The Stacks project

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 85.12.3. 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 85.16.2. By Lemma 85.15.2

On the other hand, we know that

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

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

are surjective maps of sheaves. Using the description of cartesian sheaves on $(\mathcal{C}/K)_{total}$ given in Lemma 85.12.4 and using the description of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n)$ in Lemma 85.15.3 we find that our problem can be entirely formulated¹ in terms of

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

2. 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{cosk}_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 85.12.4. Since $\{ U_2 \to (\text{cosk}_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 $(\text{cosk}_1 \text{sk}_1 U)_2$ and leaves $U_1$ and $U_0$ unchanged. Then

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

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

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\} $. $\square$

Proof. By Lemma 85.15.2 we have

where $\mathop{\mathcal{H}\! \mathit{om}}\nolimits '$ is as in Sites, Section 7.26. The boundary maps in the complex of Lemma 85.9.2 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 $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. $\square$

Proof. First, let $\mathcal{I}$ be an injective abelian sheaf on $\mathcal{C}$. Then the spectral sequence of Lemma 85.9.3 for the sheaf $a^{-1}\mathcal{I}$ degenerates as $(a^{-1}\mathcal{I})_ p = a_ p^{-1}\mathcal{I}$ is injective by Lemma 85.15.4. Thus the complex

computes $Ra_*a^{-1}\mathcal{I}$. By Lemma 85.17.2 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 85.17.1.

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

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

and

By Lemma 85.9.1 we have a quasi-isomorphism

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

By the construction in Cohomology on Sites, Section 21.35 and since $\mathcal{J}^\bullet $ is K-injective, we see that this is represented by the complex of abelian groups with terms

See Cohomology on Sites, Lemmas 21.34.6 and 21.35.1 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

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

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

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

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 18.27.10. 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 85.15.4. Since $\mathcal{J}^\bullet $ is K-injective we see that $g_ p^{-1}\mathcal{J}^\bullet $ is K-injective, see Lemma 85.3.6. Both represent the object $a_ p^{-1}E$. Hence for every $p \geq 0$ the map of complexes

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

By More on Algebra, Lemma 15.103.2 we conclude that the right vertical arrow in the commutative diagram

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

Proof. This follows from Lemma 85.17.3 because $R\Gamma ((\mathcal{C}/K)_{total}, -) = R\Gamma (\mathcal{C}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4. $\square$

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 85.12.6. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 85.17.1 and 85.17.3 and Cohomology on Sites, Lemma 21.28.5 $\square$