Lemma 75.8.4. Let $A$ be a Noetherian ring. Let $X$ be a proper algebraic space over $A$. For $L$ in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, the $A$-modules $\mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ n(K, L)$ are finite.
Proof. Recall that
\[ \mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ n(K, L) = H^ n(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, L)) = H^ n(\mathop{\mathrm{Spec}}(A), Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, L)) \]
see Cohomology on Sites, Lemma 21.35.1 and Cohomology on Sites, Section 21.14. Thus the result follows from Lemmas 75.8.3 and 75.8.2. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)