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 φ : ( R , m , k ) → ( S , m S , k ) \varphi \colon (R,\mathfrak {m},k)\to (S,\mathfrak {m} S,k) be a flat local ring homomorphism. We show that if M M is a finitely generated R R -module such that Ext R i ⁡ ( S , M ) \operatorname {Ext}_{R}^{i}(S,M) satisfies NAK (e.g. if Ext R i ⁡ ( S , M ) \operatorname {Ext}_{R}^{i}(S,M) is finitely generated over S S ) for i = 1 , … , dim R ⁡ ( M ) i=1,\ldots ,\dim _{R}(M) , then Ext R i ⁡ ( S , M ) = 0 \operatorname {Ext}_{R}^{i}(S,M)=0 for all i ≠ 0 i\neq 0 and M M has an S S -module structure that is compatible with its R R -module structure via φ \varphi . We provide explicit computations of Ext R i ⁡ ( S , M ) \operatorname {Ext}_{R}^{i}(S,M) to indicate how large it can be when M M does not have a compatible S S -module structure.