## 83.17 Cohomological descent for hypercoverings

Let $\mathcal{C}$ be a site. In this section we assume $\mathcal{C}$ has equalizers and fibre products. We let $K$ be a hypercovering as defined in Hypercoverings, Definition 25.6.1. We will study the augmentation

\[ a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]

of Section 83.16.

Lemma 83.17.1. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then

$a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,

$a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases $a_*$ provides the quasi-inverse functor.

**Proof.**
The case of abelian sheaves follows immediately from the case of sheaves of sets as the functor $a^{-1}$ commutes with products. In the rest of the proof we work with sheaves of sets. Observe that $a^{-1}\mathcal{F}$ is cartesian for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ by Lemma 83.12.3. It suffices to show that the adjunction map $\mathcal{F} \to a_*a^{-1}\mathcal{F}$ is an isomorphism $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and that for a cartesian sheaf $\mathcal{G}$ on $(\mathcal{C}/K)_{total}$ the adjunction map $a^{-1}a_*\mathcal{G} \to \mathcal{G}$ is an isomorphism.

Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Recall that $a_*a^{-1}\mathcal{F}$ is the equalizer of the two maps $a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F}$, see Lemma 83.16.2. By Lemma 83.15.2

\[ a_{0, *}a_0^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K_0)^\# , \mathcal{F}) \quad \text{and}\quad a_{1, *}a_1^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K_1)^\# , \mathcal{F}) \]

On the other hand, we know that

\[ \xymatrix{ F(K_1)^\# \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0)^\# \ar[r] & \text{final object }*\text{ of }\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) } \]

is a coequalizer diagram in sheaves of sets by definition of a hypercovering. Thus it suffices to prove that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, \mathcal{F})$ transforms coequalizers into equalizers which is immediate from the construction in Sites, Section 7.26.

Let $\mathcal{G}$ be a cartesian sheaf on $(\mathcal{C}/K)_{total}$. We will show that $\mathcal{G} = a^{-1}\mathcal{F}$ for some sheaf $\mathcal{F}$ on $\mathcal{C}$. This will finish the proof because then $a^{-1}a_*\mathcal{G} = a^{-1}a_*a^{-1}\mathcal{F} = a^{-1}\mathcal{F} = \mathcal{G}$ by the result of the previous paragraph. Set $\mathcal{K}_ n = F(K_ n)^\# $ for $n \geq 0$. Then we have maps of sheaves

\[ \xymatrix{ \mathcal{K}_2 \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 } \]

coming from the fact that $K$ is a simplicial semi-representable object. The fact that $K$ is a hypercovering means that

\[ \mathcal{K}_1 \to \mathcal{K}_0 \times \mathcal{K}_0 \quad \text{and}\quad \mathcal{K}_2 \to \left(\text{cosq}_1( \xymatrix{ \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 \ar[l] })\right)_2 \]

are surjective maps of sheaves. Using the description of cartesian sheaves on $(\mathcal{C}/K)_{total}$ given in Lemma 83.12.4 and using the description of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n)$ in Lemma 83.15.3 we find that our problem can be entirely formulated^{1} in terms of

the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

the simplicial object $\mathcal{K}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ whose terms are $\mathcal{K}_ n$.

Thus, after replacing $\mathcal{C}$ by a different site $\mathcal{C}'$ as in Sites, Lemma 7.29.5, we may assume $\mathcal{C}$ has all finite limits, the topology on $\mathcal{C}$ is subcanonical, a family $\{ V_ j \to V\} $ of morphisms of $\mathcal{C}$ is a covering if and only if $\coprod h_{V_ j} \to V$ is surjective, and there exists a simplicial object $U$ of $\mathcal{C}$ such that $\mathcal{K}_ n = h_{U_ n}$ as simplicial sheaves. Working backwards through the equivalences we may assume $K_ n = \{ U_ n\} $ for all $n$.

