Functions of One Complex Variable

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Review of Principles to Actions: Ensuring Mathematical Success for All

Commutative rings I

An introduction to noncommutative noetherian rings

【The purpose of this study is to introduce the Common Core State Standards (CCSS) for mathematics which is released on June 2, 2010 in the U.S.A. The common core state standards are aligned with college and work expectations, and include rigorous content and application of knowledge through high-order skills. The most distinguishable differences between the CCSS standards and the NCTM standards are that the CCSS standards considers the mathematical modeling as one of the mathematical content domains such as algebra, geometry, ets, and the standards are designed by working together with school leaders of all states.】

미국 Common Core State Standards for Mathematics 소개

We extend the definition of a semidualizing module to general associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule $C$. We then study the classes of $C$-flats, $C$-projectives, and $C$-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new even in the commutative, noetherian setting.

/pdf/foxby-equivalence-over-associative-rings-3izoztbi4k.pdf

Foxby equivalence over associative rings

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.

/pdf/stability-of-gorenstein-categories-2662bom8ig.pdf

Stability of Gorenstein categories

We investigate the notion of the $C$-projective dimension of a module, where $C$ is a semidualizing module. When $C=R$, this recovers the standard projective dimension. We show that three natural definitions of finite $C$-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite $C$-projective dimension. In parallel, we develop the dual theory for injective dimension and $C$-injective dimension.

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Homological aspects of semidualizing modules

We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noetherian) ring, we provide necessary and sufficient conditions for a $G_C$-approximation to be minimal.

Diana White

Papers

Foxby equivalence over associative rings

Stability of Gorenstein categories

Homological aspects of semidualizing modules

Gorenstein projective dimension with respect to a semidualizing module

Gorenstein projective dimension with respect to a semidualizing module