to X\) and a morphism \(p\colon Y \to X\), there exists a finite effective epimorphism family \(g\colon Y_j \to Y\) that factors through \(f\).

Let \(\mathcal{U}\) be a one-hypercover in a site \(\mathcal{C}\). We say \(\mathcal{U}\) is exact if taking filtered colimits preserves the multiequalizer diagram of \(\mathcal{U}\).

A sheaf \(F\) in \(\mathrm{Shv}(X_{\mathrm{proét}})\) is classical if and only if for any \(U \in X^{\mathrm{aff}}_{\mathrm{proét}}\) with presentation \(U = \lim U_i\), it holds that \(F(U) = \operatorname {colim}F(U_i)\).

Let \(A\) be a w-local ring. Let \(I \subset A\) be an ideal cutting out the set \(X^c\) of closed points in \(X = \operatorname{Spec}(A)\). Let \(B\) be a faithfully flat \(A\)-algebra. Let \(C\) be the ring \(B_{w, IB}\) constructed in Definition 1.98. Then the composition \(A \to B \to C\) is also faithfully flat.

Let \(K\) be in \(D^{+}(X_{\mathrm{et}})\). Then

For any ring \(A\), there exists an ind-étale faithfully flat \(A\)-algebra \(B\) such that every ind-étale faithfully flat map \(B \to C\) has a retraction.

Let \(F\) be a sheaf on \(\mathcal{C}\). Then the unit \(F \to \nu _{*} \nu ^{-1} F\) is an isomorphism.

Let \(P\) be a property of objects on a category \(\mathcal{A}\) that implies finitely presented. Denote by \(\mathcal{A}_P\) (resp. \(\mathcal{A}_{\mathrm{Pro}(P)}\)) the full subcategory defined by \(P\) (resp. \(\mathrm{Pro}(P)\)). Suppose \(\mathcal{A}\) has cofiltered limits, then \(\lim _{\mathcal{A}}\) induces an equivalence

Let \(\mathfrak {m}\) be a maximal ideal of \(T^{\infty }(A)\). Then \(T^{\infty }(A)_{\mathfrak {m}}\) is strictly Henselian.

Let \(A\) be a ring. The \(A\)-algebra \(A_w\) is w-local.

An object \(U\) of \(X_{\mathrm{proét}}\) is called a pro-étale affine if it can be written as \(U = \lim U_i\) over a small cofiltered diagram \(i \mapsto U_i\) of affine schemes étale over \(X\). The subcategory of pro-étale affines in \(X_{\mathrm{proét}}\) is denoted by \(X^{\mathrm{aff}}_{\mathrm{proét}}\).

Let \(X\) be affine. The affine pro-étale site of \(X\) is the category \(X^{\mathrm{aff}}_{\mathrm{proét}}\) with the covering families are the quasi-compact covers.

A sheaf \(F\) in \(\mathrm{Shv}(X_{\mathrm{proét}})\) is called classical if it lies in the essential image of \(\nu ^{*}\).

An object \(X\) of \(\mathcal{C}\) is coherent if it is compact and stable.

Let \(A\) be a ring, \(X = \operatorname{Spec}(A)\), and \(T \subset \pi _0(X)\). Define the ring \(A_T\) by

where \(W\) runs over all open and closed subsets of \(X\) containing the inverse image \(Z\) of \(T\), and \(A \to A_W\) is the localization of \(A\) at \(W\).

An object \(X\) of \(\mathcal{C}\) is compact if the top element of the poset of subobjects of \(X\) is a compact element.

Let \(X\) be a topological space. The constructible topology on \(X\) is the topology with subbase of opens the sets \(U\) and \(X - U\) for all quasi-compact opens \(U\) of \(X\).

Denote by \(\Gamma _{\mathrm{cont}}(X, -)\colon \mathrm{Ab}(X_{\mathrm{et}})^{\mathbb {N}} \to \mathrm{Ab}\) the functor

For an inverse system \((F_n)_{n \in \mathbb {N}}\), we call the cohomology groups \(H^i \mathrm{R} \Gamma _{\mathrm{cont}}\left(X, -\right)\left( \left( F_{n} \right)_{n \in \mathbb {N}} \right) \), denoted by \(H^i_{\mathrm{cont}}(X_{\mathrm{et}}, (F_{n})_{n \in \mathbb {N}})\), the continuous étale cohomology of \(X\) with coefficients in \(\left( F_n \right)_{n \in \mathbb {N}}\).

Let \(A\) be a ring. Set \(T^0(A) = A\) and for \(n \in \mathbb {N}\) set \(T^{n+1}(A) = T(T^n(A))\). We call the \(A\)-algebra \(T^{\infty }(A) = \operatorname {colim}_{n \in \mathbb {N}} T^n(A)\) the étale contraction of \(A\).

Let \(A\) be a ring. Denote by \(S(A)\) the set of finite subsets of faithfully flat étale \(A\)-algebras. We call the \(A\)-algebra \(T(A) = \operatorname {colim}_{E \in S(A)} \bigotimes _{B \in E} B\) the étale precontraction of \(A\).

A topological space \(X\) is extremally disconnected if the closure of every open subset is open.

Let \(A\) be a ring and \(E \subseteq A\) a finite subset. Let

Further, we set \(I_E \subseteq A\) to be the product of the ideals defining \(Z(E', E'')\) in \(\mathrm{Spec}(A_{\widetilde{Z(E', E'')}})\).

