Pro-etale Cohomology

Introduction

The material in this chapter and more can be found in the preprint [BS].

The goal of this chapter is to introduce the pro-étale topology and to develop the basic theory of cohomology of abelian sheaves in this topology. A secondary goal is to show how using the pro-étale topology simplifies the introduction of \(\ell\)-adic cohomology in algebraic geometry.

Here is a brief overview of the history of \(\ell\)-adic étale cohomology as we have understood it. In [SGA5, Exposés V and VI] Grothendieck et al developed a theory for dealing with \(\ell\)-adic sheaves as inverse systems of sheaves of \(\mathbf{Z}/\ell^n\mathbf{Z}\)-modules. In his second paper on the Weil conjectures ([WeilII]) Deligne introduced a derived category of \(\ell\)-adic sheaves as a certain 2-limit of categories of complexes of sheaves of \(\mathbf{Z}/\ell^n\mathbf{Z}\)-modules on the étale site of a scheme \(X\). This approach is used in the paper by Beilinson, Bernstein, and Deligne ([BBD]) as the basis for their beautiful theory of perverse sheaves. In a paper entitled “Continuous Étale Cohomology” ([Jannsen]) Uwe Jannsen discusses an important variant of the cohomology of a \(\ell\)-adic sheaf on a variety over a field. His paper is followed up by a paper of Torsten Ekedahl ([Ekedahl]) who discusses the adic formalism needed to work comfortably with derived categories defined as limits.

It turns out that, working with the pro-étale site of a scheme, one can avoid some of the technicalities these authors encountered. This comes at the expense of having to work with non-Noetherian schemes, even when one is only interested in working with \(\ell\)-adic sheaves and cohomology of such on varieties over an algebraically closed field.

A very important and remarkable feature of the (small) pro-étale site of a scheme is that it has enough quasi-compact w-contractible objects. The existence of these objects implies a number of useful and (perhaps) unusual consequences for the derived category of abelian sheaves and for inverse systems of sheaves. This is exactly the feature that will allow us to handle the intricacies of working with \(\ell\)-adic sheaves, but as we will see it has a number of other benefits as well.

Some topology

Some preliminaries. We have defined spectral spaces and spectral maps of spectral spaces in Topology, Section 08YF. The spectrum of a ring is a spectral space, see Algebra, Lemma 090M.

Lemma

Let \(X\) be a spectral space. Let \(X_0 \subset X\) be the set of closed points. The following are equivalent

1. Every open covering of \(X\) can be refined by a finite disjoint union decomposition \(X = \coprod U_i\) with \(U_i\) open and closed in \(X\).

2. The composition \(X_0 \to X \to \pi_0(X)\) is bijective.

Moreover, if \(X_0\) is closed in \(X\) and every point of \(X\) specializes to a unique point of \(X_0\), then these conditions are satisfied.

Proof

We will use without further mention that \(X_0\) is quasi-compact (Topology, Lemma 08ZM) and \(\pi_0(X)\) is profinite (Topology, Lemma 0906). Picture \[\xymatrix{ X_0 \ar[rd]_f \ar[r] & X \ar[d]^\pi \\ & \pi_0(X) }\] If (2) holds, the continuous bijective map \(f : X_0 \to \pi_0(X)\) is a homeomorphism by Topology, Lemma 08YE. Given an open covering \(X = \bigcup U_i\), we get an open covering \(\pi_0(X) = \bigcup f(X_0 \cap U_i)\). By Topology, Lemma 08ZZ we can find a finite open covering of the form \(\pi_0(X) = \coprod V_j\) which refines this covering. Since \(X_0 \to \pi_0(X)\) is bijective each connected component of \(X\) has a unique closed point, whence is equal to the set of points specializing to this closed point. Hence \(\pi^{-1}(V_j)\) is the set of points specializing to the points of \(f^{-1}(V_j)\). Now, if \(f^{-1}(V_j) \subset X_0 \cap U_i \subset U_i\), then it follows that \(\pi^{-1}(V_j) \subset U_i\) (because the open set \(U_i\) is closed under generalizations). In this way we see that the open covering \(X = \coprod \pi^{-1}(V_j)\) refines the covering we started out with. In this way we see that (2) implies (1).

Assume (1). Let \(x, y \in X\) be closed points. Then we have the open covering \(X = (X \setminus \{x\}) \cup (X \setminus \{y\})\). It follows from (1) that there exists a disjoint union decomposition \(X = U \amalg V\) with \(U\) and \(V\) open (and closed) and \(x \in U\) and \(y \in V\). In particular we see that every connected component of \(X\) has at most one closed point. By Topology, Lemma 005E every connected component (being closed) also does have a closed point. Thus \(X_0 \to \pi_0(X)\) is bijective. In this way we see that (1) implies (2).

Assume \(X_0\) is closed in \(X\) and every point specializes to a unique point of \(X_0\). Then \(X_0\) is a spectral space (Topology, Lemma 0902) consisting of closed points, hence profinite (Topology, Lemma 0905). Let \(x, y \in X_0\) be distinct. By Topology, Lemma 08ZZ we can find a disjoint union decomposition \(X_0 = U_0 \amalg V_0\) with \(U_0\) and \(V_0\) open and closed and \(x \in U_0\) and \(y \in V_0\). Let \(U \subset X\), resp. \(V \subset X\) be the set of points specializing to \(U_0\), resp. \(V_0\). Observe that \(X = U \amalg V\). By Topology, Lemma 0A31 we see that \(U\) is an intersection of quasi-compact open subsets. Hence \(U\) is closed in the constructible topology. Since \(U\) is closed under specialization, we see that \(U\) is closed by Topology, Lemma 0903. By symmetry \(V\) is closed and hence \(U\) and \(V\) are both open and closed. This proves that \(x, y\) are not in the same connected component of \(X\). In other words, \(X_0 \to \pi_0(X)\) is injective. The map is also surjective by Topology, Lemma 005E and the fact that connected components are closed. In this way we see that the final condition implies (2).

Example

Let \(T\) be a profinite space. Let \(t \in T\) be a point and assume that \(T \setminus \{t\}\) is not quasi-compact. Let \(X = T \times \{0, 1\}\). Consider the topology on \(X\) with a subbase given by the sets \(U \times \{0, 1\}\) for \(U \subset T\) open, \(X \setminus \{(t, 0)\}\), and \(U \times \{1\}\) for \(U \subset T\) open with \(t \not \in U\). The set of closed points of \(X\) is \(X_0 = T \times \{0\}\) and \((t, 1)\) is in the closure of \(X_0\). Moreover, \(X_0 \to \pi_0(X)\) is a bijection. This example shows that conditions (1) and (2) of Lemma 0968 do no imply the set of closed points is closed.

It turns out it is more convenient to work with spectral spaces which have the slightly stronger property mentioned in the final statement of Lemma 0968. We give this property a name.

Definition

A spectral space \(X\) is w-local if the set of closed points \(X_0\) is closed and every point of \(X\) specializes to a unique closed point. A continuous map \(f : X \to Y\) of w-local spaces is w-local if it is spectral and maps any closed point of \(X\) to a closed point of \(Y\).

We have seen in the proof of Lemma 0968 that in this case \(X_0 \to \pi_0(X)\) is a homeomorphism and that \(X_0 \cong \pi_0(X)\) is a profinite space. Moreover, a connected component of \(X\) is exactly the set of points specializing to a given \(x \in X_0\).

Lemma

Let \(X\) be a w-local spectral space. If \(Y \subset X\) is closed, then \(Y\) is w-local.

Proof

The subset \(Y_0 \subset Y\) of closed points is closed because \(Y_0 = X_0 \cap Y\). Since \(X\) is \(w\)-local, every \(y \in Y\) specializes to a unique point of \(X_0\). This specialization is in \(Y\), and hence also in \(Y_0\), because \(\overline{\{y\}}\subset Y\). In conclusion, \(Y\) is \(w\)-local.

Lemma

Let \(X\) be a spectral space. Let \[\xymatrix{ Y \ar[r] \ar[d] & T \ar[d] \\ X \ar[r] & \pi_0(X) }\] be a cartesian diagram in the category of topological spaces with \(T\) profinite. Then \(Y\) is spectral and \(T = \pi_0(Y)\). 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 \(X\).

Proof

Note that \(Y\) is a closed subspace of \(X \times T\) as \(\pi_0(X)\) is a profinite space hence Hausdorff (use Topology, Lemmas 0906 and 08ZH). Since \(X \times T\) is spectral (Topology, Lemma 0907) it follows that \(Y\) is spectral (Topology, Lemma 0902). Let \(Y \to \pi_0(Y) \to T\) be the canonical factorization (Topology, Lemma 08ZL). It is clear that \(\pi_0(Y) \to T\) is surjective. The fibres of \(Y \to T\) are homeomorphic to the fibres of \(X \to \pi_0(X)\). Hence these fibres are connected. It follows that \(\pi_0(Y) \to T\) is injective. We conclude that \(\pi_0(Y) \to T\) is a homeomorphism by Topology, Lemma 08YE.

Next, assume that \(X\) is w-local and let \(X_0 \subset X\) be the set of closed points. The inverse image \(Y_0 \subset Y\) of \(X_0\) in \(Y\) maps bijectively onto \(T\) as \(X_0 \to \pi_0(X)\) is a bijection by Lemma 0968. Moreover, \(Y_0\) is quasi-compact as a closed subset of the spectral space \(Y\). Hence \(Y_0 \to \pi_0(Y) = T\) is a homeomorphism by Topology, Lemma 08YE. It follows that all points of \(Y_0\) are closed in \(Y\). Conversely, if \(y \in Y\) is a closed point, then it is closed in the fibre of \(Y \to \pi_0(Y) = T\) and hence its image \(x\) in \(X\) is closed in the (homeomorphic) fibre of \(X \to \pi_0(X)\). This implies \(x \in X_0\) and hence \(y \in Y_0\). Thus \(Y_0\) is the collection of closed points of \(Y\) and for each \(y \in Y_0\) the set of generalizations of \(y\) is the fibre of \(Y \to \pi_0(Y)\). The lemma follows.

Local isomorphisms

We start with a definition.

Definition

Let \(\varphi : A \to B\) be a ring map.

1. We say \(A \to B\) is a local isomorphism if for every prime \(\mathfrak q \subset B\) there exists a \(g \in B\), \(g \not \in \mathfrak q\) such that \(A \to B_g\) induces an open immersion \(\Spec(B_g) \to \Spec(A)\).

2. We say \(A \to B\) 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.

We list some elementary properties.

Lemma

Let \(A \to B\) and \(A \to A'\) be ring maps. Let \(B' = B \otimes_A A'\) be the base change of \(B\).

1. If \(A \to B\) is a local isomorphism, then \(A' \to B'\) is a local isomorphism.

2. If \(A \to B\) identifies local rings, then \(A' \to B'\) identifies local rings.

Proof

Omitted.

Lemma

Let \(A \to B\) and \(B \to C\) be ring maps.

1. If \(A \to B\) and \(B \to C\) are local isomorphisms, then \(A \to C\) is a local isomorphism.

2. If \(A \to B\) and \(B \to C\) identify local rings, then \(A \to C\) identifies local rings.

Proof

Omitted.

Lemma

Let \(A\) be a ring. Let \(B \to C\) be an \(A\)-algebra homomorphism.

1. If \(A \to B\) and \(A \to C\) are local isomorphisms, then \(B \to C\) is a local isomorphism.

2. If \(A \to B\) and \(A \to C\) identify local rings, then \(B \to C\) identifies local rings.

Proof

Omitted.

Lemma

Let \(A \to B\) be a local isomorphism. Then

1. \(A \to B\) is étale,

2. \(A \to B\) identifies local rings,

3. \(A \to B\) is quasi-finite.

Proof

Omitted.

Lemma

Let \(A \to B\) be a local isomorphism. Then there exist \(n \geq 0\), \(g_1, \ldots, g_n \in B\), \(f_1, \ldots, f_n \in A\) such that \((g_1, \ldots, g_n) = B\) and \(A_{f_i} \cong B_{g_i}\).

Proof

Omitted.

Lemma

Let \(p : (Y, \mathcal{O}_Y) \to (X, \mathcal{O}_X)\) and \(q : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)\) be morphisms of locally ringed spaces. If \(\mathcal{O}_Y = p^{-1}\mathcal{O}_X\), then \[\Mor_{\text{LRS}/(X, \mathcal{O}_X)}((Z, \mathcal{O}_Z), (Y, \mathcal{O}_Y)) \longrightarrow \Mor_{\textit{Top}/X}(Z, Y),\quad (f, f^\sharp) \longmapsto f\] is bijective. Here \(\text{LRS}/(X, \mathcal{O}_X)\) is the category of locally ringed spaces over \(X\) and \(\textit{Top}/X\) is the category of topological spaces over \(X\).

Proof

This is immediate from the definitions.

Lemma

Let \(A\) be a ring. Set \(X = \Spec(A)\). The functor \[B \longmapsto \Spec(B)\] 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.

Proof

This follows from Lemma 096K and the fact that if \(A \to B\) identifies local rings, then the pullback of the structure sheaf of \(\Spec(A)\) via \(p : \Spec(B) \to \Spec(A)\) is equal to the structure sheaf of \(\Spec(B)\).

Ind-Zariski algebra

We start with a definition; please see Remark 0A0D for a comparison with the corresponding definition of the article [BS].

Definition

A ring map \(A \to B\) is said to be ind-Zariski if \(B\) can be written as a filtered colimit \(B = \colim B_i\) with each \(A \to B_i\) a local isomorphism.

An example of an Ind-Zariski map is a localization \(A \to S^{-1}A\), see Algebra, Lemma 00CR. The category of ind-Zariski algebras is closed under several natural operations.

Lemma

Let \(A \to B\) and \(A \to A'\) be ring maps. Let \(B' = B \otimes_A A'\) be the base change of \(B\). If \(A \to B\) is ind-Zariski, then \(A' \to B'\) is ind-Zariski.

Proof

Omitted.

Lemma

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

Proof

Omitted.

Lemma

Let \(A\) be a ring. Let \(B \to C\) be an \(A\)-algebra homomorphism. If \(A \to B\) and \(A \to C\) are ind-Zariski, then \(B \to C\) is ind-Zariski.

Proof

Omitted.

Lemma

A filtered colimit of ind-Zariski \(A\)-algebras is ind-Zariski over \(A\).

Proof

Omitted.

Lemma

Let \(A \to B\) be ind-Zariski. Then \(A \to B\) identifies local rings,

Proof

Omitted.

Constructing w-local affine schemes

An affine scheme \(X\) is called w-local if its underlying topological space is w-local (Definition 096A). It turns out given any ring \(A\) there is a canonical faithfully flat ind-Zariski ring map \(A \to A_w\) such that \(\Spec(A_w)\) is w-local. The key to constructing \(A_w\) is the following simple lemma.

Lemma

Let \(A\) be a ring. Set \(X = \Spec(A)\). Let \(Z \subset X\) be a locally closed subscheme which is of the form \(D(f) \cap V(I)\) for some \(f \in A\) and ideal \(I \subset A\). Then

1. there exists a multiplicative subset \(S \subset A\) such that \(\Spec(S^{-1}A)\) maps by a homeomorphism to the set of points of \(X\) specializing to \(Z\),

2. the \(A\)-algebra \(A_Z^\sim = S^{-1}A\) depends only on the underlying locally closed subset \(Z \subset X\),

3. \(Z\) is a closed subscheme of \(\Spec(A_Z^\sim)\),

If \(A \to A'\) is a ring map and \(Z' \subset X' = \Spec(A')\) is a locally closed subscheme of the same form which maps into \(Z\), then there is a unique \(A\)-algebra map \(A_Z^\sim \to (A')_{Z'}^\sim\).

Proof

Let \(S \subset A\) be the multiplicative set of elements which map to invertible elements of \(\Gamma(Z, \mathcal{O}_Z) = (A/I)_f\). If \(\mathfrak p\) is a prime of \(A\) which does not specialize to \(Z\), then \(\mathfrak p\) generates the unit ideal in \((A/I)_f\). Hence we can write \(f^n = g + h\) for some \(n \geq 0\), \(g \in \mathfrak p\), \(h \in I\). Then \(g \in S\) and we see that \(\mathfrak p\) is not in the spectrum of \(S^{-1}A\). Conversely, if \(\mathfrak p\) does specialize to \(Z\), say \(\mathfrak p \subset \mathfrak q \supset I\) with \(f \not \in \mathfrak q\), then we see that \(S^{-1}A\) maps to \(A_\mathfrak q\) and hence \(\mathfrak p\) is in the spectrum of \(S^{-1}A\). This proves (1).

The isomorphism class of the localization \(S^{-1}A\) depends only on the corresponding subset \(\Spec(S^{-1}A) \subset \Spec(A)\), whence (2) holds. By construction \(S^{-1}A\) maps surjectively onto \((A/I)_f\), hence (3). The final statement follows as the multiplicative subset \(S' \subset A'\) corresponding to \(Z'\) contains the image of the multiplicative subset \(S\).

Let \(A\) be a ring. Let \(E \subset A\) be a finite subset. We get a stratification of \(X = \Spec(A)\) into locally closed subschemes by looking at the vanishing behaviour of the elements of \(E\). More precisely, given a disjoint union decomposition \(E = E' \amalg E''\) we set [096W]\[\begin{equation} Z(E', E'') = \bigcap\nolimits_{f \in E'} D(f) \cap \bigcap\nolimits_{f \in E''} V(f) = D(\prod\nolimits_{f \in E'} f) \cap V( \sum\nolimits_{f \in E''} fA) \end{equation}\] The points of \(Z(E', E'')\) are exactly those \(x \in X\) such that \(f \in E'\) maps to a nonzero element in \(\kappa(x)\) and \(f \in E''\) maps to zero in \(\kappa(x)\). Thus it is clear that [096X]\[\begin{equation} X = \coprod\nolimits_{E = E' \amalg E''} Z(E', E'') \end{equation}\] set theoretically. Observe that each stratum is constructible.

Lemma

Let \(X = \Spec(A)\) as above. Given any finite stratification \(X = \coprod T_i\) by constructible subsets, there exists a finite subset \(E \subset A\) such that the stratification (096X) refines \(X = \coprod T_i\).

Proof

We may write \(T_i = \bigcup_j U_{i, j} \cap V_{i, j}^c\) as a finite union for some \(U_{i, j}\) and \(V_{i, j}\) quasi-compact open in \(X\). Then we may write \(U_{i, j} = \bigcup D(f_{i, j, k})\) and \(V_{i, j} = \bigcup D(g_{i, j, l})\). Then we set \(E = \{f_{i, j, k}\} \cup \{g_{i, j, l}\}\). This does the job, because the stratification (096X) is the one whose strata are labeled by the vanishing pattern of the elements of \(E\) which clearly refines the given stratification.

We continue the discussion. Given a finite subset \(E \subset A\) we set [096Z]\[\begin{equation} A_E = \prod\nolimits_{E = E' \amalg E''} A_{Z(E', E'')}^\sim \end{equation}\] with notation as in Lemma 096V. This makes sense because (096W) shows that each \(Z(E', E'')\) has the correct shape. We take the spectrum of this ring and denote it [0970]\[\begin{equation} X_E = \Spec(A_E) = \coprod\nolimits_{E = E' \amalg E''} X_{E', E''} \end{equation}\] with \(X_{E', E''} = \Spec(A_{Z(E', E'')}^\sim)\). Note that [0971]\[\begin{equation} Z_E = \coprod\nolimits_{E = E' \amalg E''} Z(E', E'') \longrightarrow X_E \end{equation}\] is a closed subscheme. By construction the closed subscheme \(Z_E\) contains all the closed points of the affine scheme \(X_E\) as every point of \(X_{E', E''}\) specializes to a point of \(Z(E', E'')\).

Let \(I(A)\) be the partially ordered set of all finite subsets of \(A\). This is a directed partially ordered set. For \(E_1 \subset E_2\) there is a canonical transition map \(A_{E_1} \to A_{E_2}\) of \(A\)-algebras. Namely, given a decomposition \(E_2 = E'_2 \amalg E''_2\) we set \(E'_1 = E_1 \cap E'_2\) and \(E''_1 = E_1 \cap E''_2\). Then observe that \(Z(E'_2, E''_2) \subset Z(E'_1, E''_1)\) hence a unique \(A\)-algebra map \(A_{Z(E'_1, E''_1)}^\sim \to A_{Z(E'_2, E''_2)}^\sim\) by Lemma 096V. Using these maps collectively we obtain the desired ring map \(A_{E_1} \to A_{E_2}\). Observe that the corresponding map of affine schemes [0972]\[\begin{equation} X_{E_2} \longrightarrow X_{E_1} \end{equation}\] maps \(Z_{E_2}\) into \(Z_{E_1}\). By uniqueness we obtain a system of \(A\)-algebras over \(I(A)\) and we set [0973]\[\begin{equation} A_w = \colim_{E \in I(A)} A_E \end{equation}\] This \(A\)-algebra is ind-Zariski and faithfully flat over \(A\). Finally, we set \(X_w = \Spec(A_w)\) and endow it with the closed subscheme \(Z = \lim_{E \in I(A)} Z_E\). In a formula [0974]\[\begin{equation} X_w = \lim_{E \in I(A)} X_E \supset Z = \lim_{E \in I(A)} Z_E \end{equation}\]

Lemma

Let \(X = \Spec(A)\) be an affine scheme. With \(A \to A_w\), \(X_w = \Spec(A_w)\), and \(Z \subset X_w\) as above.

1. \(A \to A_w\) is ind-Zariski and faithfully flat,

2. \(X_w \to X\) induces a bijection \(Z \to X\),

3. \(Z\) is the set of closed points of \(X_w\),

4. \(Z\) is a reduced scheme, and

5. every point of \(X_w\) specializes to a unique point of \(Z\).

In particular, \(X_w\) is w-local (Definition 096A).

Proof

The map \(A \to A_w\) is ind-Zariski by construction. For every \(E\) the morphism \(Z_E \to X\) is a bijection, hence (2). As \(Z \subset X_w\) we conclude \(X_w \to X\) is surjective and \(A \to A_w\) is faithfully flat by Algebra, Lemma 00HQ. This proves (1).

Suppose that \(y \in X_w\), \(y \not \in Z\). Then there exists an \(E\) such that the image of \(y\) in \(X_E\) is not contained in \(Z_E\). Then for all \(E \subset E'\) also \(y\) maps to an element of \(X_{E'}\) not contained in \(Z_{E'}\). Let \(T_{E'} \subset X_{E'}\) be the reduced closed subscheme which is the closure of the image of \(y\). It is clear that \(T = \lim_{E \subset E'} T_{E'}\) is the closure of \(y\) in \(X_w\). For every \(E \subset E'\) the scheme \(T_{E'} \cap Z_{E'}\) is nonempty by construction of \(X_{E'}\). Hence \(\lim T_{E'} \cap Z_{E'}\) is nonempty and we conclude that \(T \cap Z\) is nonempty. Thus \(y\) is not a closed point. It follows that every closed point of \(X_w\) is in \(Z\).

Suppose that \(y \in X_w\) specializes to \(z, z' \in Z\). We will show that \(z = z'\) which will finish the proof of (3) and will imply (5). Let \(x, x' \in X\) be the images of \(z\) and \(z'\). Since \(Z \to X\) is bijective it suffices to show that \(x = x'\). If \(x \not = x'\), then there exists an \(f \in A\) such that \(x \in D(f)\) and \(x' \in V(f)\) (or vice versa). Set \(E = \{f\}\) so that \[X_E = \Spec(A_f) \amalg \Spec(A_{V(f)}^\sim)\] Then we see that \(z\) and \(z'\) map \(x_E\) and \(x'_E\) which are in different parts of the given decomposition of \(X_E\) above. But then it impossible for \(x_E\) and \(x'_E\) to be specializations of a common point. This is the desired contradiction.

Recall that given a finite subset \(E \subset A\) we have \(Z_E\) is a disjoint union of the locally closed subschemes \(Z(E', E'')\) each isomorphic to the spectrum of \((A/I)_f\) where \(I\) is the ideal generated by \(E''\) and \(f\) the product of the elements of \(E'\). Any nilpotent element \(b\) of \((A/I)_f\) is the class of \(g/f^n\) for some \(g \in A\). Then setting \(E' = E \cup \{g\}\) the reader verifies that \(b\) is pulls back to zero under the transition map \(Z_{E'} \to Z_E\) of the system. This proves (4).

Remark

Let \(A\) be a ring. Let \(\kappa\) be an infinite cardinal bigger or equal than the cardinality of \(A\). Then the cardinality of \(A_w\) (Lemma 0975) is at most \(\kappa\). Namely, each \(A_E\) has cardinality at most \(\kappa\) and the set of finite subsets of \(A\) has cardinality at most \(\kappa\) as well. Thus the result follows as \(\kappa \otimes \kappa = \kappa\), see Sets, Section 000D.

Lemma

Let \(A\) be a ring. Let \(A \to A_w\) be the ring map constructed in Lemma 0975. For any ring map \(A \to B\) such that \(\Spec(B)\) is w-local, there is a unique factorization \(A \to A_w \to B\) such that \(\Spec(B) \to \Spec(A_w)\) is w-local.

Proof

Denote \(Y = \Spec(B)\) and \(Y_0 \subset Y\) the set of closed points. Denote \(f : Y \to X\) the given morphism. Recall that \(Y_0\) is profinite, in particular every constructible subset of \(Y_0\) is open and closed. Let \(E \subset A\) be a finite subset. Recall that \(A_w = \colim A_E\) and that the set of closed points of \(\Spec(A_w)\) is the limit of the closed subsets \(Z_E \subset X_E = \Spec(A_E)\). Thus it suffices to show there is a unique factorization \(A \to A_E \to B\) such that \(Y \to X_E\) maps \(Y_0\) into \(Z_E\). Since \(Z_E \to X = \Spec(A)\) is bijective, and since the strata \(Z(E', E'')\) are constructible we see that \[Y_0 = \coprod f^{-1}(Z(E', E'')) \cap Y_0\] is a disjoint union decomposition into open and closed subsets. As \(Y_0 = \pi_0(Y)\) we obtain a corresponding decomposition of \(Y\) into open and closed pieces. Thus it suffices to construct the factorization in case \(f(Y_0) \subset Z(E', E'')\) for some decomposition \(E = E' \amalg E''\). In this case \(f(Y)\) is contained in the set of points of \(X\) specializing to \(Z(E', E'')\) which is homeomorphic to \(X_{E', E''}\). Thus we obtain a unique continuous map \(Y \to X_{E', E''}\) over \(X\). By Lemma 096K this corresponds to a unique morphism of schemes \(Y \to X_{E', E''}\) over \(X\). This finishes the proof.

Recall that the spectrum of a ring is profinite if and only if every point is closed. There are in fact a whole slew of equivalent conditions that imply this. See Algebra, Lemma 04MG or Topology, Lemma 0905.

Lemma

Let \(A\) be a ring such that \(\Spec(A)\) is profinite. Let \(A \to B\) be a ring map. Then \(\Spec(B)\) is profinite in each of the following cases:

1. if \(\mathfrak q,\mathfrak q' \subset B\) lie over the same prime of \(A\), then neither \(\mathfrak q \subset \mathfrak q'\), nor \(\mathfrak q' \subset \mathfrak q\),

2. \(A \to B\) induces algebraic extensions of residue fields,

3. \(A \to B\) is a local isomorphism,

4. \(A \to B\) identifies local rings,

5. \(A \to B\) is weakly étale,

6. \(A \to B\) is quasi-finite,

7. \(A \to B\) is unramified,

8. \(A \to B\) is étale,

9. \(B\) is a filtered colimit of \(A\)-algebras as in (1) – (8),

10. etc.

Proof

By the references mentioned above (Algebra, Lemma 04MG or Topology, Lemma 0905) there are no specializations between distinct points of \(\Spec(A)\) and \(\Spec(B)\) is profinite if and only if there are no specializations between distinct points of \(\Spec(B)\). These specializations can only happen in the fibres of \(\Spec(B) \to \Spec(A)\). In this way we see that (1) is true.

The assumption in (2) implies all primes of \(B\) are maximal by Algebra, Lemma 00GA. Thus (2) holds. If \(A \to B\) is a local isomorphism or identifies local rings, then the residue field extensions are trivial, so (3) and (4) follow from (2). If \(A \to B\) is weakly étale, then More on Algebra, Lemma 092R tells us it induces separable algebraic residue field extensions, so (5) follows from (2). If \(A \to B\) is quasi-finite, then the fibres are finite discrete topological spaces. Hence (6) follows from (1). Hence (3) follows from (1). Cases (7) and (8) follow from this as unramified and étale ring map are quasi-finite (Algebra, Lemmas 02UR and 00U5). If \(B = \colim B_i\) is a filtered colimit of \(A\)-algebras, then \(\Spec(B) = \lim \Spec(B_i)\) in the category of topological spaces by Limits, Lemma 0CUF. Hence if each \(\Spec(B_i)\) is profinite, so is \(\Spec(B)\) by Topology, Lemma 0ET8. This proves (9).

Lemma

Let \(A\) be a ring. Let \(V(I) \subset \Spec(A)\) be a closed subset which is a profinite topological space. Then there exists an ind-Zariski ring map \(A \to B\) such that \(\Spec(B)\) is w-local, the set of closed points is \(V(IB)\), and \(A/I \cong B/IB\).

Proof

Let \(A \to A_w\) and \(Z \subset Y = \Spec(A_w)\) as in Lemma 0975. Let \(T \subset Z\) be the inverse image of \(V(I)\). Then \(T \to V(I)\) is a homeomorphism by Topology, Lemma 08YE. Let \(B = (A_w)_T^\sim\), see Lemma 096V. It is clear that \(B\) is w-local with closed points \(V(IB)\). The ring map \(A/I \to B/IB\) is ind-Zariski and induces a homeomorphism on underlying topological spaces. Hence it is an isomorphism by Lemma 096L.

