NAK for Ext, Ascent of Module Structures, and the Blindness of Extended Modules

Abstract

This dissertation investigates the interplay between properties of Ext modules and ascent of module structures along ring homomorphisms. First, we consider a flat local ring homomorphism ϕ: (R, [special characters omitted], k) → (S, [special characters omitted]S, k). We show that if M is a finitely generated R-module such that [special characters omitted](S, M) satisfies NAK (e.g. if [special characters omitted](S, M) is finitely generated over S) for i = 1,…, dimR( M), then [special characters omitted](S, M) = 0 for all i ≠ 0 and M has an S-module structure via ϕ. We also provide explicit computations of [special characters omitted](S, M) to indicate how large it can be when M does not have a compatible S-module structure.

Next, we consider the properties of an R-module M that has a compatible S-module structure via the flat local ring homomorphism ϕ. Our results in this direction show that M cannot see the difference between the rings R and S. Specifically, many homological invariants of M are the same when computed over R and over S.

Finally, we investigate these ideas in the non-local setting. We consider a faithfully flat ring homomorphism ϕ: R → S such that for all [special characters omitted] ∈ m-Spec R, the map R/[special characters omitted] → S/[special characters omitted]S is an isomorphism and the induced map ϕ*: Spec( S) → Spec(R) is such that ϕ*(m-Spec( S)) ⊆ m-Spec(R), and show that if M is a finitely generated R-module such that [special characters omitted](S, M) satisfies NAK for i = 1,…,dim R(M), then M has an S-module structure via ϕ, and obtain the same Ext vanishing as in the local case.