We denote by \(\nu = \nu _X\) the morphism of sites \(X_{\mathrm{proét}} \to X_{\mathrm{et}}\) induced by the functor from \(X_{\mathrm{et}}\) to \(X_{\mathrm{proét}}\) given by \((U \to X) \mapsto (U \to X)\).

Let \(X\) be a scheme. The fqpc site of \(X\), denoted by \(X_{\mathrm{fpqc}}\), is the \(\mathcal{P}\)-site of \(X\) with \(\mathcal{P} = \text{flat}\).

Let \(X\) be a topological space and \(Z \subseteq X\) a subset. The set

is called the generalization of \(Z\) in \(X\).

Let \(A\) be a ring and \(X = \operatorname{Spec}(A)\). Let \(A \to B\) and \(A \to C\) be two \(A\)-algebras that identifies local rings. As a consequence of Lemma 1.33, we have a bijective map

sending \(f : B \to C\) to the continuous map \(\operatorname{Spec}(f) : \operatorname{Spec}(C) \to \operatorname{Spec}(B)\) induced by \(f\).

Let \(A\) be a ring and \(X = \operatorname{Spec}(A)\). Let \(\operatorname {ILR}_A\) denote the category of \(A\)-algebras \(B\) for which \(A \to B\) identifies local rings, and let \(\operatorname {Top}_X\) denote the category of topological spaces over \(X\).

Define the functor

by sending \(B\) to \(\operatorname{Spec}(B)\).

A ring map \(A \to B\) identifies local rings if for every prime \(\mathfrak {q} \subset B\) the canonical map \(A_{\varphi ^{-1}(\mathfrak {q})} \to B_{\mathfrak {q}}\) is an isomorphism.

An \(R\)-algebra \(S\) is ind-étale if it is a filtered colimit of étale \(R\)-algebras. We say a ring homomorphism \(R \to S\) is ind-étale if \(S\) is ind-étale as an \(R\)-algebra via \(f\).

An \(R\)-algebra \(S\) is called a ind-Zariski if \(S\) can be written as a filtered colimit \(S \simeq \operatorname {colim}S_i\) with each \(R \to S_i\) a local isomorphism. A ring homomorphism \(f : R \to S\) is called a ind-Zariski if \(S\) is ind-Zariski as an \(R\)-algebra via \(f\).

We say \(A \to B\) is a local isomorphism if for every prime \(\mathfrak {q} \subset B\) there exists a \(g \in B\), \(g \notin \mathfrak {q}\) such that \(A \to B_g\) induces an open immersion \(\operatorname{Spec}(B_g) \to \operatorname{Spec}(A)\).

We say \(\mathcal{C}\) is locally weakly contractible, if each \(X\) in \(\mathcal{C}\) admits an epimorphism from the coproduct of \(Y_i\), where \(Y_i\) is coherent and weakly contractible.

Let \(A\) be a ring and let \(X = \operatorname{Spec}(A)\). Let \(T\) be profinite space and let \(f : T \to \pi _0(X)\) be a continuous map. Let \(S\) be a finite discrete quotient of \(T\) with quotient map \(q : T \to S\). Define the ring \(A^f_{q}\) as follow: let \(Z = \operatorname {Im} (T \to \pi _0(X) \times S = \pi _0(\operatorname{Spec}(A^{S})))\), where \(A^{S} = \prod _{s \in S} A\) and let

be the \(A^S\)-algebra as in Definition 1.120.

Let \(A\) be a ring and let \(X = \operatorname{Spec}(A)\). Let \(T\) be a profinite space and let \(T \to \pi _0(X)\) be a continuous map. Write \(T = \lim T_i\) as the limit of an inverse system of finite discrete spaces over a directed set. Define the ring \(A^f_{\pi _0}\) as follows:

where \(q_i : T \to T_i\) are the quotient maps and \(A^f_{q_i}\) are the rings defined in Definition 1.122 and the transition maps are those defined in Definition 1.124. By Lemma 1.125 the transition maps are compatible thus the colimit is well-defined.

Let \(A\) be a ring and let \(X = \operatorname{Spec}(A)\). Let \(T\) be profinite space and let \(f : T \to \pi _0(X)\) be a continuous map. Let \(S\) and \(S'\) be finite discrete spaces and let \(q : T \to S\) and \(q' : T \to S'\) be quotient maps. Let \(g : S' \to S\) be a map such that \(q = g \circ q'\). Let \(Z \subseteq \pi _0(\operatorname{Spec}(A^{S})) \times S\) and \(Z' \subseteq \pi _0(\operatorname{Spec}(A^{S'})) \times S'\) be the images of \(T\) under the maps induced by \(f\) and \(f'\) respectively. Then \(g\) induces a map \(Z \to Z'\) and a map \(\tilde{g} : \operatorname{Spec}(A^f_{q}) \cong X \times _{\pi _0(X)} Z \to X \times _{\pi _0(X)} Z' \cong \operatorname{Spec}(A^{f'}_{q'})\).

Since both maps \(A \to A^f_{q}\) and \(A \to A^{f'}_{q'}\) identify local rings, by the bijection in Definition 1.34 we can find a transition ring map

that induces the topological map \(\tilde{g}\) between the spectra.

Let \(F\colon \mathcal{C} \to \mathcal{D}\) be a functor between sites. We say a one-hypercover \(\mathcal{U}\) of an object \(X\) in \(\mathcal{D}\) is relatively pro-representable if it is exact and \(\mathcal{U}\) can be written as the componentwise cofiltered limit of one-hypercovers in \(\mathcal{C}\).