Lemma

Let \(A\) be a ring such that \(X = \Spec(A)\) is w-local. Let \(I \subset A\) be the radical ideal cutting out the set \(X_0\) of closed points in \(X\). Let \(A \to B\) be a ring map inducing algebraic extensions on residue fields at primes. Then

1. every point of \(Z = V(IB)\) is a closed point of \(\Spec(B)\),

2. there exists an ind-Zariski ring map \(B \to C\) such that

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

   2. the space \(Y = \Spec(C)\) is w-local,

   3. the induced map \(p : Y \to X\) is w-local, and

   4. \(p^{-1}(X_0)\) is the set of closed points of \(Y\).

Proof

By Lemma 0978 applied to \(A/I \to B/IB\) all points of \(Z = V(IB) = \Spec(B/IB)\) are closed, in fact \(\Spec(B/IB)\) is a profinite space. To finish the proof we apply Lemma 0979 to \(IB \subset B\).

Identifying local rings versus ind-Zariski

An ind-Zariski ring map \(A \to B\) identifies local rings (Lemma 096T). The converse does not hold (Examples, Section 09AN). However, it turns out that there is a kind of structure theorem for ring maps which identify local rings in terms of ind-Zariski ring maps, see Proposition 097G.

Let \(A\) be a ring. Let \(X = \Spec(A)\). The space of connected components \(\pi_0(X)\) is a profinite space by Topology, Lemma 0906 (and Algebra, Lemma 090M).

Lemma

Let \(A\) be a ring. Let \(X = \Spec(A)\). Let \(T \subset \pi_0(X)\) be a closed subset. There exists a surjective ind-Zariski ring map \(A \to B\) such that \(\Spec(B) \to \Spec(A)\) induces a homeomorphism of \(\Spec(B)\) with the inverse image of \(T\) in \(X\).

Proof

Let \(Z \subset X\) be the inverse image of \(T\). Then \(Z\) is the intersection \(Z = \bigcap Z_\alpha\) of the open and closed subsets of \(X\) containing \(Z\), see Topology, Lemma 04PL. For each \(\alpha\) we have \(Z_\alpha = \Spec(A_\alpha)\) where \(A \to A_\alpha\) is a local isomorphism (a localization at an idempotent). Setting \(B = \colim A_\alpha\) proves the lemma.

Lemma

Let \(A\) be a ring and let \(X = \Spec(A)\). Let \(T\) be a profinite space and let \(T \to \pi_0(X)\) be a continuous map. There exists an ind-Zariski ring map \(A \to B\) such that with \(Y = \Spec(B)\) the diagram \[\xymatrix{ Y \ar[r] \ar[d] & \pi_0(Y) \ar[d] \\ X \ar[r] & \pi_0(X) }\] is cartesian in the category of topological spaces and such that \(\pi_0(Y) = T\) as spaces over \(\pi_0(X)\).

Proof

Namely, write \(T = \lim T_i\) as the limit of an inverse system finite discrete spaces over a directed set (see Topology, Lemma 08ZY). For each \(i\) let \(Z_i = \Im(T \to \pi_0(X) \times T_i)\). This is a closed subset. Observe that \(X \times T_i\) is the spectrum of \(A_i = \prod_{t \in T_i} A\) and that \(A \to A_i\) is a local isomorphism. By Lemma 097C we see that \(Z_i \subset \pi_0(X \times T_i) = \pi_0(X) \times T_i\) corresponds to a surjection \(A_i \to B_i\) which is ind-Zariski such that \(\Spec(B_i) = X \times_{\pi_0(X)} Z_i\) as subsets of \(X \times T_i\). The transition maps \(T_i \to T_{i'}\) induce maps \(Z_i \to Z_{i'}\) and \(X \times_{\pi_0(X)} Z_i \to X \times_{\pi_0(X)} Z_{i'}\). Hence ring maps \(B_{i'} \to B_i\) (Lemmas 096L and 096T). Set \(B = \colim B_i\). Because \(T = \lim Z_i\) we have \(X \times_{\pi_0(X)} T = \lim X \times_{\pi_0(X)} Z_i\) and hence \(Y = \Spec(B) = \lim \Spec(B_i)\) fits into the cartesian diagram \[\xymatrix{ Y \ar[r] \ar[d] & T \ar[d] \\ X \ar[r] & \pi_0(X) }\] of topological spaces. By Lemma 096C we conclude that \(T = \pi_0(Y)\).

Example

Let \(k\) be a field. Let \(T\) be a profinite topological space. There exists an ind-Zariski ring map \(k \to A\) such that \(\Spec(A)\) is homeomorphic to \(T\). Namely, just apply Lemma 097D to \(T \to \pi_0(\Spec(k)) = \{*\}\). In fact, in this case we have \[A = \colim \text{Map}(T_i, k)\] whenever we write \(T = \lim T_i\) as a filtered limit with each \(T_i\) finite.

Lemma

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

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

2. the topological spaces \(\Spec(B)\), \(\Spec(A)\) are w-local,

3. \(\Spec(B) \to \Spec(A)\) is w-local, and

4. \(\pi_0(\Spec(B)) \to \pi_0(\Spec(A))\) is bijective.

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

Proof

Let \(X_0 \subset X = \Spec(A)\) and \(Y_0 \subset Y = \Spec(B)\) be the sets of closed points. By assumption \(Y_0\) maps into \(X_0\) and the induced map \(Y_0 \to X_0\) is a bijection. As a space \(\Spec(A)\) is the disjoint union of the spectra of the local rings of \(A\) at closed points. Similarly for \(B\). Hence \(X \to Y\) is a bijection. Since \(A \to B\) is flat we have going down (Algebra, Lemma 00HS). Thus Algebra, Lemma 00EA shows for any prime \(\mathfrak q \subset B\) lying over \(\mathfrak p \subset A\) we have \(B_\mathfrak q = B_\mathfrak p\). Since \(B_\mathfrak q = A_\mathfrak p\) by assumption, we see that \(A_\mathfrak p = B_\mathfrak p\) for all primes \(\mathfrak p\) of \(A\). Thus \(A = B\) by Algebra, Lemma 00HN.

Lemma

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

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

2. the topological spaces \(\Spec(B)\), \(\Spec(A)\) are w-local, and

3. \(\Spec(B) \to \Spec(A)\) is w-local.

Then \(A \to B\) is ind-Zariski.

Proof

Set \(X = \Spec(A)\) and \(Y = \Spec(B)\). Let \(X_0 \subset X\) and \(Y_0 \subset Y\) be the set of closed points. Let \(A \to A'\) be the ind-Zariski morphism of affine schemes such that with \(X' = \Spec(A')\) the diagram \[\xymatrix{ X' \ar[r] \ar[d] & \pi_0(X') \ar[d] \\ X \ar[r] & \pi_0(X) }\] is cartesian in the category of topological spaces and such that \(\pi_0(X') = \pi_0(Y)\) as spaces over \(\pi_0(X)\), see Lemma 097D. By Lemma 096C we see that \(X'\) is w-local and the set of closed points \(X'_0 \subset X'\) is the inverse image of \(X_0\).

We obtain a continuous map \(Y \to X'\) of underlying topological spaces over \(X\) identifying \(\pi_0(Y)\) with \(\pi_0(X')\). By Lemma 096L (and Lemma 096T) this corresponds to a morphism of affine schemes \(Y \to X'\) over \(X\). Since \(Y \to X\) maps \(Y_0\) into \(X_0\) we see that \(Y \to X'\) maps \(Y_0\) into \(X'_0\), i.e., \(Y \to X'\) is w-local. By Lemma 097E we see that \(Y \cong X'\) and we win.

The following proposition is a warm up for the type of result we will prove later.

Proposition

Let \(A \to B\) be a ring map which identifies local rings. Then there exists a faithfully flat, ind-Zariski ring map \(B \to B'\) such that \(A \to B'\) is ind-Zariski.

Proof

Let \(A \to A_w\), resp. \(B \to B_w\) be the faithfully flat, ind-Zariski ring map constructed in Lemma 0975 for \(A\), resp. \(B\). Since \(\Spec(B_w)\) is w-local, there exists a unique factorization \(A \to A_w \to B_w\) such that \(\Spec(B_w) \to \Spec(A_w)\) is w-local by Lemma 0977. Note that \(A_w \to B_w\) identifies local rings, see Lemma 096H. By Lemma 097F this means \(A_w \to B_w\) is ind-Zariski. Since \(B \to B_w\) is faithfully flat, ind-Zariski (Lemma 0975) and the composition \(A \to B \to B_w\) is ind-Zariski (Lemma 096Q) the proposition is proved.

The proposition above allows us to characterize the affine, weakly contractible objects in the pro-Zariski site of an affine scheme.

Lemma

Let \(A\) be a ring. The following are equivalent

1. every faithfully flat ring map \(A \to B\) identifying local rings has a retraction,

2. every faithfully flat ind-Zariski ring map \(A \to B\) has a retraction, and

3. \(A\) satisfies

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

   2. \(\pi_0(\Spec(A))\) is extremally disconnected.

Proof

The equivalence of (1) and (2) follows immediately from Proposition 097G.

Assume (3)(a) and (3)(b). Let \(A \to B\) be faithfully flat and ind-Zariski. We will use without further mention the fact that a flat map \(A \to B\) is faithfully flat if and only if every closed point of \(\Spec(A)\) is in the image of \(\Spec(B) \to \Spec(A)\). We will show that \(A \to B\) has a retraction.

Let \(I \subset A\) be an ideal such that \(V(I) \subset \Spec(A)\) is the set of closed points of \(\Spec(A)\). We may replace \(B\) by the ring \(C\) constructed in Lemma 097A for \(A \to B\) and \(I \subset A\). Thus we may assume \(\Spec(B)\) is w-local such that the set of closed points of \(\Spec(B)\) is \(V(IB)\).

Assume \(\Spec(B)\) is w-local and the set of closed points of \(\Spec(B)\) is \(V(IB)\). Choose a continuous section to the surjective continuous map \(V(IB) \to V(I)\). This is possible as \(V(I) \cong \pi_0(\Spec(A))\) is extremally disconnected, see Topology, Proposition 08YN. The image is a closed subspace \(T \subset \pi_0(\Spec(B)) \cong V(IB)\) mapping homeomorphically onto \(\pi_0(A)\). Replacing \(B\) by the ind-Zariski quotient ring constructed in Lemma 097C we see that we may assume \(\pi_0(\Spec(B)) \to \pi_0(\Spec(A))\) is bijective. At this point \(A \to B\) is an isomorphism by Lemma 097E.

Assume (1) or equivalently (2). Let \(A \to A_w\) be the ring map constructed in Lemma 0975. By (1) there is a retraction \(A_w \to A\). Thus \(\Spec(A)\) is homeomorphic to a closed subset of \(\Spec(A_w)\). By Lemma 096B we see (3)(a) holds. Finally, let \(T \to \pi_0(A)\) be a surjective map with \(T\) an extremally disconnected, quasi-compact, Hausdorff topological space (Topology, Lemma 090D). Choose \(A \to B\) as in Lemma 097D adapted to \(T \to \pi_0(\Spec(A))\). By (1) there is a retraction \(B \to A\). Thus we see that \(T = \pi_0(\Spec(B)) \to \pi_0(\Spec(A))\) has a section. A formal categorical argument, using Topology, Proposition 08YN, implies that \(\pi_0(\Spec(A))\) is extremally disconnected.

Lemma

Let \(A\) be a ring. There exists a faithfully flat, ind-Zariski ring map \(A \to B\) such that \(B\) satisfies the equivalent conditions of Lemma 09AZ.

Proof

We first apply Lemma 0975 to see that we may assume that \(\Spec(A)\) is w-local. Choose an extremally disconnected space \(T\) and a surjective continuous map \(T \to \pi_0(\Spec(A))\), see Topology, Lemma 090D. Note that \(T\) is profinite. Apply Lemma 097D to find an ind-Zariski ring map \(A \to B\) such that \(\pi_0(\Spec(B)) \to \pi_0(\Spec(A))\) realizes \(T \to \pi_0(\Spec(A))\) and such that \[\xymatrix{ \Spec(B) \ar[r] \ar[d] & \pi_0(\Spec(B)) \ar[d] \\ \Spec(A) \ar[r] & \pi_0(\Spec(A)) }\] is cartesian in the category of topological spaces. Note that \(\Spec(B)\) is w-local, that \(\Spec(B) \to \Spec(A)\) is w-local, and that the set of closed points of \(\Spec(B)\) is the inverse image of the set of closed points of \(\Spec(A)\), see Lemma 096C. Thus condition (3) of Lemma 09AZ holds for \(B\).

Remark

In each of Lemmas 097C, 097D, Proposition 097G, and Lemma 09B0 we find an ind-Zariski ring map with some properties. In the paper [BS] the authors use the notion of an ind-(Zariski localization) which is a filtered colimit of finite products of principal localizations. It is possible to replace ind-Zariski by ind-(Zariski localization) in each of the results listed above. However, we do not need this and the notion of an ind-Zariski homomorphism of rings as defined here has slightly better formal properties. Moreover, the notion of an ind-Zariski ring map is the natural analogue of the notion of an ind-étale ring map defined in the next section.

Ind-étale algebra

We start with a definition.

Definition

A ring map \(A \to B\) is said to be ind-étale if \(B\) can be written as a filtered colimit of étale \(A\)-algebras.

The category of ind-étale algebras is closed under a number of natural operations.

Lemma

Let \(A \to B\) and \(A \to A'\) be ring maps. Let \(B' = B \otimes_A A'\) be the base change of \(B\). If \(A \to B\) is ind-étale, then \(A' \to B'\) is ind-étale.

Proof

This is Algebra, Lemma 0BSH.

Lemma

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.

Proof

This is Algebra, Lemma 0BSI.

Lemma

A filtered colimit of ind-étale \(A\)-algebras is ind-étale over \(A\).

Proof

This is Algebra, Lemma 0BSJ.

Lemma

Let \(A\) be a ring. Let \(B \to C\) be an \(A\)-algebra map of ind-étale \(A\)-algebras. Then \(C\) is an ind-étale \(B\)-algebra.

Proof

This is Algebra, Lemma 08HS.

Lemma

Let \(A \to B\) be ind-étale. Then \(A \to B\) is weakly étale (More on Algebra, Definition 092B).

Proof

This follows from More on Algebra, Lemma 092N.

Lemma

Let \(A\) be a ring and let \(I \subset A\) be an ideal. The base change functor \[\text{ind-\'etale }A\text{-algebras} \longrightarrow \text{ind-\'etale }A/I\text{-algebras},\quad C \longmapsto C/IC\] has a fully faithful right adjoint \(v\). In particular, given an ind-étale \(A/I\)-algebra \(\overline{C}\) there exists an ind-étale \(A\)-algebra \(C = v(\overline{C})\) such that \(\overline{C} = C/IC\).

Proof

Let \(\overline{C}\) be an ind-étale \(A/I\)-algebra. Consider the category \(\mathcal{C}\) of factorizations \(A \to B \to \overline{C}\) where \(A \to B\) is étale. (We ignore some set theoretical issues in this proof.) We will show that this category is directed and that \(C = \colim_\mathcal{C} B\) is an ind-étale \(A\)-algebra such that \(\overline{C} = C/IC\).

We first prove that \(\mathcal{C}\) is directed (Categories, Definition 002V). The category is nonempty as \(A \to A \to \overline{C}\) is an object. Suppose that \(A \to B \to \overline{C}\) and \(A \to B' \to \overline{C}\) are two objects of \(\mathcal{C}\). Then \(A \to B \otimes_A B' \to \overline{C}\) is another (use Algebra, Lemma 00U2). Suppose that \(f, g : B \to B'\) are two maps between objects \(A \to B \to \overline{C}\) and \(A \to B' \to \overline{C}\) of \(\mathcal{C}\). Then a coequalizer is \(A \to B' \otimes_{f, B, g} B' \to \overline{C}\). This is an object of \(\mathcal{C}\) by Algebra, Lemmas 00U2 and 00U7. Thus the category \(\mathcal{C}\) is directed.

Write \(\overline{C} = \colim \overline{B_i}\) as a filtered colimit with \(\overline{B_i}\) étale over \(A/I\). For every \(i\) there exists \(A \to B_i\) étale with \(\overline{B_i} = B_i/IB_i\), see Algebra, Lemma 04D1. Thus \(C \to \overline{C}\) is surjective. Since \(C/IC \to \overline{C}\) is ind-étale (Lemma 097M) we see that it is flat. Hence \(\overline{C}\) is a localization of \(C/IC\) at some multiplicative subset \(S \subset C/IC\) (Algebra, Lemma 04PS). Take an \(f \in C\) mapping to an element of \(S \subset C/IC\). Choose \(A \to B \to \overline{C}\) in \(\mathcal{C}\) and \(g \in B\) mapping to \(f\) in the colimit. Then we see that \(A \to B_g \to \overline{C}\) is an object of \(\mathcal{C}\) as well. Thus \(f\) is an invertible element of \(C\). It follows that \(C/IC = \overline{C}\).

Next, we claim that for an ind-étale algebra \(D\) over \(A\) we have \[\Mor_A(D, C) = \Mor_{A/I}(D/ID, \overline{C})\] Namely, let \(D/ID \to \overline{C}\) be an \(A/I\)-algebra map. Write \(D = \colim_{i \in I} D_i\) as a colimit over a directed set \(I\) with \(D_i\) étale over \(A\). By choice of \(\mathcal{C}\) we obtain a transformation \(I \to \mathcal{C}\) and hence a map \(D \to C\) compatible with maps to \(\overline{C}\). Whence the claim.

It follows that the functor \(v\) defined by the rule \[\overline{C} \longmapsto v(\overline{C}) = \colim_{A \to B \to \overline{C}} B\] is a right adjoint to the base change functor \(u\) as required by the lemma. The functor \(v\) is fully faithful because \(u \circ v = \text{id}\) by construction, see Categories, Lemma 07RB.

Constructing ind-étale algebras

Let \(A\) be a ring. Recall that any étale ring map \(A \to B\) is isomorphic to a standard smooth ring map of relative dimension \(0\). Such a ring map is of the form \[A \longrightarrow A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)\] where the determinant of the \(n \times n\)-matrix with entries \(\partial f_i/\partial x_j\) is invertible in the quotient ring. See Algebra, Lemma 00U9.

Let \(S(A)\) be the set of all faithfully flat¹ standard smooth \(A\)-algebras of relative dimension \(0\). Let \(I(A)\) be the partially ordered (by inclusion) set of finite subsets \(E\) of \(S(A)\). Note that \(I(A)\) is a directed partially ordered set. For \(E = \{A \to B_1, \ldots, A \to B_n\}\) set \[B_E = B_1 \otimes_A \ldots \otimes_A B_n\] Observe that \(B_E\) is a faithfully flat étale \(A\)-algebra. For \(E \subset E'\), there is a canonical transition map \(B_E \to B_{E'}\) of étale \(A\)-algebras. Namely, say \(E = \{A \to B_1, \ldots, A \to B_n\}\) and \(E' = \{A \to B_1, \ldots, A \to B_{n + m}\}\) then \(B_E \to B_{E'}\) sends \(b_1 \otimes \ldots \otimes b_n\) to the element \(b_1 \otimes \ldots \otimes b_n \otimes 1 \otimes \ldots \otimes 1\) of \(B_{E'}\). This construction defines a system of faithfully flat étale \(A\)-algebras over \(I(A)\) and we set \[T(A) = \colim_{E \in I(A)} B_E\] Observe that \(T(A)\) is a faithfully flat ind-étale \(A\)-algebra (Algebra, Lemma 090N). By construction given any faithfully flat étale \(A\)-algebra \(B\) there is a (non-unique) \(A\)-algebra map \(B \to T(A)\). Namely, pick some \((A \to B_0) \in S(A)\) and an isomorphism \(B \cong B_0\). Then the canonical coprojection \[B \to B_0 \to T(A) = \colim_{E \in I(A)} B_E\] is the desired map.

Lemma

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

Proof

Set \(T^1(A) = T(A)\) and \(T^{n + 1}(A) = T(T^n(A))\). Let \[C = \colim T^n(A)\] This algebra is faithfully flat over each \(T^n(A)\) and in particular over \(A\), see Algebra, Lemma 090N. Moreover, \(C\) is ind-étale over \(A\) by Lemma 097L. If \(C \to B\) is étale, then there exists an \(n\) and an étale ring map \(T^n(A) \to B'\) such that \(B = C \otimes_{T^n(A)} B'\), see Algebra, Lemma 00U2. If \(C \to B\) is faithfully flat, then \(\Spec(B) \to \Spec(C) \to \Spec(T^n(A))\) is surjective, hence \(\Spec(B') \to \Spec(T^n(A))\) is surjective. In other words, \(T^n(A) \to B'\) is faithfully flat. By our construction, there is a \(T^n(A)\)-algebra map \(B' \to T^{n + 1}(A)\). This induces a \(C\)-algebra map \(B \to C\) which finishes the proof.

Remark

Let \(A\) be a ring. Let \(\kappa\) be an infinite cardinal bigger or equal than the cardinality of \(A\). Then the cardinality of \(T(A)\) is at most \(\kappa\). Namely, each \(B_E\) has cardinality at most \(\kappa\) and the index set \(I(A)\) has cardinality at most \(\kappa\) as well. Thus the result follows as \(\kappa \otimes \kappa = \kappa\), see Sets, Section 000D. It follows that the ring constructed in the proof of Lemma 097R has cardinality at most \(\kappa\) as well.

Remark

The construction \(A \mapsto T(A)\) is functorial in the following sense: If \(A \to A'\) is a ring map, then we can construct a commutative diagram \[\xymatrix{ A \ar[r] \ar[d] & T(A) \ar[d] \\ A' \ar[r] & T(A') }\] Namely, given \((A \to A[x_1, \ldots, x_n]/(f_1, \ldots, f_n))\) in \(S(A)\) we can use the ring map \(\varphi : A \to A'\) to obtain a corresponding element \((A' \to A'[x_1, \ldots, x_n]/(f^\varphi_1, \ldots, f^\varphi_n))\) of \(S(A')\) where \(f^\varphi\) means the polynomial obtained by applying \(\varphi\) to the coefficients of the polynomial \(f\). Moreover, there is a commutative diagram \[\xymatrix{ A \ar[r] \ar[d] & A[x_1, \ldots, x_n]/(f_1, \ldots, f_n) \ar[d] \\ A' \ar[r] & A'[x_1, \ldots, x_n]/(f^\varphi_1, \ldots, f^\varphi_n) }\] in the category of rings. For \(E \subset S(A)\) finite, set \(E' = \varphi(E)\) and define \(B_E \to B_{E'}\) in the obvious manner. Taking the colimit gives the desired map \(T(A) \to T(A')\), see Categories, Lemma 002K.

Lemma

Let \(A\) be a ring such that every faithfully flat étale ring map \(A \to B\) has a retraction. Then the same is true for every quotient ring \(A/I\).

Proof

Let \(A/I \to \overline{B}\) be faithfully flat étale. By Algebra, Lemma 04D1 we can write \(\overline{B} = B/IB\) for some étale ring map \(A \to B\). The image \(U\) of \(\Spec(B) \to \Spec(A)\) is open and contains \(V(I)\). Hence the complement \(Z = \Spec(A) \setminus U\) is quasi-compact and disjoint from \(V(I)\). Hence \(Z \subset D(f_1) \cup \ldots \cup D(f_r)\) for some \(r \geq 0\) and \(f_i \in I\). Then \(A \to B' = B \times \prod A_{f_i}\) is faithfully flat étale and \(\overline{B} = B'/IB'\). Hence the retraction \(B' \to A\) to \(A \to B'\), induces a retraction to \(A/I \to \overline{B}\).

Lemma

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.

Proof

Let \(\mathfrak m\) be a maximal ideal of \(A\). Let \(A \to B\) be an étale ring map and let \(\mathfrak q \subset B\) be a prime lying over \(\mathfrak m\). By the description of the strict henselization \(A_\mathfrak m^{sh}\) in Algebra, Lemma 04GW it suffices to show that \(A_\mathfrak m = B_\mathfrak q\). Note that there are finitely many primes \(\mathfrak q = \mathfrak q_1, \mathfrak q_2, \ldots, \mathfrak q_n\) lying over \(\mathfrak m\) and there are no specializations between them as an étale ring map is quasi-finite, see Algebra, Lemma 00U5. Thus \(\mathfrak q_i\) is a maximal ideal and we can find \(g \in \mathfrak q_2 \cap \ldots \cap \mathfrak q_n\), \(g \not \in \mathfrak q\) (Algebra, Lemma 00DS). After replacing \(B\) by \(B_g\) we see that \(\mathfrak q\) is the only prime of \(B\) lying over \(\mathfrak m\). The image \(U \subset \Spec(A)\) of \(\Spec(B) \to \Spec(A)\) is open (Algebra, Proposition 00I1). Thus the complement \(\Spec(A) \setminus U\) is closed and we can find \(f \in A\), \(f \not \in \mathfrak p\) such that \(\Spec(A) = U \cup D(f)\). The ring map \(A \to B \times A_f\) is faithfully flat and étale, hence has a retraction \(\sigma : B \times A_f \to A\) by assumption on \(A\). Observe that \(\sigma\) is étale, hence flat as a map between étale \(A\)-algebras (Algebra, Lemma 00U7). Since \(\mathfrak q\) is the only prime of \(B \times A_f\) lying over \(A\) we find that \(A_\mathfrak p \to B_\mathfrak q\) has a retraction which is also flat. Thus \(A_\mathfrak p \to B_\mathfrak q \to A_\mathfrak p\) are flat local ring maps whose composition is the identity. Since a flat local homomorphism of local rings is injective we conclude these maps are isomorphisms as desired.

Lemma

Let \(A\) be a ring such that every faithfully flat étale ring map \(A \to B\) has a retraction. Let \(Z \subset \Spec(A)\) be a closed subscheme. Let \(A \to A_Z^\sim\) be as constructed in Lemma 096V. Then every faithfully flat étale ring map \(A_Z^\sim \to C\) has a retraction.

Proof

There exists an étale ring map \(A \to B'\) such that \(C = B' \otimes_A A_Z^\sim\) as \(A_Z^\sim\)-algebras. The image \(U' \subset \Spec(A)\) of \(\Spec(B') \to \Spec(A)\) is open and contains \(V(I)\), hence we can find \(f \in I\) such that \(\Spec(A) = U' \cup D(f)\). Then \(A \to B' \times A_f\) is étale and faithfully flat. By assumption there is a retraction \(B' \times A_f \to A\). Localizing we obtain the desired retraction \(C \to A_Z^\sim\).

Lemma

Let \(A \to B\) be a ring map inducing algebraic extensions on residue fields. There exists a commutative diagram \[\xymatrix{ B \ar[r] & D \\ A \ar[r] \ar[u] & C \ar[u] }\] with the following properties:

1. \(A \to C\) is faithfully flat and ind-étale,

2. \(B \to D\) is faithfully flat and ind-étale,

3. \(\Spec(C)\) is w-local,

4. \(\Spec(D)\) is w-local,

5. \(\Spec(D) \to \Spec(C)\) is w-local,

6. the set of closed points of \(\Spec(D)\) is the inverse image of the set of closed points of \(\Spec(C)\),

7. the set of closed points of \(\Spec(C)\) surjects onto \(\Spec(A)\),

8. the set of closed points of \(\Spec(D)\) surjects onto \(\Spec(B)\),

9. for \(\mathfrak m \subset C\) maximal the local ring \(C_\mathfrak m\) is strictly henselian.

Proof

There is a faithfully flat, ind-Zariski ring map \(A \to A'\) such that \(\Spec(A')\) is w-local and such that the set of closed points of \(\Spec(A')\) maps onto \(\Spec(A)\), see Lemma 0975. Let \(I \subset A'\) be the ideal such that \(V(I)\) is the set of closed points of \(\Spec(A')\). Choose \(A' \to C'\) as in Lemma 097R. Note that the local rings \(C'_{\mathfrak m'}\) at maximal ideals \(\mathfrak m' \subset C'\) are strictly henselian by Lemma 097V. We apply Lemma 097A to \(A' \to C'\) and \(I \subset A'\) to get \(C' \to C\) with \(C'/IC' \cong C/IC\). Note that since \(A' \to C'\) is faithfully flat, \(\Spec(C'/IC')\) surjects onto the set of closed points of \(A'\) and in particular onto \(\Spec(A)\). Moreover, as \(V(IC) \subset \Spec(C)\) is the set of closed points of \(C\) and \(C' \to C\) is ind-Zariski (and identifies local rings) we obtain properties (1), (3), (7), and (9).

Denote \(J \subset C\) the ideal such that \(V(J)\) is the set of closed points of \(\Spec(C)\). Set \(D' = B \otimes_A C\). The ring map \(C \to D'\) induces algebraic residue field extensions. Keep in mind that since \(V(J) \to \Spec(A)\) is surjective the map \(T = V(JD') \to \Spec(B)\) is surjective too. Apply Lemma 097A to \(C \to D'\) and \(J \subset C\) to get \(D' \to D\) with \(D'/JD' \cong D/JD\). All of the remaining properties given in the lemma are immediate from the results of Lemma 097A.

Weakly étale versus pro-étale

Recall that a ring homomorphism \(A \to B\) is weakly étale if \(A \to B\) is flat and \(B \otimes_A B \to B\) is flat. We have proved some properties of such ring maps in More on Algebra, Section 092A. In particular, if \(A \to B\) is a local homomorphism, and \(A\) is a strictly henselian local rings, then \(A = B\), see More on Algebra, Theorem 092Z. Using this theorem and the work we’ve done above we obtain the following structure theorem for weakly étale ring maps.

Proposition

Let \(A \to B\) be a weakly étale ring map. Then there exists a faithfully flat, ind-étale ring map \(B \to B'\) such that \(A \to B'\) is ind-étale.