Let $X$ be the final object of $\mathcal{C}$. Then $\{ U_0 \to X\} $ is a covering, $\{ U_1 \to U_0 \times U_0\} $ is a covering, and $\{ U_2 \to (\text{cosq}_1 \text{sk}_1 U)_2\} $ is a covering. Let us use $d^ n_ i : U_ n \to U_{n - 1}$ and $s^ n_ j : U_ n \to U_{n + 1}$ the morphisms corresponding to $\delta ^ n_ i$ and $\sigma ^ n_ j$ as in Simplicial, Definition 14.2.1. By abuse of notation, given a morphism $c : V \to W$ of $\mathcal{C}$ we denote the morphism of topoi $c : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/W)$ by the same letter. Now $\mathcal{G}$ is given by a sheaf $\mathcal{G}_0$ on $\mathcal{C}/U_0$ and an isomorphism $\alpha : (d^1_1)^{-1}\mathcal{G}_0 \to (d^1_0)^{-1}\mathcal{G}_0$ satisfying the cocycle condition on $\mathcal{C}/U_2$ formulated in Lemma 83.12.4. Since $\{ U_2 \to (\text{cosq}_1 \text{sk}_1 U)_2\} $ is a covering, the corresponding pullback functor on sheaves is faithful (small detail omitted). Hence we may replace $U$ by $\text{cosk}_1 \text{sk}_1 U$, because this replaces $U_2$ by $(\text{cosq}_1 \text{sk}_1 U)_2$ and leaves $U_1$ and $U_0$ unchanged. Then

\[ (d^2_0, d^2_1, d^2_2) : U_2 \to U_1 \times U_1 \times U_1 \]

is a monomorphism whose its image on $T$-valued points is described in Simplicial, Lemma 14.19.6. In particular, there is a morphism $c$ fitting into a commutative diagram

\[ \xymatrix{ U_1 \times _{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1 \ar[d] \ar[rr]_ c & & U_2 \ar[d] \\ U_1 \times U_1 \ar[rr]^{(\text{pr}_1, \text{pr}_2, s^0_0 \circ d^1_1 \circ \text{pr}_1)} & & U_1 \times U_1 \times U_1 } \]

as going around the other way defines a point of $U_2$. Pulling back the cocycle condition for $\alpha $ on $U_2$ translates into the condition that the pullbacks of $\alpha $ via the projections to $U_1 \times _{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1$ are the same as the pullback of $\alpha $ via $s^0_0 \circ d^1_1 \circ \text{pr}_1$ is the identity map (namely, the pullback of $\alpha $ by $s^0_0$ is the identity). By Sites, Lemma 7.26.1 this means that $\alpha $ comes from an isomorphism

\[ \alpha ' : \text{pr}_1^{-1}\mathcal{G}_0 \to \text{pr}_2^{-1}\mathcal{G}_0 \]

of sheaves on $\mathcal{C}/U_0 \times U_0$. Then finally, the morphism $U_2 \to U_0 \times U_0 \times U_0$ is surjective on associated sheaves as is easily seen using the surjectivity of $U_1 \to U_0 \times U_0$ and the description of $U_2$ given above. Therefore $\alpha '$ satisfies the cocycle condition on $U_0 \times U_0 \times U_0$. The proof is finished by an application of Sites, Lemma 7.26.5 to the covering $\{ U_0 \to X\} $.
$\square$

Lemma 83.17.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. The Čech complex of Lemma 83.9.2 associated to $a^{-1}\mathcal{F}$

\[ a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F} \to a_{2, *}a_2^{-1}\mathcal{F} \to \ldots \]

is equal to the complex $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F})$. Here $s(\mathbf{Z}_{F(K)}^\# )$ is as in Hypercoverings, Definition 25.4.1.

**Proof.**
By Lemma 83.15.2 we have

\[ a_{n, *}a_ n^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits '(F(K_ n)^\# , \mathcal{F}) \]

where $\mathop{\mathcal{H}\! \mathit{om}}\nolimits '$ is as in Sites, Section 7.26. The boundary maps in the complex of Lemma 83.9.2 come from the simplicial structure. Thus the equality of complexes comes from the canonical identifications $\mathop{\mathcal{H}\! \mathit{om}}\nolimits '(\mathcal{G}, \mathcal{F}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathbf{Z}_\mathcal {G}, \mathcal{F})$ for $\mathcal{G}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.
$\square$

Lemma 83.17.3. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. For $E \in D(\mathcal{C})$ the map

\[ E \longrightarrow Ra_*a^{-1}E \]

is an isomorphism.

