\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

24.1 Introduction

Let $\mathcal{C}$ be a site, see Sites, Definition 7.6.2. Let $X$ be an object of $\mathcal{C}$. Given an abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ we would like to compute its cohomology groups

\[ H^ i(X, \mathcal{F}). \]

According to our general definitions (Cohomology on Sites, Section 21.3) this cohomology group is computed by choosing an injective resolution $ 0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \ldots $ and setting

\[ H^ i(X, \mathcal{F}) = H^ i( \Gamma (X, \mathcal{I}^0) \to \Gamma (X, \mathcal{I}^1) \to \Gamma (X, \mathcal{I}^2)\to \ldots ) \]

The goal of this chapter is to show that we may also compute these cohomology groups without choosing an injective resolution (in the case that $\mathcal{C}$ has fibre products). To do this we will use hypercoverings.

A hypercovering in a site is a generalization of a covering, see [Exposé V, Sec. 7, SGA4]. Given a hypercovering $K$ of an object $X$, there is a Čech to cohomology spectral sequence expressing the cohomology of an abelian sheaf $\mathcal{F}$ over $X$ in terms of the cohomology of the sheaf over the components $K_ n$ of $K$. It turns out that there are always enough hypercoverings, so that taking the colimit over all hypercoverings, the spectral sequence degenerates and the cohomology of $\mathcal{F}$ over $X$ is computed by the colimit of the Čech cohomology groups.

A more general gadget one can consider is a simplicial augmentation where one has cohomological descent, see [Exposé Vbis, SGA4]. A nice manuscript on cohomological descent is the text by Brian Conrad, see https://math.stanford.edu/~conrad/papers/hypercover.pdf. We will come back to these issue in the chapter on simplicial spaces where we will show, for example, that proper hypercoverings of “locally compact” topological spaces are of cohomological descent (Simplicial Spaces, Section 77.26). Our method of attack will be to reduce this statement to the Čech to cohomology spectral sequence constructed in this chapter.

Comments (1)

Comment #477 by a on

misspelling of cohomology:

..there is a Čech to cohomology spectral sequence expressing the cohomogy <-----TYPO of an abelian sheaf  over X in terms of the cohomology of the sheaf over the components Kn of K.

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