Proof

The ring map \(A \to B\) induces (separable) algebraic extensions of residue fields, see More on Algebra, Lemma 092R. Thus we may apply Lemma 097X and choose a diagram \[\xymatrix{ B \ar[r] & D \\ A \ar[r] \ar[u] & C \ar[u] }\] with the properties as listed in the lemma. Note that \(C \to D\) is weakly étale by More on Algebra, Lemma 092L. Pick a maximal ideal \(\mathfrak m \subset D\). By construction this lies over a maximal ideal \(\mathfrak m' \subset C\). By More on Algebra, Theorem 092Z the ring map \(C_{\mathfrak m'} \to D_\mathfrak m\) is an isomorphism. As every point of \(\Spec(C)\) specializes to a closed point we conclude that \(C \to D\) identifies local rings. Thus Proposition 097G applies to the ring map \(C \to D\). Pick \(D \to D'\) faithfully flat and ind-Zariski such that \(C \to D'\) is ind-Zariski. Then \(B \to D'\) is a solution to the problem posed in the proposition.

The V topology and the pro-h topology

The V topology was introduced in Topologies, Section 0ETA. The h topology was introduced in More on Flatness, Section 0ETQ. A kind of intermediate topology, namely the ph topology, was introduced in Topologies, Section 0DBC.

Given a topology \(\tau\) on a suitable category \(\mathcal{C}\) of schemes, we can introduce a “pro-\(\tau\) topology” on \(\mathcal{C}\) as follows. Recall that for \(X\) in \(\mathcal{C}\) we use \(h_X\) to denote the representable presheaf associated to \(X\). Let us temporarily say a morphism \(X \to Y\) of \(\mathcal{C}\) is a \(\tau\)-cover² if the \(\tau\)-sheafification of \(h_X \to h_Y\) is surjective. Then we can define the pro-\(\tau\) topology as the coarsest topology such that

1. the pro-\(\tau\) topology is finer than the \(\tau\) topology, and

2. \(X \to Y\) is a pro-\(\tau\)-cover if \(Y\) is affine and \(X = \lim X_\lambda\) is a directed limit of affine schemes \(X_\lambda\) over \(Y\) such that \(h_{X_\lambda} \to h_Y\) is a \(\tau\)-cover for all \(\lambda\).

We use this pedantic formulation because we do not want to specify a choice of pro-\(\tau\) coverings: for different \(\tau\) different choices of collections of coverings are suitable. For example, in Section 0988 we will see that in order to define the pro-étale topology looking at families of weakly étale morphisms with some finiteness property works well. More generally, the proposed construction given in this paragraph is meant mainly to motivate the results in this section and we will never implicitly define a pro-\(\tau\) topology using this method.

The following lemma tells us that the pro-V topology is equal to the V topology.

Lemma

Let \(Y\) be an affine scheme. Let \(X = \lim X_i\) be a directed limit of affine schemes over \(Y\). The following are equivalent

1. \(\{X \to Y\}\) is a standard V covering (Topologies, Definition 0ETB), and

2. \(\{X_i \to Y\}\) is a standard V covering for all \(i\).

Proof

A singleton \(\{X \to Y\}\) is a standard V covering if and only if given a morphism \(g : \Spec(V) \to Y\) there is an extension of valuation rings \(V \subset W\) and a commutative diagram \[\xymatrix{ \Spec(W) \ar[r] \ar[d] & X \ar[d] \\ \Spec(V) \ar[r]^g & Y }\] Thus (1) \(\Rightarrow\) (2) is immediate from the definition. Conversely, assume (2) and let \(g : \Spec(V) \to Y\) as above be given. Write \(\Spec(V) \times_Y X_i = \Spec(A_i)\). Since \(\{X_i \to Y\}\) is a standard V covering, we may choose a valuation ring \(W_i\) and a ring map \(A_i \to W_i\) such that the composition \(V \to A_i \to W_i\) is an extension of valuation rings. In particular, the quotient \(A'_i\) of \(A_i\) by its \(V\)-torsion is a faitfhully flat \(V\)-algebra. Flatness by More on Algebra, Lemma 0539 and surjectivity on spectra because \(A_i \to W_i\) factors through \(A'_i\). Thus \[A = \colim A'_i\] is a faithfully flat \(V\)-algebra (Algebra, Lemma 090N). Since \(\{\Spec(A) \to \Spec(V)\}\) is a standard fpqc cover, it is a standard V cover (Topologies, Lemma 0ETC) and hence we can choose \(\Spec(W) \to \Spec(A)\) such that \(V \to W\) is an extension of valuation rings. Since we can compose with the morphism \(\Spec(A) \to X = \Spec(\colim A_i)\) the proof is complete.

The following lemma tells us that the pro-h topology is equal to the pro-ph topology is equal to the V topology.

Lemma

Let \(X \to Y\) be a morphism of affine schemes. The following are equivalent

1. \(\{X \to Y\}\) is a standard V covering (Topologies, Definition 0ETB),

2. \(X = \lim X_i\) is a directed limit of affine schemes over \(Y\) such that \(\{X_i \to Y\}\) is a ph covering for each \(i\), and

3. \(X = \lim X_i\) is a directed limit of affine schemes over \(Y\) such that \(\{X_i \to Y\}\) is an h covering for each \(i\).

Proof

Proof of (2) \(\Rightarrow\) (1). Recall that a V covering given by a single arrow between affines is a standard V covering, see Topologies, Definition 0ETH and Lemma 0ETG. Recall that any ph covering is a V covering, see Topologies, Lemma 0ETK. Hence if \(X = \lim X_i\) as in (2), then \(\{X_i \to Y\}\) is a standard V covering for each \(i\). Thus by Lemma 0EVN we see that (1) is true.

Proof of (3) \(\Rightarrow\) (2). This is clear because an h covering is always a ph covering, see More on Flatness, Definition 0ETS.

Proof of (1) \(\Rightarrow\) (3). This is the interesting direction, but the interesting content in this proof is hidden in More on Flatness, Lemma 0ETR. Write \(X = \Spec(A)\) and \(Y = \Spec(R)\). We can write \(A = \colim A_i\) with \(A_i\) of finite presentation over \(R\), see Algebra, Lemma 00QN. Set \(X_i = \Spec(A_i)\). Then \(\{X_i \to Y\}\) is a standard V covering for all \(i\) by (1) and Topologies, Lemma 0ETG. Hence \(\{X_i \to Y\}\) is an h covering by More on Flatness, Definition 0ETS. This finishes the proof.

The following lemma tells us, roughly speaking, that an h sheaf which is limit preserving satisfies the sheaf condition for V coverings. Please also compare with Remark 0EVR.

Lemma

Let \(S\) be a scheme. Let \(F\) be a contravariant functor defined on the category of all schemes over \(S\). If

1. \(F\) satisfies the sheaf property for the h topology, and

2. \(F\) is limit preserving (Limits, Remark 05LX),

then \(F\) satisfies the sheaf property for the V topology.

Proof

We will prove this by verifying (1) and (2’) of Topologies, Lemma 0ETM. The sheaf property for Zariski coverings follows from the fact that \(F\) has the sheaf property for all h coverings. Finally, suppose that \(X \to Y\) is a morphism of affine schemes over \(S\) such that \(\{X \to Y\}\) is a V covering. By Lemma 0EVP we can write \(X = \lim X_i\) as a directed limit of affine schemes over \(Y\) such that \(\{X_i \to Y\}\) is an h covering for each \(i\). We obtain \[\begin{align*} & \text{Equalizer}( \xymatrix{ F(X) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X \times_Y X) } ) \\ & = \text{Equalizer}( \xymatrix{ \colim F(X_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \colim F(X_i \times_Y X_i) } ) \\ & = \colim \text{Equalizer}( \xymatrix{ F(X_i) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X_i \times_Y X_i) } ) \\ & = \colim F(Y) = F(Y) \end{align*}\] which is what we wanted to show. The first equality because \(F\) is limit preserving and \(X = \lim X_i\) and \(X \times_Y X = \lim X_i \times_Y X_i\). The second equality because filtered colimits are exact. The third equality because \(F\) satisfies the sheaf property for h coverings.

Remark

Let \(S\) be a scheme contained in a big site \(\Sch_h\). Let \(F\) be a sheaf of sets on \((\Sch/S)_h\) such that \(F(T) = \colim F(T_i)\) whenever \(T = \lim T_i\) is a directed limit of affine schemes in \((\Sch/S)_h\). In this situation \(F\) extends uniquely to a contravariant functor \(F'\) on the category of all schemes over \(S\) such that (a) \(F'\) satisfies the sheaf property for the h topology and (b) \(F'\) is limit preserving. See More on Flatness, Lemma 0EV3. In this situation Lemma 0EVQ tells us that \(F'\) satisfies the sheaf property for the V topology.

Constructing w-contractible covers

In this section we construct w-contractible covers of affine schemes.

Definition

Let \(A\) be a ring. We say \(A\) is w-contractible if every faithfully flat weakly étale ring map \(A \to B\) has a retraction.

We remark that by Proposition 097Z an equivalent definition would be to ask that every faithfully flat, ind-étale ring map \(A \to B\) has a retraction. Here is a key observation that will allow us to construct w-contractible rings.

Lemma

Let \(A\) be a ring. The following are equivalent

1. \(A\) is w-contractible,

2. every faithfully flat, ind-étale ring map \(A \to B\) has a retraction, and

3. \(A\) satisfies

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

   2. \(\pi_0(\Spec(A))\) is extremally disconnected, and

   3. for every maximal ideal \(\mathfrak m \subset A\) the local ring \(A_\mathfrak m\) is strictly henselian.

Proof

The equivalence of (1) and (2) follows immediately from Proposition 097Z.

Assume (3)(a), (3)(b), and (3)(c). Let \(A \to B\) be faithfully flat and ind-étale. We will use without further mention the fact that a flat map \(A \to B\) is faithfully flat if and only if every closed point of \(\Spec(A)\) is in the image of \(\Spec(B) \to \Spec(A)\) We will show that \(A \to B\) has a retraction.

Let \(I \subset A\) be an ideal such that \(V(I) \subset \Spec(A)\) is the set of closed points of \(\Spec(A)\). We may replace \(B\) by the ring \(C\) constructed in Lemma 097A for \(A \to B\) and \(I \subset A\). Thus we may assume \(\Spec(B)\) is w-local such that the set of closed points of \(\Spec(B)\) is \(V(IB)\). In this case \(A \to B\) identifies local rings by condition (3)(c) as it suffices to check this at maximal ideals of \(B\) which lie over maximal ideals of \(A\). Thus \(A \to B\) has a retraction by Lemma 09AZ.

Assume (1) or equivalently (2). We have (3)(c) by Lemma 097V. Properties (3)(a) and (3)(b) follow from Lemma 09AZ.

Proposition

For every ring \(A\) there exists a faithfully flat, ind-étale ring map \(A \to D\) such that \(D\) is w-contractible.

Proof

Applying Lemma 097X to \(\text{id}_A : A \to A\) we find a faithfully flat, ind-étale ring map \(A \to C\) such that \(C\) is w-local and such that every local ring at a maximal ideal of \(C\) is strictly henselian. Choose an extremally disconnected space \(T\) and a surjective continuous map \(T \to \pi_0(\Spec(C))\), see Topology, Lemma 090D. Note that \(T\) is profinite. Apply Lemma 097D to find an ind-Zariski ring map \(C \to D\) such that \(\pi_0(\Spec(D)) \to \pi_0(\Spec(C))\) realizes \(T \to \pi_0(\Spec(C))\) and such that \[\xymatrix{ \Spec(D) \ar[r] \ar[d] & \pi_0(\Spec(D)) \ar[d] \\ \Spec(C) \ar[r] & \pi_0(\Spec(C)) }\] is cartesian in the category of topological spaces. Note that \(\Spec(D)\) is w-local, that \(\Spec(D) \to \Spec(C)\) is w-local, and that the set of closed points of \(\Spec(D)\) is the inverse image of the set of closed points of \(\Spec(C)\), see Lemma 096C. Thus it is still true that the local rings of \(D\) at its maximal ideals are strictly henselian (as they are isomorphic to the local rings at the corresponding maximal ideals of \(C\)). It follows from Lemma 0982 that \(D\) is w-contractible.

Remark

Let \(A\) be a ring. Let \(\kappa\) be an infinite cardinal bigger or equal than the cardinality of \(A\). Then the cardinality of the ring \(D\) constructed in Proposition 0983 is at most \[\kappa^{2^{2^{2^\kappa}}}.\] Namely, the ring map \(A \to D\) is constructed as a composition \[A \to A_w = A' \to C' \to C \to D.\] Here the first three steps of the construction are carried out in the first paragraph of the proof of Lemma 097X. For the first step we have \(|A_w| \leq \kappa\) by Remark 0976. We have \(|C'| \leq \kappa\) by Remark 097S. Then \(|C| \leq \kappa\) because \(C\) is a localization of \((C')_w\) (it is constructed from \(C'\) by an application of Lemma 0979 in the proof of Lemma 097A). Thus \(C\) has at most \(2^\kappa\) maximal ideals. Finally, the ring map \(C \to D\) identifies local rings and the cardinality of the set of maximal ideals of \(D\) is at most \(2^{2^{2^\kappa}}\) by Topology, Remark 090E. Since \(D \subset \prod_{\mathfrak m \subset D} D_\mathfrak m\) we see that \(D\) has at most the size displayed above.

Lemma

Let \(A \to B\) be a quasi-finite and finitely presented ring map. If the residue fields of \(A\) are separably algebraically closed and \(\Spec(A)\) is Hausdorff and extremally disconnected, then \(\Spec(B)\) is extremally disconnected.

Proof

Set \(X = \Spec(A)\) and \(Y = \Spec(B)\). Choose a finite partition \(X = \coprod X_i\) and \(X'_i \to X_i\) as in Étale Cohomology, Lemma 095K. The map of topological spaces \(\coprod X_i \to X\) (where the source is the disjoint union in the category of topological spaces) has a section by Topology, Proposition 08YN. Hence we see that \(X\) is topologically the disjoint union of the strata \(X_i\). Thus we may replace \(X\) by the \(X_i\) and assume there exists a surjective finite locally free morphism \(X' \to X\) such that \((X' \times_X Y)_{red}\) is isomorphic to a finite disjoint union of copies of \(X'_{red}\). Picture \[\xymatrix{ \coprod_{i = 1, \ldots, r} X' \ar[r] \ar[d] & Y \ar[d] \\ X' \ar[r] & X }\] The assumption on the residue fields of \(A\) implies that this diagram is a fibre product diagram on underlying sets of points (details omitted). Since \(X\) is extremally disconnected and \(X'\) is Hausdorff (Lemma 0978), the continuous map \(X' \to X\) has a continuous section \(\sigma\). Then \(\coprod_{i = 1, \ldots, r} \sigma(X) \to Y\) is a bijective continuous map. By Topology, Lemma 08YE we see that it is a homeomorphism and the proof is done.

Lemma

Let \(A \to B\) be a finite and finitely presented ring map. If \(A\) is w-contractible, so is \(B\).

Proof

We will use the criterion of Lemma 0982. Set \(X = \Spec(A)\) and \(Y = \Spec(B)\) and denote \(f : Y \to X\) the induced morphism. As \(f : Y \to X\) is a finite morphism, we see that the set of closed points \(Y_0\) of \(Y\) is the inverse image of the set of closed points \(X_0\) of \(X\). Let \(y \in Y\) with image \(x \in X\). Then \(x\) specializes to a unique closed point \(x_0 \in X\). Say \(f^{-1}(\{x_0\}) = \{y_1, \ldots, y_n\}\) with \(y_i\) closed in \(Y\). Since \(R = \mathcal{O}_{X, x_0}\) is strictly henselian and since \(f\) is finite, we see that \(Y \times_{f, X} \Spec(R)\) is equal to \(\coprod_{i = 1, \ldots, n} \Spec(R_i)\) where each \(R_i\) is a local ring finite over \(R\) whose maximal ideal corresponds to \(y_i\), see Algebra, Lemma 04GG part (10). Then \(y\) is a point of exactly one of these \(\Spec(R_i)\) and we see that \(y\) specializes to exactly one of the \(y_i\). In other words, every point of \(Y\) specializes to a unique point of \(Y_0\). Thus \(Y\) is w-local. For every \(y \in Y_0\) with image \(x \in X_0\) we see that \(\mathcal{O}_{Y, y}\) is strictly henselian by Algebra, Lemma 04GH applied to \(\mathcal{O}_{X, x} \to B \otimes_A \mathcal{O}_{X, x}\). It remains to show that \(Y_0\) is extremally disconnected. To do this we look at \(X_0 \times_X Y \to X_0\) where \(X_0 \subset X\) is the reduced induced scheme structure. Note that the underlying topological space of \(X_0 \times_X Y\) agrees with \(Y_0\). Now the desired result follows from Lemma 0985.

Lemma

Let \(A\) be a ring. Let \(Z \subset \Spec(A)\) be a closed subset of the form \(Z = V(f_1, \ldots, f_r)\). Set \(B = A_Z^\sim\), see Lemma 096V. If \(A\) is w-contractible, so is \(B\).

Proof

Let \(A_Z^\sim \to B\) be a weakly étale faithfully flat ring map. Consider the ring map \[A \longrightarrow A_{f_1} \times \ldots \times A_{f_r} \times B\] this is faithful flat and weakly étale. If \(A\) is w-contractible, then there is a retraction \(\sigma\). Consider the morphism \[\Spec(A_Z^\sim) \to \Spec(A) \xrightarrow{\Spec(\sigma)} \coprod \Spec(A_{f_i}) \amalg \Spec(B)\] Every point of \(Z \subset \Spec(A_Z^\sim)\) maps into the component \(\Spec(B)\). Since every point of \(\Spec(A_Z^\sim)\) specializes to a point of \(Z\) we find a morphism \(\Spec(A_Z^\sim) \to \Spec(B)\) as desired.

The pro-étale site

In this section we only discuss the actual definition and construction of the various pro-étale sites and the morphisms between them. The existence of weakly contractible objects will be done in Section 0F4N.

The pro-étale topology is a bit like the fpqc topology (see Topologies, Section 022A) in that the topos of sheaves on the small pro-étale site of a scheme depends on the choice of the underlying category of schemes. Thus we cannot speak of the pro-étale topos of a scheme. However, it will be true that the cohomology groups of a sheaf are unchanged if we enlarge our underlying category of schemes, see Section 0F4R.

We will define pro-étale coverings using weakly étale morphisms of schemes, see More on Morphisms, Section 094N. The reason is that, on the one hand, it is somewhat awkward to define the notion of a pro-étale morphism of schemes, and on the other, Proposition 097Z assures us that we obtain the same sheaves³ with the definition that follows.

Definition

Let \(T\) be a scheme. A pro-étale covering of \(T\) is a family of morphisms \(\{f_i : T_i \to T\}_{i \in I}\) of schemes such that each \(f_i\) is weakly-étale and such that for every affine open \(U \subset T\) there exists \(n \geq 0\), a map \(a : \{1, \ldots, n\} \to I\) and affine opens \(V_j \subset T_{a(j)}\), \(j = 1, \ldots, n\) with \(\bigcup_{j = 1}^n f_{a(j)}(V_j) = U\).

To be sure this condition implies that \(T = \bigcup f_i(T_i)\). Here is a lemma that will allow us to recognize pro-étale coverings. It will also allow us to reduce many lemmas about pro-étale coverings to the corresponding results for fpqc coverings.

Lemma

Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with target \(T\). The following are equivalent

1. \(\{f_i : T_i \to T\}_{i \in I}\) is a pro-étale covering,

2. each \(f_i\) is weakly étale and \(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering,

3. each \(f_i\) is weakly étale and for every affine open \(U \subset T\) there exist quasi-compact opens \(U_i \subset T_i\) which are almost all empty, such that \(U = \bigcup f_i(U_i)\),

4. each \(f_i\) is weakly étale and there exists an affine open covering \(T = \bigcup_{\alpha \in A} U_\alpha\) and for each \(\alpha \in A\) there exist \(i_{\alpha, 1}, \ldots, i_{\alpha, n(\alpha)} \in I\) and quasi-compact opens \(U_{\alpha, j} \subset T_{i_{\alpha, j}}\) such that \(U_\alpha = \bigcup_{j = 1, \ldots, n(\alpha)} f_{i_{\alpha, j}}(U_{\alpha, j})\).

If \(T\) is quasi-separated, these are also equivalent to

1. each \(f_i\) is weakly étale, and for every \(t \in T\) there exist \(i_1, \ldots, i_n \in I\) and quasi-compact opens \(U_j \subset T_{i_j}\) such that \(\bigcup_{j = 1, \ldots, n} f_{i_j}(U_j)\) is a (not necessarily open) neighbourhood of \(t\) in \(T\).

Proof

The equivalence of (1) and (2) is immediate from the definitions. Hence the lemma follows from Topologies, Lemma 03L7.

Lemma

Any étale covering and any Zariski covering is a pro-étale covering.

Proof

This follows from the corresponding result for fpqc coverings (Topologies, Lemma 022C), Lemma 098A, and the fact that an étale morphism is a weakly étale morphism, see More on Morphisms, Lemma 094X.

Lemma

Let \(T\) be a scheme.

1. If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is a pro-étale covering of \(T\).

2. If \(\{T_i \to T\}_{i\in I}\) is a pro-étale covering and for each \(i\) we have a pro-étale covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is a pro-étale covering.

3. If \(\{T_i \to T\}_{i\in I}\) is a pro-étale covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is a pro-étale covering.

Proof

This follows from the fact that composition and base changes of weakly étale morphisms are weakly étale (More on Morphisms, Lemmas 094T and 094U), Lemma 098A, and the corresponding results for fpqc coverings, see Topologies, Lemma 022D.

Lemma

Let \(T\) be an affine scheme. Let \(\{T_i \to T\}_{i \in I}\) be a pro-étale covering of \(T\). Then there exists a pro-étale covering \(\{U_j \to T\}_{j = 1, \ldots, n}\) which is a refinement of \(\{T_i \to T\}_{i \in I}\) such that each \(U_j\) is an affine scheme. Moreover, we may choose each \(U_j\) to be open affine in one of the \(T_i\).

Proof

This follows directly from the definition.

Thus we define the corresponding standard coverings of affines as follows.

Definition

Let \(T\) be an affine scheme. A standard pro-étale covering of \(T\) is a family \(\{f_i : T_i \to T\}_{i = 1, \ldots, n}\) where each \(T_j\) is affine, each \(f_i\) is weakly étale, and \(T = \bigcup f_i(T_i)\).

We follow the general outline given in Topologies, Section 020M for constructing the big pro-étale site we will be working with. However, because we need a bit larger rings to accommodate for the size of certain constructions we modify the constructions slightly.

Definition

A big pro-étale site is any site \(\Sch_\proetale\) as in Sites, Definition 00VH constructed as follows:

1. Choose any set of schemes \(S_0\), and any set of pro-étale coverings \(\text{Cov}_0\) among these schemes.

2. Change the function \(Bound\) of Sets, Equation (046U) into \[Bound(\kappa) = \max\{\kappa^{2^{2^{2^\kappa}}}, \kappa^{\aleph_0}, \kappa^+\}.\]

3. As underlying category take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\) and the function \(Bound\).

4. Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of pro-étale coverings, and the set \(\text{Cov}_0\) chosen above.

See the remarks following Topologies, Definition 020S for motivation and explanation regarding the definition of big sites.

It will turn out, see Lemma 098J, that the topology on a big pro-étale site \(\Sch_\proetale\) is in some sense induced from the pro-étale topology on the category of all schemes.

Definition

Let \(S\) be a scheme. Let \(\Sch_\proetale\) be a big pro-étale site containing \(S\).

1. The big pro-étale site of \(S\), denoted \((\Sch/S)_\proetale\), is the site \(\Sch_\proetale/S\) introduced in Sites, Section 00XZ.

2. The small pro-étale site of \(S\), which we denote \(S_\proetale\), is the full subcategory of \((\Sch/S)_\proetale\) whose objects are those \(U/S\) such that \(U \to S\) is weakly étale. A covering of \(S_\proetale\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_\proetale\) with \(U \in \Ob(S_\proetale)\).

3. The big affine pro-étale site of \(S\), denoted \((\textit{Aff}/S)_\proetale\), is the full subcategory of \((\Sch/S)_\proetale\) whose objects are affine \(U/S\). A covering of \((\textit{Aff}/S)_\proetale\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_\proetale\) which is a standard pro-étale covering.

It is not completely clear that the small pro-étale site and the big affine pro-étale site are sites. We check this now.

Lemma

Let \(S\) be a scheme. Let \(\Sch_\proetale\) be a big pro-étale site containing \(S\). Both \(S_\proetale\) and \((\textit{Aff}/S)_\proetale\) are sites.

Proof

Let us show that \(S_\proetale\) is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 00VH. Since \((\Sch/S)_\proetale\) is a site, it suffices to prove that given any covering \(\{U_i \to U\}\) of \((\Sch/S)_\proetale\) with \(U \in \Ob(S_\proetale)\) we also have \(U_i \in \Ob(S_\proetale)\). This follows from the definitions as the composition of weakly étale morphisms is weakly étale.

To show that \((\textit{Aff}/S)_\proetale\) is a site, reasoning as above, it suffices to show that the collection of standard pro-étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 00VH. This follows from Lemma 098A and the corresponding result for standard fpqc coverings (Topologies, Lemma 03LA).

Lemma

Let \(S\) be a scheme. Let \(\Sch_\proetale\) be a big pro-étale site containing \(S\). Let \(\Sch\) be the category of all schemes.

1. The categories \(\Sch_\proetale\), \((\Sch/S)_\proetale\), \(S_\proetale\), and \((\textit{Aff}/S)_\proetale\) have fibre products agreeing with fibre products in \(\Sch\).

2. The categories \(\Sch_\proetale\), \((\Sch/S)_\proetale\), \(S_\proetale\) have equalizers agreeing with equalizers in \(\Sch\).

3. The categories \((\Sch/S)_\proetale\), and \(S_\proetale\) both have a final object, namely \(S/S\).

4. The category \(\Sch_\proetale\) has a final object agreeing with the final object of \(\Sch\), namely \(\Spec(\mathbf{Z})\).

Proof

The category \(\Sch_\proetale\) contains \(\Spec(\mathbf{Z})\) and is closed under products and fibre products by construction, see Sets, Lemma 000R. Suppose we have \(U \to S\), \(V \to U\), \(W \to U\) morphisms of schemes with \(U, V, W \in \Ob(\Sch_\proetale)\). The fibre product \(V \times_U W\) in \(\Sch_\proetale\) is a fibre product in \(\Sch\) and is the fibre product of \(V/S\) with \(W/S\) over \(U/S\) in the category of all schemes over \(S\), and hence also a fibre product in \((\Sch/S)_\proetale\). This proves the result for \((\Sch/S)_\proetale\). If \(U \to S\), \(V \to U\) and \(W \to U\) are weakly étale then so is \(V \times_U W \to S\) (see More on Morphisms, Section 094N) and hence we get fibre products for \(S_\proetale\). If \(U, V, W\) are affine, so is \(V \times_U W\) and hence we get fibre products for \((\textit{Aff}/S)_\proetale\).

Let \(a, b : U \to V\) be two morphisms in \(\Sch_\proetale\). In this case the equalizer of \(a\) and \(b\) (in the category of schemes) is \[V \times_{\Delta_{V/\Spec(\mathbf{Z})}, V \times_{\Spec(\mathbf{Z})} V, (a, b)} (U \times_{\Spec(\mathbf{Z})} U)\] which is an object of \(\Sch_\proetale\) by what we saw above. Thus \(\Sch_\proetale\) has equalizers. If \(a\) and \(b\) are morphisms over \(S\), then the equalizer (in the category of schemes) is also given by \[V \times_{\Delta_{V/S}, V \times_S V, (a, b)} (U \times_S U)\] hence we see that \((\Sch/S)_\proetale\) has equalizers. Moreover, if \(U\) and \(V\) are weakly-étale over \(S\), then so is the equalizer above as a fibre product of schemes weakly étale over \(S\). Thus \(S_\proetale\) has equalizers. The statements on final objects is clear.

Next, we check that the big affine pro-étale site defines the same topos as the big pro-étale site.

Lemma

Let \(S\) be a scheme. Let \(\Sch_\proetale\) be a big pro-étale site containing \(S\). The functor \((\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale\) is a special cocontinuous functor. Hence it induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_\proetale)\) to \(\Sh((\Sch/S)_\proetale)\).

Proof

The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale\). Being cocontinuous just means that any pro-étale covering of \(T/S\), \(T\) affine, can be refined by a standard pro-étale covering of \(T\). This is the content of Lemma 098D. Hence (1) holds. We see \(u\) is continuous simply because a standard pro-étale covering is a pro-étale covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that \(u\) is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering.

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site. Let \(f : T \to S\) be a morphism in \(\Sch_\proetale\). The functor \(T_\proetale \to (\Sch/S)_\proetale\) is cocontinuous and induces a morphism of topoi \[i_f : \Sh(T_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale)\] For a sheaf \(\mathcal{G}\) on \((\Sch/S)_\proetale\) we have the formula \((i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)\). The functor \(i_f^{-1}\) also has a left adjoint \(i_{f, !}\) which commutes with fibre products and equalizers.

Proof

Denote the functor \(u : T_\proetale \to (\Sch/S)_\proetale\). In other words, given a weakly étale morphism \(j : U \to T\) corresponding to an object of \(T_\proetale\) we set \(u(U \to T) = (f \circ j : U \to S)\). This functor commutes with fibre products, see Lemma 098M. Moreover, \(T_\proetale\) has equalizers and \(u\) commutes with them by Lemma 098M. It is clearly cocontinuous. It is also continuous as \(u\) transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas 00XR and 00XS.

Lemma

Let \(S\) be a scheme. Let \(\Sch_\proetale\) be a big pro-étale site containing \(S\). The inclusion functor \(S_\proetale \to (\Sch/S)_\proetale\) satisfies the hypotheses of Sites, Lemma 00XU and hence induces a morphism of sites \[\pi_S : (\Sch/S)_\proetale \longrightarrow S_\proetale\] and a morphism of topoi \[i_S : \Sh(S_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale)\] such that \(\pi_S \circ i_S = \text{id}\). Moreover, \(i_S = i_{\text{id}_S}\) with \(i_{\text{id}_S}\) as in Lemma 098P. In particular the functor \(i_S^{-1} = \pi_{S, *}\) is described by the rule \(i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)\).

