The Stacks project

Lemma 114.20.4. In Quot, Situation 98.5.1 assume that $S$ is a locally Noetherian scheme and $S = B$. Let $\mathcal{X} = \textit{Coh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition 97.13.1).

Proof (sketch). Let $U \to S$ be of finite type morphism of schemes, $x$ an object of $\mathcal{X}$ over $U$ and $u_0 \in U$ a finite type point such that $x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$ is a closed point (Morphisms, Lemma 29.16.1) and $U = \mathop{\mathrm{Spec}}(A)$ with $U \to S$ mapping into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$ of $S$. We will use Artin's Axioms, Lemma 97.24.4 to prove the lemma. Let $\mathcal{F}$ be the coherent module on $X_ A = \mathop{\mathrm{Spec}}(A) \times _ S X$ flat over $A$ corresponding to the given object $x$.

Choose $E(\mathcal{F})$ and $e_\mathcal {F}$ as in Remark 114.20.1. The description of the cohomology sheaves of $E(\mathcal{F})$ shows that

\[ \mathop{\mathrm{Ext}}\nolimits ^1(E(\mathcal{F}), \mathcal{F} \otimes _ A M) = \mathop{\mathrm{Ext}}\nolimits ^1(\mathcal{F}, \mathcal{F} \otimes _ A M) \]

for any $A$-module $M$. Using this and using Deformation Theory, Lemma 90.11.7 we have an isomorphism of functors

\[ T_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A M) \]

By Lemma 114.20.3 given any surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have an obstruction class

\[ \xi _{A'} \in \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A I) \]

Apply Derived Categories of Spaces, Lemma 74.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We omit the verification that $E(\mathcal{F})$ is in $D^-_{\textit{Coh}}$; hint: use Cotangent, Lemma 91.5.4. We find a perfect object $K \in D(A)$ and functorial isomorphisms

\[ H^ i(K \otimes _ A^\mathbf {L} M) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A M) \]

for $i \leq m$ compatible with boundary maps. This object $K$, together with the displayed identifications above gives us a datum as in Artin's Axioms, Situation 97.24.2. Finally, condition (iv) of Artin's Axioms, Lemma 97.24.3 holds by a variant of Deformation Theory, Lemma 90.12.5 whose formulation and proof we omit. Thus Artin's Axioms, Lemma 97.24.4 applies and the lemma is proved. $\square$


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