Section 20.43 (0BQP): Ext sheaves

20.43 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 20.42.1 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).

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

The Stacks project

20.43 Ext sheaves

Comments (0)

Post a comment