The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

20.37 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 20.36.1 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 20.29.9 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)


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BQP. Beware of the difference between the letter 'O' and the digit '0'.