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Tag 01FY

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 [SGA4, Exposé V, Sec. 7]. 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 [SGA4, Exposé Vbis]. A nice manuscript on cohomological descent is the text by Brian Conrad, see http://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 75.26). Our method of attack will be to reduce this statement to the Čech to cohomology spectral sequence constructed in this chapter.

    The code snippet corresponding to this tag is a part of the file hypercovering.tex and is located in lines 17–102 (see updates for more information).

    \section{Introduction}
    \label{section-introduction}
    
    \noindent
    Let $\mathcal{C}$ be a site, see Sites, Definition \ref{sites-definition-site}.
    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
    \ref{sites-cohomology-section-cohomology-sheaves})
    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.
    
    \medskip\noindent
    A hypercovering in a site is a generalization of a covering, see
    \cite[Expos\'e V, Sec. 7]{SGA4}. Given a hypercovering $K$ of an object
    $X$, there is a {\v C}ech 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 {\v C}ech cohomology groups.
    
    \medskip\noindent
    A more general gadget one can consider is a simplicial augmentation where
    one has cohomological descent, see \cite[Expos\'e Vbis]{SGA4}. A nice
    manuscript on cohomological descent is the text by Brian Conrad, see
    \url{http://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
    \ref{spaces-simplicial-section-proper-hypercovering}).
    Our method of attack will be to reduce this statement to the {\v C}ech to
    cohomology spectral sequence constructed in this chapter.

    Comments (1)

    Comment #477 by a on March 23, 2014 a 11:17 pm UTC

    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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