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Tag 01ES

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

Lemma 20.12.5. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) $$ converging to $H^{p + q}(U, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.

Proof. This is a Grothendieck spectral sequence (see Derived Categories, Lemma 13.22.2) for the functors $$ i : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X) \quad\text{and}\quad \check{H}^0(\mathcal{U}, - ) : \textit{PMod}(\mathcal{O}_X) \to \text{Mod}_{\mathcal{O}_X(U)}. $$ Namely, we have $\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)$ by Lemma 20.10.2. We have that $i(\mathcal{I})$ is Čech acyclic by Lemma 20.12.1. And we have that $\check{H}^p(\mathcal{U}, -) = R^p\check{H}^0(\mathcal{U}, -)$ as functors on $\textit{PMod}(\mathcal{O}_X)$ by Lemma 20.11.5. Putting everything together gives the lemma. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 1481–1492 (see updates for more information).

    \begin{lemma}
    \label{lemma-cech-spectral-sequence}
    Let $X$ be a ringed space.
    Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering.
    For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ there
    is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with
    $$
    E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))
    $$
    converging to $H^{p + q}(U, \mathcal{F})$.
    This spectral sequence is functorial in $\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    This is a Grothendieck spectral sequence
    (see
    Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence})
    for the functors
    $$
    i :  \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)
    \quad\text{and}\quad
    \check{H}^0(\mathcal{U}, - ) : \textit{PMod}(\mathcal{O}_X)
    \to \text{Mod}_{\mathcal{O}_X(U)}.
    $$
    Namely, we have $\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)$
    by Lemma \ref{lemma-cech-h0}. We have that $i(\mathcal{I})$ is
    {\v C}ech acyclic by Lemma \ref{lemma-injective-trivial-cech}. And we
    have that $\check{H}^p(\mathcal{U}, -) = R^p\check{H}^0(\mathcal{U}, -)$
    as functors on $\textit{PMod}(\mathcal{O}_X)$
    by Lemma \ref{lemma-cech-cohomology-derived-presheaves}.
    Putting everything together gives the lemma.
    \end{proof}

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