A morphism \(f\colon X \to Y\) is \(P\)-Henselian, if for every factorisation

with \(u\) satisfying \(P\), there exists a retraction of \(u\).

Let \(X\) be a topological space. Let \(\pi _0(X)\) be the set of connected components of \(X\). Let \(X \to \pi _0(X)\) be the map which sends \(x \in X\) to the connected component of \(X\) passing through \(x\). Endow \(\pi _0(X)\) with the quotient topology. Then any continuous map \(X \to Y\) from \(X\) to a totally disconnected space \(Y\) factors through \(\pi _0(X)\).

Let \(X\) be a scheme. The pro-étale site of \(X\), denoted by \(X_{\mathrm{proét}}\), is the \(\mathcal{P}\)-site of \(X\) with \(\mathcal{P} = \text{weakly étale}\).

A jointly surjective family \((f_i \colon Y_i \to X)_{i \in I}\) of morphisms of schemes is quasi-compact if for every affine open \(U\) of \(X\) there exist quasi-compact opens \(V_i\) in \(Y_i\) such that \(U = \bigcup _{i \in I} f_i(V_i)\).

We say \(\mathcal{C}\) is replete if epimorphisms are stable under sequential limits, i.e., if \((F_n)_{n \in \mathbb {N}}\) is an inverse system in \(\mathcal{C}\) such that \(F_{n+1} \to F_n\) is an epimorphism for all \(n \in \mathbb {N}\), then \(\mathrm{lim}\; F_n \to F_n\) is an epimorphism for all \(n \in \mathbb {N}\).

Let \(\mathcal{P}\) be a morphism property of schemes and \(X\) be a scheme. The small \(\mathcal{P}\)-site of \(X\) is the category of \(X\)-schemes with structure morphism satisfying \(\mathcal{P}\) and with covers given by quasi-compact, jointly surjective families of morphisms satisfying \(\mathcal{P}\).

An object \(X\) of \(\mathcal{C}\) is stable if for all compact \(Y\) and morphisms \(Y \to X\), the fiber product \(Y \times _{X} Y\) is compact.

Let \(X\) be a topological space. The Stone-Čech compactification of \(X\) is the profinite space \(\beta (X)\) such that

1. \(X\) is dense in \(\beta (X)\);

2. every continuous map \(f: X \to Y\) to a quasi-compact Hausdorff space \(Y\) extends uniquely to a continuous map \(\beta (f): \beta (X) \to Y\).

We denote the Stone-Čech compactification of \(X\) by \(\beta (X)\).

A local ring \((R, \mathfrak {m}, \kappa )\) is strictly henselian if the composition \(R \to \kappa \to \kappa ^{\mathrm{sep}}\) is étale-Henselian for a separable closure \(\kappa ^{\mathrm{sep}}\).

Let \(E, F \subseteq A\) be subsets. We define

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 \(A_{\widetilde{(I, f)}}\) to be the \(A\)-algebra \(S_{f, I}^{-1}\) and \(\widetilde{(I, f)} \subseteq A_{\widetilde{(I, f)}}\) to be the kernel of the induced map \(A_{\widetilde{(I, f)}} \to (A / I)_{f}\).

A ring \(A\) is w-contractible if it is w-strictly local and \(\pi _0(\mathrm{Spec}(A))\) is extremally disconnected.

A ring \(A\) is w-local if Spec(A) is w-local.

A ring map \(f: A \to B\) between w-local rings is w-local if the induced map of w-local spaces \(\operatorname{Spec}(f) : \operatorname{Spec}(B) \to \operatorname{Spec}(A)\) is w-local.

A topological space \(X\) is w-local if it satisfies:

1. \(X\) is spectral,

2. every point specializes to a unique closed point,

3. the subspace \(X^c\) of closed points is closed.

Let \(X\) and \(Y\) be w-local spaces. A morphism \(f: X \to Y\) is w-local if it is spectral and the image of closed points \(f(X^c) \subseteq Y^c\).

Let \(I(A)\) be the partially ordered set of all finite subsets of \(A\). We define the \(A\)-algebra \(A_{w}\) as

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 \(A_w\).

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 \(A_{w,I}\).

A ring \(A\) is w-strictly local if \(A\) is w-local, and every local ring of \(A\) at a maximal ideal is strictly henselian.

An object \(F\) of \(\mathcal{C}\) is called weakly contractible if every epimorphism \(G \to F\) has a section. We say \(\mathcal{C}\) has enough weakly contractible objects, if every object admits an epimorphism from a weakly contractible one.

An \(R\)-algebra \(S\) is weakly étale if it is flat over \(R\) and flat over \(S \otimes _{R} S\). A ring homorphism \(f \colon R \to S\) is weakly étale if \(S\) is weakly étale as an \(R\)-algebra.

A map \(f \colon Y \to X\) is weakly étale if \(f\) is flat and \(\Delta _f\colon Y \to Y \times _{X} Y\) is flat.

The topos \(\mathrm{Shv}(X_{\mathrm{et}})\) is generated by the affine étale site of \(X\), i.e., the full subcategory of \(X_{\mathrm{et}}\) consisting of affine, étale schemes over \(X\).

The topos \(\mathrm{Shv}(X_{\mathrm{proét}})\) is generated by \(X^{\mathrm{aff}}_{\mathrm{proét}}\).

