Questions about a construction

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 ⊗LA M —> Y ⊗LA M by functoriality.

Let ξ ∈ Ext^1_B(X, Y). I claim there is an element in Ext^2_A(X ⊗LA M, Y ⊗LA 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 ⊗LA M —> Y ⊗LA M[2] as desired.

(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 ⊗LA M, Y ⊗LA M)?
(d) If you have a reference, could you please let us know?.


2 thoughts on “Questions about a construction

  1. 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.

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