Proof

In this case the functor \(u : S_\proetale \to (\Sch/S)_\proetale\), in addition to the properties seen in the proof of Lemma 098P above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 00XU.

Definition

In the situation of Lemma 098Q the functor \(i_S^{-1} = \pi_{S, *}\) is often called the restriction to the small pro-étale site, and for a sheaf \(\mathcal{F}\) on the big pro-étale site we denote \(\mathcal{F}|_{S_\proetale}\) this restriction.

With this notation in place we have for a sheaf \(\mathcal{F}\) on the big site and a sheaf \(\mathcal{G}\) on the big site that \[\begin{align*} \Mor_{\Sh(S_\proetale)}(\mathcal{F}|_{S_\proetale}, \mathcal{G}) & = \Mor_{\Sh((\Sch/S)_\proetale)}(\mathcal{F}, i_{S, *}\mathcal{G}) \\ \Mor_{\Sh(S_\proetale)}(\mathcal{G}, \mathcal{F}|_{S_\proetale}) & = \Mor_{\Sh((\Sch/S)_\proetale)}(\pi_S^{-1}\mathcal{G}, \mathcal{F}) \end{align*}\] Moreover, we have \((i_{S, *}\mathcal{G})|_{S_\proetale} = \mathcal{G}\) and we have \((\pi_S^{-1}\mathcal{G})|_{S_\proetale} = \mathcal{G}\).

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site. Let \(f : T \to S\) be a morphism in \(\Sch_\proetale\). The functor \[u : (\Sch/T)_\proetale \longrightarrow (\Sch/S)_\proetale, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_\proetale \longrightarrow (\Sch/T)_\proetale, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale)\] We have \(f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)\). We have \(f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)\). Also, \(f_{big}^{-1}\) has a left adjoint \(f_{big!}\) which commutes with fibre products and equalizers.

Proof

The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers (details omitted; compare with proof of Lemma 098P). Hence Sites, Lemmas 00XR and 00XS apply and we deduce the formula for \(f_{big}^{-1}\) and the existence of \(f_{big!}\). Moreover, the functor \(v\) is a right adjoint because given \(U/T\) and \(V/S\) we have \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site. Let \(f : T \to S\) be a morphism in \(\Sch_\proetale\).

1. We have \(i_f = f_{big} \circ i_T\) with \(i_f\) as in Lemma 098P and \(i_T\) as in Lemma 098Q.

2. The functor \(S_\proetale \to T_\proetale\), \((U \to S) \mapsto (U \times_S T \to T)\) is continuous and induces a morphism of topoi \[f_{small} : \Sh(T_\proetale) \longrightarrow \Sh(S_\proetale).\] We have \(f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)\).

3. We have a commutative diagram of morphisms of sites \[\xymatrix{ T_\proetale \ar[d]_{f_{small}} & (\Sch/T)_\proetale \ar[d]^{f_{big}} \ar[l]^{\pi_T}\\ S_\proetale & (\Sch/S)_\proetale \ar[l]_{\pi_S} }\] so that \(f_{small} \circ \pi_T = \pi_S \circ f_{big}\) as morphisms of topoi.

4. We have \(f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f\).

Proof

The equality \(i_f = f_{big} \circ i_T\) follows from the equality \(i_f^{-1} = i_T^{-1} \circ f_{big}^{-1}\) which is clear from the descriptions of these functors above. Thus we see (1).

The functor \(u : S_\proetale \to T_\proetale\), \(u(U \to S) = (U \times_S T \to T)\) transforms coverings into coverings and commutes with fibre products, see Lemmas 098C and 098M. Moreover, both \(S_\proetale\), \(T_\proetale\) have final objects, namely \(S/S\) and \(T/T\) and \(u(S/S) = T/T\). Hence by Sites, Proposition 00X6 the functor \(u\) corresponds to a morphism of sites \(T_\proetale \to S_\proetale\). This in turn gives rise to the morphism of topoi, see Sites, Lemma 00XC. The description of the pushforward is clear from these references.

Part (3) follows because \(\pi_S\) and \(\pi_T\) are given by the inclusion functors and \(f_{small}\) and \(f_{big}\) by the base change functors \(U \mapsto U \times_S T\).

Statement (4) follows from (3) by precomposing with \(i_T\).

In the situation of the lemma, using the terminology of Definition 098R we have: for \(\mathcal{F}\) a sheaf on the big pro-étale site of \(T\) [0F60]\[\begin{equation} (f_{big, *}\mathcal{F})|_{S_\proetale} = f_{small, *}(\mathcal{F}|_{T_\proetale}), \end{equation}\] This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small pro-étale site of \(T\), resp. \(S\) is given by \(\pi_{T, *}\), resp. \(\pi_{S, *}\). A similar formula involving pullbacks and restrictions is false.

Lemma

Given schemes \(X\), \(Y\), \(Y\) in \(\Sch_\proetale\) and morphisms \(f : X \to Y\), \(g : Y \to Z\) we have \(g_{big} \circ f_{big} = (g \circ f)_{big}\) and \(g_{small} \circ f_{small} = (g \circ f)_{small}\).

Proof

This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 098S. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 098T.

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site. Consider a cartesian diagram \[\xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S }\] in \(\Sch_\proetale\). Then \(i_g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}\) and \(g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}\).

Proof

Since the diagram is cartesian, we have for \(U'/S'\) that \(U' \times_{S'} T' = U' \times_S T\). Hence both \(i_g^{-1} \circ f_{big, *}\) and \(f'_{small, *} \circ (i_{g'})^{-1}\) send a sheaf \(\mathcal{F}\) on \((\Sch/T)_\proetale\) to the sheaf \(U' \mapsto \mathcal{F}(U' \times_{S'} T')\) on \(S'_\proetale\) (use Lemmas 098P and 098S). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 03CF.

We can think about a sheaf on the big pro-étale site of \(S\) as a collection of sheaves on the small pro-étale site on schemes over \(S\).

Lemma

Let \(S\) be a scheme contained in a big pro-étale site \(\Sch_\proetale\). A sheaf \(\mathcal{F}\) on the big pro-étale site \((\Sch/S)_\proetale\) is given by the following data:

1. for every \(T/S \in \Ob((\Sch/S)_\proetale)\) a sheaf \(\mathcal{F}_T\) on \(T_\proetale\),

2. for every \(f : T' \to T\) in \((\Sch/S)_\proetale\) a map \(c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}\).

These data are subject to the following conditions:

1. given any \(f : T' \to T\) and \(g : T'' \to T'\) in \((\Sch/S)_\proetale\) the composition \(c_g \circ g_{small}^{-1}c_f\) is equal to \(c_{f \circ g}\), and

2. if \(f : T' \to T\) in \((\Sch/S)_\proetale\) is weakly étale then \(c_f\) is an isomorphism.

Proof

Identical to the proof of Topologies, Lemma 021K.

Lemma

Let \(S\) be a scheme. Let \(S_{affine, \proetale}\) denote the full subcategory of \(S_\proetale\) consisting of affine objects. A covering of \(S_{affine, \proetale}\) will be a standard pro-étale covering, see Definition 098E. Then restriction \[\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}}\] defines an equivalence of topoi \(\Sh(S_\proetale) \cong \Sh(S_{affine, \proetale})\).

Proof

This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 03A0 by checking that the inclusion functor \(S_{affine, \proetale} \to S_\proetale\) is a special cocontinuous functor (see Sites, Definition 03CG).

Lemma

Let \(S\) be an affine scheme. Let \(S_{app}\) denote the full subcategory of \(S_\proetale\) consisting of affine objects \(U\) such that \(\mathcal{O}(S) \to \mathcal{O}(U)\) is ind-étale. A covering of \(S_{app}\) will be a standard pro-étale covering, see Definition 098E. Then restriction \[\mathcal{F} \longmapsto \mathcal{F}|_{S_{app}}\] defines an equivalence of topoi \(\Sh(S_\proetale) \cong \Sh(S_{app})\).

Proof

By Lemma 098W we may replace \(S_\proetale\) by \(S_{affine, \proetale}\). The lemma follows from Sites, Lemma 03A0 by checking that the inclusion functor \(S_{app} \to S_{affine, \proetale}\) is a special cocontinuous functor, see Sites, Definition 03CG. The conditions of Sites, Lemma 03A0 follow immediately from the definition and the facts (a) any object \(U\) of \(S_{affine, \proetale}\) has a covering \(\{V \to U\}\) with \(V\) ind-étale over \(X\) (Proposition 097Z) and (b) the functor \(u\) is fully faithful.

Lemma

Let \(S\) be a scheme. The topology on each of the pro-étale sites \(\Sch_\proetale\), \(S_\proetale\), \((\Sch/S)_\proetale\), \(S_{affine, \proetale}\), and \((\textit{Aff}/S)_\proetale\) is subcanonical.

Proof

Combine Lemma 098A and Descent, Lemma 023Q.

Weakly contractible objects

In this section we prove the key fact that our pro-étale sites contain many weakly contractible objects. In fact, the proof of Lemma 098I is the reason for the shape of the function \(Bound\) in Definition 098G (although for readers who are ignoring set theoretical questions, this information is without content).

We first express the notion of w-contractible rings in terms of pro-étale coverings.

Lemma

Let \(T = \Spec(A)\) be an affine scheme. The following are equivalent

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

2. every pro-étale covering of \(T\) can be refined by a Zariski covering of the form \(T = \coprod_{i = 1, \ldots, n} U_i\).

Proof

Assume \(A\) is w-contractible. By Lemma 098D it suffices to prove we can refine every standard pro-étale covering \(\{f_i : T_i \to T\}_{i = 1, \ldots, n}\) by a Zariski covering of \(T\). The morphism \(\coprod T_i \to T\) is a surjective weakly étale morphism of affine schemes. Hence by Definition 0981 there exists a morphism \(\sigma : T \to \coprod T_i\) over \(T\). Then the Zariski covering \(T = \coprod \sigma^{-1}(T_i)\) refines \(\{f_i : T_i \to T\}\).

Conversely, assume (2). If \(A \to B\) is faithfully flat and weakly étale, then \(\{\Spec(B) \to T\}\) is a pro-étale covering. Hence there exists a Zariski covering \(T = \coprod U_i\) and morphisms \(U_i \to \Spec(B)\) over \(T\). Since \(T = \coprod U_i\) we obtain \(T \to \Spec(B)\), i.e., an \(A\)-algebra map \(B \to A\). This means \(A\) is w-contractible.

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site as in Definition 098G. Let \(T = \Spec(A)\) be an affine object of \(\Sch_\proetale\). The following are equivalent

1. \(A\) is w-contractible,

2. \(T\) is a weakly contractible (Sites, Definition 090L) object of \(\Sch_\proetale\), and

3. every pro-étale covering of \(T\) can be refined by a Zariski covering of the form \(T = \coprod_{i = 1, \ldots, n} U_i\).

Proof

We have seen the equivalence of (1) and (3) in Lemma 098F.

Assume (3) and let \(\mathcal{F} \to \mathcal{G}\) be a surjection of sheaves on \(\Sch_\proetale\). Let \(s \in \mathcal{G}(T)\). To prove (2) we will show that \(s\) is in the image of \(\mathcal{F}(T) \to \mathcal{G}(T)\). We can find a covering \(\{T_i \to T\}\) of \(\Sch_\proetale\) such that \(s\) lifts to a section of \(\mathcal{F}\) over \(T_i\) (Sites, Definition 00WM). By (3) we may assume we have a finite covering \(T = \coprod_{j = 1, \ldots, m} U_j\) by open and closed subsets and we have \(t_j \in \mathcal{F}(U_j)\) mapping to \(s|_{U_j}\). Since Zariski coverings are coverings in \(\Sch_\proetale\) (Lemma 098B) we conclude that \(\mathcal{F}(T) = \prod \mathcal{F}(U_j)\). Thus \(t = (t_1, \ldots, t_m) \in \mathcal{F}(T)\) is a section mapping to \(s\).

Assume (2). Let \(A \to D\) be as in Proposition 0983. Then \(\{V \to T\}\) is a covering of \(\Sch_\proetale\). (Note that \(V = \Spec(D)\) is an object of \(\Sch_\proetale\) by Remark 0984 combined with our choice of the function \(Bound\) in Definition 098G and the computation of the size of affine schemes in Sets, Lemma 000Q.) Since the topology on \(\Sch_\proetale\) is subcanonical (Lemma 098Z) we see that \(h_V \to h_T\) is a surjective map of sheaves (Sites, Lemma 00WT). Since \(T\) is assumed weakly contractible, we see that there is an element \(f \in h_V(T) = \Mor(T, V)\) whose image in \(h_T(T)\) is \(\text{id}_T\). Thus \(A \to D\) has a retraction \(\sigma : D \to A\). Now if \(A \to B\) is faithfully flat and weakly étale, then \(D \to D \otimes_A B\) has the same properties, hence there is a retraction \(D \otimes_A B \to D\) and combined with \(\sigma\) we get a retraction \(B \to D \otimes_A B \to D \to A\) of \(A \to B\). Thus \(A\) is w-contractible and (1) holds.

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site as in Definition 098G. For every object \(T\) of \(\Sch_\proetale\) there exists a covering \(\{T_i \to T\}\) in \(\Sch_\proetale\) with each \(T_i\) affine and the spectrum of a w-contractible ring. In particular, \(T_i\) is weakly contractible in \(\Sch_\proetale\).

Proof

For those readers who do not care about set-theoretical issues this lemma is a trivial consequence of Lemma 098H and Proposition 0983. Here are the details. Choose an affine open covering \(T = \bigcup U_i\). Write \(U_i = \Spec(A_i)\). Choose faithfully flat, ind-étale ring maps \(A_i \to D_i\) such that \(D_i\) is w-contractible as in Proposition 0983. The family of morphisms \(\{\Spec(D_i) \to T\}\) is a pro-étale covering. If we can show that \(\Spec(D_i)\) is isomorphic to an object, say \(T_i\), of \(\Sch_\proetale\), then \(\{T_i \to T\}\) will be combinatorially equivalent to a covering of \(\Sch_\proetale\) by the construction of \(\Sch_\proetale\) in Definition 098G and more precisely the application of Sets, Lemma 000X in the last step. To prove \(\Spec(D_i)\) is isomorphic to an object of \(\Sch_\proetale\), it suffices to prove that \(|D_i| \leq Bound(\text{size}(T))\) by the construction of \(\Sch_\proetale\) in Definition 098G and more precisely the application of Sets, Lemma 000J in step (3). Since \(|A_i| \leq \text{size}(U_i) \leq \text{size}(T)\) by Sets, Lemmas 000P and 04T7 we get \(|D_i| \leq \kappa^{2^{2^{2^\kappa}}}\) where \(\kappa = \text{size}(T)\) by Remark 0984. Thus by our choice of the function \(Bound\) in Definition 098G we win.

Lemma

Let \(S\) be a scheme. The pro-étale sites \(S_\proetale\), \((\Sch/S)_\proetale\), \(S_{affine, \proetale}\), and \((\textit{Aff}/S)_\proetale\) and if \(S\) is affine \(S_{app}\) have enough (affine) quasi-compact, weakly contractible objects, see Sites, Definition 090L.

Proof

Follows immediately from Lemma 098I.

Lemma

Let \(S\) be a scheme. The pro-étale sites \(\Sch_\proetale\), \(S_\proetale\), \((\Sch/S)_\proetale\) have the following property: for any object \(U\) there exists a covering \(\{V \to U\}\) with \(V\) a weakly contractible object. If \(U\) is quasi-compact, then we may choose \(V\) affine and weakly contractible.

Proof

Suppose that \(V = \coprod_{j \in J} V_j\) is an object of \((\Sch/S)_\proetale\) which is the disjoint union of weakly contractible objects \(V_j\). Since a disjoint union decomposition is a pro-étale covering we see that \(\mathcal{F}(V) = \prod_{j \in J} \mathcal{F}(V_j)\) for any pro-étale sheaf \(\mathcal{F}\). Let \(\mathcal{F} \to \mathcal{G}\) be a surjective map of sheaves of sets. Since \(V_j\) is weakly contractible, the map \(\mathcal{F}(V_j) \to \mathcal{G}(V_j)\) is surjective, see Sites, Definition 090L. Thus \(\mathcal{F}(V) \to \mathcal{G}(V)\) is surjective as a product of surjective maps of sets and we conclude that \(V\) is weakly contractible.

Choose a covering \(\{U_i \to U\}_{i \in I}\) with \(U_i\) affine and weakly contractible as in Lemma 098I. Take \(V = \coprod_{i \in I} U_i\) (there is a set theoretic issue here which we will address below). Then \(\{V \to U\}\) is the desired pro-étale covering by a weakly contractible object (to check it is a covering use Lemma 098A). If \(U\) is quasi-compact, then it follows immediately from Lemma 098A that we can choose a finite subset \(I' \subset I\) such that \(\{U_i \to U\}_{i \in I'}\) is still a covering and then \(\{\coprod_{i \in I'} U_i \to U\}\) is the desired covering by an affine and weakly contractible object.

In this paragraph, which we urge the reader to skip, we address set theoretic problems. In order to know that the disjoint union lies in our partial universe, we need to bound the cardinality of the index set \(I\). It is seen immediately from the construction of the covering \(\{U_i \to U\}_{i \in I}\) in the proof of Lemma 098I that \(|I| \leq \text{size}(U)\) where the size of a scheme is as defined in Sets, Section 000H. Moreover, for each \(i\) we have \(\text{size}(U_i) \leq Bound(\text{size}(U))\); this follows for the bound of the cardinality of \(\Gamma(U_i, \mathcal{O}_{U_i})\) in the proof of Lemma 098I and Sets, Lemma 000P. Thus \(\text{size}(\coprod_{i \in I} U_i)) \leq Bound(\text{size}(U))\) by Sets, Lemma 000Q. Hence by construction of the big pro-étale site through Sets, Lemma 000J we see that \(\coprod_{i \in I} U_i\) is isomorphic to an object of our site and the proof is complete.

Weakly contractible hypercoverings

The results of Section 0F4N leads to the existence of hypercoverings made up out weakly contractible objects.

Lemma

Let \(X\) be a scheme.

1. For every object \(U\) of \(X_\proetale\) there exists a hypercovering \(K\) of \(U\) in \(X_\proetale\) such that each term \(K_n\) consists of a single weakly contractible object of \(X_\proetale\) covering \(U\).

2. For every quasi-compact and quasi-separated object \(U\) of \(X_\proetale\) there exists a hypercovering \(K\) of \(U\) in \(X_\proetale\) such that each term \(K_n\) consists of a single affine and weakly contractible object of \(X_\proetale\) covering \(U\).

Proof

Let \(\mathcal{B} \subset \Ob(X_\proetale)\) be the set of weakly contractible objects of \(X_\proetale\). Every object \(T\) of \(X_\proetale\) has a covering \(\{T_i \to T\}_{i \in I}\) with \(I\) finite and \(T_i \in \mathcal{B}\) by Lemma 0F4P. By Hypercoverings, Lemma 094K we get a hypercovering \(K\) of \(U\) such that \(K_n = \{U_{n, i}\}_{i \in I_n}\) with \(I_n\) finite and \(U_{n, i}\) weakly contractible. Then we can replace \(K\) by the hypercovering of \(U\) given by \(\{U_n\}\) in degree \(n\) where \(U_n = \coprod_{i \in I_n} U_{n, i}\) This is allowed by Hypercoverings, Remark 0DB2.

Let \(X_{qcqs, \proetale} \subset X_\proetale\) be the full subcategory consisting of quasi-compact and quasi-separated objects. A covering of \(X_{qcqs, \proetale}\) will be a finite pro-étale covering. Then \(X_{qcqs, \proetale}\) is a site, has fibre products, and the inclusion functor \(X_{qcqs, \proetale} \to X_\proetale\) is continuous and commutes with fibre products. In particular, if \(K\) is a hypercovering of an object \(U\) in \(X_{qcqs, \proetale}\) then \(K\) is a hypercovering of \(U\) in \(X_\proetale\) by Hypercoverings, Lemma 0DAZ. Let \(\mathcal{B} \subset \Ob(X_{qcqs, \proetale})\) be the set of affine and weakly contractible objects. By Lemma 098I and the fact that finite unions of affines are affine, for every object \(U\) of \(X_{qcqs, \proetale}\) there exists a covering \(\{V \to U\}\) of \(X_{qcqs, \proetale}\) with \(V \in \mathcal{B}\). By Hypercoverings, Lemma 094K we get a hypercovering \(K\) of \(U\) such that \(K_n = \{U_{n, i}\}_{i \in I_n}\) with \(I_n\) finite and \(U_{n, i}\) affine and weakly contractible. Then we can replace \(K\) by the hypercovering of \(U\) given by \(\{U_n\}\) in degree \(n\) where \(U_n = \coprod_{i \in I_n} U_{n, i}\). This is allowed by Hypercoverings, Remark 0DB2.

In the following lemma we use the Čech complex \(s(\mathcal{F}(K))\) associated to a hypercovering \(K\) in a site. See Hypercoverings, Section 01GU. If \(K\) is a hypercovering of \(U\) and \(K_n = \{U_n \to U\}\), then the Čech complex looks like this: \[s(\mathcal{F}(K)) = \left( \mathcal{F}(U_0) \to \mathcal{F}(U_1) \to \mathcal{F}(U_2) \to \ldots \right)\] where \(s(\mathcal{F}(U_n))\) is placed in cohomological degree \(n\).

Lemma

Let \(X\) be a scheme. Let \(E \in D^+(X_\proetale)\) be represented by a bounded below complex \(\mathcal{E}^\bullet\) of abelian sheaves. Let \(K\) be a hypercovering of \(U \in \Ob(X_\proetale)\) with \(K_n = \{U_n \to U\}\) where \(U_n\) is a weakly contractible object of \(X_\proetale\). Then \[R\Gamma(U, E) = \text{Tot}(s(\mathcal{E}^\bullet(K)))\] in \(D(\textit{Ab})\).

Proof

If \(\mathcal{E}\) is an abelian sheaf on \(X_\proetale\), then the spectral sequence of Hypercoverings, Lemma 01GY implies that \[R\Gamma(X_\proetale, \mathcal{E}) = s(\mathcal{E}(K))\] because the higher cohomology groups of any sheaf over \(U_n\) vanish, see Cohomology on Sites, Lemma 0946.

If \(\mathcal{E}^\bullet\) is bounded below, then we can choose an injective resolution \(\mathcal{E}^\bullet \to \mathcal{I}^\bullet\) and consider the map of complexes \[\text{Tot}(s(\mathcal{E}^\bullet(K))) \longrightarrow \text{Tot}(s(\mathcal{I}^\bullet(K)))\] For every \(n\) the map \(\mathcal{E}^\bullet(U_n) \to \mathcal{I}^\bullet(U_n)\) is a quasi-isomorphism because taking sections over \(U_n\) is exact. Hence the displayed map is a quasi-isomorphism by one of the spectral sequences of Homology, Lemma 0132. Using the result of the first paragraph we see that for every \(p\) the complex \(s(\mathcal{I}^p(K))\) is acyclic in degrees \(n > 0\) and computes \(\mathcal{I}^p(U)\) in degree \(0\). Thus the other spectral sequence of Homology, Lemma 0132 shows \(\text{Tot}(s(\mathcal{I}^\bullet(K)))\) computes \(R\Gamma(U, E) = \mathcal{I}^\bullet(U)\).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. The functor \(R\Gamma(X, -) : D^+(X_\proetale) \to D(\textit{Ab})\) commutes with direct sums and homotopy colimits.

Proof

The statement means the following: Suppose we have a family of objects \(E_i\) of \(D^+(X_\proetale)\) such that \(\bigoplus E_i\) is an object of \(D^+(X_\proetale)\). Then \(R\Gamma(X, \bigoplus E_i) = \bigoplus R\Gamma(X, E_i)\). To see this choose a hypercovering \(K\) of \(X\) with \(K_n = \{U_n \to X\}\) where \(U_n\) is an affine and weakly contractible scheme, see Lemma 09A1. Let \(N\) be an integer such that \(H^p(E_i) = 0\) for \(p < N\). Choose a complex of abelian sheaves \(\mathcal{E}_i^\bullet\) representing \(E_i\) with \(\mathcal{E}_i^p = 0\) for \(p < N\). The termwise direct sum \(\bigoplus \mathcal{E}_i^\bullet\) represents \(\bigoplus E_i\) in \(D(X_\proetale)\), see Injectives, Lemma 07D9. By Lemma 09A2 we have \[R\Gamma(X, \bigoplus E_i) = \text{Tot}(s((\bigoplus \mathcal{E}^\bullet_i)(K)))\] and \[R\Gamma(X, E_i) = \text{Tot}(s(\mathcal{E}^\bullet_i(K)))\] Since each \(U_n\) is quasi-compact we see that \[\text{Tot}(s((\bigoplus \mathcal{E}^\bullet_i)(K))) = \bigoplus \text{Tot}(s(\mathcal{E}^\bullet_i(K)))\] by Modules on Sites, Lemma 0935. The statement on homotopy colimits is a formal consequence of the fact that \(R\Gamma\) is an exact functor of triangulated categories and the fact (just proved) that it commutes with direct sums.

Remark

Let \(X\) be a scheme. Because \(X_\proetale\) has enough weakly contractible objects for all \(K\) in \(D(X_\proetale)\) we have \(K = R\lim \tau_{\geq -n}K\) by Cohomology on Sites, Proposition 0947. Since \(R\Gamma\) commutes with \(R\lim\) by Injectives, Lemma 08U1 we see that \[R\Gamma(X, K) = R\lim R\Gamma(X, \tau_{\geq -n}K)\] in \(D(\textit{Ab})\). This will sometimes allow us to extend results from bounded below complexes to all complexes.

Compact generation

In this section we prove that various derived categories associated to our pro-étale sites are compactly generated as defined in Derived Categories, Definition 09SQ.

Lemma

Let \(S\) be a scheme. Let \(\Lambda\) be a ring.

1. \(D(S_\proetale)\) is compactly generated,

2. \(D(S_\proetale, \Lambda)\) is compactly generated,

3. \(D(S_\proetale, \mathcal{A})\) is compactly generated for any sheaf of rings \(\mathcal{A}\) on \(S_\proetale\),

4. \(D((\Sch/S)_\proetale)\) is compactly generated,

5. \(D((\Sch/S)_\proetale, \Lambda)\) is compactly generated, and

6. \(D((\Sch/S)_\proetale, \mathcal{A})\) is compactly generated for any sheaf of rings \(\mathcal{A}\) on \((\Sch/S)_\proetale\),

Proof

Proof of (3). Let \(U\) be an affine object of \(S_\proetale\) which is weakly contractible. Then \(j_{U!}\mathcal{A}_U\) is a compact object of the derived category \(D(S_\proetale, \mathcal{A})\), see Cohomology on Sites, Lemma 094E. Choose a set \(I\) and for each \(i \in I\) an affine weakly contractible object \(U_i\) of \(S_\proetale\) such that every affine weakly contractible object of \(S_\proetale\) is isomorphic to one of the \(U_i\). This is possible because \(\Ob(S_\proetale)\) is a set. To finish the proof of (3) it suffices to show that \(\bigoplus j_{U_i, !}\mathcal{A}_{U_i}\) is a generator of \(D(S_\proetale, \mathcal{A})\), see Derived Categories, Definition 09SJ. To see this, let \(K\) be a nonzero object of \(D(S_\proetale, \mathcal{A})\). Then there exists an object \(T\) of our site \(S_\proetale\) and a nonzero element \(\xi\) of \(H^n(K)(T)\). In other words, \(\xi\) is a nonzero section of the \(n\)th cohomology sheaf of \(K\). We may assume \(K\) is represented by a complex \(\mathcal{K}^\bullet\) of sheaves of \(\mathcal{A}\)-modules and \(\xi\) is the class of a section \(s \in \mathcal{K}^n(T)\) with \(\text{d}(s) = 0\). Namely, \(\xi\) is locally represented as the class of a section (so you get the result after replacing \(T\) by a member of a covering of \(T\)). Next, we choose a covering \(\{T_j \to T\}_{j \in J}\) as in Lemma 098I. Since \(H^n(K)\) is a sheaf, we see that for some \(j\) the restriction \(\xi|_{T_j}\) remains nonzero. Thus \(s|_{T_j}\) defines a nonzero map \(j_{T_j, !}\mathcal{A}_{T_j} \to K\) in \(D(S_\proetale, \mathcal{A})\). Since \(T_j \cong U_i\) for some \(i \in I\) we conclude.

The exact same argument works for the big pro-étale site of \(S\).

Comparing topologies

This section is the analogue of Étale Cohomology, Section 09XL.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a presheaf of sets on \(X_\proetale\) which sends finite disjoint unions to products. Then \(\mathcal{F}^\#(W) = \mathcal{F}(W)\) if \(W\) is an affine weakly contractible object of \(X_\proetale\).

Proof

Recall that \(\mathcal{F}^\#\) is equal to \((\mathcal{F}^+)^+\), see Sites, Theorem 00WB, where \(\mathcal{F}^+\) is the presheaf which sends an object \(U\) of \(X_\proetale\) to \(\colim H^0(\mathcal{U}, \mathcal{F})\) where the colimit is over all pro-étale coverings \(\mathcal{U}\) of \(U\). Thus it suffices to prove that (a) \(\mathcal{F}^+\) sends finite disjoint unions to products and (b) sends \(W\) to \(\mathcal{F}(W)\). If \(U = U_1 \amalg U_2\), then given a pro-étale covering \(\mathcal{U} = \{f_j : V_j \to U\}\) of \(U\) we obtain pro-étale coverings \(\mathcal{U}_i = \{f_j^{-1}(U_i) \to U_i\}\) and we clearly have \[H^0(\mathcal{U}, \mathcal{F}) = H^0(\mathcal{U}_1, \mathcal{F}) \times H^0(\mathcal{U}_2, \mathcal{F})\] because \(\mathcal{F}\) sends finite disjoint unions to products (this includes the condition that \(\mathcal{F}\) sends the empty scheme to the singleton). This proves (a). Finally, any pro-étale covering of \(W\) can be refined by a finite disjoint union decomposition \(W = W_1 \amalg \ldots W_n\) by Lemma 098H. Hence \(\mathcal{F}^+(W) = \mathcal{F}(W)\) exactly because the value of \(\mathcal{F}\) on \(W\) is the product of the values of \(\mathcal{F}\) on the \(W_j\). This proves (b).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a sheaf of sets on \(Y_\proetale\). If \(W\) is an affine weakly contractible object of \(X_\proetale\), then \[f_{small}^{-1}\mathcal{F}(W) = \colim_{W \to V} \mathcal{F}(V)\] where the colimit is over morphisms \(W \to V\) over \(Y\) with \(V \in Y_\proetale\).

