The Stacks project

Definition 98.22.1. Let $S$ be a locally Noetherian base. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. An obstruction theory is given by the following data

1. for every $S$-algebra $A$ such that $\mathop{\mathrm{Spec}}(A) \to S$ maps into an affine open and every object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$ an $A$-linear functor

   $$\mathcal{O}_ x : \text{Mod}_ A \to \text{Mod}_ A$$

   of obstruction modules,

2. for $(x, A)$ as in (1), a ring map $A \to B$, $M \in \text{Mod}_ A$, $N \in \text{Mod}_ B$, and an $A$-linear map $M \to N$ an induced $A$-linear map $\mathcal{O}_ x(M) \to \mathcal{O}_ y(N)$ where $y = x|_{\mathop{\mathrm{Spec}}(B)}$, and

3. for every deformation situation $(x, A' \to A)$ an obstruction element $o_ x(A') \in \mathcal{O}_ x(I)$ where $I = \mathop{\mathrm{Ker}}(A' \to A)$.

These data are subject to the following conditions

1. the functoriality maps turn the obstruction modules into a functor from the category of triples $(x, A, M)$ to sets,

2. for every morphism of deformation situations $(y, B' \to B) \to (x, A' \to A)$ the element $o_ x(A')$ maps to $o_ y(B')$, and

3. we have

   $$\text{Lift}(x, A') \not= \emptyset \Leftrightarrow o_ x(A') = 0$$

   for every deformation situation $(x, A' \to A)$.