The Stacks project

20.41 Ext sheaves

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L \in D(\mathcal{O}_ X)$. Using the construction of the internal hom in the derived category we obtain a well defined sheaves of $\mathcal{O}_ X$-modules

\[ \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ n(K, L) = H^ n(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)) \]

by taking the $n$th cohomology sheaf of the object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ of $D(\mathcal{O}_ X)$. We will sometimes write $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ n_{\mathcal{O}_ X}(K, L)$ for this object. By Lemma 20.40.1 we see that this $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ n$-sheaf is the sheafification of the rule

\[ U \longmapsto \mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ U)}(K|_ U, L|_ U) \]

By Example 20.29.3 there is always a spectral sequence

\[ E_2^{p, q} = H^ p(X, \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(K, L)) \]

converging to $\mathop{\mathrm{Ext}}\nolimits ^{p + q}_{D(\mathcal{O}_ X)}(K, L)$ in favorable situations (for example if $L$ is bounded below and $K$ is bounded above).

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