The Stacks project

Remark 97.21.5. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $A$ be an $S$-algebra. There is a notion of a short exact sequence

\[ (x, A_1' \to A) \to (x, A_2' \to A) \to (x, A_3' \to A) \]

of deformation situations: we ask the corresponding maps between the kernels $I_ i = \mathop{\mathrm{Ker}}(A_ i' \to A)$ give a short exact sequence

\[ 0 \to I_3 \to I_2 \to I_1 \to 0 \]

of $A$-modules. Note that in this case the map $A_3' \to A_1'$ factors through $A$, hence there is a canonical isomorphism $A_1' = A[I_1]$.


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