Example 21.42.2 (08S0): Computing Exts

Example 21.42.2 (Computing Exts). In Example 21.39.1 assume we are moreover given a sheaf of rings $\mathcal{O}$ on $\mathcal{C}$. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}$-modules. Consider the complex $K^\bullet (\mathcal{G}, \mathcal{F})$ with degree $n$ term

\[ \prod \nolimits _{U_0 \to U_1 \to \ldots \to U_ n} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U_ n)}(\mathcal{G}(U_ n), \mathcal{F}(U_0)) \]

and transition map given by

\[ (\varphi _{U_0 \to U_1}) \longmapsto ((U_0 \to U_1 \to U_2) \mapsto \varphi _{U_0 \to U_1} \circ \rho ^{U_2}_{U_1} - \varphi _{U_0 \to U_2} + \rho ^{U_1}_{U_0} \circ \varphi _{U_1 \to U_2} \]

and similarly in other degrees. Here the $\rho $'s indicate restriction maps. By construction

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{F}) = H^0(K^\bullet (\mathcal{G}, \mathcal{F})) \]

for all pairs of $\mathcal{O}$-modules $\mathcal{F}, \mathcal{G}$. The assignment $(\mathcal{G}, \mathcal{F}) \mapsto K^\bullet (\mathcal{G}, \mathcal{F})$ is a bifunctor which transforms direct sums in the first variable into products and commutes with products in the second variable. We claim that

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {O}(\mathcal{G}, \mathcal{F}) = H^ i(K^\bullet (\mathcal{G}, \mathcal{F})) \]

for $i \geq 0$ provided either

$\mathcal{G}(U)$ is a projective $\mathcal{O}(U)$-module for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, or
$\mathcal{F}(U)$ is an injective $\mathcal{O}(U)$-module for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

Namely, case (1) the functor $K^\bullet (\mathcal{G}, -)$ is an exact functor from the category of $\mathcal{O}$-modules to the category of cochain complexes of abelian groups. Thus, arguing as in Example 21.42.1, it suffices to show that $K^\bullet (\mathcal{G}, \mathcal{F})$ is acyclic in positive degrees when $\mathcal{F}$ is $p_{U, *}A$ for an $\mathcal{O}(U)$-module $A$. Choose a short exact sequence

21.42.2.1

\begin{equation} \label{sites-cohomology-equation-split} 0 \to \mathcal{G}' \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \to \mathcal{G} \to 0 \end{equation}

see Modules on Sites, Lemma 18.28.8. Since (1) holds for the middle and right sheaves, it also holds for $\mathcal{G}'$ and evaluating (21.42.2.1) on an object of $\mathcal{C}$ gives a split exact sequence of modules. We obtain a short exact sequence of complexes

\[ 0 \to K^\bullet (\mathcal{G}, \mathcal{F}) \to \prod K^\bullet (j_{U_ i!}\mathcal{O}_{U_ i}, \mathcal{F}) \to K^\bullet (\mathcal{G}', \mathcal{F}) \to 0 \]

for any $\mathcal{F}$, in particular $\mathcal{F} = p_{U, *}A$. On $H^0$ we obtain

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}, p_{U, *}A) \to \mathop{\mathrm{Hom}}\nolimits (\prod j_{U_ i!}\mathcal{O}_{U_ i}, p_{U, *}A) \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}', p_{U, *}A) \to 0 \]

which is exact as $\mathop{\mathrm{Hom}}\nolimits (\mathcal{H}, p_{U, *}A) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U)}(\mathcal{H}(U), A)$ and the sequence of sections of (21.42.2.1) over $U$ is split exact. Thus we can use dimension shifting to see that it suffices to prove $K^\bullet (j_{U'!}\mathcal{O}_{U'}, p_{U, *}A)$ is acyclic in positive degrees for all $U, U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. In this case $K^ n(j_{U'!}\mathcal{O}_{U'}, p_{U, *}A)$ is equal to

\[ \prod \nolimits _{U \to U_0 \to U_1 \to \ldots \to U_ n \to U'} A \]

In other words, $K^\bullet (j_{U'!}\mathcal{O}_{U'}, p_{U, *}A)$ is the complex with terms $\text{Map}(X_\bullet , A)$ where

\[ X_ n = \coprod \nolimits _{U_0 \to \ldots \to U_{n - 1} \to U_ n} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, U_0) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_ n, U') \]

This simplicial set is homotopy equivalent to the constant simplicial set on a singleton $\{ *\} $ as can be proved in exactly the same way as the corresponding statement in Example 21.42.1. This finishes the proof of the claim.

The argument in case (2) is similar (but dual).

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