Lemma 93.12.2. Consider a commutative diagram of Noetherian rings
\[ \xymatrix{ A' \ar[d] \ar[r] & P' \ar[d] \ar[r] & Q' \ar[d] \\ A \ar[r] & P \ar[r] & Q } \]
with cartesian squares, with flat horizontal arrows, and with surjective vertial arrows whose kernels are nilpotent. Let $J' \subset P'$ be an ideal such that $P'/J' = Q'/J'Q'$. Let $M$ be an $A$-flat $P$-module. Assume for all $g \in J'$ there exists an $A'$-flat $(P')_ g$-module lifting $M_ g$. Then the following are equivalent
$M$ has an $A'$-flat lift to a $P'$-module, and
$M \otimes _ P Q$ has an $A'$-flat lift to a $Q'$-module.
Proof.
Let $I = \mathop{\mathrm{Ker}}(A' \to A)$. By induction on the integer $n > 1$ such that $I^ n = 0$ we reduce to the case where $I$ is an ideal of square zero; details omitted. We translate the condition of liftability of $M$ into the problem of finding an object of $D^-(P')$ as in Lemma 93.12.1. The obstruction to doing this is the element
\[ \omega (M) \in \text{Ext}^2_ P(M, M \otimes _ P^\mathbf {L} IP) = \text{Ext}^2_ P(M, M \otimes _ P IP) \]
constructed in Deformation Theory, Lemma 91.15.1. The equality in the displayed formula holds as $M \otimes _ P^\mathbf {L} IP = M \otimes _ P IP$ since $M$ and $P$ are $A$-flat1. The obstruction for lifting $M \otimes _ P Q$ is similarly the element
\[ \omega (M \otimes _ P Q) \in \text{Ext}^2_ Q(M \otimes _ P Q, (M \otimes _ P Q) \otimes _ Q IQ) \]
which is the image of $\omega (M)$ by the functoriality of the construction $\omega (-)$ of Deformation Theory, Lemma 91.15.1. By More on Algebra, Lemma 15.99.2 we have
\[ \text{Ext}^2_ Q(M \otimes _ P Q, (M \otimes _ P Q) \otimes _ Q IQ) = \text{Ext}^2_ P(M, M \otimes _ P IP) \otimes _ P Q \]
here we use that $P$ is Noetherian and $M$ finite. Our assumption on $P' \to Q'$ guarantees that for an $P$-module $E$ the map $E \to E \otimes _ P Q$ is bijective on $J'$-power torsion, see More on Algebra, Lemma 15.89.3. Thus we conclude that it suffices to show $\omega (M)$ is $J'$-power torsion. In other words, it suffices to show that $\omega (M)$ dies in
\[ \text{Ext}^2_ P(M, M \otimes _ P IP)_ g = \text{Ext}^2_{P_ g}(M_ g, M_ g \otimes _{P_ g} IP_ g) \]
for all $g \in J'$. Howeover, by the compatibility of formation of $\omega (M)$ with base change again, we conclude that this is true as $M_ g$ is assumed to have a lift (of course you have to use the whole string of equivalences again).
$\square$
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