The category \(X^{\mathrm{aff}}_{\mathrm{proét}}\) admits limits indexed by a connected diagram and the forgetful functor to \(\mathrm{Sch} / X\) preserves them. If \(X\) is affine, \(X^{\mathrm{aff}}_{\mathrm{proét}}\) has all small limits.

Let \(F\) be a presheaf on \(X_{\mathrm{et}}\). Then \((\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\) preserves filtered colimits.

The topology on \(X^{\mathrm{aff}}_{\mathrm{proét}}\) is generated by relatively pro-representable one-hypercovers with respect to the inclusion \(X^{\mathrm{aff}}_{\mathrm{ét}} \to X^{\mathrm{aff}}_{\mathrm{proét}}\).

Let \(I \subseteq A\) be an ideal such that \(V(I) \subseteq A\) only contains closed points. Then

1. the set of closed points of \(A_{w, I}\) is \(V(IA_{w, I})\), and

2. the map \(A / I \to A_{w, I} / I A_{w, I}\) is an isomorphism.

Let \(X = \lim _i X_i\) be a limit of topological spaces over some index category \(I\) and let \(s \subseteq X\) be a subset of \(X\). Denote by \(s_i\) the image of \(s\) in \(X_i\). Then \(\overline{s} = \bigcap _i \pi _i^{-1} (\overline{s_i})\), where \(\pi _i : X \to X_i\) is the projection map.

Let \(K\) be in \(D^+(X_{\mathrm{et}})\) and \(U = \lim _i U_i\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then \(R \Gamma (U, \nu ^*K) = \operatorname {colim}_i R\Gamma (U_i, K)\).

Let \(f\colon X \to Y\) be an embedding of topological spaces. Then \(f\) preserves and reflects specializations, i.e. for \(a, b \in X\), \(a\) specializes to \(b\) if and only if \(f(a)\) specializes to \(f(b)\).

A morphism \(f\colon X \to Y\) in \(S_{\mathrm{et}}\) is an effective epimorphism if and only if it is surjective.

The category \(X_{\mathrm{et}}\) is precoherent.

\(T^{\infty }(A)\) is ind-étale and faithfully flat over \(A\).

Let \(A = \operatorname {colim}_i A_i\) be a filtered colimit of \(A\)-algebras and let \(B\) be an étale \(A\)-algebra. Then there exists an \(i\) and an étale \(A_i\)-algebra \(B'\) such that

is a pushout diagram.

\(T(A)\) is ind-étale and faithfully flat over \(A\).

For every faithfully flat étale ring map \(A \to B\), there exists an \(A\)-algebra map \(B \to T(A)\).

Etale morphisms are weakly étale.

Every étale algebra is weakly étale.

Let \(k\) be a field and \(A\) an ind-étale, local \(k\)-algebra. Then \(A\) is a separable algebraic field extension of \(k\).

Let \(E \subseteq A\) be a finite subset. Then every point of \(\mathrm{Spec}(A_E)\) specializes to a point in \(V(I_E)\). In particular, \(V(I_E)\) contains all closed points of \(\mathrm{Spec}(A_E)\).

Let \(E \subseteq A\) be a finite subset. Then \(V(I_E) = \coprod _{E = E' \coprod E''} Z(E', E'')\). In particular, the map \(\mathrm{Spec}(A_E) \to \mathrm{Spec}(A)\) induces a bijection \(V(I_E) \to \mathrm{Spec}(A)\).

Let \(E_1, E_2 \subseteq A\) be finite subsets with \(E_1 \subseteq E_2\). The induced map \(\mathrm{Spec}(A_{E_2}) \to \mathrm{Spec}(A_{E_1})\) maps \(V(I_{E_2})\) into \(V(I_{E_1})\).

The morphism of sites \(\nu \) is continuous.

Let \(X\) be w-local and \(Z \subseteq X\) a subset closed under specialization. Then the generalization \(Z^g\) in \(X\) is still closed under specialization.

Let \(Z \subseteq X\) be a subset of a topological space. Then \(Z^g\) is closed under generalization.

Let \(B\) be an \(A\)-algebra satisfying going down. If every closed point of \(\mathrm{Spec}(A)\) has a preimage in \(\mathrm{Spec}(B)\), the map \(\mathrm{Spec}(B) \to \mathrm{Spec}(A)\) is surjective.

Let \(B = \operatorname {colim}_i B_i\) be a filtered colimit of (faithfully) flat \(A\)-algebras. Then \(B\) is (faithfully) flat over \(A\).

Let \(B = \operatorname {colim}_i B_i\) be a filtered colimit of ind-étale \(A\)-algebras. Then \(B\) is an ind-étale \(A\)-algebra.

A filtered colimit of ind-Zariski algebras is ind-Zariski.

Let \(A\) be a ring and \(S \subseteq A\) a multiplicative subset. Then \(S^{-1}A\) is ind-Zariski over \(A\).

A finite product of ind-Zariski algebras is ind-Zariski.

Let \(f \in A\) be an element and \(I \subseteq A\) an ideal.

1. The map \(\mathrm{Spec}(A_{\widetilde{(I, f)}}) \to \mathrm{Spec}(A)\) induces a homeomorphism onto the generalization of \(D(f) \cap V(g)\).

2. The closed subscheme \(V(\widetilde{(I, f)})\) induces an isomorphism onto \(D(f) \cap V(I)\).

3. If \(A'\) is a ring with an element \(f' \in A\) and ideal \(I' \subseteq A'\) and \(A \to A'\) is a ring map that maps \(D(f') \cap V(I')\) into \(D(f) \cap V(I)\), then there is a unique \(A\)-algebra map \(A_{\widetilde{(I, f)}} \to A’_{\widetilde{(I', f')}}\) making the obvious diagram with \(A \to A'\) commute.

