The Stacks project

60.18 General remarks on cohomology

In this section we do a bit of work to translate the cohomology of modules on the cristalline site of an affine scheme into an algebraic question.

Lemma 60.18.1. In Situation 60.7.5. Let $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then we have

\[ H^ p((U, T, \delta ), \mathcal{F}) = 0 \]

for all $p > 0$ and all $(U, T, \delta )$ with $T$ or $U$ affine.

Proof. As $U \to T$ is a thickening we see that $U$ is affine if and only if $T$ is affine, see Limits, Lemma 32.11.1. Having said this, let us apply Cohomology on Sites, Lemma 21.10.9 to the collection $\mathcal{B}$ of affine objects $(U, T, \delta )$ and the collection $\text{Cov}$ of affine open coverings $\mathcal{U} = \{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} $. The Čech complex ${\check C}^*(\mathcal{U}, \mathcal{F})$ for such a covering is simply the Čech complex of the quasi-coherent $\mathcal{O}_ T$-module $\mathcal{F}_ T$ (here we are using the assumption that $\mathcal{F}$ is locally quasi-coherent) with respect to the affine open covering $\{ T_ i \to T\} $ of the affine scheme $T$. Hence the Čech cohomology is zero by Cohomology of Schemes, Lemma 30.2.6 and 30.2.2. Thus the hypothesis of Cohomology on Sites, Lemma 21.10.9 are satisfied and we win. $\square$

Lemma 60.18.2. In Situation 60.7.5. Assume moreover $X$ and $S$ are affine schemes. Consider the full subcategory $\mathcal{C} \subset \text{Cris}(X/S)$ consisting of divided power thickenings $(X, T, \delta )$ endowed with the chaotic topology (see Sites, Example 7.6.6). For any locally quasi-coherent $\mathcal{O}_{X/S}$-module $\mathcal{F}$ we have

\[ R\Gamma (\mathcal{C}, \mathcal{F}|_\mathcal {C}) = R\Gamma (\text{Cris}(X/S), \mathcal{F}) \]

Proof. Denote $\text{AffineCris}(X/S)$ the fully subcategory of $\text{Cris}(X/S)$ consisting of those objects $(U, T, \delta )$ with $U$ and $T$ affine. We turn this into a site by saying a family of morphisms $\{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} _{i \in I}$ of $\text{AffineCris}(X/S)$ is a covering if and only if it is a covering of $\text{Cris}(X/S)$. With this definition the inclusion functor

\[ \text{AffineCris}(X/S) \longrightarrow \text{Cris}(X/S) \]

is a special cocontinuous functor as defined in Sites, Definition 7.29.2. The proof of this is exactly the same as the proof of Topologies, Lemma 34.3.10. Thus we see that the topos of sheaves on $\text{Cris}(X/S)$ is the same as the topos of sheaves on $\text{AffineCris}(X/S)$ via restriction by the displayed inclusion functor. Therefore we have to prove the corresponding statement for the inclusion $\mathcal{C} \subset \text{AffineCris}(X/S)$.

We will use without further mention that $\mathcal{C}$ and $\text{AffineCris}(X/S)$ have products and fibre products (details omitted, see Lemma 60.8.2). The inclusion functor $u : \mathcal{C} \to \text{AffineCris}(X/S)$ is fully faithful, continuous, and commutes with products and fibre products. We claim it defines a morphism of ringed sites

\[ f : (\text{AffineCris}(X/S), \mathcal{O}_{X/S}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_{X/S}|_\mathcal {C}) \]

To see this we will use Sites, Lemma 7.14.6. Note that $\mathcal{C}$ has fibre products and $u$ commutes with them so the categories $\mathcal{I}^ u_{(U, T, \delta )}$ are disjoint unions of directed categories (by Sites, Lemma 7.5.1 and Categories, Lemma 4.19.8). Hence it suffices to show that $\mathcal{I}^ u_{(U, T, \delta )}$ is connected. Nonempty follows from Lemma 60.5.6: since $U$ and $T$ are affine that lemma says there is at least one object $(X, T', \delta ')$ of $\mathcal{C}$ and a morphism $(U, T, \delta ) \to (X, T', \delta ')$ of divided power thickenings. Connectedness follows from the fact that $\mathcal{C}$ has products and that $u$ commutes with them (compare with the proof of Sites, Lemma 7.5.2).

