Lemma 98.5.11. In Situation 98.5.1 assume that $S$ is a locally Noetherian scheme, $S = B$, and $f : X \to B$ is flat. Let $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition 97.13.1).

First proof. This proof is based on the criterion of Artin's Axioms, Lemma 97.24.4. 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 $S$. 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 $x$.

According to Deformation Theory, Lemma 90.12.1 we have an isomorphism of functors

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

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

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

This uses that for any $A' \to A$ as above the base change $X_{A'} = \mathop{\mathrm{Spec}}(A') \times _ B X$ is flat over $A'$. Moreover, the construction of the obstruction class is functorial in the surjection $A' \to A$ (for fixed $A$) by Deformation Theory, Lemma 90.12.3. Apply Derived Categories of Spaces, Lemma 74.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms

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

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 Deformation Theory, Lemma 90.12.5. Thus Artin's Axioms, Lemma 97.24.4 does indeed apply and the lemma is proved. $\square$

Second proof. This proof is based on Artin's Axioms, Lemma 97.22.2. Conditions (1), (2), and (3) of that lemma correspond to Lemmas 98.5.3, 98.5.7, and 98.5.6.

We have constructed an obstruction theory in the chapter on deformation theory. Namely, given an $S$-algebra $A$ and an object $x$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $\mathop{\mathrm{Spec}}(A)$ given by $\mathcal{F}$ on $X_ A$ we set $\mathcal{O}_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ and if $A' \to A$ is a surjection with kernel $I$, then as obstruction element we take the element

$o_ x(A') = o(\mathcal{F}, \mathcal{F} \otimes _ A I, 1) \in \mathcal{O}_ x(I) = \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A I)$

of Deformation Theory, Lemma 90.12.1. All properties of an obstruction theory as defined in Artin's Axioms, Definition 97.22.1 follow from this lemma except for functoriality of obstruction classes as formulated in condition (ii) of the definition. But as stated in the footnote to assumption (4) of Artin's Axioms, Lemma 97.22.2 it suffices to check functoriality of obstruction classes for a fixed $A$ which follows from Deformation Theory, Lemma 90.12.3. Deformation Theory, Lemma 90.12.1 also tells us that $T_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for any $A$-module $M$.

To finish the proof it suffices to show that $T_ x(\prod M_ n) = \prod T_ x(M_ n)$ and $\mathcal{O}_ x(\prod M_ n) = \prod \mathcal{O}_ x(M)$. Apply Derived Categories of Spaces, Lemma 74.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms

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

for $i = 1, 2$. A straightforward argument shows that

$H^ i(K \otimes _ A^\mathbf {L} \prod M_ n) = \prod H^ i(K \otimes _ A^\mathbf {L} M_ n)$

whenever $K$ is a pseudo-coherent object of $D(A)$. In fact, this property (for all $i$) characterizes pseudo-coherent complexes, see More on Algebra, Lemma 15.65.5. $\square$

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