In particular, every point in \(\mathrm{Spec}(A_{\widetilde{(I, f)}})\) specializes to a point in \(V(\widetilde{(I, f)})\).

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.

Let \(B\) be an \(A\)-algebra and \(\mathfrak {m}\) a maximal ideal of of \(R\). Let \(\mathfrak {q}\) be a prime ideal lying over \(\mathfrak {m}\) such that \(\kappa (\mathfrak {q})\) is algebraic over \(\kappa (\mathfrak {m})\). Then \(\mathfrak {q}\) is maximal.

Let \(X\) be a profinite topological space and \(x, y \in X\) be distinct. Then there exist \(U, V \subseteq X\) open and closed such that \(X = U \coprod V\) and \(x \in U\), \(y \in V\).

\(\nu ^*\) is fully faithful.

Let \(F\) be a sheaf on \(X_{\mathrm{et}}\) and \(U = \lim _i U_i\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then \(\nu ^*F(U) = \operatorname {colim}_i F(U_i)\).

For any \(F \in \mathrm{Shv}(X_{\mathrm{et}})\), the unit \(F \to \nu _{*} \nu ^{*} F\) is an isomorphism.

Let \((f_i \colon Y_i \to X)_{i \in I}\) be a jointly-surjective family of morphisms of schemes. If \(f_i\) is flat and locally of finite presentation for every \(i \in I\), the family is quasi-compact.

Let \(\mathcal{C}\) be replete and let \((F_n)_{n \in \mathbb {N}} \to (G_n)_{n \in \mathbb {N}}\) be a morphism of inverse systems in \(\mathcal{C}\). If \(F_n \to G_n\) and \(F_{n+1} \to F_{n} \times _{G_n} G_{n+1}\) are epimorphisms for each \(n \in \mathbb {N}\), then \(\lim F_n \to \lim G_n\) is an epimorphism.

Let \(A\) be a ring such that every faithfully flat étale ring map \(A \to B\) has a retraction. Then every local ring of \(A\) at a maximal ideal is strictly henselian.

The category of schemes is finitary extensive.

Let \(X\) be a spectral space such that every point is closed. Then \(X\) is profinite.

Let \(X\) be a spectral space and \(E \subseteq X\) a subset closed in the constructible topology.

1. If \(x \in \overline{E}\), then \(x\) is the specialization of a point in \(E\).

2. If \(E\) is stable under specialization, then \(E\) is closed.

Let \(X\) be a spectral space. Then \(X\) is quasi-compact in the constructible topology.

Let \(X\) be a spectral space and \(E \subseteq X\) a quasi-compact subset. Then the generalization \(E^g\) is an intersection of quasi-compact open subsets.

Let \(A\) be a ring, \(E \subseteq A\) a finite subset. Then

set theoretically.

Let \(A\) be a ring, \(E, F \subseteq A\) finite subsets. Then

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 \(Z(E_2, F_2) \subseteq Z(E_1, F_1)\).

Let \(f \in A\) be an element and \(I \subseteq A\) an ideal. Then \(A_{\widetilde{(I, f)}}\) is ind-Zariski over \(A\).

Let \(X\) be a w-local space. Then the set of closed points \(X^c\) of \(X\) is profinite.

Let \(X\) be a w-local space and \(Z \subseteq X\) a closed subset. Then \(Z^g\) is closed.

Let \(X\) be w-local and \(Z \subseteq X\) a subset closed under specialization. If \(Z\) is spectral, it is w-local and the inclusion \(Z \to X\) maps closed points to closed points.

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 \(V(IA_{\widetilde{V(I)}})\).

Let \(A\) be a ring.

1. The composition \(V(I_w) \to \mathrm{Spec}(A_w) \to \mathrm{Spec}(A)\) is a bijection.

2. Every point of \(\mathrm{Spec}(A_w)\) specializes to a unique point of \(V(I_w)\).

3. \(V(I_w)\) is the set of closed points of \(\mathrm{Spec}(A_w)\).

The \(A\)-algebra \(A_w\) is faithfully flat.

Let \(A\) be a ring and \(I \subseteq A\) an ideal. The ring \(A_{w,I}\) is w-local.

The \(A\)-algebra \(A_{w}\) is ind-Zariski.

Let \(\mathcal{C}\) have enough weakly contractible objects and \(F \to G\) be a morphism. Then \(F \to G\) is an epimorphism if and only if for every weakly contractible \(Y\) in \(\mathcal{C}\), the induced map \(F(Y) = \mathrm{Hom}(Y, F) \to \mathrm{Hom}(Y, G) = G(Y)\) is surjective.

\(f \colon Y \to X\) is weakly étale if and only if for every affine open \(U\) of \(Y\) and every affine open \(V\) of \(X\) with \(f(U) \subset V\), the induced ring homomorphism \(\Gamma (X, V) \to \Gamma (Y, U)\) is weakly étale.

Weakly étale is stable under composition and base change.

Let \(T^{\infty }(A) \to B\) be a faithfully flat and étale \(A\)-algebra map. Then it has a retraction.

Let \(F\) be a presheaf on the category of schemes. Then \(F\) is a sheaf in the fpqc topology if and only if it is a sheaf in the Zariski topology and satisfies the sheaf property for \(\{ V \to U\} \) with \(V\), \(U\) affine and \(V \to U\) faithfully flat.

