The Stacks project

Lemma 115.20.3. Notation and assumptions as in Lemma 115.20.2. Consider the object

\[ L = L(\Lambda , X, A, \mathcal{G}) = L\pi _!(Li^*(i_*(\underline{\mathcal{G}}))) \]

of $D(\mathcal{O}_ X \otimes _\Lambda A)$. Given a surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have

  1. The category $\textit{Lift}(\mathcal{G}, A')$ is nonempty if and only if a certain class $\xi \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X \otimes A}(L, \mathcal{G} \otimes _ A I)$ is zero.

  2. If $\textit{Lift}(\mathcal{G}, A')$ is nonempty, then $\text{Lift}(\mathcal{G}, A')$ is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X \otimes A}(L, \mathcal{G} \otimes _ A I)$.

  3. Given a lift $\mathcal{G}'$, the set of automorphisms of $\mathcal{G}'$ which pull back to $\text{id}_\mathcal {G}$ is canonically isomorphic to $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X \otimes A}(L, \mathcal{G} \otimes _ A I)$.

Proof. FIXME. $\square$

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Your email address will not be published.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 08W2. Beware of the difference between the letter 'O' and the digit '0'.