Note that $f_*\mathcal{F} = \mathcal{F}|_\mathcal {C}$. Hence the lemma follows if $R^ pf_*\mathcal{F} = 0$ for $p > 0$, see Cohomology on Sites, Lemma 21.14.6. By Cohomology on Sites, Lemma 21.7.4 it suffices to show that $H^ p(\text{AffineCris}(X/S)/(X, T, \delta ), \mathcal{F}) = 0$ for all $(X, T, \delta )$. This follows from Lemma 60.18.1 because the topos of the site $\text{AffineCris}(X/S)/(X, T, \delta )$ is equivalent to the topos of the site $\text{Cris}(X/S)/(X, T, \delta )$ used in the lemma. $\square$

Lemma 60.18.3. In Situation 60.5.1. Set $\mathcal{C} = (\text{Cris}(C/A))^{opp}$ and $\mathcal{C}^\wedge = (\text{Cris}^\wedge (C/A))^{opp}$ endowed with the chaotic topology, see Remark 60.5.4 for notation. There is a morphism of topoi

\[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}^\wedge ) \]

such that if $\mathcal{F}$ is a sheaf of abelian groups on $\mathcal{C}$, then

\[ R^ pg_*\mathcal{F}(B \to C, \delta ) = \left\{ \begin{matrix} \mathop{\mathrm{lim}}\nolimits _ e \mathcal{F}(B_ e \to C, \delta ) & \text{if }p = 0 \\ R^1\mathop{\mathrm{lim}}\nolimits _ e \mathcal{F}(B_ e \to C, \delta ) & \text{if }p = 1 \\ 0 & \text{else} \end{matrix} \right. \]

where $B_ e = B/p^ eB$ for $e \gg 0$.

Proof. Any functor between categories defines a morphism between chaotic topoi in the same direction, for example because such a functor can be considered as a cocontinuous functor between sites, see Sites, Section 7.21. Proof of the description of $g_*\mathcal{F}$ is omitted. Note that in the statement we take $(B_ e \to C, \delta )$ is an object of $\text{Cris}(C/A)$ only for $e$ large enough. Let $\mathcal{I}$ be an injective abelian sheaf on $\mathcal{C}$. Then the transition maps

\[ \mathcal{I}(B_ e \to C, \delta ) \leftarrow \mathcal{I}(B_{e + 1} \to C, \delta ) \]

are surjective as the morphisms

\[ (B_ e \to C, \delta ) \longrightarrow (B_{e + 1} \to C, \delta ) \]

are monomorphisms in the category $\mathcal{C}$. Hence for an injective abelian sheaf both sides of the displayed formula of the lemma agree. Taking an injective resolution of $\mathcal{F}$ one easily obtains the result (sheaves are presheaves, so exactness is measured on the level of groups of sections over objects). $\square$

Lemma 60.18.4. Let $\mathcal{C}$ be a category endowed with the chaotic topology. Let $X$ be an object of $\mathcal{C}$ such that every object of $\mathcal{C}$ has a morphism towards $X$. Assume that $\mathcal{C}$ has products of pairs. Then for every abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ the total cohomology $R\Gamma (\mathcal{C}, \mathcal{F})$ is represented by the complex

\[ \mathcal{F}(X) \to \mathcal{F}(X \times X) \to \mathcal{F}(X \times X \times X) \to \ldots \]

associated to the cosimplicial abelian group $[n] \mapsto \mathcal{F}(X^ n)$.

Proof. Note that $H^ q(X^ p, \mathcal{F}) = 0$ for all $q > 0$ as any presheaf is a sheaf on $\mathcal{C}$. The assumption on $X$ is that $h_ X \to *$ is surjective. Using that $H^ q(X, \mathcal{F}) = H^ q(h_ X, \mathcal{F})$ and $H^ q(\mathcal{C}, \mathcal{F}) = H^ q(*, \mathcal{F})$ we see that our statement is a special case of Cohomology on Sites, Lemma 21.13.2. $\square$

Comments (4)

Comment #6254 by Hao Peng on

I think the proof of tag07JK cannot go through where you claim that is non-empty: the tag07HP cannot be used because may not be affine so is only locally nilpotent on , thus may not exist an that on .

However, the same argument in fact shows the is surjective as sheaves. So I think this tag and the introduction of the chaotic site is in fact unnecessary: as this tag is used for the tag07JN, we can prove the tag07JN directly using tag079Z for instead of using tag07JM. The shifified product are in fact represeted by as an ind-object and the hypothesis of tag07JN is strong enough to prove the spectral sequence of tag079Z degenrate at , thus we are done.

Comment #6255 by on

OK, I think the lemma is correct, but you are correct in saying that the argument does not work as given. Thanks for pointing this out! I do want to keep this lemma as I think it is very useful psychologically! The fact that one can use the chaotic topology is sort of marvelous and plays a role als in other similar settings (prismatic cohomology, etc).

To fix the proof we can look at the full subcategory of consisting of the objects with and affine and endowed with the Zariski topology (defined in the obvious manner). Then we first prove that and define the same topos by the usual method, see for example Lemma 34.4.11. Finally, we prove the inclusion functor does work by the arguments given in the proof of lemma. OK?

(In fact, I think I replaced Cris by AffineCris in my head when I wrote the proof of this lemma.)

Comment #6256 by Hao Peng on

Thanks! I think this will do. Btw chaotic site simplifies things so much, and it can be used for the the prismatic cohomology is for the same reason that higher cohomology of and all vanishes. But I'm not sure when we consider -modules with structures(if someday) will still be so lucky. For example Scholze's calculus of condensed cohomology used hypercovering in an essential way... But I will take care of the simplification of cohomologies.

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