Proof

Recall that \(f_{small}^{-1}\mathcal{F}\) is the sheaf associated to the presheaf \[u_p\mathcal{F} : U \mapsto \colim_{U \to V} \mathcal{F}(V)\] on \(X_\etale\), see Sites, Sections 00X0 and 00WU; we’ve suppressed from the notation that the colimit is over the opposite of the category \(\{U \to V, V \in Y_\proetale\}\). By Lemma 0F63 it suffices to prove that \(u_p\mathcal{F}\) sends finite disjoint unions to products. Suppose that \(U = U_1 \amalg U_2\) is a disjoint union of open and closed subschemes. There is a functor \[\{U_1 \to V_1\} \times \{U_2 \to V_2\} \longrightarrow \{U \to V\},\quad (U_1 \to V_1, U_2 \to V_2) \longmapsto (U \to V_1 \amalg V_2)\] which is initial (Categories, Definition 09WP). Hence the corresponding functor on opposite categories is cofinal and by Categories, Lemma 04E7 we see that \(u_p\mathcal{F}\) on \(U\) is the colimit of the values \(\mathcal{F}(V_1 \amalg V_2)\) over the product category. Since \(\mathcal{F}\) is a sheaf it sends disjoint unions to products and we conclude \(u_p\mathcal{F}\) does too.

Lemma

Let \(S\) be a scheme. Consider the morphism \[\pi_S : (\Sch/S)_\proetale \longrightarrow S_\proetale\] of Lemma 098Q. Let \(\mathcal{F}\) be a sheaf on \(S_\proetale\). Then \(\pi_S^{-1}\mathcal{F}\) is given by the rule \[(\pi_S^{-1}\mathcal{F})(T) = \Gamma(T_\proetale, f_{small}^{-1}\mathcal{F})\] where \(f : T \to S\). Moreover, \(\pi_S^{-1}\mathcal{F}\) satisfies the sheaf condition with respect to fpqc coverings.

Proof

Observe that we have a morphism \(i_f : \Sh(T_\proetale) \to \Sh(\Sch/S)_\proetale)\) such that \(\pi_S \circ i_f = f_{small}\) as morphisms \(T_\proetale \to S_\proetale\), see Lemma 098P. Since pullback is transitive we see that \(i_f^{-1} \pi_S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}\) as desired.

Let \(\{g_i : T_i \to T\}_{i \in I}\) be an fpqc covering. The final statement means the following: Given a sheaf \(\mathcal{G}\) on \(T_\proetale\) and given sections \(s_i \in \Gamma(T_i, g_{i, small}^{-1}\mathcal{G})\) whose pullbacks to \(T_i \times_T T_j\) agree, there is a unique section \(s\) of \(\mathcal{G}\) over \(T\) whose pullback to \(T_i\) agrees with \(s_i\). We will prove this statement when \(T\) is affine and the covering is given by a single surjective flat morphism \(T' \to T\) of affines and omit the reduction of the general case to this case.

Let \(g : T' \to T\) be a surjective flat morphism of affines and let \(s' \in g_{small}^{-1}\mathcal{G}(T')\) be a section with \(\text{pr}_0^*s' = \text{pr}_1^*s'\) on \(T' \times_T T'\). Choose a surjective weakly étale morphism \(W \to T'\) with \(W\) affine and weakly contractible, see Lemma 0F4P. By Lemma 0F64 the restriction \(s'|_W\) is an element of \(\colim_{W \to U} \mathcal{G}(U)\). Choose \(\phi : W \to U_0\) and \(s_0 \in \mathcal{G}(U_0)\) corresponding to \(s'\). Choose a surjective weakly étale morphism \(V \to W \times_T W\) with \(V\) affine and weakly contractible. Denote \(a, b : V \to W\) the induced morphisms. Since \(a^*(s'|_W) = b^*(s'|_W)\) and since the category \(\{V \to U, U \in T_\proetale\}\) is cofiltered (this is clear but see Sites, Lemma 00X5 if in doubt), we see that the two morphisms \(\phi \circ a , \phi \circ b : V \to U_0\) have to be equal. By the results in Descent, Section 023P (especially Descent, Lemma 023Q) it follows there is a unique morphism \(T \to U_0\) such that \(\phi\) is the composition of this morphism with the structure morphism \(W \to T\) (small detail omitted). Then we can let \(s\) be the pullback of \(s_0\) by this morphism. We omit the verification that \(s\) pulls back to \(s'\) on \(T'\).

Comparing big and small topoi

This section is the analogue of Étale Cohomology, Section 0757. In the following we will often denote \(\mathcal{F} \mapsto \mathcal{F}|_{S_\proetale}\) the pullback functor \(i_S^{-1}\) corresponding to the morphism of topoi \(i_S : \Sh(S_\proetale) \to \Sh((\Sch/S)_\proetale)\) of Lemma 098Q.

Lemma

Let \(S\) be a scheme. Let \(T\) be an object of \((\Sch/S)_\proetale\).

1. If \(\mathcal{I}\) is injective in \(\textit{Ab}((\Sch/S)_\proetale)\), then

   1. \(i_f^{-1}\mathcal{I}\) is injective in \(\textit{Ab}(T_\proetale)\),

   2. \(\mathcal{I}|_{S_\proetale}\) is injective in \(\textit{Ab}(S_\proetale)\),

2. If \(\mathcal{I}^\bullet\) is a K-injective complex in \(\textit{Ab}((\Sch/S)_\proetale)\), then

   1. \(i_f^{-1}\mathcal{I}^\bullet\) is a K-injective complex in \(\textit{Ab}(T_\proetale)\),

   2. \(\mathcal{I}^\bullet|_{S_\proetale}\) is a K-injective complex in \(\textit{Ab}(S_\proetale)\),

Proof

Proof of (1)(a) and (2)(a): \(i_f^{-1}\) is a right adjoint of an exact functor \(i_{f, !}\). Namely, recall that \(i_f\) corresponds to a cocontinuous functor \(u : T_\proetale \to (\Sch/S)_\proetale\) which is continuous and commutes with fibre products and equalizers, see Lemma 098P and its proof. Hence we obtain \(i_{f, !}\) by Modules on Sites, Lemma 04BG. It is shown in Modules on Sites, Lemma 04BH that it is exact. Then we conclude (1)(a) and (2)(a) hold by Homology, Lemma 015Z and Derived Categories, Lemma 08BJ.

Parts (1)(b) and (2)(b) are special cases of (1)(a) and (2)(a) as \(i_S = i_{\text{id}_S}\).

Lemma

Let \(f : T \to S\) be a morphism of schemes. For \(K\) in \(D((\Sch/T)_\proetale)\) we have \[(Rf_{big, *}K)|_{S_\proetale} = Rf_{small, *}(K|_{T_\proetale})\] in \(D(S_\proetale)\). More generally, let \(S' \in \Ob((\Sch/S)_\proetale)\) with structure morphism \(g : S' \to S\). Consider the fibre product \[\xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S }\] Then for \(K\) in \(D((\Sch/T)_\proetale)\) we have \[i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)\] in \(D(S'_\proetale)\) and \[g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)\] in \(D((\Sch/S')_\proetale)\).

Proof

The first equality follows from Lemma 0F67 and (0F60) on choosing a K-injective complex of abelian sheaves representing \(K\). The second equality follows from Lemma 0F67 and Lemma 0F61 on choosing a K-injective complex of abelian sheaves representing \(K\). The third equality follows similarly from Cohomology on Sites, Lemmas 03F3 and 08FI and Lemma 0F61 on choosing a K-injective complex of abelian sheaves representing \(K\).

Let \(S\) be a scheme and let \(\mathcal{H}\) be an abelian sheaf on \((\Sch/S)_\proetale\). Recall that \(H^n_\proetale(U, \mathcal{H})\) denotes the cohomology of \(\mathcal{H}\) over an object \(U\) of \((\Sch/S)_\proetale\).

Lemma

Let \(f : T \to S\) be a morphism of schemes. For \(K\) in \(D(S_\proetale)\) we have \[H^n_\proetale(S, \pi_S^{-1}K) = H^n(S_\proetale, K)\] and \[H^n_\proetale(T, \pi_S^{-1}K) = H^n(T_\proetale, f_{small}^{-1}K).\] For \(M\) in \(D((\Sch/S)_\proetale)\) we have \[H^n_\proetale(T, M) = H^n(T_\proetale, i_f^{-1}M).\]

Proof

To prove the last equality represent \(M\) by a K-injective complex of abelian sheaves and apply Lemma 0F67 and work out the definitions. The second equality follows from this as \(i_f^{-1} \circ \pi_S^{-1} = f_{small}^{-1}\). The first equality is a special case of the second one.

Lemma

Let \(S\) be a scheme. For \(K \in D(S_\proetale)\) the map \[K \longrightarrow R\pi_{S, *}\pi_S^{-1}K\] is an isomorphism.

Proof

This is true because both \(\pi_S^{-1}\) and \(\pi_{S, *} = i_S^{-1}\) are exact functors and the composition \(\pi_{S, *} \circ \pi_S^{-1}\) is the identity functor.

Points of the pro-étale site

We first apply Deligne’s criterion to show that there are enough points.

Lemma

Let \(S\) be a scheme. The pro-étale sites \(\Sch_\proetale\), \(S_\proetale\), \((\Sch/S)_\proetale\), \(S_{affine, \proetale}\), and \((\textit{Aff}/S)_\proetale\) have enough points.

Proof

The big pro-étale topos of \(S\) is equivalent to the topos defined by \((\textit{Aff}/S)_\proetale\), see Lemma 098N. The topos of sheaves on \(S_\proetale\) is equivalent to the topos associated to \(S_{affine, \proetale}\), see Lemma 098W. The result for the sites \((\textit{Aff}/S)_\proetale\) and \(S_{affine, \proetale}\) follows immediately from Deligne’s result Sites, Lemma 0DW0. The case \(\Sch_\proetale\) is handled because it is equal to \((\Sch/\Spec(\mathbf{Z}))_\proetale\).

Let \(S\) be a scheme. Let \(\overline{s} : \Spec(k) \to S\) be a geometric point. We define a pro-étale neighbourhood of \(\overline{s}\) to be a commutative diagram \[\xymatrix{ \Spec(k) \ar[r]_-{\overline{u}} \ar[rd]_{\overline{s}} & U \ar[d] \\ & S }\] with \(U \to S\) weakly étale.

Lemma

Let \(S\) be a scheme and let \(\overline{s} : \Spec(k) \to S\) be a geometric point. The category of pro-étale neighbourhoods of \(\overline{s}\) is cofiltered.

Proof

The proof is identitical to the proof of Étale Cohomology, Lemma 03PQ but using the corresponding facts about weakly étale morphisms proven in More on Morphisms, Lemmas 094T, 094U, and 0951.

Lemma

Let \(S\) be a scheme. Let \(\overline{s}\) be a geometric point of \(S\). Let \(\mathcal{U} = \{\varphi_i : S_i \to S\}_{i\in I}\) be a pro-étale covering. Then there exist \(i \in I\) and geometric point \(\overline{s}_i\) of \(S_i\) mapping to \(\overline{s}\).

Proof

Immediate from the fact that \(\coprod \varphi_i\) is surjective and that residue field extensions induced by weakly étale morphisms are separable algebraic (see for example More on Morphisms, Lemma 094Z.

Let \(S\) be a scheme and let \(\overline{s}\) be a geometric point of \(S\). For \(\mathcal{F}\) in \(\Sh(S_\proetale)\) define the stalk of \(\mathcal{F}\) at \(\overline{s}\) by the formula \[\mathcal{F}_{\overline{s}} = \colim_{(U, \overline{u})} \mathcal{F}(U)\] where the colimit is over all pro-étale neighbourhoods \((U, \overline{u})\) of \(\overline{s}\) with \(U \in \Ob(S_\proetale)\). It follows from the two lemmas above that the functor \[S_\proetale \textit{Sets},\quad U \longmapsto \{\overline{u}\text{ geometric point of }U\text{ mapping to }\overline{s}\}\] defines a point of the site \(S_\proetale\), see Sites, Definition 00Y5 and Lemma 0F4E. Hence the functor \(\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}\) defines a point of the topos \(\Sh(S_\proetale)\), see Sites, Definition 00Y4 and Lemma 00YA. In particular this functor is exact and commutes with arbitrary colimits. In fact, this functor has another description.

Lemma

In the situation above the scheme \(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})\) is an object of \(X_\proetale\) and there is a canonical isomorphism \[\mathcal{F}(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})) = \mathcal{F}_{\overline{s}}\] functorial in \(\mathcal{F}\).

Proof

The first statement is clear from the construction of the strict henselization as a filtered colimit of étale algebras over \(S\), or by the characterization of weakly étale morphisms of More on Morphisms, Lemma 094Z. The second statement follows as by Olivier’s theorem (More on Algebra, Theorem 092Z) the scheme \(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})\) is an initial object of the category of pro-étale neighbourhoods of \(\overline{s}\).

Contrary to the situation with the étale topos of \(S\) it is not true that every point of \(\Sh(S_\proetale)\) is of this form, and it is not true that the collection of points associated to geometric points is conservative. Namely, suppose that \(S = \Spec(k)\) where \(k\) is an algebraically closed field. Let \(A\) be a nonzero abelian group. Consider the sheaf \(\mathcal{F}\) on \(S_\proetale\) defined by the \[\mathcal{F}(U) = \frac{\{\text{functions }U \to A\}}{\{\text{locally constant functions}\}}\] for \(U\) affine and by sheafification in general, see Example 0F6D. Then \(\mathcal{F}(U) = 0\) if \(U = S = \Spec(k)\) but in general \(\mathcal{F}\) is not zero. Namely, \(S_\proetale\) contains affine objects with infinitely many points. For example, let \(E = \lim E_n\) be an inverse limit of finite sets with surjective transition maps, e.g., \(E = \mathbf{Z}_p = \lim \mathbf{Z}/p^n\mathbf{Z}\). The scheme \(U = \Spec(\colim \text{Map}(E_n, k))\) is an object of \(S_\proetale\) because \(\colim \text{Map}(E_n, k)\) is weakly étale (even ind-Zariski) over \(k\). Thus \(\mathcal{F}(U)\) is nonzero as there exist maps \(E \to A\) which aren’t locally constant. Thus \(\mathcal{F}\) is a nonzero abelian sheaf whose stalk at the unique geometric point of \(S\) is zero. Since we know that \(S_\proetale\) has enough points, we conclude there must be a point of the pro-étale site which does not come from the construction explained above.

The replacement for arguments using points, is to use affine weakly contractible objects. First, there are enough affine weakly contractible objects by Lemma 0990. Second, if \(W \in \Ob(S_\proetale)\) is affine weakly contractible, then the functor \[\Sh(S_\proetale) \longrightarrow \textit{Sets},\quad \mathcal{F} \longmapsto \mathcal{F}(W)\] is an exact functor \(\Sh(S_\proetale) \to \textit{Sets}\) which commutes with all limits. The functor \[\textit{Ab}(S_\proetale) \longrightarrow \textit{Ab},\quad \mathcal{F} \longmapsto \mathcal{F}(W)\] is exact and commutes with direct sums (as \(W\) is quasi-compact, see Sites, Lemma 0738), hence commutes with all limits and colimits. Moreover, we can check exactness of a complex of abelian sheaves by evaluation at these affine weakly contractible objects of \(S_\proetale\), see Cohomology on Sites, Proposition 0947.

A final remark is that the functor \(\mathcal{F} \mapsto \mathcal{F}(W)\) for \(W\) affine weakly contractible in general isn’t a stalk functor of a point of \(S_\proetale\) because it doesn’t preserve coproducts of sheaves of sets if \(W\) is disconnected. And in fact, \(W\) is disconnected as soon as \(W\) has more than \(1\) closed point, i.e., when \(W\) is not the spectrum of a strictly henselian local ring (which is the special case discussed above).

Comparison with the étale site

Let \(X\) be a scheme. With suitable choices of sites⁴ the functor \(u : X_\etale \to X_\proetale\) sending \(U/X\) to \(U/X\) defines a morphism of sites \[\epsilon : X_\proetale \longrightarrow X_\etale\] This follows from Sites, Proposition 00X6.

Lemma

With notation as above. Let \(\mathcal{F}\) be a sheaf on \(X_\etale\). The rule \[X_\proetale \longrightarrow \textit{Sets},\quad (f : Y \to X) \longmapsto \Gamma(Y_\etale, f_\etale^{-1}\mathcal{F})\] is a sheaf and is equal to \(\epsilon^{-1}\mathcal{F}\). Here \(f_\etale : Y_\etale \to X_\etale\) is the morphism of small étale sites constructed in Étale Cohomology, Section 04I0.

Proof

By Lemma 098A any pro-étale covering is an fpqc covering. Hence the formula defines a sheaf on \(X_\proetale\) by Étale Cohomology, Lemma 09XN. Let \(a : \Sh(X_\etale) \to \Sh(X_\proetale)\) be the functor sending \(\mathcal{F}\) to the sheaf given by the formula in the lemma. To show that \(a = \epsilon^{-1}\) it suffices to show that \(a\) is a left adjoint to \(\epsilon_*\).

Let \(\mathcal{G}\) be an object of \(\Sh(X_\proetale)\). Recall that \(\epsilon_*\mathcal{G}\) is simply given by the restriction of \(\mathcal{G}\) to the full subcategory \(X_\etale\). Let \(f : Y \to X\) be an object of \(X_\proetale\). We view \(Y_\etale\) as a subcategory of \(X_\proetale\). The restriction maps of the sheaf \(\mathcal{G}\) define a map \[\epsilon_*\mathcal{G} = \mathcal{G}|_{X_\etale} \longrightarrow f_{\etale, *}(\mathcal{G}|_{Y_\etale})\] Namely, for \(U\) in \(X_\etale\) the value of \(f_{\etale, *}(\mathcal{G}|_{Y_\etale})\) on \(U\) is \(\mathcal{G}(Y \times_X U)\) and there is a restriction map \(\mathcal{G}(U) \to \mathcal{G}(Y \times_X U)\). By adjunction this determines a map \[f_\etale^{-1}(\epsilon_*\mathcal{G}) \to \mathcal{G}|_{Y_\etale}\] Putting these together for all \(f : Y \to X\) in \(X_\proetale\) we obtain a canonical map \(a(\epsilon_*\mathcal{G}) \to \mathcal{G}\).

Let \(\mathcal{F}\) be an object of \(\Sh(X_\etale)\). It is immediately clear that \(\mathcal{F} = \epsilon_*a(\mathcal{F})\).

We claim the maps \(\mathcal{F} \to \epsilon_*a(\mathcal{F})\) and \(a(\epsilon_*\mathcal{G}) \to \mathcal{G}\) are the unit and counit of the adjunction (see Categories, Section 0036). To see this it suffices to show that the corresponding maps \[\Mor_{\Sh(X_\proetale)}(a(\mathcal{F}), \mathcal{G}) \to \Mor_{\Sh(X_\etale)}(\mathcal{F}, \epsilon^{-1}\mathcal{G})\] and \[\Mor_{\Sh(X_\etale)}(\mathcal{F}, \epsilon^{-1}\mathcal{G}) \to \Mor_{\Sh(X_\proetale)}(a(\mathcal{F}), \mathcal{G})\] are mutually inverse. We omit the detailed verification.

Lemma

Let \(X\) be a scheme. For every sheaf \(\mathcal{F}\) on \(X_\etale\) the adjunction map \(\mathcal{F} \to \epsilon_*\epsilon^{-1}\mathcal{F}\) is an isomorphism, i.e., \(\epsilon^{-1}\mathcal{F}(U) = \mathcal{F}(U)\) for \(U\) in \(X_\etale\).

Proof

Follows immediately from the description of \(\epsilon^{-1}\) in Lemma 0GLZ.

Lemma

Let \(X\) be a scheme. Let \(Y = \lim Y_i\) be the limit of a directed inverse system of quasi-compact and quasi-separated objects of \(X_\proetale\) with affine transition morphisms. For any sheaf \(\mathcal{F}\) on \(X_\etale\) we have \[\epsilon^{-1}\mathcal{F}(Y) = \colim \epsilon^{-1}\mathcal{F}(Y_i)\] Moreover, if \(Y_i\) is in \(X_\etale\) we have \(\epsilon^{-1}\mathcal{F}(Y) = \colim \mathcal{F}(Y_i)\).

Proof

By the description of \(\epsilon^{-1}\mathcal{F}\) in Lemma 0GLZ, the displayed formula is a special case of Étale Cohomology, Theorem 09YQ. (When \(X\), \(Y\), and the \(Y_i\) are all affine, see the easier to parse Étale Cohomology, Lemma 03Q6.) The final statement follows immediately from this and Lemma 099T.

Lemma

Let \(X\) be an affine scheme. For injective abelian sheaf \(\mathcal{I}\) on \(X_\etale\) we have \(H^p(X_\proetale, \epsilon^{-1}\mathcal{I}) = 0\) for \(p > 0\).

Proof

We are going to use Cohomology on Sites, Lemma 03F9 to prove this. Let \(\mathcal{B} \subset \Ob(X_\proetale)\) be the set of affine objects \(U\) of \(X_\proetale\) such that \(\mathcal{O}(X) \to \mathcal{O}(U)\) is ind-étale. Let \(\text{Cov}\) be the set of pro-étale coverings \(\{U_i \to U\}_{i = 1, \ldots, n}\) with \(U \in \mathcal{B}\) such that \(\mathcal{O}(U) \to \mathcal{O}(U_i)\) is ind-étale for \(i = 1, \ldots, n\). Properties (1) and (2) of Cohomology on Sites, Lemma 03F9 hold for \(\mathcal{B}\) and \(\text{Cov}\) by Lemmas 097K, 097J, and 098D and Proposition 097Z.

To check condition (3) suppose that \(\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}\) is an element of \(\text{Cov}\). We have to show that the higher Cech cohomology groups of \(\epsilon^{-1}\mathcal{I}\) with respect to \(\mathcal{U}\) are zero. First we write \(U_i = \lim_{a \in A_i} U_{i, a}\) as a directed inverse limit with \(U_{i, a} \to U\) étale and \(U_{i, a}\) affine. We think of \(A_1 \times \ldots \times A_n\) as a direct set with ordering \((a_1, \ldots, a_n) \geq (a_1', \ldots, a_n')\) if and only if \(a_i \geq a_i'\) for \(i = 1, \ldots, n\). Observe that \(\mathcal{U}_{(a_1, \ldots, a_n)} = \{U_{i, a_i} \to U\}_{i = 1, \ldots, n}\) is an étale covering for all \(a_1, \ldots, a_n \in A_1 \times \ldots \times A_n\). Observe that \[U_{i_0} \times_U U_{i_1} \times_U \ldots \times_U U_{i_p} = \lim_{(a_1, \ldots, a_n) \in A_1 \times \ldots \times A_n} U_{i_0, a_{i_0}} \times_U U_{i_1, a_{i_1}} \times_U \ldots \times_U U_{i_p, a_{i_p}}\] for all \(i_0, \ldots, i_p \in \{1, \ldots, n\}\) because limits commute with fibred products. Hence by Lemma 099S and exactness of filtered colimits we have \[\check{H}^p(\mathcal{U}, \epsilon^{-1}\mathcal{I}) = \colim \check{H}^p(\mathcal{U}_{(a_1, \ldots, a_n)}, \epsilon^{-1}\mathcal{I})\] Thus it suffices to prove the vanishing for étale coverings of \(U\)!

Let \(\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}\) be an étale covering with \(U_i\) affine. Write \(U = \lim_{b \in B} U_b\) as a directed inverse limit with \(U_b\) affine and \(U_b \to X\) étale. By Limits, Lemmas 01ZM, 01Z6, and 07RP we can choose a \(b_0 \in B\) such that for \(i = 1, \ldots, n\) there is an étale morphism \(U_{i, b_0} \to U_{b_0}\) of affines such that \(U_i = U \times_{U_{b_0}} U_{i, b_0}\). Set \(U_{i, b} = U_b \times_{U_{b_0}} U_{i, b_0}\) for \(b \geq b_0\). For \(b\) large enough the family \(\mathcal{U}_b = \{U_{i, b} \to U_b\}_{i = 1, \ldots, n}\) is an étale covering, see Limits, Lemma 07RR. Exactly as before we find that \[\check{H}^p(\mathcal{U}, \epsilon^{-1}\mathcal{I}) = \colim \check{H}^p(\mathcal{U}_b, \epsilon^{-1}\mathcal{I}) = \colim \check{H}^p(\mathcal{U}_b, \mathcal{I})\] the final equality by Lemma 099T. Since each of the Čech complexes on the right hand side is acyclic in positive degrees (Cohomology on Sites, Lemma 03AW) it follows that the one on the left is too. This proves condition (3) of Cohomology on Sites, Lemma 03F9. Since \(X \in \mathcal{B}\) the lemma follows.

Lemma

Let \(X\) be a scheme.

1. For an abelian sheaf \(\mathcal{F}\) on \(X_\etale\) we have \(R\epsilon_*(\epsilon^{-1}\mathcal{F}) = \mathcal{F}\).

2. For \(K \in D^+(X_\etale)\) the map \(K \to R\epsilon_*\epsilon^{-1}K\) is an isomorphism.

Proof

Let \(\mathcal{I}\) be an injective abelian sheaf on \(X_\etale\). Recall that \(R^q\epsilon_*(\epsilon^{-1}\mathcal{I})\) is the sheaf associated to \(U \mapsto H^q(U_\proetale, \epsilon^{-1}\mathcal{I})\), see Cohomology on Sites, Lemma 072W. By Lemma 099U we see that this is zero for \(q > 0\) and \(U\) affine and étale over \(X\). Since every object of \(X_\etale\) has a covering by affine objects, it follows that \(R^q\epsilon_*(\epsilon^{-1}\mathcal{I}) = 0\) for \(q > 0\).

Let \(K \in D^+(X_\etale)\). Choose a bounded below complex \(\mathcal{I}^\bullet\) of injective abelian sheaves on \(X_\etale\) representing \(K\). Then \(\epsilon^{-1}K\) is represented by \(\epsilon^{-1}\mathcal{I}^\bullet\). By Leray’s acyclicity lemma (Derived Categories, Lemma 015E) we see that \(R\epsilon_*\epsilon^{-1}K\) is represented by \(\epsilon_*\epsilon^{-1}\mathcal{I}^\bullet\). By Lemma 099T we conclude that \(R\epsilon_*\epsilon^{-1}\mathcal{I}^\bullet = \mathcal{I}^\bullet\) and the proof of (2) is complete. Part (1) is a special case of (2).

Lemma

Let \(X\) be a scheme.

1. For an abelian sheaf \(\mathcal{F}\) on \(X_\etale\) we have \[H^i(X_\etale, \mathcal{F}) = H^i(X_\proetale, \epsilon^{-1}\mathcal{F})\] for all \(i\).

2. For \(K \in D^+(X_\etale)\) we have \[R\Gamma(X_\etale, K) = R\Gamma(X_\proetale, \epsilon^{-1}K)\]

Proof

Immediate consequence of Lemma 099V and the Leray spectral sequence (Cohomology on Sites, Lemma 0733).

Lemma

Let \(X\) be a scheme. Let \(\mathcal{G}\) be a sheaf of (possibly noncommutative) groups on \(X_\etale\). We have \[H^1(X_\etale, \mathcal{G}) = H^1(X_\proetale, \epsilon^{-1}\mathcal{G})\] where \(H^1\) is defined as the set of isomorphism classes of torsors (see Cohomology on Sites, Section 03AG).

Proof

Since the functor \(\epsilon^{-1}\) is fully faithful by Lemma 099T it is clear that the map \(H^1(X_\etale, \mathcal{G}) \to H^1(X_\proetale, \epsilon^{-1}\mathcal{G})\) is injective. To show surjectivity it suffices to show that any \(\epsilon^{-1}\mathcal{G}\)-torsor \(\mathcal{F}\) is étale locally trivial. To do this we may assume that \(X\) is affine. Thus we reduce to proving surjectivity for \(X\) affine.

Choose a covering \(\{U \to X\}\) with (a) \(U\) affine, (b) \(\mathcal{O}(X) \to \mathcal{O}(U)\) ind-étale, and (c) \(\mathcal{F}(U)\) nonempty. We can do this by Proposition 097Z and the fact that standard pro-étale coverings of \(X\) are cofinal among all pro-étale coverings of \(X\) (Lemma 098D). Write \(U = \lim U_i\) as a limit of affine schemes étale over \(X\). Pick \(s \in \mathcal{F}(U)\). Let \(g \in \epsilon^{-1}\mathcal{G}(U \times_X U)\) be the unique section such that \(g \cdot \text{pr}_1^*s = \text{pr}_2^*s\) in \(\mathcal{F}(U \times_X U)\). Then \(g\) satisfies the cocycle condition \[\text{pr}_{12}^*g \cdot \text{pr}_{23}^*g = \text{pr}_{13}^*g\] in \(\epsilon^{-1}\mathcal{G}(U \times_X U \times_X U)\). By Lemma 099S we have \[\epsilon^{-1}\mathcal{G}(U \times_X U) = \colim \mathcal{G}(U_i \times_X U_i)\] and \[\epsilon^{-1}\mathcal{G}(U \times_X U \times_X U) = \colim \mathcal{G}(U_i \times_X U_i \times_X U_i)\] hence we can find an \(i\) and an element \(g_i \in \mathcal{G}(U_i \times_X U_i)\) mapping to \(g\) satisfying the cocycle condition. The cocycle \(g_i\) then defines a torsor for \(\mathcal{G}\) on \(X_\etale\) whose pullback is isomorphic to \(\mathcal{F}\) by construction. Some details omitted (namely, the relationship between torsors and 1-cocycles which should be added to the chapter on cohomology on sites).

