题目二：Cotorsion pairs, Gorenstein dimensions and triangle-equivalences

主讲：江苏理工学院数学系胡江胜教授

时间：2017年7月6日10:30-11:30

地点：1教436

报告摘要：Let (A, B) be a complete hereditary cotorsion pair in ModR. Yang and Ding made a general study of B dimensions of complexes. In this paper, we define the notion of Gorenstein B dimensions for complexes by applying the model structure induced by (A, B), which can be used to describe how Gorenstein dimensions of complexes should work for any complete hereditary cotorsion pair. Characterizations of the finiteness of Gorenstein B dimensions for complexes are given. As a consequence, we study relative cohomology groups for complexes with finite Gorenstein B dimensions. Moreover, the relationships between Gorenstein B dimensions and B dimensions for complexes are given. Next we get two triangle-equivalences between the homotopy category of a hereditary abelian model structure, the singularity category of an exact category and the stable category of a Forbenius category. As applications, some necessary and sufficient conditions for the validity of the Finitistic Dimension Conjecture are given. In particular, we show that the Finitistic Dimension Conjecture is ture for an Artin algebra R if and only if the homotopy category H0(M) of the hereditary abelian model structureM= (X, cores b Y<1, G(Y)) is triangle-equivalent to the stable category X \ G(Y) if and only if there is a triangle-equivalence Dsg(X)_=X \ G(Y), where (X;Y) is the cotorsion pair cogenerated by the classs of finitely generated modules with finite projective dimension in ModR and Dsg(X) := Db(X)=Kb(X \ Y) is the singularity category of X. This is a joint work with N.Q. Ding, H.H. Li, J.Q. Wei and X.Y. Yang.

报告人简介：胡江胜教授博士毕业于南京大学数学系，在同调代数方面做出了非常出色的工作，已在《J. Pure Appl. Algebra》、《Algebr. Represent. Theory》等重要SCI杂志上发表论文多篇，并主持（完成）多项国家自然科学基金。