**Proof.**
First, let $\mathcal{I}$ be an injective abelian sheaf on $\mathcal{C}$. Then the spectral sequence of Lemma 83.9.3 for the sheaf $a^{-1}\mathcal{I}$ degenerates as $(a^{-1}\mathcal{I})_ p = a_ p^{-1}\mathcal{I}$ is injective by Lemma 83.15.4. Thus the complex

\[ a_{0, *}a_0^{-1}\mathcal{I} \to a_{1, *}a_1^{-1}\mathcal{I} \to a_{2, *}a_2^{-1}\mathcal{I} \to \ldots \]

computes $Ra_*a^{-1}\mathcal{I}$. By Lemma 83.17.2 this is equal to the complex $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (s(\mathbf{Z}_{F(K)}^\# ), \mathcal{I})$. Because $K$ is a hypercovering, we see that $s(\mathbf{Z}_{F(K)}^\# )$ is exact in degrees $> 0$ by Hypercoverings, Lemma 25.4.4 applied to the simplicial presheaf $F(K)$. Since $\mathcal{I}$ is injective, the functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, \mathcal{I})$ is exact and we conclude that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (s(\mathbf{Z}_{F(K)}^\# ), \mathcal{I})$ is exact in positive degrees. We conclude that $R^ pa_*a^{-1}\mathcal{I} = 0$ for $p > 0$. On the other hand, we have $\mathcal{I} = a_*a^{-1}\mathcal{I}$ by Lemma 83.17.1.

Bounded case. Let $E \in D^+(\mathcal{C})$. Choose a bounded below complex $\mathcal{I}^\bullet $ of injectives representing $E$. By the result of the first paragraph and Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) $Ra_*a^{-1}\mathcal{I}^\bullet $ is computed by the complex $a_*a^{-1}\mathcal{I}^\bullet = \mathcal{I}^\bullet $ and we conclude the lemma is true in this case.

Unbounded case. We urge the reader to skip this, since the argument is the same as above, except that we use explicit representation by double complexes to get around convergence issues. Let $E \in D(\mathcal{C})$. To show the map $E \to Ra_*a^{-1}E$ is an isomorphism, it suffices to show for every object $U$ of $\mathcal{C}$ that

\[ R\Gamma (U, E) = R\Gamma (U, Ra_*a^{-1}E) \]

We will compute both sides and show the map $E \to Ra_*a^{-1}E$ induces an isomorphism. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $E$. Choose a quasi-isomorphism $a^{-1}\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ for some K-injective complex $\mathcal{J}^\bullet $ on $(\mathcal{C}/K)_{total}$. We have

\[ R\Gamma (U, E) = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ U^\# , E) \]

and

\[ R\Gamma (U, Ra_*a^{-1}E) = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ U^\# , Ra_*a^{-1}E) = R\mathop{\mathrm{Hom}}\nolimits (a^{-1}\mathbf{Z}_ U^\# , a^{-1}E) \]

By Lemma 83.9.1 we have a quasi-isomorphism

\[ \Big(\ldots \to g_{2!}(a_2^{-1}\mathbf{Z}_ U^\# ) \to g_{1!}(a_1^{-1}\mathbf{Z}_ U^\# ) \to g_{0!}(a_0^{-1}\mathbf{Z}_ U^\# )\Big) \longrightarrow a^{-1}\mathbf{Z}_ U^\# \]

Hence $R\mathop{\mathrm{Hom}}\nolimits (a^{-1}\mathbf{Z}_ U^\# , a^{-1}E)$ is equal to

\[ R\Gamma ((\mathcal{C}/K)_{total}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ( \ldots \to g_{2!}(a_2^{-1}\mathbf{Z}_ U^\# ) \to g_{1!}(a_1^{-1}\mathbf{Z}_ U^\# ) \to g_{0!}(a_0^{-1}\mathbf{Z}_ U^\# ), \mathcal{J}^\bullet )) \]

By the construction in Cohomology on Sites, Section 21.34 and since $\mathcal{J}^\bullet $ is K-injective, we see that this is represented by the complex of abelian groups with terms

\[ \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits (g_{p!}(a_ p^{-1}\mathbf{Z}_ U^\# ), \mathcal{J}^ q) = \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , g_ p^{-1}\mathcal{J}^ q) \]