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a ring.

1. The essential image of the fully faithful functor \(\epsilon^{-1} : \textit{Mod}(X_\etale, \Lambda) \to \textit{Mod}(X_\proetale, \Lambda)\) is a weak Serre subcategory \(\mathcal{C}\).

2. The functor \(\epsilon^{-1}\) defines an equivalence of categories of \(D^+(X_\etale, \Lambda)\) with \(D^+_\mathcal{C}(X_\proetale, \Lambda)\) with quasi-inverse given by \(R\epsilon_*\).

Proof

To prove (1) we will prove conditions (1) – (4) of Homology, Lemma 0754. Since \(\epsilon^{-1}\) is fully faithful (Lemma 099T) and exact, everything is clear except for condition (4). However, if \[0 \to \epsilon^{-1}\mathcal{F}_1 \to \mathcal{G} \to \epsilon^{-1}\mathcal{F}_2 \to 0\] is a short exact sequence of sheaves of \(\Lambda\)-modules on \(X_\proetale\), then we get \[0 \to \epsilon_*\epsilon^{-1}\mathcal{F}_1 \to \epsilon_*\mathcal{G} \to \epsilon_*\epsilon^{-1}\mathcal{F}_2 \to R^1\epsilon_*\epsilon^{-1}\mathcal{F}_1\] which by Lemma 099V is the same as a short exact sequence \[0 \to \mathcal{F}_1 \to \epsilon_*\mathcal{G} \to \mathcal{F}_2 \to 0\] Pulling pack we find that \(\mathcal{G} = \epsilon^{-1}\epsilon_*\mathcal{G}\). This proves (1).

Part (2) follows from part (1) and Cohomology on Sites, Lemma 0D7U.

Let \(\Lambda\) be a ring. In Modules on Sites, Section 093P we have defined the notion of a locally constant sheaf of \(\Lambda\)-modules on a site. If \(M\) is a \(\Lambda\)-module, then \(\underline{M}\) is of finite presentation as a sheaf of \(\underline{\Lambda}\)-modules if and only if \(M\) is a finitely presented \(\Lambda\)-module, see Modules on Sites, Lemma 093N.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a ring. The functor \(\epsilon^{-1}\) defines an equivalence of categories \[\left\{ \begin{matrix} \text{locally constant sheaves}\\ \text{of }\Lambda\text{-modules on }X_\etale\\ \text{of finite presentation} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{locally constant sheaves}\\ \text{of }\Lambda\text{-modules on }X_\proetale\\ \text{of finite presentation} \end{matrix} \right\}\]

Proof

Let \(\mathcal{F}\) be a locally constant sheaf of \(\Lambda\)-modules on \(X_\proetale\) of finite presentation. Choose a pro-étale covering \(\{U_i \to X\}\) such that \(\mathcal{F}|_{U_i}\) is constant, say \(\mathcal{F}|_{U_i} \cong \underline{M_i}_{U_i}\). Observe that \(U_i \times_X U_j\) is empty if \(M_i\) is not isomorphic to \(M_j\). For each \(\Lambda\)-module \(M\) let \(I_M = \{i \in I \mid M_i \cong M\}\). As pro-étale coverings are fpqc coverings and by Descent, Lemma 03N0 we see that \(U_M = \bigcup_{i \in I_M} \Im(U_i \to X)\) is an open subset of \(X\). Then \(X = \coprod U_M\) is a disjoint open covering of \(X\). We may replace \(X\) by \(U_M\) for some \(M\) and assume that \(M_i = M\) for all \(i\).

Consider the sheaf \(\mathcal{I} = \mathit{Isom}(\underline{M}, \mathcal{F})\). This sheaf is a torsor for \(\mathcal{G} = \mathit{Isom}(\underline{M}, \underline{M})\). By Modules on Sites, Lemma 093T we have \(\mathcal{G} = \underline{G}\) where \(G = \mathit{Isom}_\Lambda(M, M)\). Since torsors for the étale topology and the pro-étale topology agree by Lemma 099X it follows that \(\mathcal{I}\) has sections étale locally on \(X\). Thus \(\mathcal{F}\) is étale locally a constant sheaf which is what we had to show.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. Let \(D_{flc}(X_\etale, \Lambda)\), resp. \(D_{flc}(X_\proetale, \Lambda)\) be the full subcategory of \(D(X_\etale, \Lambda)\), resp. \(D(X_\proetale, \Lambda)\) consisting of those complexes whose cohomology sheaves are locally constant sheaves of \(\Lambda\)-modules of finite type. Then \[\epsilon^{-1} : D_{flc}^+(X_\etale, \Lambda) \longrightarrow D_{flc}^+(X_\proetale, \Lambda)\] is an equivalence of categories.

Proof

The categories \(D_{flc}(X_\etale, \Lambda)\) and \(D_{flc}(X_\proetale, \Lambda)\) are strictly full, saturated, triangulated subcategories of \(D(X_\etale, \Lambda)\) and \(D(X_\proetale, \Lambda)\) by Modules on Sites, Lemma 093U and Derived Categories, Section 06UP. The statement of the lemma follows by combining Lemmas 09B1 and 099Y.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. Let \(K\) be an object of \(D(X_\proetale, \Lambda)\). Set \(K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\). If \(K_1\) is

1. in the essential image of \(\epsilon^{-1} :D(X_\etale, \Lambda/I) \to D(X_\proetale, \Lambda/I)\), and

2. has tor amplitude in \([a,\infty)\) for some \(a \in \mathbf{Z}\),

then (1) and (2) hold for \(K_n\) as an object of \(D(X_\proetale, \Lambda/I^n)\).

Proof

Assertion (2) for \(K_n\) follows from the more general Cohomology on Sites, Lemma 0942. Assertion (1) for \(K_n\) follows by induction on \(n\) from the distinguished triangles \[K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}} \to K_{n + 1} \to K_n \to K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}}[1]\] and the isomorphism \[K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}} = K_1 \otimes_{\Lambda/I}^\mathbf{L} \underline{I^n/I^{n + 1}}\] and the fact proven in Lemma 09B1 that the essential image of \(\epsilon^{-1}\) is a triangulated subcategory of \(D^+(X_\proetale, \Lambda/I^n)\).

Example

Let \(X\) be a scheme. Let \(A\) be an abelian group. Denote \(fun(-, A)\) the sheaf on \(X_\proetale\) which maps \(U\) to the set of all maps \(U \to A\) (of sets of points). Consider the sequence of sheaves \[0 \to \underline{A} \to fun(-, A) \to \mathcal{F} \to 0\] on \(X_\proetale\). Since the constant sheaf is the pullback from the final topos we see that \(\underline{A} = \epsilon^{-1}\underline{A}\). However, if \(A\) has more than one element, then neither \(fun(-, A)\) nor \(\mathcal{F}\) are pulled back from the étale site of \(X\). To work out the values of \(\mathcal{F}\) in some cases, assume that all points of \(X\) are closed with separably closed residue fields and \(U\) is affine. Then all points of \(U\) are closed with separably closed residue fields and we have \[H^1_\proetale(U, \underline{A}) = H^1_\etale(U, \underline{A}) = 0\] by Lemma 099W and Étale Cohomology, Lemma 09AY. Hence in this case we have \[\mathcal{F}(U) = fun(U, A)/\underline{A}(U)\]

Derived completion in the constant Noetherian case

We continue the discussion started in Algebraic and Formal Geometry, Section 0995; we assume the reader has read at least some of that section.

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Recall from Modules on Sites, Lemma 093M that \[\underline{\Lambda}^\wedge = \lim \underline{\Lambda/I^n}\] is a flat \(\underline{\Lambda}\)-algebra and that the map \(\underline{\Lambda} \to \underline{\Lambda}^\wedge\) identifies quotients by \(I\). Hence Algebraic and Formal Geometry, Lemma 099I tells us that \[D_{comp}(\mathcal{C}, \Lambda) = D_{comp}(\mathcal{C}, \underline{\Lambda}^\wedge)\] In particular the cohomology sheaves \(H^i(K)\) of an object \(K\) of \(D_{comp}(\mathcal{C}, \Lambda)\) are sheaves of \(\underline{\Lambda}^\wedge\)-modules. For notational convenience we often work with \(D_{comp}(\mathcal{C}, \Lambda)\).

Lemma

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. The left adjoint to the inclusion functor \(D_{comp}(\mathcal{C}, \Lambda) \to D(\mathcal{C}, \Lambda)\) of Algebraic and Formal Geometry, Proposition 099F sends \(K\) to \[K^\wedge = R\lim(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n})\] In particular, \(K\) is derived complete if and only if \(K = R\lim(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n})\).

Proof

Choose generators \(f_1, \ldots, f_r\) of \(I\). By Algebraic and Formal Geometry, Lemma 0A0E we have \[K^\wedge = R\lim (K \otimes_\Lambda^\mathbf{L} \underline{K_n})\] where \(K_n = K(\Lambda, f_1^n, \ldots, f_r^n)\). In More on Algebra, Lemma 0921 we have seen that the pro-systems \(\{K_n\}\) and \(\{\Lambda/I^n\}\) of \(D(\Lambda)\) are isomorphic. Thus the lemma follows.

Lemma

Let \(\Lambda\) be a Noetherian ring. Let \(I \subset \Lambda\) be an ideal. Let \(f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})\) be a morphism of topoi. Then

1. \(Rf_*\) sends \(D_{comp}(\mathcal{D}, \Lambda)\) into \(D_{comp}(\mathcal{C}, \Lambda)\),

2. the map \(Rf_* : D_{comp}(\mathcal{D}, \Lambda) \to D_{comp}(\mathcal{C}, \Lambda)\) has a left adjoint \(Lf_{comp}^* : D_{comp}(\mathcal{C}, \Lambda) \to D_{comp}(\mathcal{D}, \Lambda)\) which is \(Lf^*\) followed by derived completion,

3. \(Rf_*\) commutes with derived completion,

4. for \(K\) in \(D_{comp}(\mathcal{D}, \Lambda)\) we have \(Rf_*K = R\lim Rf_*(K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n})\).

5. for \(M\) in \(D_{comp}(\mathcal{C}, \Lambda)\) we have \(Lf^*_{comp}M = R\lim Lf^*(M \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n})\).

Proof

We have seen (1) and (2) in Algebraic and Formal Geometry, Lemma 099K. Part (3) follows from Algebraic and Formal Geometry, Lemma 0A0G. For (4) let \(K\) be derived complete. Then \[Rf_*K = Rf_*( R\lim K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n}) = R\lim Rf_*(K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n})\] the first equality by Lemma 099M and the second because \(Rf_*\) commutes with \(R\lim\) (Cohomology on Sites, Lemma 0A07). This proves (4). To prove (5), by Lemma 099M we have \[Lf_{comp}^*M = R\lim ( Lf^*M \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n})\] Since \(Lf^*\) commutes with derived tensor product by Cohomology on Sites, Lemma 07A4 and since \(Lf^*\underline{\Lambda/I^n} = \underline{\Lambda/I^n}\) we get (5).

Derived completion and weakly contractible objects

We continue the discussion in Section 099L. In this section we will see how the existence of weakly contractible objects simplifies the study of derived complete modules.

Let \(\mathcal{C}\) be a site. Let \(\Lambda\) be a Noetherian ring. Let \(I \subset \Lambda\) be an ideal. Although the general theory concerning \(D_{comp}(\mathcal{C}, \Lambda)\) is quite satisfactory it is hard to explicitly give examples of derived complete complexes. We know that

1. every object \(M\) of \(D(\mathcal{C}, \Lambda/I^n)\) restricts to a derived complete object of \(D(\mathcal{C}, \Lambda)\), and

2. for every \(K \in D(\mathcal{C}, \Lambda)\) the derived completion \(K^\wedge = R\lim (K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n})\) is derived complete.

The first type of objects are trivially complete and perhaps not interesting. The problem with (2) is that derived completion in general is somewhat mysterious, even in case \(K = \underline{\Lambda}\). Namely, by definition of homotopy limits there is a distinguished triangle \[R\lim(\underline{\Lambda/I^n}) \to \prod \underline{\Lambda/I^n} \to \prod \underline{\Lambda/I^n} \to R\lim(\underline{\Lambda/I^n})[1]\] in \(D(\mathcal{C}, \Lambda)\) where the products are in \(D(\mathcal{C}, \Lambda)\). These are computed by taking products of injective resolutions (Injectives, Lemma 07D9), so we see that the sheaf \(H^p(\prod \underline{\Lambda/I^n})\) is the sheafification of the presheaf \[U \longmapsto \prod H^p(U, \Lambda/I^n).\] As an explicit example, if \(X = \Spec(\mathbf{C}[t, t^{-1}])\), \(\mathcal{C} = X_\etale\), \(\Lambda = \mathbf{Z}\), \(I = (2)\), and \(p = 1\), then we get the sheafification of the presheaf \[U \mapsto \prod H^1(U_\etale, \mathbf{Z}/2^n\mathbf{Z})\] for \(U\) étale over \(X\). Note that \(H^1(X_\etale, \mathbf{Z}/m\mathbf{Z})\) is cyclic of order \(m\) with generator \(\alpha_m\) given by the finite étale \(\mathbf{Z}/m\mathbf{Z}\)-covering given by the equation \(t = s^m\) (see Étale Cohomology, Section 03N8). Then the section \[\alpha = (\alpha_{2^n}) \in \prod H^1(X_\etale, \mathbf{Z}/2^n\mathbf{Z})\] of the presheaf above does not restrict to zero on any nonempty étale scheme over \(X\), whence the sheaf associated to the presheaf is not zero.

However, on the pro-étale site this phenomenon does not occur. The reason is that we have enough (quasi-compact) weakly contractible objects. In the following proposition we collect some results about derived completion in the Noetherian constant case for sites having enough weakly contractible objects (see Sites, Definition 090L).

Proposition

Let \(\mathcal{C}\) be a site. Assume \(\mathcal{C}\) has enough weakly contractible objects. Let \(\Lambda\) be a Noetherian ring. Let \(I \subset \Lambda\) be an ideal.

1. The category of derived complete sheaves \(\Lambda\)-modules is a weak Serre subcategory of \(\textit{Mod}(\mathcal{C}, \Lambda)\).

2. A sheaf \(\mathcal{F}\) of \(\Lambda\)-modules satisfies \(\mathcal{F} = \lim \mathcal{F}/I^n\mathcal{F}\) if and only if \(\mathcal{F}\) is derived complete and \(\bigcap I^n\mathcal{F} = 0\).

3. The sheaf \(\underline{\Lambda}^\wedge\) is derived complete.

4. If \(\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\) is an inverse system of derived complete sheaves of \(\Lambda\)-modules, then \(\lim \mathcal{F}_n\) is derived complete.

5. An object \(K \in D(\mathcal{C}, \Lambda)\) is derived complete if and only if each cohomology sheaf \(H^p(K)\) is derived complete.

6. An object \(K \in D_{comp}(\mathcal{C}, \Lambda)\) is bounded above if and only if \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}\) is bounded above.

7. An object \(K \in D_{comp}(\mathcal{C}, \Lambda)\) is bounded if \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}\) has finite tor dimension.

Proof

Let \(\mathcal{B} \subset \Ob(\mathcal{C})\) be a subset such that every \(U \in \mathcal{B}\) is weakly contractible and every object of \(\mathcal{C}\) has a covering by elements of \(\mathcal{B}\). We will use the results of Cohomology on Sites, Lemma 0946 and Proposition 0947 without further mention.

Recall that \(R\lim\) commutes with \(R\Gamma(U, -)\), see Injectives, Lemma 08U1. Let \(f \in I\). Recall that \(T(K, f)\) is the homotopy limit of the system \[\ldots \xrightarrow{f} K \xrightarrow{f} K \xrightarrow{f} K\] in \(D(\mathcal{C}, \Lambda)\). Thus \[R\Gamma(U, T(K, f)) = T(R\Gamma(U, K), f).\] Since we can test isomorphisms of maps between objects of \(D(\mathcal{C}, \Lambda)\) by evaluating at \(U \in \mathcal{B}\) we conclude an object \(K\) of \(D(\mathcal{C}, \Lambda)\) is derived complete if and only if for every \(U \in \mathcal{B}\) the object \(R\Gamma(U, K)\) is derived complete as an object of \(D(\Lambda)\).

The remark above implies that items (1), (5) follow from the corresponding results for modules over rings, see More on Algebra, Lemmas 091P and 091U. In the same way (2) can be deduced from More on Algebra, Proposition 091T as \((I^n\mathcal{F})(U) = I^n \cdot \mathcal{F}(U)\) for \(U \in \mathcal{B}\) (by exactness of evaluating at \(U\)).

Proof of (4). The homotopy limit \(R\lim \mathcal{F}_n\) is in \(D_{comp}(X, \Lambda)\) (see discussion following Algebraic and Formal Geometry, Definition 0999). By part (5) just proved we conclude that \(\lim \mathcal{F}_n = H^0(R\lim \mathcal{F}_n)\) is derived complete. Part (3) is a special case of (4).

Proof of (6) and (7). Follows from Lemma 099M and Cohomology on Sites, Lemma 0942 and the computation of homotopy limits in Cohomology on Sites, Proposition 0947.

Cohomology of a point

Let \(\Lambda\) be a Noetherian ring complete with respect to an ideal \(I \subset \Lambda\). Let \(k\) be a field. In this section we “compute” \[H^i(\Spec(k)_\proetale, \underline{\Lambda}^\wedge)\] where \(\underline{\Lambda}^\wedge = \lim_m \underline{\Lambda/I^m}\) as before. Let \(k^{sep}\) be a separable algebraic closure of \(k\). Then \[\mathcal{U} = \{\Spec(k^{sep}) \to \Spec(k)\}\] is a pro-étale covering of \(\Spec(k)\). We will use the Čech to cohomology spectral sequence with respect to this covering. Set \(U_0 = \Spec(k^{sep})\) and \[\begin{align*} U_n & = \Spec(k^{sep}) \times_{\Spec(k)} \Spec(k^{sep}) \times_{\Spec(k)} \ldots \times_{\Spec(k)} \Spec(k^{sep}) \\ & = \Spec(k^{sep} \otimes_k k^{sep} \otimes_k \ldots \otimes_k k^{sep}) \end{align*}\] (\(n + 1\) factors). Note that the underlying topological space \(|U_0|\) of \(U_0\) is a singleton and for \(n \geq 1\) we have \[|U_n| = G \times \ldots \times G\quad (n\text{ factors})\] as profinite spaces where \(G = \text{Gal}(k^{sep}/k)\). Namely, every point of \(U_n\) has residue field \(k^{sep}\) and we identify \((\sigma_1, \ldots, \sigma_n)\) with the point corresponding to the surjection \[k^{sep} \otimes_k k^{sep} \otimes_k \ldots \otimes_k k^{sep} \longrightarrow k^{sep}, \quad \lambda_0 \otimes \lambda_1 \otimes \ldots \lambda_n \longmapsto \lambda_0 \sigma_1(\lambda_1) \ldots \sigma_n(\lambda_n)\] Then we compute \[\begin{align*} R\Gamma((U_n)_\proetale, \underline{\Lambda}^\wedge) & = R\lim_m R\Gamma((U_n)_\proetale, \underline{\Lambda/I^m}) \\ & = R\lim_m R\Gamma((U_n)_\etale, \underline{\Lambda/I^m}) \\ & = \lim_m H^0(U_n, \underline{\Lambda/I^m}) \\ & = \text{Maps}_{cont}(G \times \ldots \times G, \Lambda) \end{align*}\] The first equality because \(R\Gamma\) commutes with derived limits and as \(\Lambda^\wedge\) is the derived limit of the sheaves \(\underline{\Lambda/I^m}\) by Proposition 099Q. The second equality by Lemma 099W. The third equality by Étale Cohomology, Lemma 09AY. The fourth equality uses Étale Cohomology, Remark 03P5 to identify sections of the constant sheaf \(\underline{\Lambda/I^m}\). Then it uses the fact that \(\Lambda\) is complete with respect to \(I\) and hence equal to \(\lim_m \Lambda/I^m\) as a topological space, to see that \(\lim_m \text{Map}_{cont}(G, \Lambda/I^m) = \text{Map}_{cont}(G, \Lambda)\) and similarly for higher powers of \(G\). At this point Cohomology on Sites, Lemmas 03AX and 03F7 tell us that \[\Lambda \to \text{Maps}_{cont}(G, \Lambda) \to \text{Maps}_{cont}(G \times G, \Lambda) \to \ldots\] computes the pro-étale cohomology. In other words, we see that \[H^i(\Spec(k)_\proetale, \underline{\Lambda}^\wedge) = H^i_{cont}(G, \Lambda)\] where the right hand side is Tate’s continuous cohomology, see Étale Cohomology, Section 0DVG. Of course, this is as it should be.

Lemma

Let \(k\) be a field. Let \(G = \text{Gal}(k^{sep}/k)\) be its absolute Galois group. Further,

1. let \(M\) be a profinite abelian group with a continuous \(G\)-action, or

2. let \(\Lambda\) be a Noetherian ring and \(I \subset \Lambda\) an ideal an let \(M\) be an \(I\)-adically complete \(\Lambda\)-module with continuous \(G\)-action.

Then there is a canonical sheaf \(\underline{M}^\wedge\) on \(\Spec(k)_\proetale\) associated to \(M\) such that \[H^i(\Spec(k), \underline{M}^\wedge) = H^i_{cont}(G, M)\] as abelian groups or \(\Lambda\)-modules.

Proof

Proof in case (2). Set \(M_n = M/I^nM\). Then \(M = \lim M_n\) as \(M\) is assumed \(I\)-adically complete. Since the action of \(G\) is continuous we get continuous actions of \(G\) on \(M_n\). By Étale Cohomology, Theorem 03QT this action corresponds to a (locally constant) sheaf \(\underline{M_n}\) of \(\Lambda/I^n\)-modules on \(\Spec(k)_\etale\). Pull back to \(\Spec(k)_\proetale\) by the comparison morphism \(\epsilon\) and take the limit \[\underline{M}^\wedge = \lim \epsilon^{-1}\underline{M_n}\] to get the sheaf promised in the lemma. Exactly the same argument as given in the introduction of this section gives the comparison with Tate’s continuous Galois cohomology.

Functoriality of the pro-étale site

Let \(f : X \to Y\) be a morphism of schemes. The functor \(Y_\proetale \to X_\proetale\), \(V \mapsto X \times_Y V\) induces a morphism of sites \(f_\proetale : X_\proetale \to Y_\proetale\), see Sites, Proposition 00X6. In fact, we obtain a commutative diagram of morphisms of sites \[\xymatrix{ X_\proetale \ar[r]_\epsilon \ar[d]_{f_\proetale} & X_\etale \ar[d]^{f_\etale} \\ Y_\proetale \ar[r]^\epsilon & Y_\etale }\] where \(\epsilon\) is as in Section 099R. In particular we have \(\epsilon^{-1} f_\etale^{-1} = f_\proetale^{-1} \epsilon^{-1}\). Here is the corresponding result for pushforward.

Lemma

Let \(f : X \to Y\) be a morphism of schemes which is quasi-compact and quasi-separated.

1. Let \(\mathcal{F}\) be a sheaf of sets on \(X_\etale\). Then we have \(f_{\proetale, *}\epsilon^{-1}\mathcal{F} = \epsilon^{-1}f_{\etale, *}\mathcal{F}\).

2. Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). Then we have \(Rf_{\proetale, *}\epsilon^{-1}\mathcal{F} = \epsilon^{-1}Rf_{\etale, *}\mathcal{F}\).

Proof

Proof of (1). Let \(\mathcal{F}\) be a sheaf of sets on \(X_\etale\). There is a canonical map \(\epsilon^{-1}f_{\etale, *}\mathcal{F} \to f_{\proetale, *}\epsilon^{-1}\mathcal{F}\), see Sites, Section 06UM. To show it is an isomorphism we may work (Zariski) locally on \(Y\), hence we may assume \(Y\) is affine. In this case every object of \(Y_\proetale\) has a covering by objects \(V = \lim V_i\) which are limits of affine schemes \(V_i\) étale over \(Y\) (by Proposition 097Z for example). Evaluating the map \(\epsilon^{-1}f_{\etale, *}\mathcal{F} \to f_{\proetale, *}\epsilon^{-1}\mathcal{F}\) on \(V\) we obtain a map \[\colim \Gamma(X \times_Y V_i, \mathcal{F}) \longrightarrow \Gamma(X \times_Y V, \epsilon^*\mathcal{F})\] See Lemma 099S for the left hand side. By Lemma 0GLZ we have \[\Gamma(X \times_Y V, \epsilon^*\mathcal{F}) = \Gamma(X \times_Y V, g_\etale^{-1}\mathcal{F})\] where \(g : X \times_Y V \to X\) is the projection. Hence the result holds by Étale Cohomology, Lemma 03Q6.

Proof of (2). Arguing in exactly the same manner as above we see that it suffices to show that \[\colim H^i_\etale(X \times_Y V_i, \mathcal{F}) \longrightarrow H^i_\etale(X \times_Y V, \mathcal{F})\] which follows once more from Étale Cohomology, Lemma 03Q6.

Finite morphisms and pro-étale sites

It is not clear that a finite morphism of schemes determines an exact pushforward on abelian pro-étale sheaves.

Lemma

Let \(f : Z \to X\) be a finite morphism of schemes which is locally of finite presentation. Then \(f_{\proetale, *} : \textit{Ab}(Z_\proetale) \to \textit{Ab}(X_\proetale)\) is exact.

Proof

The prove this we may work (Zariski) locally on \(X\) and assume that \(X\) is affine, say \(X = \Spec(A)\). Then \(Z = \Spec(B)\) for some finite \(A\)-algebra \(B\) of finite presentation. The construction in the proof of Proposition 0983 produces a faithfully flat, ind-étale ring map \(A \to D\) with \(D\) w-contractible. We may check exactness of a sequence of sheaves by evaluating on \(U = \Spec(D)\) be such an object. Then \(f_{\proetale, *}\mathcal{F}\) evaluated at \(U\) is equal to \(\mathcal{F}\) evaluated at \(V = \Spec(D \otimes_A B)\). Since \(D \otimes_A B\) is w-contractible by Lemma 0986 evaluation at \(V\) is exact.

Closed immersions and pro-étale sites

It is not clear (and likely false) that a closed immersion of schemes determines an exact pushforward on abelian pro-étale sheaves.

Lemma

Let \(i : Z \to X\) be a closed immersion morphism of affine schemes. Denote \(X_{app}\) and \(Z_{app}\) the sites introduced in Lemma 098X. The base change functor \[u : X_{app} \to Z_{app},\quad U \longmapsto u(U) = U \times_X Z\] is continuous and has a fully faithful left adjoint \(v\). For \(V\) in \(Z_{app}\) the morphism \(V \to v(V)\) is a closed immersion identifying \(V\) with \(u(v(V)) = v(V) \times_X Z\) and every point of \(v(V)\) specializes to a point of \(V\). The functor \(v\) is cocontinuous and sends coverings to coverings.

Proof

The existence of the adjoint follows immediately from Lemma 097P and the definitions. It is clear that \(u\) is continuous from the definition of coverings in \(X_{app}\).

Write \(X = \Spec(A)\) and \(Z = \Spec(A/I)\). Let \(V = \Spec(\overline{C})\) be an object of \(Z_{app}\) and let \(v(V) = \Spec(C)\). We have seen in the statement of Lemma 097P that \(V\) equals \(v(V) \times_X Z = \Spec(C/IC)\). Any \(g \in C\) which maps to an invertible element of \(C/IC = \overline{C}\) is invertible in \(C\). Namely, we have the \(A\)-algebra maps \(C \to C_g \to C/IC\) and by adjointness we obtain an \(C\)-algebra map \(C_g \to C\). Thus every point of \(v(V)\) specializes to a point of \(V\).

Suppose that \(\{V_i \to V\}\) is a covering in \(Z_{app}\). Then \(\{v(V_i) \to v(V)\}\) is a finite family of morphisms of \(Z_{app}\) such that every point of \(V \subset v(V)\) is in the image of one of the maps \(v(V_i) \to v(V)\). As the morphisms \(v(V_i) \to v(V)\) are flat (since they are weakly étale) we conclude that \(\{v(V_i) \to v(V)\}\) is jointly surjective. This proves that \(v\) sends coverings to coverings.

Let \(V\) be an object of \(Z_{app}\) and let \(\{U_i \to v(V)\}\) be a covering in \(X_{app}\). Then we see that \(\{u(U_i) \to u(v(V)) = V\}\) is a covering of \(Z_{app}\). By adjointness we obtain morphisms \(v(u(U_i)) \to U_i\). Thus the family \(\{v(u(U_i)) \to v(V)\}\) refines the given covering and we conclude that \(v\) is cocontinuous.

Lemma

Let \(Z \to X\) be a closed immersion morphism of affine schemes. The corresponding morphism of topoi \(i = i_\proetale\) is equal to the morphism of topoi associated to the fully faithful cocontinuous functor \(v : Z_{app} \to X_{app}\) of Lemma 09BK. It follows that

1. \(i^{-1}\mathcal{F}\) is the sheaf associated to the presheaf \(V \mapsto \mathcal{F}(v(V))\),

2. for a weakly contractible object \(V\) of \(Z_{app}\) we have \(i^{-1}\mathcal{F}(V) = \mathcal{F}(v(V))\),

3. \(i^{-1} : \Sh(X_\proetale) \to \Sh(Z_\proetale)\) has a left adjoint \(i^{Sh}_!\),

4. \(i^{-1} : \textit{Ab}(X_\proetale) \to \textit{Ab}(Z_\proetale)\) has a left adjoint \(i_!\),

5. \(\text{id} \to i^{-1}i^{Sh}_!\), \(\text{id} \to i^{-1}i_!\), and \(i^{-1}i_* \to \text{id}\) are isomorphisms, and

6. \(i_*\), \(i^{Sh}_!\) and \(i_!\) are fully faithful.

