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Tag 01ER

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

Lemma 20.12.4. Let $X$ be a ringed space. Consider the functor $i : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$. It is a left exact functor with right derived functors given by $$ R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F}) $$ see discussion in Section 20.8.

Proof. It is clear that $i$ is left exact. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. By definition $R^pi$ is the $p$th cohomology presheaf of the complex $\mathcal{I}^\bullet$. In other words, the sections of $R^pi(\mathcal{F})$ over an open $U$ are given by $$ \frac{\text{Ker}(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\text{Im}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}. $$ which is the definition of $H^p(U, \mathcal{F})$. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 1455–1466 (see updates for more information).

    \begin{lemma}
    \label{lemma-include}
    Let $X$ be a ringed space.
    Consider the functor
    $i : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$.
    It is a left exact functor with right derived functors given by
    $$
    R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) :
    U \longmapsto H^p(U, \mathcal{F})
    $$
    see discussion in Section \ref{section-locality}.
    \end{lemma}
    
    \begin{proof}
    It is clear that $i$ is left exact.
    Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$.
    By definition $R^pi$ is the $p$th cohomology {\it presheaf}
    of the complex $\mathcal{I}^\bullet$. In other words, the
    sections of $R^pi(\mathcal{F})$ over an open $U$ are given by
    $$
    \frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))}
    {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.
    $$
    which is the definition of $H^p(U, \mathcal{F})$.
    \end{proof}

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