Sean Sather-Wagstaff

Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a commutative ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective R-modules such that the complexes HomR(G, H) and HomR(H, G) are exact for each Gorenstein projective R-module H, the module Coker() is Gorenstein projective. The proof of this result hinges upon our analysis of Gorenstein subcategories of abelian categories.

REFLEXIVITY AND RING HOMOMORPHISMS OF FINITE FLAT DIMENSION

In this article we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. Most of these results are new even when the ring homomorphism is local. The main tool for these analyses is a nonlocal version of the amplitude inequality of Iversen, Foxby, and Iyengar. We provide numerous examples demonstrating the need for certain hypotheses and the strictness of many inequalities.

Relations between semidualizing complexes

We study the following question: Given two semidualizing com- plexes B and C over a commutative noetherian ring R, does the vanishing of Ext n(B, C) for n ≫ 0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext n(B, C) = 0 for all n ≫ 0. We introduce and investigate an equivalence relation ≈ on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characteriza- tions of this relation, each of which includes the condition Ext n(B, C) = 0 for all n ≫ 0. Finally, we answer our original question in some special cases.

Support and adic finiteness for complexes

ABSTRACT Let X be a chain complex over a commutative noetherian ring R, that is, an object in the derived category 𝒟(R). We investigate the small support and co-support of X, introduced by Foxby and Benson, Iyengar, and Krause. We show that the derived functors and RHomR(M,−) can detect isomorphisms in 𝒟(R) between complexes with restrictions on their supports or co-supports. In particular, the derived local (co)homology functors RΓ𝔞(−) and LΛ𝔞(−) with respect to an ideal 𝔞⊊R have the same ability. Furthermore, we give reprove some results of Benson, Iyengar, and Krause in our setting, with more direct proofs. Also, we include some computations of co-supports, since this construction is still quite mysterious. Lastly, we investigate “𝔞-adically finite” R-complexes, that is, the X∈𝒟b(R) that are 𝔞-cofinite à la Hartshorne. For instance, we characterize these complexes in terms of a finiteness condition on LΛ𝔞(X).

A Cohen-Macaulay algebra has only finitely many semidualizing modules

We prove the result stated in the title, which answers the equicharacteristic case of a question of Vasconcelos.

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let R be a commutative noetherian ring and α an ideal in the Jacobson radical of R. Let Ra be the α-adic completion of R. If M is a finitely generated R-module such that Ext i R (M) = 0 for all i ≠ 0, then M is α-adically complete.

A Somewhat Gentle Introduction to Differential Graded Commutative Algebra

Differential graded (DG) commutative algebra provides powerful techniques for proving theorems about modules over commutative rings. These notes are a somewhat colloquial introduction to these techniques. In order to provide some motivation for commutative algebraists who are wondering about the benefits of learning and using these techniques, we present them in the context of a recent result of Nasseh and Sather-Wagstaff. These notes were used for the course “Differential Graded Commutative Algebra” that was part of the Workshop on Connections Between Algebra and Geometry at the University of Regina, May 29–June 1, 2012.

Testing for the Gorenstein property

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring

Adic Finiteness: Bounding Homology and Applications

(R,\mathfrak m)