Let \(f\colon A \to B\) be ind-étale. Then \(f\) induces separable algebraic extensions on residue fields.

Let \(\mathcal{C}\) be replete. If \((F_n)_{n \in \mathbb {N}}\) is an inverse system in \(\mathrm{Ab}(\mathcal{C})\) with epimorphic transition maps, then \(\mathrm{lim} \; F_n \cong \mathrm{R}\; \mathrm{lim} \; F_n\).

Let \(F\) be a sheaf on \(X_{\mathrm{et}}\). Then \((\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\) is a sheaf. In particular, \((\nu ^*F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}} \cong (\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\).

Let \(\mathcal{A}\) be a category with cofiltered limits and suppose there exists a fully faithful functor \(i\colon \mathcal{C} \to \mathcal{A}\) such that \(i(X)\) is cocompact for all \(X \in \mathcal{C}\). Then

is fully faithful.

If \(\mathcal{C}\) is precoherent, also \(\mathrm{Pro}(\mathcal{C})\) is precoherent.

Let \(\mathcal{D}\) be a category with cofiltered limits. The restriction functor

admits a left-adjoint that induces an equivalence on the full subcategory of \(\mathrm{Fun}(\mathrm{Pro}(\mathcal{C}), \mathcal{D})\) on functors preserving cofiltered limits.

The category of sheaves on \(X_{\mathrm{proét}}\) is replete.

The site \(X_{\mathrm{proét}}\) is subcanonical.

The category of sheaves on \(X_{\mathrm{proét}}\) has enough weakly contractible objects.

For any \(K\) in \(D^{+}(X_{\mathrm{et}})\) the unit \(K \to R \nu _{*} \nu ^* K\) is an isomorphism.

If \(\mathcal{C}\) is replete, countable products are exact.

Let \(F \colon \mathrm{Pro}(\mathcal{C})^{\mathrm{op}} \to \mathrm{Set}\) be a presheaf. Suppose \(F|_{\mathcal{C}^{\mathrm{op}}}\) is a sheaf and \(F\) preserves filtered colimits. Then \(F\) is a sheaf.

Let \(F \colon \mathrm{Pro}(\mathcal{C})^{\mathrm{op}} \to \mathrm{Set}\) be a sheaf. Then \(F|_{\mathcal{C}^{\mathrm{op}}}\) is a sheaf.

Suppose the topology on \(\mathcal{D}\) is generated by relatively pro-representable one-hypercovers and let \(P\) be a presheaf on \(\mathcal{D}\). If \(P \circ F^{\mathrm{op}}\) is a sheaf on \(\mathcal{C}\) and \(P\) preserves filtered colimits, then \(P\) is a sheaf on \(\mathcal{D}\).

If \(\mathcal{C}\) has enough weakly contractible objects, it is replete.

Let \(X\) be a spectral space. Let

be a cartesian diagram in the category of topological spaces with \(T\) profinite. Then \(Y\) is spectral and the canonical map \(\pi _0(Y) \to T\) defined in Definition 1.13 is an isomorphism.

Let \(X\) be a spectral space. Let

be a cartesian diagram in the category of topological spaces with \(T\) profinite. If moreover \(X\) is w-local, then \(Y\) is w-local, \(Y \to X\) is w-local, and the set of closed points of \(Y\) is the inverse image of the set of closed points of \(X\).

Let \(X\) be a w-local space. Then the composition \(X^c \to X \to \pi _0(X)\) is a homeomorphism, where \(X^c\) denotes the closed points of \(X\).

Let \(A\) be a w-local ring. Let \(I \subset A\) be an ideal cutting out the set \(X^c\) of closed points in \(X = \operatorname{Spec}(A)\). 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 1.98. Then

1. \(B/IB \to C/IC\) is an isomorphism,

2. \(C\) is w-local,

3. the map \(p: Y = \operatorname{Spec}(C) \to X\) induced by the ring map \(A \to B \to C\) is w-local, and

4. \(p^{-1}(X^c)\) is the set of closed points of \(Y\), i.e. V(IC) is the set of closed points of \(Y\).

Let \(X\) be a spectral space. Let \(Y \subset X\) be a closed subspace. Then \(Y\) is spectral.

Let \(X\) be a w-local space. Then every closed subspace \(Y \subseteq X\) is w-local. Moreover, the inclusion map \(Y \to X\) is w-local.

Let \(X = \lim _i X_i\) be a limit of topological spaces over some index category \(I\) and let \(s \subseteq X\) be a subset of \(X\). Denote by \(s_i\) the image of \(s\) in \(X_i\). Then for any morphism \(f : i \to j\) in \(I\) with transition map \(\varphi _{f} : X_i \to X_j\), we have \(\varphi _{f}(\overline{s_i}) \subseteq \overline{s_j}\).

Let \(X\) and \(Y\) be topological spaces. Let \((x, y) \in X \times Y\) be a point. Then

Let \((A_i, I_i)_{i \in I}\) be a filtered system of rings with ideals. Set \(A = \operatorname {colim}_{i \in I} A_i\) and \(I = \operatorname {colim}_{i \in I} I_i = \operatorname {colim}_{i \in I} I_i A\). Then

Let \(A\) be a ring, \(X = \operatorname{Spec}(A)\), and \(T \subset \pi _0(X)\) a closed subset. Then the map \(A \to A_T\) from Definition 1.120 is a surjective ind-Zariski ring map, and the induced morphism \(\operatorname{Spec}(A_T) \to \operatorname{Spec}(A)\) restricts to a homeomorphism of \(\operatorname{Spec}(A_T)\) onto the inverse image of \(T\) in \(X\).

