25.6 Hypercoverings a la Verdier
The astute reader will have noticed that all we need in order to get the Čech to cohomology spectral sequence for a hypercovering of an object X, is the conclusion of Lemma 25.3.9. Therefore the following definition makes sense.
Definition 25.6.1. Let \mathcal{C} be a site. Assume \mathcal{C} has equalizers and fibre products. Let \mathcal{G} be a presheaf of sets. A hypercovering of \mathcal{G} is a simplicial object K of \text{SR}(\mathcal{C}) endowed with an augmentation F(K) \to \mathcal{G} such that
F(K_0) \to \mathcal{G} becomes surjective after sheafification,
F(K_1) \to F(K_0) \times _\mathcal {G} F(K_0) becomes surjective after sheafification, and
F(K_{n + 1}) \longrightarrow F((\text{cosk}_ n \text{sk}_ n K)_{n + 1}) for n \geq 1 becomes surjective after sheafification.
We say that a simplicial object K of \text{SR}(\mathcal{C}) is a hypercovering if K is a hypercovering of the final object * of \textit{PSh}(\mathcal{C}).
The assumption that \mathcal{C} has fibre products and equalizers guarantees that \text{SR}(\mathcal{C}) has fibre products and equalizers and F commutes with these (Lemma 25.2.3) which suffices to define the coskeleton functors used (see Simplicial, Remark 14.19.11 and Categories, Lemma 4.18.2). If \mathcal{C} is general, we can replace the condition (3) by the condition that F(K_{n + 1}) \longrightarrow ((\text{cosk}_ n \text{sk}_ n F(K))_{n + 1}) for n \geq 1 becomes surjective after sheafification and the results of this section remain valid.
Let \mathcal{F} be an abelian sheaf on \mathcal{C}. In the previous section, we defined the Čech complex of \mathcal{F} with respect to a simplicial object K of \text{SR}(\mathcal{C}). Next, given a presheaf \mathcal{G} we set
H^0(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}^\# , \mathcal{F}) = H^0(\mathcal{G}^\# , \mathcal{F})
with notation as in Cohomology on Sites, Section 21.13. This is a left exact functor and its higher derived functors (briefly studied in Cohomology on Sites, Section 21.13) are denoted H^ i(\mathcal{G}, \mathcal{F}). We will show that given a hypercovering K of \mathcal{G}, there is a Čech to cohomology spectral sequence converging to the cohomology H^ i(\mathcal{G}, \mathcal{F}). Note that if \mathcal{G} = *, then H^ i(*, \mathcal{F}) = H^ i(\mathcal{C}, \mathcal{F}) recovers the cohomology of \mathcal{F} on the site \mathcal{C}.
Lemma 25.6.2. Let \mathcal{C} be a site with equalizers and fibre products. Let \mathcal{G} be a presheaf on \mathcal{C}. Let K be a hypercovering of \mathcal{G}. Let \mathcal{F} be a sheaf of abelian groups on \mathcal{C}. Then \check{H}^0(K, \mathcal{F}) = H^0(\mathcal{G}, \mathcal{F}).
Proof.
This follows from the definition of H^0(\mathcal{G}, \mathcal{F}) and the fact that
\xymatrix{ F(K_1) \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0) \ar[r] & \mathcal{G} }
becomes an coequalizer diagram after sheafification.
\square
Lemma 25.6.3. Let \mathcal{C} be a site with equalizers and fibre products. Let \mathcal{G} be a presheaf on \mathcal{C}. Let K be a hypercovering of \mathcal{G}. Let \mathcal{I} be an injective sheaf of abelian groups on \mathcal{C}. Then
\check{H}^ p(K, \mathcal{I}) = \left\{ \begin{matrix} H^0(\mathcal{G}, \mathcal{I})
& \text{if}
& p = 0
\\ 0
& \text{if}
& p > 0
\end{matrix} \right.
Proof.
By (25.5.2.1) we have
s(\mathcal{F}(K)) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(\mathcal{C})}(s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F})
The complex s(\mathbf{Z}_{F(K)}^\# ) is quasi-isomorphic to \mathbf{Z}_\mathcal {G}^\# , see Lemma 25.4.4. We conclude that if \mathcal{I} is an injective abelian sheaf, then the complex s(\mathcal{I}(K)) is acyclic except possibly in degree 0. In other words, we have \check{H}^ i(K, \mathcal{I}) = 0 for i > 0. Combined with Lemma 25.6.2 the lemma is proved.
\square
Lemma 25.6.4. Let \mathcal{C} be a site with equalizers and fibre products. Let \mathcal{G} be a presheaf on \mathcal{C}. Let K be a hypercovering of \mathcal{G}. Let \mathcal{F} be a sheaf of abelian groups on \mathcal{C}. There is a map
s(\mathcal{F}(K)) \longrightarrow R\Gamma (\mathcal{G}, \mathcal{F})
in D^{+}(\textit{Ab}) functorial in \mathcal{F}, which induces a natural transformation
\check{H}^ i(K, -) \longrightarrow H^ i(\mathcal{G}, -)
of 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}(\mathcal{G}, \mathcal{F}). This spectral sequence is functorial in \mathcal{F} and in the hypercovering K.
Proof.
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. 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.
By Lemma 25.6.3 the complexes s(\mathcal{I}^ p(K)) are acyclic in positive degrees and have H^0 equal to H^0(\mathcal{G}, \mathcal{I}^ p). Hence by Homology, Lemma 12.25.4 and its proof the spectral sequence ({}'E_ r, {}'d_ r) degenerates, and the natural map
H^0(\mathcal{G}, \mathcal{I}^\bullet ) \longrightarrow \text{Tot}(A^{\bullet , \bullet })
is a quasi-isomorphism of complexes of abelian groups. The map s(\mathcal{F}(K)) \longrightarrow R\Gamma (\mathcal{G}, \mathcal{F}) of the lemma is the composition of the natural map s(\mathcal{F}(K)) \to \text{Tot}(A^{\bullet , \bullet }) followed by the inverse of the displayed quasi-isomorphism above. This works because H^0(\mathcal{G}, \mathcal{I}^\bullet ) is a representative of R\Gamma (\mathcal{G}, \mathcal{F}).
Consider the spectral sequence ({}''E_ r, {}''d_ r)_{r \geq 0}. By Homology, Lemma 12.25.1 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})). Since this spectral sequence converges to the cohomology of \text{Tot}(A^{\bullet , \bullet }) the proof is finished.
\square
Lemma 25.6.5. Let \mathcal{C} be a site with equalizers and fibre products. Let K be a hypercovering. Let \mathcal{F} be an abelian sheaf. 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 the global cohomology groups H^{p + q}(\mathcal{F}).
Proof.
This is a special case of Lemma 25.6.4.
\square
Comments (0)