Proof

By Lemma 098X we may describe \(i_\proetale\) in terms of the morphism of sites \(u : X_{app} \to Z_{app}\), \(V \mapsto V \times_X Z\). The first statement of the lemma follows from Sites, Lemmas 00XX and 00XY (but with the roles of \(u\) and \(v\) reversed).

Proof of (1). By the description of \(i\) as the morphism of topoi associated to \(v\) this holds by the construction, see Sites, Lemma 00XO.

Proof of (2). Since the functor \(v\) sends coverings to coverings by Lemma 09BK we see that the presheaf \(\mathcal{G} : V \mapsto \mathcal{F}(v(V))\) is a separated presheaf (Sites, Definition 00WA). Hence the sheafification of \(\mathcal{G}\) is \(\mathcal{G}^+\), see Sites, Theorem 00WB. Next, let \(V\) be a weakly contractible object of \(Z_{app}\). Let \(\mathcal{V} = \{V_i \to V\}_{i = 1, \ldots, n}\) be any covering in \(Z_{app}\). Set \(\mathcal{V}' = \{\coprod V_i \to V\}\). Since \(v\) commutes with finite disjoint unions (as a left adjoint or by the construction) and since \(\mathcal{F}\) sends finite disjoint unions into products, we see that \[H^0(\mathcal{V}, \mathcal{G}) = H^0(\mathcal{V}', \mathcal{G})\] (notation as in Sites, Section 00W1; compare with Étale Cohomology, Lemma 03OZ). Thus we may assume the covering is given by a single morphism, like so \(\{V' \to V\}\). Since \(V\) is weakly contractible, this covering can be refined by the trivial covering \(\{V \to V\}\). It therefore follows that the value of \(\mathcal{G}^+ = i^{-1}\mathcal{F}\) on \(V\) is simply \(\mathcal{F}(v(V))\) and (2) is proved.

Proof of (3). Every object of \(Z_{app}\) has a covering by weakly contractible objects (Lemma 0990). By the above we see that we would have \(i^{Sh}_!h_V = h_{v(V)}\) for \(V\) weakly contractible if \(i^{Sh}_!\) existed. The existence of \(i^{Sh}_!\) then follows from Sites, Lemma 09YX.

Proof of (4). Existence of \(i_!\) follows in the same way by setting \(i_!\mathbf{Z}_V = \mathbf{Z}_{v(V)}\) for \(V\) weakly contractible in \(Z_{app}\), using similar for direct sums, and applying Homology, Lemma 0793. Details omitted.

Proof of (5). Let \(V\) be a contractible object of \(Z_{app}\). Then \(i^{-1}i^{Sh}_!h_V = i^{-1}h_{v(V)} = h_{u(v(V))} = h_V\). (It is a general fact that \(i^{-1}h_U = h_{u(U)}\).) Since the sheaves \(h_V\) for \(V\) contractible generate \(\Sh(Z_{app})\) (Sites, Lemma 00WS) we conclude \(\text{id} \to i^{-1}i^{Sh}_!\) is an isomorphism. Similarly for the map \(\text{id} \to i^{-1}i_!\). Then \((i^{-1}i_*\mathcal{H})(V) = i_*\mathcal{H}(v(V)) = \mathcal{H}(u(v(V))) = \mathcal{H}(V)\) and we find that \(i^{-1}i_* \to \text{id}\) is an isomorphism.

The fully faithfulness statements of (6) now follow from Categories, Lemma 07RB.

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. Then

1. \(i_\proetale^{-1}\) commutes with limits,

2. \(i_{\proetale, *}\) is fully faithful, and

3. \(i_\proetale^{-1}i_{\proetale, *} \cong \text{id}_{\Sh(Z_\proetale)}\).

Proof

Assertions (2) and (3) are equivalent by Sites, Lemma 04D6. Parts (1) and (3) are (Zariski) local on \(X\), hence we may assume that \(X\) is affine. In this case the result follows from Lemma 09BL.

Lemma

Let \(i : Z \to X\) be an integral universally injective and surjective morphism of schemes. Then \(i_{\proetale, *}\) and \(i_\proetale^{-1}\) are quasi-inverse equivalences of categories of pro-étale topoi.

Proof

There is an immediate reduction to the case that \(X\) is affine. Then \(Z\) is affine too. Set \(A = \mathcal{O}(X)\) and \(B = \mathcal{O}(Z)\). Then the categories of étale algebras over \(A\) and \(B\) are equivalent, see Étale Cohomology, Theorem 04DZ and Remark 05YX. Thus the categories of ind-étale algebras over \(A\) and \(B\) are equivalent. In other words the categories \(X_{app}\) and \(Z_{app}\) of Lemma 098X are equivalent. We omit the verification that this equivalence sends coverings to coverings and vice versa. Thus the result as Lemma 098X tells us the pro-étale topos is the topos of sheaves on \(X_{app}\).

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. Let \(U \to X\) be an object of \(X_\proetale\) such that

1. \(U\) is affine and weakly contractible, and

2. every point of \(U\) specializes to a point of \(U \times_X Z\).

Then \(i_\proetale^{-1}\mathcal{F}(U \times_X Z) = \mathcal{F}(U)\) for all abelian sheaves on \(X_\proetale\).

Proof

Since pullback commutes with restriction, we may replace \(X\) by \(U\). Thus we may assume that \(X\) is affine and weakly contractible and that every point of \(X\) specializes to a point of \(Z\). By Lemma 09BL part (1) it suffices to show that \(v(Z) = X\) in this case. Thus we have to show: If \(A\) is a w-contractible ring, \(I \subset A\) an ideal contained in the Jacobson radical of \(A\) and \(A \to B \to A/I\) is a factorization with \(A \to B\) ind-étale, then there is a unique retraction \(B \to A\) compatible with maps to \(A/I\). Observe that \(B/IB = A/I \times R\) as \(A/I\)-algebras. After replacing \(B\) by a localization we may assume \(B/IB = A/I\). Note that \(\Spec(B) \to \Spec(A)\) is surjective as the image contains \(V(I)\) and hence all closed points and is closed under specialization. Since \(A\) is w-contractible there is a retraction \(B \to A\). Since \(B/IB = A/I\) this retraction is compatible with the map to \(A/I\). We omit the proof of uniqueness (hint: use that \(A\) and \(B\) have isomorphic local rings at maximal ideals of \(A\)).

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. If \(X \setminus i(Z)\) is a retrocompact open of \(X\), then \(i_{\proetale, *}\) is exact.

Proof

The question is local on \(X\) hence we may assume \(X\) is affine. Say \(X = \Spec(A)\) and \(Z = \Spec(A/I)\). There exist \(f_1, \ldots, f_r \in I\) such that \(Z = V(f_1, \ldots, f_r)\) set theoretically, see Algebra, Lemma 00F6. By Lemma 09AB we may assume that \(Z = \Spec(A/(f_1, \ldots, f_r))\). In this case the functor \(i_{\proetale, *}\) is exact by Lemma 09A8.

Extension by zero

The general material in Modules on Sites, Section 03DH allows us to make the following definition.

Definition

Let \(j : U \to X\) be a weakly étale morphism of schemes.

1. The restriction functor \(j^{-1} : \Sh(X_\proetale) \to \Sh(U_\proetale)\) has a left adjoint \(j_!^{Sh} : \Sh(X_\proetale) \to \Sh(U_\proetale)\).

2. The restriction functor \(j^{-1} : \textit{Ab}(X_\proetale) \to \textit{Ab}(U_\proetale)\) has a left adjoint which is denoted \(j_! : \textit{Ab}(U_\proetale) \to \textit{Ab}(X_\proetale)\) and called extension by zero.

3. Let \(\Lambda\) be a ring. The functor \(j^{-1} : \textit{Mod}(X_\proetale, \Lambda) \to \textit{Mod}(U_\proetale, \Lambda)\) has a left adjoint \(j_! : \textit{Mod}(U_\proetale, \Lambda) \to \textit{Mod}(X_\proetale, \Lambda)\) and called extension by zero.

As usual we compare this to what happens in the étale case.

Lemma

Let \(j : U \to X\) be an étale morphism of schemes. Let \(\mathcal{G}\) be an abelian sheaf on \(U_\etale\). Then \(\epsilon^{-1} j_!\mathcal{G} = j_!\epsilon^{-1}\mathcal{G}\) as sheaves on \(X_\proetale\).

Proof

This is true because both functors are left adjoint to \(j_\proetale^{-1} \epsilon_* = \epsilon_* j_\etale^{-1}\). The equality holds by the discussion in Section 09A5.

Lemma

Let \(j : U \to X\) be a weakly étale morphism of schemes. Let \(i : Z \to X\) be a closed immersion such that \(U \times_X Z = \emptyset\). Let \(V \to X\) be an affine object of \(X_\proetale\) such that every point of \(V\) specializes to a point of \(V_Z = Z \times_X V\). Then \(j_!\mathcal{F}(V) = 0\) for all abelian sheaves on \(U_\proetale\).

Proof

Let \(\{V_i \to V\}\) be a pro-étale covering. The lemma follows if we can refine this covering to a covering where the members have no morphisms into \(U\) over \(X\) (see construction of \(j_!\) in Modules on Sites, Section 03DH). First refine the covering to get a finite covering with \(V_i\) affine. For each \(i\) let \(V_i = \Spec(A_i)\) and let \(Z_i \subset V_i\) be the inverse image of \(Z\). Set \(W_i = \Spec(A_{i, Z_i}^\sim)\) with notation as in Lemma 096V. Then \(\coprod W_i \to V\) is weakly étale and the image contains all points of \(V_Z\). Hence the image contains all points of \(V\) by our assumption on specializations. Thus \(\{W_i \to V\}\) is a pro-étale covering refining the given one. But each point in \(W_i\) specializes to a point lying over \(Z\), hence there are no morphisms \(W_i \to U\) over \(X\).

Lemma

Let \(j : U \to X\) be an open immersion of schemes. Then \(\text{id} \cong j^{-1}j_!\) and \(j^{-1}j_* \cong \text{id}\) and the functors \(j_!\) and \(j_*\) are fully faithful.

Proof

See Modules on Sites, Lemma 0F6Z (and Sites, Lemma 00Y2 for the case of sheaves of sets) and Categories, Lemma 07RB.

Here is the relationship between extension by zero and restriction to the complementary closed subscheme.

Lemma

Let \(X\) be a scheme. Let \(Z \subset X\) be a closed subscheme and let \(U \subset X\) be the complement. Denote \(i : Z \to X\) and \(j : U \to X\) the inclusion morphisms. Assume that \(j\) is a quasi-compact morphism. For every abelian sheaf on \(X_\proetale\) there is a canonical short exact sequence \[0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0\] on \(X_\proetale\) where all the functors are for the pro-étale topology.

Proof

We obtain the maps by the adjointness properties of the functors involved. It suffices to show that \(X_\proetale\) has enough objects (Sites, Definition 090L) on which the sequence evaluates to a short exact sequence. Let \(V = \Spec(A)\) be an affine object of \(X_\proetale\) such that \(A\) is w-contractible (there are enough objects of this type). Then \(V \times_X Z\) is cut out by an ideal \(I \subset A\). The assumption that \(j\) is quasi-compact implies there exist \(f_1, \ldots, f_r \in I\) such that \(V(I) = V(f_1, \ldots, f_r)\). We obtain a faithfully flat, ind-Zariski ring map \[A \longrightarrow A_{f_1} \times \ldots \times A_{f_r} \times A_{V(I)}^\sim\] with \(A_{V(I)}^\sim\) as in Lemma 096V. Since \(V_i = \Spec(A_{f_i}) \to X\) factors through \(U\) we have \[j_!j^{-1}\mathcal{F}(V_i) = \mathcal{F}(V_i) \quad\text{and}\quad i_*i^{-1}\mathcal{F}(V_i) = 0\] On the other hand, for the scheme \(V^\sim = \Spec(A_{V(I)}^\sim)\) we have \[j_!j^{-1}\mathcal{F}(V^\sim) = 0 \quad\text{and}\quad \mathcal{F}(V^\sim) = i_*i^{-1}\mathcal{F}(V^\sim)\] the first equality by Lemma 09AG and the second by Lemmas 09AC and 0987. Thus the sequence evaluates to an exact sequence on \(\Spec(A_{f_1} \times \ldots \times A_{f_r} \times A_{V(I)}^\sim)\) and the lemma is proved.

Lemma

Let \(j : U \to X\) be a quasi-compact open immersion morphism of schemes. The functor \(j_! : \textit{Ab}(U_\proetale) \to \textit{Ab}(X_\proetale)\) commutes with limits.

Proof

Since \(j_!\) is exact it suffices to show that \(j_!\) commutes with products. The question is local on \(X\), hence we may assume \(X\) affine. Let \(\mathcal{G}\) be an abelian sheaf on \(U_\proetale\). We have \(j^{-1}j_*\mathcal{G} = \mathcal{G}\). Hence applying the exact sequence of Lemma 09AH we get \[0 \to j_!\mathcal{G} \to j_*\mathcal{G} \to i_*i^{-1}j_*\mathcal{G} \to 0\] where \(i : Z \to X\) is the inclusion of the reduced induced scheme structure on the complement \(Z = X \setminus U\). The functors \(j_*\) and \(i_*\) commute with products as right adjoints. The functor \(i^{-1}\) commutes with products by Lemma 09AA. Hence \(j_!\) does because on the pro-étale site products are exact (Cohomology on Sites, Proposition 0947).

Constructible sheaves on the pro-étale site

We stick to constructible sheaves of \(\Lambda\)-modules for a Noetherian ring. In the future we intend to discuss constructible sheaves of sets, groups, etc.

Definition

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. A sheaf of \(\Lambda\)-modules on \(X_\proetale\) is constructible if for every affine open \(U \subset X\) there exists a finite decomposition of \(U\) into constructible locally closed subschemes \(U = \coprod_i U_i\) such that \(\mathcal{F}|_{U_i}\) is of finite type and locally constant for all \(i\).

Again this does not give anything “new”.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. The functor \(\epsilon^{-1}\) defines an equivalence of categories \[\left\{ \begin{matrix} \text{constructible sheaves of}\\ \Lambda\text{-modules on }X_\etale\\ \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{constructible sheaves of}\\ \Lambda\text{-modules on }X_\proetale\\ \end{matrix} \right\}\] between constructible sheaves of \(\Lambda\)-modules on \(X_\etale\) and constructible sheaves of \(\Lambda\)-modules on \(X_\proetale\).

Proof

By Lemma 099T the functor \(\epsilon^{-1}\) is fully faithful and commutes with pullback (restriction) to the strata. Hence \(\epsilon^{-1}\) of a constructible étale sheaf is a constructible pro-étale sheaf. To finish the proof let \(\mathcal{F}\) be a constructible sheaf of \(\Lambda\)-modules on \(X_\proetale\) as in Definition 09AJ. There is a canonical map \[\epsilon^{-1}\epsilon_*\mathcal{F} \longrightarrow \mathcal{F}\] We will show this map is an isomorphism. This will prove that \(\mathcal{F}\) is in the essential image of \(\epsilon^{-1}\) and finish the proof (details omitted).

Since it suffices to prove this locally on \(X\) we may assume \(X\) is quasi-compact and quasi-separated and that we have a finite partition \(X = \coprod_{i = 1, \ldots, n} X_i\) by constructible locally closed strata such that \(\mathcal{F}|_{X_i}\) is locally constant of finite type. We will use induction on \(n\). The base case \(n = 1\) follows from Lemma 099Y. Take a point \(x \in X\); then \(x \in X_k\) for some \(k\). It suffices to show the displayed map is an isomorphism in an open neighbourhood of \(x\). Hence we may assume \(X_k\) is closed in \(X\). Set \(Z = X_k\) and denote \(U \subset X\) the complement of \(Z\). Observe that the induction hypothesis applies to the restriction of \(\mathcal{F}\) to \(Z\) and to \(U\). By Lemma 09AH we have a short exact sequence \[0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0\] on \(X_\proetale\). Functoriality gives a commutative diagram \[\xymatrix{ 0 \ar[r] & \epsilon^{-1}\epsilon_*j_!j^{-1}\mathcal{F} \ar[r] \ar[d] & \epsilon^{-1}\epsilon_*\mathcal{F} \ar[r] \ar[d] & \epsilon^{-1}\epsilon_*i_*i^{-1}\mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & j_!j^{-1}\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & i_*i^{-1}\mathcal{F} \ar[r] & 0 }\] By induction we know that on the one hand \(\epsilon^{-1}\epsilon_*i^{-1}\mathcal{F} \to i^{-1}\mathcal{F}\) and \(\epsilon^{-1}\epsilon_*j^{-1}\mathcal{F} \to j^{-1}\mathcal{F}\) are isomorphisms and on the other that \(i^{-1}\mathcal{F} = \epsilon^{-1}\mathcal{A}\) and \(j^{-1}\mathcal{F} = \epsilon^{-1}\mathcal{B}\) for some constructible sheaves of \(\Lambda\)-modules \(\mathcal{A}\) on \(Z_\etale\) and \(\mathcal{B}\) on \(U_\etale\). Then \[\epsilon^{-1}\epsilon_*j_!j^{-1}\mathcal{F} = \epsilon^{-1}\epsilon_*j_!\epsilon^{-1}\mathcal{B} = \epsilon^{-1}\epsilon_*\epsilon^{-1}j_!\mathcal{B} = \epsilon^{-1}j_!\mathcal{B} = j_!\epsilon^{-1}\mathcal{B} = j_!j^{-1}\mathcal{F}\] the second equality by Lemma 09AF, the third equality by Lemma 099T, and the fourth equality by Lemma 09AF again. Similarly, we have \[\epsilon^{-1}\epsilon_*i_*i^{-1}\mathcal{F} = \epsilon^{-1}\epsilon_*i_*\epsilon^{-1}\mathcal{A} = \epsilon^{-1}\epsilon_*\epsilon^{-1}i_*\mathcal{A} = \epsilon^{-1}i_*\mathcal{A} = i_*\epsilon^{-1}\mathcal{A} = i_*i^{-1}\mathcal{F}\] this time using Lemma 09A6. By the five lemma we conclude the vertical map in the middle of the big diagram is an isomorphism.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. The category of constructible sheaves of \(\Lambda\)-modules on \(X_\proetale\) is a weak Serre subcategory of \(\textit{Mod}(X_\proetale, \Lambda)\).

Proof

This is a formal consequence of Lemmas 09AK and 09B1 and the result for the étale site (Étale Cohomology, Lemma 03RZ).

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. Let \(D_c(X_\etale, \Lambda)\), resp. \(D_c(X_\proetale, \Lambda)\) be the full subcategory of \(D(X_\etale, \Lambda)\), resp. \(D(X_\proetale, \Lambda)\) consisting of those complexes whose cohomology sheaves are constructible sheaves of \(\Lambda\)-modules. Then \[\epsilon^{-1} : D_c^+(X_\etale, \Lambda) \longrightarrow D_c^+(X_\proetale, \Lambda)\] is an equivalence of categories.

Proof

The categories \(D_c(X_\etale, \Lambda)\) and \(D_c(X_\proetale, \Lambda)\) are strictly full, saturated, triangulated subcategories of \(D(X_\etale, \Lambda)\) and \(D(X_\proetale, \Lambda)\) by Étale Cohomology, Lemma 03RZ and Lemma 09B5 and Derived Categories, Section 06UP. The statement of the lemma follows by combining Lemmas 09B1 and 09AK.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. Let \(K, L \in D_c^-(X_\proetale, \Lambda)\). Then \(K \otimes_\Lambda^\mathbf{L} L\) is in \(D_c^-(X_\proetale, \Lambda)\).

Proof

Note that \(H^i(K \otimes_\Lambda^\mathbf{L} L)\) is the same as \(H^i(\tau_{\geq i - 1}K \otimes_\Lambda^\mathbf{L} \tau_{\geq i - 1}L)\). Thus we may assume \(K\) and \(L\) are bounded. In this case we can apply Lemma 09AL to reduce to the case of the étale site, see Étale Cohomology, Lemma 0961.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring. Let \(I \subset \Lambda\) be an ideal. Let \(K\) be an object of \(D(X_\proetale, \Lambda)\). Set \(K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\). If \(K_1\) is in \(D^-_c(X_\proetale, \Lambda/I)\), then \(K_n\) is in \(D^-_c(X_\proetale, \Lambda/I^n)\) for all \(n\).

Proof

Consider the distinguished triangles \[K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}} \to K_{n + 1} \to K_n \to K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}}[1]\] and the isomorphisms \[K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}} = K_1 \otimes_{\Lambda/I}^\mathbf{L} \underline{I^n/I^{n + 1}}\] By Lemma 09BQ we see that this tensor product has constructible cohomology sheaves (and vanishing when \(K_1\) has vanishing cohomology). Hence by induction on \(n\) using Lemma 09B5 we see that each \(K_n\) has constructible cohomology sheaves.

Constructible adic sheaves

In this section we define the notion of a constructible \(\Lambda\)-sheaf as well as some variants.

Definition

Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(X\) be a scheme. Let \(\mathcal{F}\) be a sheaf of \(\Lambda\)-modules on \(X_\proetale\).

1. We say \(\mathcal{F}\) is a constructible \(\Lambda\)-sheaf if \(\mathcal{F} = \lim \mathcal{F}/I^n\mathcal{F}\) and each \(\mathcal{F}/I^n\mathcal{F}\) is a constructible sheaf of \(\Lambda/I^n\)-modules.

2. If \(\mathcal{F}\) is a constructible \(\Lambda\)-sheaf, then we say \(\mathcal{F}\) is lisse if each \(\mathcal{F}/I^n\mathcal{F}\) is locally constant.

3. We say \(\mathcal{F}\) is adic lisse⁵ if there exists a \(I\)-adically complete \(\Lambda\)-module \(M\) with \(M/IM\) finite such that \(\mathcal{F}\) is locally isomorphic to \[\underline{M}^\wedge = \lim \underline{M/I^nM}.\]

4. We say \(\mathcal{F}\) is adic constructible⁶ if for every affine open \(U \subset X\) there exists a decomposition \(U = \coprod U_i\) into constructible locally closed subschemes such that \(\mathcal{F}|_{U_i}\) is adic lisse.

The definition of a constructible \(\Lambda\)-sheaf is equivalent to the one in [SGA5, Exposé VI, Definition 1.1.1] when \(\Lambda = \mathbf{Z}_\ell\) and \(I = (\ell)\). It is clear that we have the implications \[\xymatrix{ \text{lisse adic} \ar@{=>}[r] \ar@{=>}[d] & \text{adic constructible} \ar@{=>}[d] \\ \text{lisse constructible }\Lambda\text{-sheaf} \ar@{=>}[r] & \text{constructible }\Lambda\text{-sheaf} }\] The vertical arrows can be inverted in some cases (see Lemmas 09BU and 09BX). In general neither the category of adic constructible sheaves nor the category of constructible \(\Lambda\)-sheaves is closed under kernels and cokernels.

Namely, let \(X\) be an affine scheme whose underlying topological space \(|X|\) is homeomorphic to \(\Lambda = \mathbf{Z}_\ell\), see Example 09BJ. Denote \(f : |X| \to \mathbf{Z}_\ell = \Lambda\) a homeomorphism. We can think of \(f\) as a section of \(\underline{\Lambda}^\wedge\) over \(X\) and multiplication by \(f\) then defines a two term complex \[\underline{\Lambda}^\wedge \xrightarrow{f} \underline{\Lambda}^\wedge\] on \(X_\proetale\). The sheaf \(\underline{\Lambda}^\wedge\) is adic lisse. However, the cokernel of the map above, is not adic constructible, as the isomorphism type of the stalks of this cokernel attains infinitely many values: \(\mathbf{Z}/\ell^n\mathbf{Z}\) and \(\mathbf{Z}_\ell\). The cokernel is a constructible \(\mathbf{Z}_\ell\)-sheaf. However, the kernel is not even a constructible \(\mathbf{Z}_\ell\)-sheaf as it is zero a non-quasi-compact open but not zero.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(\mathcal{F}\) be a constructible \(\Lambda\)-sheaf on \(X_\proetale\). Then there exists a finite partition \(X = \coprod X_i\) by locally closed subschemes such that the restriction \(\mathcal{F}|_{X_i}\) is lisse.

Proof

Let \(R = \bigoplus I^n/I^{n + 1}\). Observe that \(R\) is a Noetherian ring. Since each of the sheaves \(\mathcal{F}/I^n\mathcal{F}\) is a constructible sheaf of \(\Lambda/I^n\Lambda\)-modules also \(I^n\mathcal{F}/I^{n + 1}\mathcal{F}\) is a constructible sheaf of \(\Lambda/I\)-modules and hence the pullback of a constructible sheaf \(\mathcal{G}_n\) on \(X_\etale\) by Lemma 09AK. Set \(\mathcal{G} = \bigoplus \mathcal{G}_n\). This is a sheaf of \(R\)-modules on \(X_\etale\) and the map \[\mathcal{G}_0 \otimes_{\Lambda/I} \underline{R} \longrightarrow \mathcal{G}\] is surjective because the maps \[\mathcal{F}/I\mathcal{F} \otimes \underline{I^n/I^{n + 1}} \to I^n\mathcal{F}/I^{n + 1}\mathcal{F}\] are surjective. Hence \(\mathcal{G}\) is a constructible sheaf of \(R\)-modules by Étale Cohomology, Proposition 09BH. Choose a partition \(X = \coprod X_i\) such that \(\mathcal{G}|_{X_i}\) is a locally constant sheaf of \(R\)-modules of finite type (Étale Cohomology, Lemma 095E). We claim this is a partition as in the lemma. Namely, replacing \(X\) by \(X_i\) we may assume \(\mathcal{G}\) is locally constant. It follows that each of the sheaves \(I^n\mathcal{F}/I^{n + 1}\mathcal{F}\) is locally constant. Using the short exact sequences \[0 \to I^n\mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^n\mathcal{F} \to 0\] induction and Modules on Sites, Lemma 093U the lemma follows.

Lemma

Let \(X\) be a weakly contractible affine scheme. Let \(\Lambda\) be a Noetherian ring and \(I \subset \Lambda\) be an ideal. Let \(\mathcal{F}\) be a sheaf of \(\Lambda\)-modules on \(X_\proetale\) such that

1. \(\mathcal{F} = \lim \mathcal{F}/I^n\mathcal{F}\),

2. \(\mathcal{F}/I^n\mathcal{F}\) is a constant sheaf of \(\Lambda/I^n\)-modules,

3. \(\mathcal{F}/I\mathcal{F}\) is of finite type.

Then \(\mathcal{F} \cong \underline{M}^\wedge\) where \(M\) is a finite \(\Lambda^\wedge\)-module.

Proof

Pick a \(\Lambda/I^n\)-module \(M_n\) such that \(\mathcal{F}/I^n\mathcal{F} \cong \underline{M_n}\). Since we have the surjections \(\mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^n\mathcal{F}\) we conclude that there exist surjections \(M_{n + 1} \to M_n\) inducing isomorphisms \(M_{n + 1}/I^nM_{n + 1} \to M_n\). Fix a choice of such surjections and set \(M = \lim M_n\). Then \(M\) is an \(I\)-adically complete \(\Lambda\)-module with \(M/I^nM = M_n\), see Algebra, Lemma 09B8. Since \(M_1\) is a finite type \(\Lambda\)-module (Modules on Sites, Lemma 093N) we see that \(M\) is a finite \(\Lambda^\wedge\)-module. Consider the sheaves \[\mathcal{I}_n = \mathit{Isom}(\underline{M_n}, \mathcal{F}/I^n\mathcal{F})\] on \(X_\proetale\). Modding out by \(I^n\) defines a transition map \[\mathcal{I}_{n + 1} \longrightarrow \mathcal{I}_n\] By our choice of \(M_n\) the sheaf \(\mathcal{I}_n\) is a torsor under \[\mathit{Isom}(\underline{M_n}, \underline{M_n}) = \underline{\text{Isom}_\Lambda(M_n, M_n)}\] (Modules on Sites, Lemma 093T) since \(\mathcal{F}/I^n\mathcal{F}\) is (étale) locally isomorphic to \(\underline{M_n}\). It follows from More on Algebra, Lemma 09BB that the system of sheaves \((\mathcal{I}_n)\) is Mittag-Leffler. For each \(n\) let \(\mathcal{I}'_n \subset \mathcal{I}_n\) be the image of \(\mathcal{I}_N \to \mathcal{I}_n\) for all \(N \gg n\). Then \[\ldots \to \mathcal{I}'_3 \to \mathcal{I}'_2 \to \mathcal{I}'_1 \to *\] is a sequence of sheaves of sets on \(X_\proetale\) with surjective transition maps. Since \(*(X)\) is a singleton (not empty) and since evaluating at \(X\) transforms surjective maps of sheaves of sets into surjections of sets, we can pick \(s \in \lim \mathcal{I}'_n(X)\). The sections define isomorphisms \(\underline{M}^\wedge \to \lim \mathcal{F}/I^n\mathcal{F} = \mathcal{F}\) and the proof is done.

Lemma

Let \(X\) be a connected scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. If \(\mathcal{F}\) is a lisse constructible \(\Lambda\)-sheaf on \(X_\proetale\), then \(\mathcal{F}\) is adic lisse.

