The Stacks project

Remark 115.21.2 (Construction of obstruction class). With notation as in Remark 115.21.1 let $i : U \to U'$ be a first order thickening of $U$ over $B$. Let $\mathcal{I} \subset \mathcal{O}_{U'}$ be the quasi-coherent sheaf of ideals cutting out $B$ in $B'$. The fundamental triangle

\[ Li^*L_{U'/B} \to L_{U/B} \to L_{U/U'} \to Li^*L_{U'/B}[1] \]

together with the map $L_{U/U'} \to \mathcal{I}[1]$ determine a map $e_{U'} : L_{U/B} \to \mathcal{I}[1]$. Combined with the map $e_\mathcal {F}$ of the previous remark we obtain

\[ (\text{id}_\mathcal {F} \otimes Lq^*e_{U'}) \cup e_\mathcal {F} : E(\mathcal{F}) \longrightarrow \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}} q^*\mathcal{I}[2] \]

(we have also composed with the map from the derived tensor product to the usual tensor product). In other words, we obtain an element

\[ \xi _{U'} \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_{X \times _ B U}}( E(\mathcal{F}), \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}} q^*\mathcal{I}) \]

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Your email address will not be published.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 09DP. Beware of the difference between the letter 'O' and the digit '0'.