Lemma 98.23.4 (07YN)—The Stacks project

Lemma 98.23.4. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ satisfying (RS*). 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 $U$. Let $\xi : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$ be a morphism of $D^{-}(A)$. Assume

for every deformation situation $(x, A' \to A)$ we have: $x$ lifts to $\mathop{\mathrm{Spec}}(A')$ if and only if $E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'}$ is zero,
there is an isomorphism of functors $T_ x(-) \to \mathop{\mathrm{Ext}}\nolimits ^0_ A(E, -)$ such that $E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \Omega ^1_{A/\Lambda }$ corresponds to the canonical element (see Remark 98.21.8),
the cohomology groups of $E$ are finite $A$-modules.

If $x$ is versal at a closed point $u_0 \in U$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.

Proof. Let $C$ be the cone of $\xi $ so that we have a distinguished triangle

\[ E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to C \to E[1] \]

in $D^{-}(A)$. By Lemma 98.23.3 the assumption that $x$ is versal at $u_0$ implies that $H^{-1}(C \otimes ^\mathbf {L} k) = 0$. By More on Algebra, Lemma 15.76.4 there exists an $f \in A$ not contained in the prime corresponding to $u_0$ such that $H^{-1}(C \otimes ^\mathbf {L}_ A M) = 0$ for any $A_ f$-module $M$. Using Lemma 98.23.3 again we see that we have versality for all finite type points of the open $D(f) \subset U$. $\square$

The Stacks project

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