Proof

By Lemma 099Y we have \(\mathcal{F}/I^n\mathcal{F} = \epsilon^{-1}\mathcal{G}_n\) for some locally constant sheaf \(\mathcal{G}_n\) of \(\Lambda/I^n\)-modules. By Étale Cohomology, Lemma 09BF there exists a finite \(\Lambda/I^n\)-module \(M_n\) such that \(\mathcal{G}_n\) is locally isomorphic to \(\underline{M_n}\). Choose a covering \(\{W_t \to X\}_{t \in T}\) with each \(W_t\) affine and weakly contractible. Then \(\mathcal{F}|_{W_t}\) satisfies the assumptions of Lemma 09BV and hence \(\mathcal{F}|_{W_t} \cong \underline{N_t}^\wedge\) for some finite \(\Lambda^\wedge\)-module \(N_t\). Note that \(N_t/I^nN_t \cong M_n\) for all \(t\) and \(n\). Hence \(N_t \cong N_{t'}\) for all \(t, t' \in T\), see More on Algebra, Lemma 09BC. This proves that \(\mathcal{F}\) is adic lisse.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(\mathcal{F}\) be a constructible \(\Lambda\)-sheaf on \(X_\proetale\). Then \(\mathcal{F}\) is adic constructible.

Proof

This is a consequence of Lemmas 09BU and 09BW, the fact that a Noetherian scheme is locally connected (Topology, Lemma 04MF), and the definitions.

It will be useful to identify the constructible \(\Lambda\)-sheaves inside the category of derived complete sheaves of \(\Lambda\)-modules. It turns out that the naive analogue of More on Algebra, Lemma 09BA is wrong in this setting. However, here is the analogue of More on Algebra, Lemma 09B9.

Lemma

Let \(X\) be a scheme. Let \(\Lambda\) be a ring and let \(I \subset \Lambda\) be a finitely generated ideal. Let \(\mathcal{F}\) be a sheaf of \(\Lambda\)-modules on \(X_\proetale\). If \(\mathcal{F}\) is derived complete and \(\mathcal{F}/I\mathcal{F} = 0\), then \(\mathcal{F} = 0\).

Proof

Assume that \(\mathcal{F}/I\mathcal{F}\) is zero. Let \(I = (f_1, \ldots, f_r)\). Let \(i < r\) be the largest integer such that \(\mathcal{G} = \mathcal{F}/(f_1, \ldots, f_i)\mathcal{F}\) is nonzero. If \(i\) does not exist, then \(\mathcal{F} = 0\) which is what we want to show. Then \(\mathcal{G}\) is derived complete as a cokernel of a map between derived complete modules, see Proposition 099Q. By our choice of \(i\) we have that \(f_{i + 1} : \mathcal{G} \to \mathcal{G}\) is surjective. Hence \[\lim (\ldots \to \mathcal{G} \xrightarrow{f_{i + 1}} \mathcal{G} \xrightarrow{f_{i + 1}} \mathcal{G})\] is nonzero, contradicting the derived completeness of \(\mathcal{G}\).

Lemma

Let \(X\) be a weakly contractible affine scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(\mathcal{F}\) be a derived complete sheaf of \(\Lambda\)-modules on \(X_\proetale\) with \(\mathcal{F}/I\mathcal{F}\) a locally constant sheaf of \(\Lambda/I\)-modules of finite type. Then there exists an integer \(t\) and a surjective map \[(\underline{\Lambda}^\wedge)^{\oplus t} \to \mathcal{F}\]

Proof

Since \(X\) is weakly contractible, there exists a finite disjoint open covering \(X = \coprod U_i\) such that \(\mathcal{F}/I\mathcal{F}|_{U_i}\) is isomorphic to the constant sheaf associated to a finite \(\Lambda/I\)-module \(M_i\). Choose finitely many generators \(m_{ij}\) of \(M_i\). We can find sections \(s_{ij} \in \mathcal{F}(X)\) restricting to \(m_{ij}\) viewed as a section of \(\mathcal{F}/I\mathcal{F}\) over \(U_i\). Let \(t\) be the total number of \(s_{ij}\). Then we obtain a map \[\alpha : \underline{\Lambda}^{\oplus t} \longrightarrow \mathcal{F}\] which is surjective modulo \(I\) by construction. By Lemma 099M the derived completion of \(\underline{\Lambda}^{\oplus t}\) is the sheaf \((\underline{\Lambda}^\wedge)^{\oplus t}\). Since \(\mathcal{F}\) is derived complete we see that \(\alpha\) factors through a map \[\alpha^\wedge : (\underline{\Lambda}^\wedge)^{\oplus t} \longrightarrow \mathcal{F}\] Then \(\mathcal{Q} = \Coker(\alpha^\wedge)\) is a derived complete sheaf of \(\Lambda\)-modules by Proposition 099Q. By construction \(\mathcal{Q}/I\mathcal{Q} = 0\). It follows from Lemma 09BY that \(\mathcal{Q} = 0\) which is what we wanted to show.

A suitable derived category

Let \(X\) be a scheme. It will turn out that for many schemes \(X\) a suitable derived category of \(\ell\)-adic sheaves can be gotten by considering the derived complete objects \(K\) of \(D(X_\proetale, \Lambda)\) with the property that \(K \otimes_\Lambda^\mathbf{L} \mathbf{F}_\ell\) is bounded with constructible cohomology sheaves. Here is the general definition.

Definition

Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(X\) be a scheme. An object \(K\) of \(D(X_\proetale, \Lambda)\) is called constructible if

1. \(K\) is derived complete with respect to \(I\),

2. \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}\) has constructible cohomology sheaves and locally has finite tor dimension.

We denote \(D_{cons}(X, \Lambda)\) the full subcategory of constructible \(K\) in \(D(X_\proetale, \Lambda)\).

Recall that with our conventions a complex of finite tor dimension is bounded (Cohomology on Sites, Definition 08FZ). In fact, let’s collect everything proved so far in a lemma.

Lemma

In the situation above suppose \(K\) is in \(D_{cons}(X, \Lambda)\) and \(X\) is quasi-compact. Set \(K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\). There exist \(a, b\) such that

1. \(K = R\lim K_n\) and \(H^i(K) = 0\) for \(i \not \in [a, b]\),

2. each \(K_n\) has tor amplitude in \([a, b]\),

3. each \(K_n\) has constructible cohomology sheaves,

4. each \(K_n = \epsilon^{-1}L_n\) for some \(L_n \in D_{ctf}(X_\etale, \Lambda/I^n)\) (Étale Cohomology, Definition 03TQ).

Proof

Since \(K\) is derived complete, we have \(K = R\lim K_n\) by Lemma 099M. By definition of locally having finite tor dimension, we can find \(a, b\) such that \(K_1\) has tor amplitude in \([a, b]\). Part (2) follows from Cohomology on Sites, Lemma 0942. Then (1) follows as \(K\) is derived complete by the description of limits in Cohomology on Sites, Proposition 0947 and the fact that \(H^b(K_{n + 1}) \to H^b(K_n)\) is surjective as \(K_n = K_{n + 1} \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n}\). Part (3) follows from Lemma 09BR, Part (4) follows from Lemma 09AL and the fact that \(L_n\) has finite tor dimension because \(K_n\) does (small argument omitted).

Lemma

Let \(X\) be a weakly contractible affine scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(K\) be an object of \(D_{cons}(X, \Lambda)\) such that the cohomology sheaves of \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}\) are locally constant. Then there exists a finite disjoint open covering \(X = \coprod U_i\) and for each \(i\) a finite collection of finite projective \(\Lambda^\wedge\)-modules \(M^a, \ldots, M^b\) such that \(K|_{U_i}\) is represented by a complex \[(\underline{M^a})^\wedge \to \ldots \to (\underline{M^b})^\wedge\] in \(D(U_{i, \proetale}, \Lambda)\) for some maps of sheaves of \(\Lambda\)-modules \((\underline{M^i})^\wedge \to (\underline{M^{i + 1}})^\wedge\).

Proof

We freely use the results of Lemma 09C2. Choose \(a, b\) as in that lemma. We will prove the lemma by induction on \(b - a\). Let \(\mathcal{F} = H^b(K)\). Note that \(\mathcal{F}\) is a derived complete sheaf of \(\Lambda\)-modules by Proposition 099Q. Moreover \(\mathcal{F}/I\mathcal{F}\) is a locally constant sheaf of \(\Lambda/I\)-modules of finite type. Apply Lemma 09BZ to get a surjection \(\rho : (\underline{\Lambda}^\wedge)^{\oplus t} \to \mathcal{F}\).

If \(a = b\), then \(K = \mathcal{F}[-b]\). In this case we see that \[\mathcal{F} \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I} = \mathcal{F}/I\mathcal{F}\] As \(X\) is weakly contractible and \(\mathcal{F}/I\mathcal{F}\) locally constant, we can find a finite disjoint union decomposition \(X = \coprod U_i\) by affine opens \(U_i\) and \(\Lambda/I\)-modules \(\overline{M}_i\) such that \(\mathcal{F}/I\mathcal{F}\) restricts to \(\underline{\overline{M}_i}\) on \(U_i\). After refining the covering we may assume the map \[\rho|_{U_i} \bmod I : \underline{\Lambda/I}^{\oplus t} \longrightarrow \underline{\overline{M}_i}\] is equal to \(\underline{\alpha_i}\) for some surjective module map \(\alpha_i : \Lambda/I^{\oplus t} \to \overline{M}_i\), see Modules on Sites, Lemma 093S. Note that each \(\overline{M}_i\) is a finite \(\Lambda/I\)-module. Since \(\mathcal{F}/I\mathcal{F}\) has tor amplitude in \([0, 0]\) we conclude that \(\overline{M}_i\) is a flat \(\Lambda/I\)-module. Hence \(\overline{M}_i\) is finite projective (Algebra, Lemma 00NX). Hence we can find a projector \(\overline{p}_i : (\Lambda/I)^{\oplus t} \to (\Lambda/I)^{\oplus t}\) whose image maps isomorphically to \(\overline{M}_i\) under the map \(\alpha_i\). We can lift \(\overline{p}_i\) to a projector \(p_i : (\Lambda^\wedge)^{\oplus t} \to (\Lambda^\wedge)^{\oplus t}\)⁷. Then \(M_i = \Im(p_i)\) is a finite \(I\)-adically complete \(\Lambda^\wedge\)-module with \(M_i/IM_i = \overline{M}_i\). Over \(U_i\) consider the maps \[\underline{M_i}^\wedge \to (\underline{\Lambda}^\wedge)^{\oplus t} \to \mathcal{F}|_{U_i}\] By construction the composition induces an isomorphism modulo \(I\). The source and target are derived complete, hence so are the cokernel \(\mathcal{Q}\) and the kernel \(\mathcal{K}\). We have \(\mathcal{Q}/I\mathcal{Q} = 0\) by construction hence \(\mathcal{Q}\) is zero by Lemma 09BY. Then \[0 \to \mathcal{K}/I\mathcal{K} \to \underline{\overline{M}_i} \to \mathcal{F}/I\mathcal{F} \to 0\] is exact by the vanishing of \(\text{Tor}_1\) see at the start of this paragraph; also use that \(\underline{\Lambda}^\wedge/I\overline{\Lambda}^\wedge\) by Modules on Sites, Lemma 093M to see that \(\underline{M_i}^\wedge/I\underline{M_i}^\wedge = \underline{\overline{M}_i}\). Hence \(\mathcal{K}/I\mathcal{K} = 0\) by construction and we conclude that \(\mathcal{K} = 0\) as before. This proves the result in case \(a = b\).

If \(b > a\), then we lift the map \(\rho\) to a map \[\tilde \rho : (\underline{\Lambda}^\wedge)^{\oplus t}[-b] \longrightarrow K\] in \(D(X_\proetale, \Lambda)\). This is possible as we can think of \(K\) as a complex of \(\underline{\Lambda}^\wedge\)-modules by discussion in the introduction to Section 099L and because \(X_\proetale\) is weakly contractible hence there is no obstruction to lifting the elements \(\rho(e_s) \in H^0(X, \mathcal{F})\) to elements of \(H^b(X, K)\). Fitting \(\tilde \rho\) into a distinguished triangle \[(\underline{\Lambda}^\wedge)^{\oplus t}[-b] \to K \to L \to (\underline{\Lambda}^\wedge)^{\oplus t}[-b + 1]\] we see that \(L\) is an object of \(D_{cons}(X, \Lambda)\) such that \(L \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}\) has tor amplitude contained in \([a, b - 1]\) (details omitted). By induction we can describe \(L\) locally as stated in the lemma, say \(L\) is isomorphic to \[(\underline{M^a})^\wedge \to \ldots \to (\underline{M^{b - 1}})^\wedge\] The map \(L \to (\underline{\Lambda}^\wedge)^{\oplus t}[-b + 1]\) corresponds to a map \((\underline{M^{b - 1}})^\wedge \to (\underline{\Lambda}^\wedge)^{\oplus t}\) which allows us to extend the complex by one. The corresponding complex is isomorphic to \(K\) in the derived category by the properties of triangulated categories. This finishes the proof.

Motivated by what happens for constructible \(\Lambda\)-sheaves we introduce the following notion.

Definition

Let \(X\) be a scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(K \in D(X_\proetale, \Lambda)\).

1. We say \(K\) is adic lisse⁸ if there exists a finite complex of finite projective \(\Lambda^\wedge\)-modules \(M^\bullet\) such that \(K\) is locally isomorphic to \[\underline{M^a}^\wedge \to \ldots \to \underline{M^b}^\wedge\]

2. We say \(K\) is adic constructible⁹ if for every affine open \(U \subset X\) there exists a decomposition \(U = \coprod U_i\) into constructible locally closed subschemes such that \(K|_{U_i}\) is adic lisse.

The difference between the local structure obtained in Lemma 09C3 and the structure of an adic lisse complex is that the maps \(\underline{M^i}^\wedge \to \underline{M^{i + 1}}^\wedge\) in Lemma 09C3 need not be constant, whereas in the definition above they are required to be constant.

Lemma

Let \(X\) be a weakly contractible affine scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(K\) be an object of \(D_{cons}(X, \Lambda)\) such that \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\) is isomorphic in \(D(X_\proetale, \Lambda/I^n)\) to a complex of constant sheaves of \(\Lambda/I^n\)-modules. Then \[H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^n)\] has the Mittag-Leffler condition.

Proof

Say \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\) is isomorphic to \(\underline{E_n}\) for some object \(E_n\) of \(D(\Lambda/I^n)\). Since \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}\) has finite tor dimension and has finite type cohomology sheaves we see that \(E_1\) is perfect (see More on Algebra, Lemma 0658). The transition maps \[K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^{n + 1}} \to K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\] locally come from (possibly many distinct) maps of complexes \(E_{n + 1} \to E_n\) in \(D(\Lambda/I^{n + 1})\) see Cohomology on Sites, Lemma 09BE. For each \(n\) choose one such map and observe that it induces an isomorphism \(E_{n + 1} \otimes_{\Lambda/I^{n + 1}}^\mathbf{L} \Lambda/I^n \to E_n\) in \(D(\Lambda/I^n)\). By More on Algebra, Lemma 09AW we can find a finite complex \(M^\bullet\) of finite projective \(\Lambda^\wedge\)-modules and isomorphisms \(M^\bullet/I^nM^\bullet \to E_n\) in \(D(\Lambda/I^n)\) compatible with the transition maps.

Now observe that for each finite collection of indices \(n > m > k\) the triple of maps \[H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^n) \to H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^m) \to H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^k)\] is isomorphic to \[H^0(X, \underline{M^\bullet/I^nM^\bullet}) \to H^0(X, \underline{M^\bullet/I^mM^\bullet}) \to H^0(X, \underline{M^\bullet/I^kM^\bullet})\] Namely, choose any isomorphism \[\underline{M^\bullet/I^nM^\bullet} \to K \otimes_\Lambda^\mathbf{L} \Lambda/I^n\] induces similar isomorphisms module \(I^m\) and \(I^k\) and we see that the assertion is true. Thus to prove the lemma it suffices to show that the system \(H^0(X, \underline{M^\bullet/I^nM^\bullet})\) has Mittag-Leffler. Since taking sections over \(X\) is exact, it suffices to prove that the system of \(\Lambda\)-modules \[H^0(M^\bullet/I^nM^\bullet)\] has Mittag-Leffler. Set \(A = \Lambda^\wedge\) and consider the spectral sequence \[\text{Tor}_{-p}^A(H^q(M^\bullet), A/I^nA) \Rightarrow H^{p + q}(M^\bullet/I^nM^\bullet)\] By More on Algebra, Lemma 0911 the pro-systems \(\{\text{Tor}_{-p}^A(H^q(M^\bullet), A/I^nA)\}\) are zero for \(p > 0\). Thus the pro-system \(\{H^0(M^\bullet/I^nM^\bullet)\}\) is equal to the pro-system \(\{H^0(M^\bullet)/I^nH^0(M^\bullet)\}\) and the lemma is proved.

Lemma

Let \(X\) be a connected scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. If \(K\) is in \(D_{cons}(X, \Lambda)\) such that \(K \otimes_\Lambda \underline{\Lambda/I}\) has locally constant cohomology sheaves, then \(K\) is adic lisse (Definition 09C4).

Proof

Write \(K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\). We will use the results of Lemma 09C2 without further mention. By Cohomology on Sites, Lemma 094I we see that \(K_n\) has locally constant cohomology sheaves for all \(n\). We have \(K_n = \epsilon^{-1}L_n\) some \(L_n\) in \(D_{ctf}(X_\etale, \Lambda/I^n)\) with locally constant cohomology sheaves. By Étale Cohomology, Lemma 09BI there exist perfect \(M_n \in D(\Lambda/I^n)\) such that \(L_n\) is étale locally isomorphic to \(\underline{M_n}\). The maps \(L_{n + 1} \to L_n\) corresponding to \(K_{n + 1} \to K_n\) induces isomorphisms \(L_{n + 1} \otimes_{\Lambda/I^{n + 1}}^\mathbf{L} \underline{\Lambda/I^n} \to L_n\). Looking locally on \(X\) we conclude that there exist maps \(M_{n + 1} \to M_n\) in \(D(\Lambda/I^{n + 1})\) inducing isomorphisms \(M_{n + 1} \otimes_{\Lambda/I^{n + 1}} \Lambda/I^n \to M_n\), see Cohomology on Sites, Lemma 09BE. Fix a choice of such maps. By More on Algebra, Lemma 09AW we can find a finite complex \(M^\bullet\) of finite projective \(\Lambda^\wedge\)-modules and isomorphisms \(M^\bullet/I^nM^\bullet \to M_n\) in \(D(\Lambda/I^n)\) compatible with the transition maps. To finish the proof we will show that \(K\) is locally isomorphic to \[\underline{M^\bullet}^\wedge = \lim \underline{M^\bullet/I^nM^\bullet} = R\lim \underline{M^\bullet/I^nM^\bullet}\] Let \(E^\bullet\) be the dual complex to \(M^\bullet\), see More on Algebra, Lemma 07VI and its proof. Consider the objects \[H_n = R\SheafHom_{\Lambda/I^n}(\underline{M^\bullet/I^nM^\bullet}, K_n) = \underline{E^\bullet/I^nE^\bullet} \otimes_{\Lambda/I^n}^\mathbf{L} K_n\] of \(D(X_\proetale, \Lambda/I^n)\). Modding out by \(I^n\) defines a transition map \(H_{n + 1} \to H_n\). Set \(H = R\lim H_n\). Then \(H\) is an object of \(D_{cons}(X, \Lambda)\) (details omitted) with \(H \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n} = H_n\). Choose a covering \(\{W_t \to X\}_{t \in T}\) with each \(W_t\) affine and weakly contractible. By our choice of \(M^\bullet\) we see that \[\begin{align*} H_n|_{W_t} & \cong R\SheafHom_{\Lambda/I^n}(\underline{M^\bullet/I^nM^\bullet}, \underline{M^\bullet/I^nM^\bullet}) \\ & = \underline{ \text{Tot}(E^\bullet/I^nE^\bullet \otimes_{\Lambda/I^n} M^\bullet/I^nM^\bullet) } \end{align*}\] Thus we may apply Lemma 09C5 to \(H = R\lim H_n\). We conclude the system \(H^0(W_t, H_n)\) satisfies Mittag-Leffler. Since for all \(n \gg 1\) there is an element of \(H^0(W_t, H_n)\) which maps to an isomorphism in \[H^0(W_t, H_1) = \Hom(\underline{M^\bullet/IM^\bullet}, K_1)\] we find an element \((\varphi_{t, n})\) in the inverse limit which produces an isomorphism mod \(I\). Then \[R\lim \varphi_{t, n} : \underline{M^\bullet}^\wedge|_{W_t} = R\lim \underline{M^\bullet/I^nM^\bullet}|_{W_t} \longrightarrow R\lim K_n|_{W_t} = K|_{W_t}\] is an isomorphism. This finishes the proof.

Proposition

Let \(X\) be a Noetherian scheme. Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal. Let \(K\) be an object of \(D_{cons}(X, \Lambda)\). Then \(K\) is adic constructible (Definition 09C4).

Proof

This is a consequence of Lemma 09C6 and the fact that a Noetherian scheme is locally connected (Topology, Lemma 04MF), and the definitions.

Proper base change

In this section we explain how to prove the proper base change theorem for derived complete objects on the pro-étale site using the proper base change theorem for étale cohomology following the general theme that we use the pro-étale topology only to deal with “limit issues” and we use results proved for the étale topology to handle everything else.

Theorem

Let \(f : X \to Y\) be a proper morphism of schemes. Let \(g : Y' \to Y\) be a morphism of schemes giving rise to the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] Let \(\Lambda\) be a Noetherian ring and let \(I \subset \Lambda\) be an ideal such that \(\Lambda/I\) is torsion. Let \(K\) be an object of \(D(X_\proetale)\) such that

1. \(K\) is derived complete, and

2. \(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}\) is bounded below with cohomology sheaves coming from \(X_\etale\),

3. \(\Lambda/I^n\) is a perfect \(\Lambda\)-module¹⁰.

Then the base change map \[Lg_{comp}^*Rf_*K \longrightarrow Rf'_*L(g')^*_{comp}K\] is an isomorphism.

Proof

We omit the construction of the base change map (this uses only formal properties of derived pushforward and completed derived pullback, compare with Cohomology on Sites, Remark 07A7). Write \(K_n = K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n}\). By Lemma 099M we have \(K = R\lim K_n\) because \(K\) is derived complete. By Lemmas 099N and 099M we can unwind the left hand side \[Lg_{comp}^* Rf_* K = R\lim Lg^*(Rf_*K)\otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n} = R\lim Lg^* Rf_* K_n\] the last equality because \(\Lambda/I^n\) is a perfect module and the projection formula (Cohomology on Sites, Lemma 0944). Using Lemma 099N we can unwind the right hand side \[Rf'_* L(g')^*_{comp} K = Rf'_* R\lim L(g')^* K_n = R\lim Rf'_* L(g')^* K_n\] the last equality because \(Rf'_*\) commutes with \(R\lim\) (Cohomology on Sites, Lemma 0A07). Thus it suffices to show the maps \[Lg^* Rf_* K_n \longrightarrow Rf'_* L(g')^* K_n\] are isomorphisms. By Lemma 09B1 and our second condition we can write \(K_n = \epsilon^{-1}L_n\) for some \(L_n \in D^+(X_\etale, \Lambda/I^n)\). By Lemma 09A6 and the fact that \(\epsilon^{-1}\) commutes with pullbacks we obtain \[Lg^* Rf_* K_n = Lg^* Rf_* \epsilon^*L_n = Lg^* \epsilon^{-1} Rf_* L_n = \epsilon^{-1} Lg^* Rf_* L_n\] and \[Rf'_* L(g')^* K_n = Rf'_* L(g')^* \epsilon^{-1} L_n = Rf'_* \epsilon^{-1} L(g')^* L_n = \epsilon^{-1} Rf'_* L(g')^* L_n\] (this also uses that \(L_n\) is bounded below). Finally, by the proper base change theorem for étale cohomology (Étale Cohomology, Theorem 095T) we have \[Lg^* Rf_* L_n = Rf'_* L(g')^* L_n\] (again using that \(L_n\) is bounded below) and the theorem is proved.

Change of partial universe

We advise the reader to skip this section: here we show that cohomology of sheaves in the pro-étale topology is independent of the choice of partial universe. Namely, the functor \(g_*\) of Lemma 098Y below is an embedding of small pro-étale topoi which does not change cohomology. For big pro-étale sites we have Lemmas 0F4S and 0F4T saying essentially the same thing.

But first, as promised in Section 0988 we prove that the topology on a big pro-étale site \(\Sch_\proetale\) is in some sense induced from the pro-étale topology on the category of all schemes.

Lemma

Let \(\Sch_\proetale\) be a big pro-étale site as in Definition 098G. Let \(T \in \Ob(\Sch_\proetale)\). Let \(\{T_i \to T\}_{i \in I}\) be an arbitrary pro-étale covering of \(T\). There exists a covering \(\{U_j \to T\}_{j \in J}\) of \(T\) in the site \(\Sch_\proetale\) which refines \(\{T_i \to T\}_{i \in I}\).

Proof

Namely, we first let \(\{V_k \to T\}\) be a covering as in Lemma 098I. Then the pro-étale coverings \(\{T_i \times_T V_k \to V_k\}\) can be refined by a finite disjoint open covering \(V_k = V_{k, 1} \amalg \ldots \amalg V_{k, n_k}\), see Lemma 098F. Then \(\{V_{k, i} \to T\}\) is a covering of \(\Sch_\proetale\) which refines \(\{T_i \to T\}_{i \in I}\).

We first state and prove the comparison for the small pro-étale sites. Note that we are not claiming that the small pro-étale topos of a scheme is independent of the choice of partial universe; this isn’t true in contrast with the case of the small étale topos (Étale Cohomology, Lemma 0958).

Lemma

Let \(S\) be a scheme. Let \(S_\proetale \subset S_\proetale'\) be two small pro-étale sites of \(S\) as constructed in Definition 098K. Then the inclusion functor satisfies the assumptions of Sites, Lemma 00XU. Hence there exist morphisms of topoi \[\xymatrix{ \Sh(S_\proetale) \ar[r]^g & \Sh(S_\proetale') \ar[r]^f & \Sh(S_\proetale) }\] whose composition is isomorphic to the identity and with \(f_* = g^{-1}\). Moreover,

1. for \(\mathcal{F}' \in \textit{Ab}(S_\proetale')\) we have \(H^p(S_\proetale', \mathcal{F}') = H^p(S_\proetale, g^{-1}\mathcal{F}')\),

2. for \(\mathcal{F} \in \textit{Ab}(S_\proetale)\) we have \[H^p(S_\proetale, \mathcal{F}) = H^p(S_\proetale', g_*\mathcal{F}) = H^p(S_\proetale', f^{-1}\mathcal{F}).\]

Proof

The inclusion functor is fully faithful and continuous. We have seen that \(S_\proetale\) and \(S_\proetale'\) have fibre products and final objects and that our functor commutes with these (Lemma 098M). It follows from Lemma 098J that the inclusion functor is cocontinuous. Hence the existence of \(f\) and \(g\) follows from Sites, Lemma 00XU. The equality in (1) is Cohomology on Sites, Lemma 03YU. Part (2) follows from (1) as \(\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}\).

Next, we prove a corresponding result for the big pro-étale topoi.

Lemma

Suppose given big sites \(\Sch_\proetale\) and \(\Sch'_\proetale\) as in Definition 098G. Assume that \(\Sch_\proetale\) is contained in \(\Sch'_\proetale\). The inclusion functor \(\Sch_\proetale \to \Sch'_\proetale\) satisfies the assumptions of Sites, Lemma 00XU. There are morphisms of topoi \[\begin{eqnarray*} g : \Sh(\Sch_\proetale) & \longrightarrow & \Sh(\Sch'_\proetale) \\ f : \Sh(\Sch'_\proetale) & \longrightarrow & \Sh(\Sch_\proetale) \end{eqnarray*}\] such that \(f \circ g \cong \text{id}\). For any object \(S\) of \(\Sch_\proetale\) the inclusion functor \((\Sch/S)_\proetale \to (\Sch'/S)_\proetale\) satisfies the assumptions of Sites, Lemma 00XU also. Hence similarly we obtain morphisms \[\begin{eqnarray*} g : \Sh((\Sch/S)_\proetale) & \longrightarrow & \Sh((\Sch'/S)_\proetale) \\ f : \Sh((\Sch'/S)_\proetale) & \longrightarrow & \Sh((\Sch/S)_\proetale) \end{eqnarray*}\] with \(f \circ g \cong \text{id}\).

Proof

Assumptions (b), (c), and (e) of Sites, Lemma 00XU are immediate for the functors \(\Sch_\proetale \to \Sch'_\proetale\) and \((\Sch/S)_\proetale \to (\Sch'/S)_\proetale\). Property (a) holds by Lemma 098J. Property (d) holds because fibre products in the categories \(\Sch_\proetale\), \(\Sch'_\proetale\) exist and are compatible with fibre products in the category of schemes.

Lemma

Let \(S\) be a scheme. Let \((\Sch/S)_\proetale\) and \((\Sch'/S)_\proetale\) be two big pro-étale sites of \(S\) as in Definition 098K. Assume that the first is contained in the second. In this case

1. for any abelian sheaf \(\mathcal{F}'\) defined on \((\Sch'/S)_\proetale\) and any object \(U\) of \((\Sch/S)_\proetale\) we have \[H^p(U, \mathcal{F}'|_{(\Sch/S)_\proetale}) = H^p(U, \mathcal{F}')\] In words: the cohomology of \(\mathcal{F}'\) over \(U\) computed in the bigger site agrees with the cohomology of \(\mathcal{F}'\) restricted to the smaller site over \(U\).

2. for any abelian sheaf \(\mathcal{F}\) on \((\Sch/S)_\proetale\) there is an abelian sheaf \(\mathcal{F}'\) on \((\Sch/S)_\proetale'\) whose restriction to \((\Sch/S)_\proetale\) is isomorphic to \(\mathcal{F}\).

Proof

By Lemma 0F4S the inclusion functor \((\Sch/S)_\proetale \to (\Sch'/S)_\proetale\) satisfies the assumptions of Sites, Lemma 00XU. This implies (2) and (1) follows from Cohomology on Sites, Lemma 03YU.