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Tag 0B8R

Chapter 20: Cohomology of Sheaves > Section 20.12: Čech cohomology and cohomology

Lemma 20.12.3. Let $X$ be a topological space. Let $\mathcal{H}$ be an abelian sheaf on $X$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. The map $$ \check{H}^1(\mathcal{U}, \mathcal{H}) \longrightarrow H^1(X, \mathcal{H}) $$ is injective and identifies $\check{H}^1(\mathcal{U}, \mathcal{H})$ via the bijection of Lemma 20.5.3 with the set of isomorphism classes of $\mathcal{H}$-torsors which restrict to trivial torsors over each $U_i$.

Proof. To see this we construct an inverse map. Namely, let $\mathcal{F}$ be a $\mathcal{H}$-torsor whose restriction to $U_i$ is trivial. By Lemma 20.5.2 this means there exists a section $s_i \in \mathcal{F}(U_i)$. On $U_{i_0} \cap U_{i_1}$ there is a unique section $s_{i_0i_1}$ of $\mathcal{H}$ such that $s_{i_0i_1} \cdot s_{i_0}|_{U_{i_0} \cap U_{i_1}} = s_{i_1}|_{U_{i_0} \cap U_{i_1}}$. A computation shows that $s_{i_0i_1}$ is a Čech cocycle and that its class is well defined (i.e., does not depend on the choice of the sections $s_i$). The inverse maps the isomorphism class of $\mathcal{F}$ to the cohomology class of the cocycle $(s_{i_0i_1})$. We omit the verification that this map is indeed an inverse. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 1429–1441 (see updates for more information).

    \begin{lemma}
    \label{lemma-cech-h1}
    Let $X$ be a topological space. Let $\mathcal{H}$ be an abelian sheaf
    on $X$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering.
    The map
    $$
    \check{H}^1(\mathcal{U}, \mathcal{H}) \longrightarrow H^1(X, \mathcal{H})
    $$
    is injective and identifies $\check{H}^1(\mathcal{U}, \mathcal{H})$ via
    the bijection of Lemma \ref{lemma-torsors-h1}
    with the set of isomorphism classes of $\mathcal{H}$-torsors
    which restrict to trivial torsors over each $U_i$.
    \end{lemma}
    
    \begin{proof}
    To see this we construct an inverse map. Namely, let $\mathcal{F}$ be a
    $\mathcal{H}$-torsor whose restriction to $U_i$ is trivial. By
    Lemma \ref{lemma-trivial-torsor} this means there
    exists a section $s_i \in \mathcal{F}(U_i)$. On $U_{i_0} \cap U_{i_1}$
    there is a unique section $s_{i_0i_1}$ of $\mathcal{H}$ such that
    $s_{i_0i_1} \cdot s_{i_0}|_{U_{i_0} \cap U_{i_1}} =
    s_{i_1}|_{U_{i_0} \cap U_{i_1}}$. A computation shows
    that $s_{i_0i_1}$ is a {\v C}ech cocycle and that its class is well
    defined (i.e., does not depend on the choice of the sections $s_i$).
    The inverse maps the isomorphism class of $\mathcal{F}$ to the cohomology
    class of the cocycle $(s_{i_0i_1})$.
    We omit the verification that this map is indeed an inverse.
    \end{proof}

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