## 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 cohomologypresheafof the complex $\mathcal{I}^\bullet$. In other words, the sections of $R^pi(\mathcal{F})$ over an open $U$ are given by $$ \frac{\mathop{\rm Ker}(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\mathop{\rm Im}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}. $$ 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 1458–1469 (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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