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 vertical 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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