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83.23 Unbounded cohomological descent for hypercoverings

In this section we discuss unbounded cohomological descent. The results themselves will be immediate consequences of our results on bounded cohomological descent in the previous sections and Cohomology on Sites, Lemmas 21.27.6 and/or 21.27.7; the real work lies in setting up notation and choosing appropriate assumptions. Our discussion is motivated by the discussion in [six-I] although the details are a good bit different.

Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. Assume given for every object $U$ of $\mathcal{C}$ a weak Serre subcategory $\mathcal{A}_ U \subset \textit{Mod}(\mathcal{O}_ U)$ satisfying the following properties

  1. given a morphism $U \to V$ of $\mathcal{C}$ the restriction functor $\textit{Mod}(\mathcal{O}_ V) \to \textit{Mod}(\mathcal{O}_ U)$ sends $\mathcal{A}_ V$ into $\mathcal{A}_ U$,

  2. given a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_ U)$ is in $\mathcal{A}_ U$ if and only if the restriction of $\mathcal{F}$ to $\mathcal{C}/U_ i$ is in $\mathcal{A}_{U_ i}$ for all $i \in I$.

  3. there exists a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that

    1. every object of $\mathcal{C}$ has a covering whose members are in $\mathcal{B}$, and

    2. for every $V \in \mathcal{B}$ there exists an integer $d_ V$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that

      \[ H^ p(V_ i, \mathcal{F}) = 0 \text{ for } \{ V_ i \to V\} \in \text{Cov}_ V,\ p > d_ V, \text{ and } \mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}_ V) \]

Note that we require this to be true for $\mathcal{F}$ in $\mathcal{A}_ V$ and not just for “global” objects (and thus it is stronger than the condition imposed in Cohomology on Sites, Situation 21.24.1). In this situation, there is a weak Serre subcategory $\mathcal{A} \subset \textit{Mod}(\mathcal{O}_\mathcal {C})$ consisting of objects whose restriction to $\mathcal{C}/U$ is in $\mathcal{A}_ U$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Moreover, there are derived categories $D_\mathcal {A}(\mathcal{O}_\mathcal {C})$ and $D_{\mathcal{A}_ U}(\mathcal{O}_ U)$ and the restriction functors send these into each other.

Example 83.23.1. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{C} = X_{spaces, {\acute{e}tale}}$ be the étale site on the category of algebraic spaces étale over $X$, see Properties of Spaces, Definition 64.18.2. Denote $\mathcal{O}_\mathcal {C}$ the structure sheaf, i.e., the sheaf given by the rule $U \mapsto \Gamma (U, \mathcal{O}_ U)$. Denote $\mathcal{A}_ U$ the category of quasi-coherent $\mathcal{O}_ U$-modules. Let $\mathcal{B} = \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for $V \in \mathcal{B}$ set $d_ V = 0$ and let $\text{Cov}_ V$ denote the coverings $\{ V_ i \to V\} $ with $V_ i$ affine for all $i$. Then the assumptions (1), (2), (3) are satisfied. See Properties of Spaces, Lemmas 64.29.2 and 64.29.7 for properties (1) and (2) and the vanishing in (3) follows from Cohomology of Schemes, Lemma 30.2.2 and the discussion in Cohomology of Spaces, Section 67.3.

Example 83.23.2. Let $S$ be one of the following types of schemes

  1. the spectrum of a finite field,

  2. the spectrum of a separably closed field,

  3. the spectrum of a strictly henselian Noetherian local ring,

  4. the spectrum of a henselian Noetherian local ring with finite residue field,

  5. add more here.

Let $\Lambda $ be a finite ring whose order is invertible on $S$. Let $\mathcal{C} \subset (\mathit{Sch}/S)_{\acute{e}tale}$ be the full subcategory consisting of schemes locally of finite type over $S$ endowed with the étale topology. Let $\mathcal{O}_\mathcal {C} = \underline{\Lambda }$ be the constant sheaf. Set $\mathcal{A}_ U = \textit{Mod}(\mathcal{O}_ U)$, in other words, we consider all étale sheaves of $\Lambda $-modules. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be the set of quasi-compact objects. For $V \in \mathcal{B}$ set

\[ d_ V = 1 + 2\dim (S) + \sup \nolimits _{v \in V}(\text{trdeg}_{\kappa (s)}(\kappa (v)) + 2 \dim \mathcal{O}_{V, v}) \]

and let $\text{Cov}_ V$ denote the étale coverings $\{ V_ i \to V\} $ with $V_ i$ quasi-compact for all $i$. Our choice of bound $d_ V$ comes from Gabber's theorem on cohomological dimension. To see that condition (3) holds with this choice, use [Exposé VIII-A, Corollary 1.2 and Lemma 2.2, Traveaux] plus elementary arguments on cohomological dimensions of fields. We add $1$ to the formula because our list contains cases where we allow $S$ to have finite residue field. We will come back to this example later (insert future reference).

Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. Assume given weak Serre subcategories $\mathcal{A}_ U \subset \textit{Mod}(\mathcal{O}_ U)$ satisfying condition (1). Then

  1. given a semi-representable object $K = \{ U_ i\} _{i \in I}$ we get a weak Serre subcategory $\mathcal{A}_ K \subset \textit{Mod}(\mathcal{O}_ K)$ by taking $\prod \mathcal{A}_{U_ i} \subset \prod \textit{Mod}(\mathcal{O}_{U_ i}) = \textit{Mod}(\mathcal{O}_ K)$, and

  2. given a morphism of semi-representable objects $f : K \to L$ the pullback map $f^* : \textit{Mod}(\mathcal{O}_ L) \to \textit{Mod}(\mathcal{O}_ L)$ sends $\mathcal{A}_ L$ into $\mathcal{A}_ K$.

See Remark 83.15.6 for notation and explanation. In particular, given a simplicial semi-representable object $K$ it is unambiguous to say what it means for an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O})$ as in Remark 83.16.5 to have restrictions $\mathcal{F}_ n$ in $\mathcal{A}_{K_ n}$ for all $n$.

Lemma 83.23.3. Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. Assume given weak Serre subcategories $\mathcal{A}_ U \subset \textit{Mod}(\mathcal{O}_ U)$ satisfying conditions (1), (2), and (3) above. Assume $\mathcal{C}$ has equalizers and fibre products and let $K$ be a hypercovering. Let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in Remark 83.16.5. Let $\mathcal{A}_{total} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules $\mathcal{F}$ whose restriction $\mathcal{F}_ n$ is in $\mathcal{A}_{K_ n}$ for all $n$ (as defined above). Then the functor $La^*$ defines an equivalence

\[ D_\mathcal {A}(\mathcal{O}_\mathcal {C}) \longrightarrow D_{\mathcal{A}_{total}}(\mathcal{O}) \]

with quasi-inverse $Ra_*$.

Proof. The cartesian $\mathcal{O}$-modules form a weak Serre subcategory by Lemma 83.12.6 (the required hypotheses hold by the discussion in Remark 83.16.5). Since the restriction functor $g_ n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ n)$ are exact, it follows that $\mathcal{A}_{total}$ is a weak Serre subcategory.

Let us show that $a^* : \mathcal{A} \to \mathcal{A}_{total}$ is an equivalence of categories with inverse given by $La_*$. We already know that $La_*a^*\mathcal{F} = \mathcal{F}$ by the bounded version (Lemma 83.18.4). It is clear that $a^*\mathcal{F}$ is in $\mathcal{A}_{total}$ for $\mathcal{F}$ in $\mathcal{A}$. Conversely, assume that $\mathcal{G} \in \mathcal{A}_{total}$. Because $\mathcal{G}$ is cartesian we see that $\mathcal{G} = a^*\mathcal{F}$ for some $\mathcal{O}_\mathcal {C}$-module $\mathcal{F}$ by Lemma 83.18.1. We want to show that $\mathcal{F}$ is in $\mathcal{A}$. Take $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We have to show that the restriction of $\mathcal{F}$ to $\mathcal{C}/U$ is in $\mathcal{A}_ U$. As usual, write $K_0 = \{ U_{0, i}\} _{i \in I_0}$. Since $K$ is a hypercovering, the map $\coprod _{i \in I_0} h_{U_{0, i}} \to *$ becomes surjective after sheafification. This implies there is a covering $\{ U_ j \to U\} _{j \in J}$ and a map $\tau : J \to I_0$ and for each $j \in J$ a morphism $\varphi _ j : U_ j \to U_{0, \tau (j)}$. Since $\mathcal{G}_0 = a_0^*\mathcal{F}$ we find that the restriction of $\mathcal{F}$ to $\mathcal{C}/U_ j$ is equal to the restriction of the $\tau (j)$th component of $\mathcal{G}_0$ to $\mathcal{C}/U_ j$ via the morphism $\varphi _ j : U_ j \to U_{0, \tau (i)}$. Hence by (1) we find that $\mathcal{F}|_{\mathcal{C}/U_ j}$ is in $\mathcal{A}_{U_ j}$ and in turn by (2) we find that $\mathcal{F}|_{\mathcal{C}/U}$ is in $\mathcal{A}_ U$.

In particular the statement of the lemma makes sense. The lemma now follows from Cohomology on Sites, Lemma 21.27.6. Assumption (1) is clear (see Remark 83.16.5). Assumptions (2) and (3) we proved in the preceding paragraph. Assumption (4) is immediate from (3). For assumption (5) let $\mathcal{B}_{total}$ be the set of objects $U/U_{n, i}$ of the site $(\mathcal{C}/K)_{total}$ such that $U \in \mathcal{B}$ where $\mathcal{B}$ is as in (3). Here we use the description of the site $(\mathcal{C}/K)_{total}$ given in Section 83.16. Moreover, we set $\text{Cov}_{U/U_{n, i}}$ equal to $\text{Cov}_ U$ and $d_{U/U_{n, i}}$ equal $d_ U$ where $\text{Cov}_ U$ and $d_ U$ are given to us by (3). Then we claim that condition (5) holds with these choices. This follows immediately from Lemma 83.16.3 and the fact that $\mathcal{F} \in \mathcal{A}_{total}$ implies $\mathcal{F}_ n \in \mathcal{A}_{K_ n}$ and hence $\mathcal{F}_{n, i} \in \mathcal{A}_{U_{n, i}}$. (The reader who worries about the difference between cohomology of abelian sheaves versus cohomology of sheaves of modules may consult Cohomology on Sites, Lemma 21.12.4.) $\square$

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