The Stacks project

Situation 98.24.2. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume that $\mathcal{X}$ has (RS*) so that we can speak of the functor $T_ x(-)$, see Lemma 98.21.2. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$. Let $x$ be an object of $\mathcal{X}$ over $U$. Assume we are given

  1. a complex of $A$-modules $K^\bullet $,

  2. a transformation of functors $T_ x(-) \to H^1(K^\bullet \otimes _ A^\mathbf {L} -)$,

  3. for every deformation situation $(x, A' \to A)$ with kernel $I = \mathop{\mathrm{Ker}}(A' \to A)$ an element $o_ x(A') \in H^2(K^\bullet \otimes _ A^\mathbf {L} I)$

satisfying the following (minimal) conditions

  1. the transformation $T_ x(-) \to H^1(K^\bullet \otimes _ A^\mathbf {L} -)$ is an isomorphism,

  2. given a morphism $(x, A'' \to A) \to (x, A' \to A)$ of deformation situations the element $o_ x(A')$ maps to the element $o_ x(A'')$ via the map $H^2(K^\bullet \otimes _ A^\mathbf {L} I) \to H^2(K^\bullet \otimes _ A^\mathbf {L} I')$ where $I' = \mathop{\mathrm{Ker}}(A'' \to A)$, and

  3. $x$ lifts to an object over $\mathop{\mathrm{Spec}}(A')$ if and only if $o_ x(A') = 0$.

It is possible to incorporate infinitesimal automorphisms as well, but we refrain from doing so in order to get the sharpest possible result.

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