NAK for Ext and ascent of module structures

Abstract:

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