NAK for Ext and Ascent of module structures

TL;DR

This paper explores how Ext modules influence the ascent of module structures along flat local ring homomorphisms, establishing conditions under which modules acquire compatible structures and analyzing the size of Ext^1(S,M).

Contribution

It provides new criteria linking Ext module properties to the ascent of module structures, including explicit computations of Ext^1(S,M).

Findings

Ext^i(S,M)=0 for all i≠0 under NAK conditions

Modules with Ext^i(S,M) satisfying NAK admit compatible S-module structures

Explicit bounds on the size of Ext^1(S,M) when no compatible structure exists

Abstract

We investigate the interplay between properties of Ext modules and ascent of module structures along local ring homomorphisms. Specifically, let f: (R,m,k) -> (S,mS,k) be a flat local ring homomorphism. We show that if M is a finitely generated R-module such that Ext^i(S,M) satisfies NAK (e.g. if Ext^i(S,M) is finitely generated over S) for i=1,...,dim_R(M), then Ext^i(S,M)=0 for all i\neq 0 and M has an S-module structure that is compatible with its R-module structure via f. We provide explicit computations of Ext^1(S,M) to indicate how large it can be when M does not have a compatible S-module structure.