The Stacks project

Example 97.22.3. Let $S = \mathop{\mathrm{Spec}}(\Lambda )$ for some Noetherian ring $\Lambda $. Let $W \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ W$-module flat over $S$. Consider the functor

\[ F : (\mathit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T/S \longrightarrow H^0(W_ T, \mathcal{F}_ T) \]

where $W_ T = T \times _ S W$ is the base change and $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ to $W_ T$. If $T = \mathop{\mathrm{Spec}}(A)$ we will write $W_ T = W_ A$, etc. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be the category fibred in groupoids associated to $F$. Then $\mathcal{X}$ has an obstruction theory. Namely,

  1. given $A$ over $\Lambda $ and $x \in H^0(W_ A, \mathcal{F}_ A)$ we set $\mathcal{O}_ x(M) = H^1(W_ A, \mathcal{F}_ A \otimes _ A M)$,

  2. given a deformation situation $(x, A' \to A)$ we let $o_ x(A') \in \mathcal{O}_ x(A)$ be the image of $x$ under the boundary map

    \[ H^0(W_ A, \mathcal{F}_ A) \longrightarrow H^1(W_ A, \mathcal{F}_ A \otimes _ A I) \]

    coming from the short exact sequence of modules

    \[ 0 \to \mathcal{F}_ A \otimes _ A I \to \mathcal{F}_{A'} \to \mathcal{F}_ A \to 0. \]

We have omitted some details, in particular the construction of the short exact sequence above (it uses that $W_ A$ and $W_{A'}$ have the same underlying topological space) and the explanation for why flatness of $\mathcal{F}$ over $S$ implies that the sequence above is short exact.

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