An Auslander-Buchsbaum identity for semidualizing modules

Foxby equivalence over associative rings

We extend the definition of a semidualizing module to 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 in the commutative, noetherian setting.

Adic Foxby Classes

We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to $F\otimes^{\mathbf{L}}_R-$ where $F$ is an $R$-complex finite flat dimension, including special conver…

On reduced G-perfection and horizontal linkage relative to a semidualizing module

In their investigation of horizontal linkage of modules of finite Gorenstein dimension over a commutative, Noetherian, semiperfect (e.g., local) ring, Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making use of Bass' concept of reduced grade. A few years later, the same authors extended this class by considering the relative property of reduced G$_C$-perfection, where $C$ is a semidualizing module, and studied linkage even further. In the present paper, we contribute to their theory and also generalize results of Auslander and Bridger as well as of Martsinkovsky and Str…

关于准射影维数的Auslander深度公式的一些注记

Remarks on Auslander's depth formula for quasi-projective dimension

本文研究了在M具有有限准射影维数、q有限且Tor_q^R(M,N)的深度不超过1的情况下，Auslander深度公式仍成立的问题，并展示了该结果在射影维数依赖公式改进中的应用。

Gorenstein projective dimension with respect to a semidualizing module

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 Jørgensen'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 (noethe…

Auslander-Reiten猜想、Hom的有限C-内射维数与Ext的消失

The Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\operatorname{Hom}$, and vanishing of $\operatorname{Ext}$

本文研究Noether局部环上Ext消失与Hom的有限C-内射维数，给出Auslander-Reiten猜想的肯定回答，并得到半对偶模成为典范模的判别准则。

Classifying Resolving Subcategories

We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving subcategories containing maximal Cohen-Macaulay modules are in bijection with grade consistent functions and thus are the precisely the dominant resolving subcategories.

基于 $G_{C}$-维数的对偶复形的若干刻画

Some characterizations of dualizing complexes in terms of $G_{C}$-dimension

本文研究了局部环上对偶复形的刻画问题，基于 $G_{C}$-维数，证明了若干关于半对偶化复形成为对偶复形或同构于环本身的充要条件。

The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I.Barany and its generalizations by G.Kalai and R.Meshulam translates to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular gives a syzygetic version of Hellys theorem.