The Stacks project

Lemma 73.23.2. Assumption and notation as in Lemma 73.22.3. Then there are functorial isomorphisms

\[ H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).

Proof. As in the proof of Lemma 73.22.3 let $E^\vee $ be the dual perfect complex and recall that $K = Rf_*(E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet )$. Since we also have

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) = H^ i(X, E^\vee \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) \]

by construction of $E^\vee $, the existence of the isomorphisms follows from Lemma 73.23.1 applied to $E^\vee $ and $\mathcal{G}^\bullet $. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ then the isomorphisms fit into commutative diagrams

\[ \xymatrix{ H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_3) \ar[r] \ar[d]_\delta & \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3) \ar[d]^\delta \\ H^{i + 1}(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1) \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1) } \]

where the boundary maps come from the distinguished triangle

\[ K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1[1] \]

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

\[ 0 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3 \to 0 \]

of complexes. This sequence is exact because $\mathcal{G}^ n$ is flat over $B$. We omit the verification of the commutativity of the displayed diagram. $\square$


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Your email address will not be published.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 08JN. Beware of the difference between the letter 'O' and the digit '0'.