Let \((F_n)_{n \in \mathbb {N}}\) be an inverse system of abelian sheaves on \(X_{\mathrm{et}}\) with epimorphic transition maps. Then there exists a canonical identification

In particular, for every \(i \ge 0\) we have

Let \(X\) be a topological space. Assume \(X\) is quasi-compact, quasi-separated and prespectral. Then for any \(x \in X\) the connected component of \(X\) containing \(x\) is the intersection of all open and closed subsets of \(X\) containing \(x\).

Let \(A\) be a strictly Henselian local ring and \(f : A \to B\) an étale ring map. Suppose \({\mathfrak {n}}\) is a maximal ideal of \(B\) lying over the maximal ideal \({\mathfrak {m}}\) of \(A\). Then the induced map \(A \to B_{\mathfrak {n}}\) is an isomorphism.

Let \(X\) be a quasi-compact Hausdorff space. There exists a continuous surjection \(X' \to X\) with \(X'\) quasi-compact, Hausdorff, and extremally disconnected.

Let \(X\) be a quasi-compact Hausdorff topological space. Then \(X\) is extremally disconnected if and only if it is a projective object in the category of quasi-compact Hausdorff spaces, i.e., for every continuous surjection \(f : Y \to Z\) of quasi-compact Hausdorff spaces and every continuous map \(g : X \to Z\), there exists a continuous map \(h : X \to Y\) such that \(f \circ h = g\).

Let \(A\) be a ring. If the following two conditions hold:

1. \(A\) is w-local, and

2. \(\pi _0(\operatorname {Spec}(A))\) is extremally disconnected.

Then every faithfully flat ring map \(A \to B\) identifying local rings with \(B\) w-local and whose closed points of \(\operatorname{Spec}(B)\) are exactly \(V(IB)\) has a retraction.

The map \(A \to \prod _{i = 1}^{n} A_i \) identifies local rings, if \(A \to A_i\) identifies local rings for all \(i\).

The fpqc site is subcanonical, i.e., every representable presheaf is a sheaf.

Let \(X\) be a topological space. Then \(X\) is Hausdorff if and only if the diagonal map \(\Delta : X \to X \times X\) is a closed embedding.

Let \(X\) and \(Y\) be topological spaces. If \(X\) is spectral (resp. T0, Hausdorff, Sober, quasi-separated, quasi-compact, prespectral) and there is a homeomorphism \(X \cong Y\), then \(Y\) is spectral (resp. T0, Hausdorff, Sober, quasi-separated, quasi-compact, prespectral).

[ Sta18 , Tag 096L ] Let \(A\) be a ring. Set \(X = \operatorname{Spec}(A)\). The functor \(F\) constructed in Definition 1.32

from the category of \(A\)-algebras \(B\) such that \(A \to B\) identifies local rings to the category of topological spaces over \(X\) is fully faithful.

Let \(A \to B\) and \(B \to C\) be ring maps that identify local rings. Then the composition \(A \to C\) also identifies local rings.

Let \(A \to B\) be an ind-étale ring map. For any ring map \(A \to C\), the base change \(C \to B \otimes _A C\) is ind-étale.

Let \(A \to B\) and \(B \to C\) be ring maps. If \(A \to B\) and \(B \to C\) are ind-étale, then \(A \to C\) is ind-étale.

Let \(A\) be w-contractible and \(I \subseteq A\) an ideal cutting out the set \(X^c\) of closed points in \(X = \mathrm{Spec}(A)\). Then every faithfully flat ind-étale map \(A \to B\) with \(B\) w-local and whose closed points of \(\operatorname{Spec}(B)\) are exactly \(V(IB)\) has a retraction.

Let \(A\) be a strictly henselian local ring. Let \(A \to B\) be an ind-étale ring map. Let \({\mathfrak {n}}\) be an maximal ideal of \(B\) lying over the maximal ideal \({\mathfrak {m}}\) of \(A\). Then the map \(A \to B_{\mathfrak {n}}\) is isomorphism.

For any ring \(A\), there exists an ind-étale faithfully flat \(A\)-algebra \(D\) with \(D\) w-contractible.

If a ring \(C\) is w-strictly local, then there exists an ind-Zariski faithfully flat \(C\)-algebra \(D\) with \(D\) w-contractible.

For any ring \(A\), there exists an ind-étale faithfully flat \(A\)-algebra \(B\) with \(B\) w-strictly local.

Let \(A \to B\) and \(B \to C\) be ind-Zariski ring maps. Then the composition \(A \to C\) is also ind-Zariski.

Let \(A \to B\) be an ind-Zariski ring map. Then it identifies local rings, i.e. for every prime \(\mathfrak {q} \subset B\) the canonical map \(A_{\varphi ^{-1}(\mathfrak {q})} \to B_{\mathfrak {q}}\) is an isomorphism.

Let \(A \to B\) be an ind-Zariski ring map. Then \(A \to B\) is flat.

Let \(A \to B\) be an ind-Zariski ring map. Then \(A \to B\) is ind-étale.

Let \(X\) be a topological space. Assume \(X\) is quasi-compact, quasi-separated and prespectral. Then for a subset \(T \subset X\) the following are equivalent:

1. \(T\) is an intersection of open and closed subsets of \(X\), and

2. \(T\) is closed in \(X\) and is a union of connected components of \(X\).

