Lemma 98.23.3. 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, and
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).
Let $u_0 \in U$ be a finite type point with residue field $k = \kappa (u_0)$. Consider the following statements
$x$ is versal at $u_0$, and
$\xi : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$ induces a surjection $H^{-1}(E \otimes _ A^{\mathbf{L}} k) \to H^{-1}(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^{\mathbf{L}} k)$ and an injection $H^0(E \otimes _ A^{\mathbf{L}} k) \to H^0(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^{\mathbf{L}} k)$.
Then we always have (2) $\Rightarrow $ (1) and we have (1) $\Rightarrow $ (2) if $u_0$ is a closed point.
Proof.
Let $\mathfrak p = \mathop{\mathrm{Ker}}(A \to k)$ be the prime corresponding to $u_0$.
Assume that $x$ versal at $u_0$ and that $u_0$ is a closed point of $U$. If $H^{-1}(\xi \otimes _ A^{\mathbf{L}} k)$ is not surjective, then let $A' \to A$ be an extension with kernel $I$ as in Lemma 98.23.2. Because $u_0$ is a closed point, we see that $I$ is a finite $A$-module, hence that $A'$ is a finite type $\Lambda $-algebra (this fails if $u_0$ is not closed). In particular $A'$ is Noetherian. By property (c) for $A'$ and (i) for $\xi $ we see that $x$ lifts to an object $x'$ over $A'$. Let $\mathfrak p' \subset A'$ be kernel of the surjective map to $k$. By Artin-Rees (Algebra, Lemma 10.51.2) there exists an $n > 1$ such that $(\mathfrak p')^ n \cap I = 0$. Then we see that
\[ B' = A'/(\mathfrak p')^ n \longrightarrow A/\mathfrak p^ n = B \]
is a small, essential extension of local Artinian rings, see Formal Deformation Theory, Lemma 90.3.12. On the other hand, as $x$ is versal at $u_0$ and as $x'|_{\mathop{\mathrm{Spec}}(B')}$ is a lift of $x|_{\mathop{\mathrm{Spec}}(B)}$, there exists an integer $m \geq n$ and a map $q : A/\mathfrak p^ m \to B'$ such that the composition $A/\mathfrak p^ m \to B' \to B$ is the quotient map. Since the maximal ideal of $B'$ has $n$th power equal to zero, this $q$ factors through $B$ which contradicts the fact that $B' \to B$ is an essential surjection. This contradiction shows that $H^{-1}(\xi \otimes _ A^{\mathbf{L}} k)$ is surjective.
Assume that $x$ versal at $u_0$. By Lemma 98.23.1 the map $H^0(\xi \otimes _ A^{\mathbf{L}} k)$ is dual to the map $\mathop{\mathrm{Ext}}\nolimits ^0_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, k) \to \text{Ext}^0_ A(E, k)$. Note that
\[ \mathop{\mathrm{Ext}}\nolimits ^0_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, k) = \text{Der}_\Lambda (A, k) \quad \text{and}\quad T_ x(k) = \mathop{\mathrm{Ext}}\nolimits ^0_ A(E, k) \]
Condition (ii) assures us the map $\mathop{\mathrm{Ext}}\nolimits ^0_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, k) \to \text{Ext}^0_ A(E, k)$ sends a tangent vector $\theta $ to $U$ at $u_0$ to the corresponding infinitesimal deformation of $x_0$, see Remark 98.21.8. Hence if $x$ is versal, then this map is surjective, see Formal Deformation Theory, Lemma 90.13.2. Hence $H^0(\xi \otimes _ A^{\mathbf{L}} k)$ is injective. This finishes the proof of (1) $\Rightarrow $ (2) in case $u_0$ is a closed point.
For the rest of the proof assume $H^{-1}(E \otimes _ A^\mathbf {L} k) \to H^{-1}(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^\mathbf {L} k)$ is surjective and $H^0(E \otimes _ A^\mathbf {L} k) \to H^0(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^\mathbf {L} k)$ injective. Set $R = A_\mathfrak p^\wedge $ and let $\eta $ be the formal object over $R$ associated to $x|_{\mathop{\mathrm{Spec}}(R)}$. The map $d\underline{\eta }$ on tangent spaces is surjective because it is identified with the dual of the injective map $H^0(E \otimes _ A^{\mathbf{L}} k) \to H^0(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^{\mathbf{L}} k)$ (see previous paragraph). According to Formal Deformation Theory, Lemma 90.13.2 it suffices to prove the following: Let $C' \to C$ be a small extension of finite type Artinian local $\Lambda $-algebras with residue field $k$. Let $R \to C$ be a $\Lambda $-algebra map compatible with identifications of residue fields. Let $y = x|_{\mathop{\mathrm{Spec}}(C)}$ and let $y'$ be a lift of $y$ to $C'$. To show: we can lift the $\Lambda $-algebra map $R \to C$ to $R \to C'$.
Observe that it suffices to lift the $\Lambda $-algebra map $A \to C$. Let $I = \mathop{\mathrm{Ker}}(C' \to C)$. Note that $I$ is a $1$-dimensional $k$-vector space. The obstruction $ob$ to lifting $A \to C$ is an element of $\mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, I)$, see Example 98.22.4. By Lemma 98.23.1 and our assumption the map $\xi $ induces an injection
\[ \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, I) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_ A(E, I) \]
By the construction of $ob$ and (i) the image of $ob$ in $\mathop{\mathrm{Ext}}\nolimits ^1_ A(E, I)$ is the obstruction to lifting $x$ to $A \times _ C C'$. By (RS*) the fact that $y/C$ lifts to $y'/C'$ implies that $x$ lifts to $A \times _ C C'$. Hence $ob = 0$ and we are done.
$\square$
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