Let B —> A be a surjection of rings. Let M, X, Y be a A-modules.

If φ : X —> Y is a B-module map, then φ is an A-module map. We obtain X ⊗_A M —> Y ⊗_A M and X ⊗^L_A M —> Y ⊗^L_A M by functoriality.

Let ξ ∈ Ext^1_B(X, Y). I claim there is an element in Ext^2_A(X ⊗^L_A M, Y ⊗^L_A M) associated to ξ. Here is my construction. Choose a complex of free A-modules F_* resolving M. Choose a sequence (not a complex) of free B-modules F’_* such that F’_* ⊗_B A is isomorphic to F_*. Let 0 —> Y —> E —> X —> 0 be the short exact sequence representing ξ. Then consider the composition

F’_{n + 2} ⊗_B E —> F’_{n + 1} ⊗_B E —> F’_n ⊗_B E

Clearly this factors through a map

F_{n + 2} ⊗_A X = F’_{n + 2} ⊗_B X —> F’_n ⊗_B Y = F_n ⊗_A Y

The collection of these map gives X ⊗^L_A M —> Y ⊗^L_A M[2] as desired.

Questions:
(a) Does this actually work?
(b) What is a “better” description of this construction?
(c) Is there a similar map Ext^2_B(X, Y) —> Ext^3_A(X ⊗^L_A M, Y ⊗^L_A M)?
(d) If you have a reference, could you please let us know?.

Thanks!