The Stacks project

Proposition 115.20.4. With $\Lambda $, $X$, $A$, $\mathcal{F}$ as above. There exists a canonical object $L = L(\Lambda , X, A, \mathcal{F})$ of $D(X_ A)$ such that given a surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have

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

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

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