5.3 Cohomological descent
Let a\colon X_\bullet\to S be an augmented simplicial topological space. For every sheaf F on S, we have a morphism \varphi\colon F \to a_*a^*F from the adjunction a^*\dashv a_*. This morphism derives to a morphism of functors from \mathrm{D}^+(S) to \mathrm{D}^+(S), namely \varphi\colon\mathrm{Id} \to \mathrm{R}a_*a^*. \tag{5.3.1.1}
We say that a is of cohomological descent if, for every abelian sheaf F on S, we have F\xrightarrow{\sim}\operatorname{Ker}({a_0}_*{a_0}^*F\to{a_1}_*{a_1}^*F) and \mathrm{R}^i a_*a^*F = 0 \quad\text{for }i>0. This is equivalent to asking that (5.3.1.1) be an isomorphism.
If a is of cohomological descent, then, for K\in\operatorname{Ob}\mathrm{D}^+(S), the canonical map \mathrm{R}\Gamma(S,K) \to \mathrm{R}\Gamma(S,\mathrm{R}a_*a^*K) \simeq \mathrm{R}\Gamma(X_\bullet,a^*K) \tag{5.3.3.1} is an isomorphism. In particular, for F an abelian sheaf on S, we have a spectral sequence (5.2.3.2)) E_1^{pq} = \operatorname{H}^q(X_p,a_p^*F) \Rightarrow\operatorname{H}^{p+q}(S,F). \tag{5.3.3.2}
For a complex, we again have, in hypercohomology, a spectral sequence E_1^{pq} = \operatorname{\mathbb{H}}^q(X_p,a_p^*K) \Rightarrow\operatorname{\mathbb{H}}^{p+q}(S,K). \tag{5.3.3.3}
In both cases, the E_1^{\bullet q} (for fixed q) form a simplicial group, and d_1 = \sum_i (-1)^i\delta_i \colon E_1^{p,q} \to E_1^{p+1,q}.
A continuous map a\colon X\to S is of cohomological descent if the augmentation morphism of \operatorname{cosq}(X\to S), namely \big((X/S)^{\Delta_n}\big)_{n\geqslant0} \to S, is of cohomological descent. We say that a is of universal cohomological descent if, for every u\colon S'\to S, the continuous map a'\colon X\times_S S'\to S' is of cohomological descent.
The fundamental results, proven in (Saint-Donat 1972), are the following.
- The continuous maps of universal cohomological descent form a Grothendieck topology on the category of topological spaces, which we call the universal cohomological descent topology.
- A proper (0.1) surjective map is of universal cohomological descent.
- A map a\colon X\to S that admits sections locally on S is of universal cohomological descent.
- Let a\colon X_\bullet\to S be a k-truncated augmented simplicial space (with -1\leqslant k\leqslant\infty). For k\geqslant n\geqslant-1, let \varphi_n\colon\operatorname{cosq}X_\bullet\to\operatorname{cosq}\operatorname{sq}_n X_\bullet be the evident map. We say that X_\bullet is a k-truncated hypercover of S, for the universal cohomological descent topology, if the maps (\varphi_n)_{n+1} \colon X_{n+1} \to (\operatorname{cosq}\operatorname{sq}_n X_\bullet)_{n+1} \tag{5.3.5.1} (for -1\leqslant n\leqslant k-1) are of universal cohomological descent. If X_\bullet is such a hypercover, then the simplicial space \operatorname{cosq}(X_\bullet) augmented over S is of cohomological descent.
- Let a be a morphism of simplicial topological spaces augmented over S. \begin{CD} X_\bullet @>a>> Y_\bullet \\@VxVV @VVyV \\S @= S \end{CD} We say that a is a hypercover for the universal cohomological descent topology if the evident maps X_n\to(\operatorname{cosq}_{n-1}^{Y_\bullet}\operatorname{sq}_{n-1}X_\bullet)_n are of universal cohomological descent. If a is such a hypercover then, for every K\in\operatorname{Ob}\mathrm{D}^+(S), a^*\colon \mathrm{R}y_*y^*K \xrightarrow{\sim}\mathrm{R}x_*x^*K.
For k=\infty, (iv) of (5.3.5) implies that the a\colon X_\bullet\to S are of cohomological descent if the (\varphi_n){n+1}} are of universal cohomological descent. For n=-1,0, these maps are (\varphi_n)_{n+1} = \begin{cases} X_0\xrightarrow{a}S &\text{if }n=-1 \\X_1\xrightarrow{(\delta_0,\delta_1)}X_0\times_S X_0 &\text{if }n=0. \end{cases} For n=1, (\operatorname{cosq}\operatorname{sq}_1(X_\bullet))_1 is the subspace of X_1\times_S X_1\times_S X_1 consisting of the triples (x,y,z) such that \delta_0x=\delta_0y, \delta_1x=\delta_0z, and \delta_1y=\delta_1z. The map (\varphi_1)_2 is x\mapsto(\delta_0x,\delta_1x,\delta_2x).
Let \mathscr{U}=(U_i)_{i\in I} be an open cover, or a finite locally closed cover, of S. Let X=\coprod_{i\in I}U_i. Then a\colon X\to S is of cohomological descent. The spectral sequence in (5.3.3.2) for X_\bullet=\operatorname{cosq}(X\to S) is then exactly the Leray spectral sequence of the cover \mathscr{U}.
Let a\colon X_\bullet\to S be as in (iv) of (5.3.5). We say that X_\bullet is a proper k-truncated hypercover of S if the arrows in (5.3.5.1) are proper and surjective. For k=\infty, we simply say “proper hypercover”.