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. $\square$

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