Yu, Bin Zhu, Xiaosheng Zhou, Yanbo
Published in
Algebras and Representation Theory

We compare relative cohomology theories arising from using different proper resolutions of modules. Criteria for the vanishing of such distinctions are given in certain cases, and we show that this is related to the generalized Tate cohomology theory. We also demonstrate that the two balance properties admitted by the two different cohomology theor...

Li, Huanhuan Xu, Yunge
Published in
Frontiers of Mathematics in China

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are...

Selvaraj, C. Saravanan, R.
Published in
Bollettino dell'Unione Matematica Italiana

In this paper, we introduce and study the notions of Gorenstein n-FP-injective and Gorenstein n-flat complexes, which are special cases of Gorenstein FP-injective and Gorenstein flat complexes respectively. We prove that over a left n-coherent ring R the class of all Gorenstein n-FP-injective (resp., Gorenstein n-flat) complexes is injectively (res...

Selvaraj, Chelliah Saravanan, Raja
Published in
Vietnam Journal of Mathematics

In this paper, we extend the notion of strongly Gorenstein flat (SG-flat) modules to SG-flat complexes which are special types of Gorenstein flat complexes, and we discuss the relationship between the SG-injective and SG-flat complexes. Also, we generalize the SG-flat complexes to n-SG-flat complexes and characterize such complexes.

Mao, Lixin
Published in
Frontiers of Mathematics in China

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.

Ma, Jun Fan, Feifei Wang, Xiangjun
Published in
Chinese Annals of Mathematics, Series B

This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using Čech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.

Hu, Jiangsheng Geng, Yuxian Jiang, Qinghua
Published in
Frontiers of Mathematics in China

We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with fini...

Zhang, Chao Han, Yang
Published in
Algebras and Representation Theory

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of a complex (an algebra) are introduced. Cohomological range leads to the concepts of derived bounded algebra and strongly derived unbounded algebra naturally. The first and second Brauer-Thrall type theorems for the bounded deriv...

Huisgen-Zimmermann, Birge
Published in
Algebras and Representation Theory

We show that even within the class of special biserial algebras, one of the most thoroughly studied classes of representation-tame finite dimensional algebras, the (left) big finitistic dimension may be strictly larger than the little. In fact, we find that the discrepancies Fin dim Λ – fin dim Λ fail to be bounded as Λ traces the special biserial ...

Goto, Shiro Takahashi, Ryo
Published in
Algebras and Representation Theory

We refine a well-known theorem of Auslander and Reiten about the extension closedness of nth syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre’s condition (Sn) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of A...