The Stacks project

Lemma 24.5.3. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. There is a map

\[ s(\mathcal{F}(K)) \longrightarrow R\Gamma (X, \mathcal{F}) \]

in $D^{+}(\textit{Ab})$ functorial in $\mathcal{F}$, which induces natural transformations

\[ \check{H}^ i(K, -) \longrightarrow H^ i(X, -) \]

as functors $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$. Moreover, there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]

converging to $H^{p + q}(X, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$ and in the hypercovering $K$.

Proof. We could prove this by the same method as employed in the corresponding lemma in the chapter on cohomology. Instead let us prove this by a double complex argument.

Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ in the category of abelian sheaves on $\mathcal{C}$. Consider the double complex $A^{\bullet , \bullet }$ with terms

\[ A^{p, q} = \mathcal{I}^ q(K_ p) \]

where the differential $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ is the one coming from the differential $\mathcal{I}^ p \to \mathcal{I}^{p + 1}$ and the differential $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ is the one coming from the differential on the complex $s(\mathcal{I}^ p(K))$ associated to the cosimplicial abelian group $\mathcal{I}^ p(K)$ as explained above. As usual we denote $sA^\bullet $ the simple complex associated to the double complex $A^{\bullet , \bullet }$. We will use the two spectral sequences $({}'E_ r, {}'d_ r)$ and $({}''E_ r, {}''d_ r)$ associated to this double complex, see Homology, Section 12.22.

By Lemma 24.5.2 the complexes $s(\mathcal{I}^ p(K))$ are acyclic in positive degrees and have $H^0$ equal to $\mathcal{I}^ p(X)$. Hence by Homology, Lemma 12.22.7 and its proof the spectral sequence $({}'E_ r, {}'d_ r)$ degenerates, and the natural map

\[ \mathcal{I}^\bullet (X) \longrightarrow sA^\bullet \]

is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that $H^ n(sA^\bullet ) = H^ n(X, \mathcal{F})$.

The map $s(\mathcal{F}(K)) \longrightarrow R\Gamma (X, \mathcal{F})$ of the lemma is the composition of the natural map $s(\mathcal{F}(K)) \to sA^\bullet $ followed by the inverse of the displayed quasi-isomorphism above. This works because $\mathcal{I}^\bullet (X)$ is a representative of $R\Gamma (X, \mathcal{F})$.

Consider the spectral sequence $({}''E_ r, {}''d_ r)_{r \geq 0}$. By Homology, Lemma 12.22.4 we see that

\[ {}''E_2^{p, q} = H^ p_{II}(H^ q_ I(A^{\bullet , \bullet })) \]

In other words, we first take cohomology with respect to $d_1$ which gives the groups ${}''E_1^{p, q} = \underline{H}^ p(\mathcal{F})(K_ q)$. Hence it is indeed the case (by the description of the differential ${}''d_1$) that ${}''E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F}))$. And by the other spectral sequence above we see that this one converges to $H^ n(X, \mathcal{F})$ as desired.

We omit the proof of the statements regarding the functoriality of the above constructions in the abelian sheaf $\mathcal{F}$ and the hypercovering $K$. $\square$


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