Homological algebra modulo exact zero-divisors

Auslander class, $\g_C$ and $C$--projective modules modulo exact zero-divisors

For a semidualizing module $C$ over a ring $R$, we study the following classes modulo exact zero divisors: $\g_C$--projectives, $\mathcal G_C$; the Auslander class $\mathcal A_C$; the Bass class $\mathcal B_C$; $\mathcal{P}_C$--projective; $ {\mathcal F}_C$--projective; and ${\mathcal I}_C$--injective dimensions.

DG 模结构与模精确零因子的极小自由分解

DG module structures and minimal free resolutions modulo an exact zero-divisor

该研究针对满足fM=0的有限Q-模，证明其在Q上的极小自由分解具有特定微分分次代数上的DG模结构，并借助该结构刻画了模精确零因子时的极小自由分解。

关于模 $2^n$ 的高斯整数环的零因子图

On zero-divisor graph of the ring of Gaussian integers modulo $2^n$

本文研究了模 $2^n$ 的高斯整数环的零因子图结构，通过因子的相伴类确定了图的大小、色数、团数、独立数和匹配数，并求得了相关拓扑指标。

Hom的有限同调维数、Ext的消失及其对因子类群的应用

Finite homological dimension of Hom, vanishing of Ext, and applications to divisor class group

本文通过有限个Ext模的消失和Hom模的有限同调维数给出了检测模自由性的充分条件，推广了Ghosh-Takahashi的结果，并应用于Auslander-Reiten猜想、I-Ulrich模的新刻画以及n-半对偶化模的两个问题。

Almost Buchsbaumness of some rings arising from complexes with isolated singularities

We study properties of the Stanley-Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley-Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of non-homologically isolated singularities, providing many examples in which the Stanley-Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.

Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread

Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tool…