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!

Isn’t this the Kodaira-Spencer map?

This is very similar to the construction of Eisenbud in his paper “Homological algebra on a complete intersection, with an application to group representations”. He mentions a thesis by Mehta which may contain this construction exactly, but I haven’t found it.