See Cohomology on Sites, Lemmas 21.33.6 and 21.34.1 for more information. Thus we find that $R\Gamma (U, Ra_*a^{-1}E)$ is computed by the product total complex $\text{Tot}_\pi (B^{\bullet , \bullet })$ with $B^{p, q} = \mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , g_ p^{-1}\mathcal{J}^ q)$. For the other side we argue similarly. First we note that

\[ s(\mathbf{Z}_{F(K)}^\# ) \longrightarrow \mathbf{Z} \]

is a quasi-isomorphism of complexes on $\mathcal{C}$ by Hypercoverings, Lemma 25.4.4. Since $\mathbf{Z}_ U^\# $ is a flat sheaf of $\mathbf{Z}$-modules we see that

\[ s(\mathbf{Z}_{F(K)}^\# ) \otimes _\mathbf {Z} \mathbf{Z}_ U^\# \longrightarrow \mathbf{Z}_ U^\# \]

is a quasi-isomorphism. Therefore $R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ U^\# , E)$ is equal to

\[ R\Gamma (\mathcal{C}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ( s(\mathbf{Z}_{F(K)}^\# ) \otimes _\mathbf {Z} \mathbf{Z}_ U^\# , \mathcal{I}^\bullet )) \]

By the construction of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ and since $\mathcal{I}^\bullet $ is K-injective, this is represented by the complex of abelian groups with terms

\[ \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}^\# _{K_ p} \otimes _\mathbf {Z} \mathbf{Z}_ U^\# , \mathcal{I}^ q) = \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , a_ p^{-1}\mathcal{I}^ q) \]

The equality of terms follows from the fact that $\mathbf{Z}^\# _{K_ p} \otimes _\mathbf {Z} \mathbf{Z}_ U^\# = a_{p!}a_ p^{-1}\mathbf{Z}_ U^\# $ by Modules on Sites, Remark 18.27.8. Thus we find that $R\Gamma (U, E)$ is computed by the product total complex $\text{Tot}_\pi (A^{\bullet , \bullet })$ with $A^{p, q} = \mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , a_ p^{-1}\mathcal{I}^ q)$.

Since $\mathcal{I}^\bullet $ is K-injective we see that $a_ p^{-1}\mathcal{I}^\bullet $ is K-injective, see Lemma 83.15.4. Since $\mathcal{J}^\bullet $ is K-injective we see that $g_ p^{-1}\mathcal{J}^\bullet $ is K-injective, see Lemma 83.3.6. Both represent the object $a_ p^{-1}E$. Hence for every $p \geq 0$ the map of complexes

\[ A^{p, \bullet } = \mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , a_ p^{-1}\mathcal{I}^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , g_ p^{-1}\mathcal{J}^\bullet ) = B^{p, \bullet } \]

induced by $g_ p^{-1}$ applied to the given map $a^{-1}\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ is a quasi-isomorphisms as these complexes both compute

\[ R\mathop{\mathrm{Hom}}\nolimits (a_ p^{-1}\mathbf{Z}_ U^\# , a_ p^{-1}E) \]

By More on Algebra, Lemma 15.95.1 we conclude that the right vertical arrow in the commutative diagram

\[ \xymatrix{ R\Gamma (U, E) \ar[r] \ar[d] & \text{Tot}_\pi (A^{\bullet , \bullet }) \ar[d] \\ R\Gamma (U, Ra_*a^{-1}E) \ar[r] & \text{Tot}_\pi (B^{\bullet , \bullet }) } \]

is a quasi-isomorphism. Since we saw above that the horizontal arrows are quasi-isomorphisms, so is the left vertical arrow.
$\square$

Lemma 83.17.4. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then we have a canonical isomorphism

\[ R\Gamma (\mathcal{C}, E) = R\Gamma ((\mathcal{C}/K)_{total}, a^{-1}E) \]

for $E \in D(\mathcal{C})$.

**Proof.**
This follows from Lemma 83.17.3 because $R\Gamma ((\mathcal{C}/K)_{total}, -) = R\Gamma (\mathcal{C}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4.
$\square$

Lemma 83.17.5. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence

\[ D^+(\mathcal{C}) \longrightarrow D_\mathcal {A}^+((\mathcal{C}/K)_{total}) \]

with quasi-inverse $Ra_*$.

**Proof.**
Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 83.12.6. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 83.17.1 and 83.17.3 and Cohomology on Sites, Lemma 21.27.5
$\square$

We urge the reader to skip the following remark.

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