Processing math: 100%

The Stacks project

Lemma 69.6.3. Let S be a scheme. Let f : U \to X be a surjective étale morphism of algebraic spaces over S. Let \mathcal{F} be an object of \textit{Ab}(X_{\acute{e}tale}). There exists a canonical map

\check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F}) \longrightarrow R\Gamma (X, \mathcal{F})

in D(\textit{Ab}). Moreover, there is a spectral sequence with E_1-page

E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits _{\textit{Ab}(X_{\acute{e}tale})}^ q(K^ p, \mathcal{F})

converging to H^{p + q}(X, \mathcal{F}) where K^ p = \wedge ^{p + 1}f_!\underline{\mathbf{Z}}.

Proof. Recall that we have the quasi-isomorphism K^\bullet \to \underline{\mathbf{Z}}[0], see (69.6.1.1). Choose an injective resolution \mathcal{F} \to \mathcal{I}^\bullet in \textit{Ab}(X_{\acute{e}tale}). Consider the double complex \mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet ) with terms \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^ q). The differential d_1^{p, q} : A^{p, q} \to A^{p + 1, q} is the one coming from the differential K^{p + 1} \to K^ p and the differential d_2^{p, q} : A^{p, q} \to A^{p, q + 1} is the one coming from the differential \mathcal{I}^ q \to \mathcal{I}^{q + 1}. Denote \text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet )) the associated total complex, see Homology, Section 12.18. We will use the two spectral sequences ({}'E_ r, {}'d_ r) and ({}''E_ r, {}''d_ r) associated to this double complex, see Homology, Section 12.25.

Because K^\bullet is a resolution of \underline{\mathbf{Z}} we see that the complexes

\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^ q) : \mathop{\mathrm{Hom}}\nolimits (K^0, \mathcal{I}^ q) \to \mathop{\mathrm{Hom}}\nolimits (K^1, \mathcal{I}^ q) \to \mathop{\mathrm{Hom}}\nolimits (K^2, \mathcal{I}^ q) \to \ldots

are acyclic in positive degrees and have H^0 equal to \Gamma (X, \mathcal{I}^ q). Hence by Homology, Lemma 12.25.4 the natural map

\mathcal{I}^\bullet (X) \longrightarrow \text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet ))

is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that H^ n(\text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet ))) = H^ n(X, \mathcal{F}).

The map \check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F}) \to R\Gamma (X, \mathcal{F}) of the lemma is the composition of \check{\mathcal{C}}^\bullet _{alt}(f, \mathcal{F}) \to \text{Tot}(\mathop{\mathrm{Hom}}\nolimits (K^\bullet , \mathcal{I}^\bullet )) with the inverse of the displayed quasi-isomorphism.

Finally, consider the spectral sequence ({}'E_ r, {}'d_ r). We have

E_1^{p, q} = q\text{th cohomology of } \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^0) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^1) \to \mathop{\mathrm{Hom}}\nolimits (K^ p, \mathcal{I}^2) \to \ldots

This proves the lemma. \square


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.