Let \(A \to B\) be a ring map such that

1. \(A \to B\) identifies local rings, and

2. \(\operatorname{Spec}(B) \to \operatorname{Spec}(A)\) is bijective.

Then \(A \to B\) is an isomorphism.

Let \(A \to B\) be a w-local ring map between w-local rings such that

1. \(A \to B\) identifies local rings, and

2. \(\pi _0(\operatorname{Spec}(B)) \to \pi _0(\operatorname{Spec}(A))\) is bijective.

Then \(A \to B\) is an isomorphism.

Let \(A \to B\) be a ring map which has going down. Let \({\mathfrak {p}}\subset A\) be a prime ideal and let \({\mathfrak {q}}\subset B\) be a prime ideal lying over \({\mathfrak {p}}\). Assume that and there is at most one prime of \(B\) above every prime of \(A\). Then the natural map \(B_{{\mathfrak {p}}B} \to B_{{\mathfrak {q}}}\) is an isomorphism.

Let \(A\) be a ring and let \(X = \operatorname{Spec}(A)\). Let \(T\) be profinite space and let \(f : T \to \pi _0(X)\) be a continuous map. Let \(S\) be a finite discrete quotient of \(T\) with quotient map \(q : T \to S\). Then the \(A^S\)-algebra defined in Definition 1.122 viewed as \(A\)-algebra through \(A \to A^S \to A^f_{q}\) is ind-Zariski. In particular, it identifies local rings. The map \(A^S \to A^f_{q}\) induces a homeomorphism of \(\operatorname{Spec}(A^f_{q})\) onto \((X \times S) \times _{\pi _0(X) \times S} Z = X \times _{\pi _0(X)} Z\).

Let \(A\) be a ring and let \(X = \operatorname{Spec}(A)\). Let \(T\) be a profinite space and let \(T \to \pi _0(X)\) be a continuous map. Then the \(A\)-algebra \(A^f_{\pi _0}\) defined in Definition 1.126 is ind-Zariski. Let \(Y = \operatorname{Spec}(A^f_{\pi _0})\). Then \(\pi _0(Y) \cong T\) and the diagram

is cartesian in the category of topological spaces.

Let \(A\) be a ring and let \(X = \operatorname{Spec}(A)\). Let \(T\) be profinite space and let \(f : T \to \pi _0(X)\) be a continuous map. Let \(S\), \(S'\) and \(S''\) be finite discrete spaces and let \(q : T \to S\), \(q' : T \to S'\) and \(q'' : T \to S''\) be quotient maps. Let \(g : S' \to S\) and \(g' : S'' \to S'\) be maps such that \(q = g \circ q'\) and \(q' = g' \circ q''\). Then the transition maps defined in Definition 1.124 satisfy

Let \(X\) be a totally disconnected topological space. Then \(X \simeq \pi _0 (X)\) as topological spaces.

Let \(X, Y\) be topological spaces. Then we have \(\pi _0 (X \times Y) \simeq \pi _0(X) \times \pi _0(Y)\) as topological spaces.

Let \(X\) be a spectral space. Then \(\pi _0(X)\) is a profinite space.

Let \(X\) be a topological space. Let \(\pi _0(X)\) be the topological space of connected components of \(X\). Let \(Y\) be a totally disconnected space. If every fiber of \(X \to Y\) is connected, then the map \(\pi _0(X) \to Y\) is injective.

Let \(X\) be a profinite space. Then \(X\) is spectral.

Let \(X\), \(Y\), and \(S\) be topological spaces. Assume that \(S\) is Hausdorff. Let \(f : X \to S\) and \(g : Y \to S\) be continuous maps. The natural map \(X \times _S Y \to X \to Y\) is a closed embedding.

Let \(X\) and \(Y\) be spectral spaces. Let \(S\) be a Hausdorff topological space. Let \(f : X \to S\) and \(g : Y \to S\) be continuous maps. Then the fiber product \(X \times _S Y\) is spectral.

Let \(X\) and \(Y\) be spectral spaces. Then the product \(X \times Y\) is spectral.

Let \(X\) be a topological space. Then the Stone-Čech compactification \(\beta (X)\) is extremally disconnected.

Let \(A, {\mathfrak {m}}\) be a strictly henselian local ring. Let \(f : A \to B\) a étale ring map with \({\mathfrak {n}}\) a maximal ideal of \(B\) lying over \({\mathfrak {m}}\). Then there exists a retraction \(s : B \to A\) of \(A \to B\) such that \({\mathfrak {n}}= s^{-1}({\mathfrak {m}})\).

If \(A \to B\) is a surjective ring map and \(A\) is w-local, then \(B\) is w-local.

If \(A\) is w-contractible, every ind-étale, faithfully flat map \(A \to B\) has a retraction.

Let \(f: X \to Y\) and \(g: Y \to Z\) be w-local maps between w-local spaces. Then the composition \(g \circ f: X \to Z\) is w-local.

Let \(A\) be a w-local ring. Then \(\operatorname{Spec}(A)\) is set theoretically the disjoint union of the spectra of the local rings at closed points.

Let \(A\) be a ring and \(I \subseteq A\) an ideal. The ring map \(A \to A_{w, I}\) is ind-Zariski.

Let \(R \to S\) be weakly étale. Then there exists a faithfully flat ind-étale morphism \(S \to T\) such that the composition \(R \to T\) is ind-étale.