Gorenstein投射覆盖、预包络与相对上同调

发布时间：2018-01-01 01:05

本文关键词：Gorenstein投射覆盖、预包络与相对上同调　出处：《南京大学》2016年博士论文　论文类型：学位论文

【摘要】：(预)覆盖和(预)包络模类在相对同调代数中发挥着重要作用。它们可视为内射和投射模类的推广,基于这些不同的模类,我们能够对任意模构造不同于内射和投射的恰当分解。通过证明这些模类的广泛存在性,在2000年左右,Enochs和Jenda等人给肇始于Eilenberg和Moore的相对同调代数理论奠定了更稳固的基础。应用这些恰当分解,我们能得出更为一般的维数、导出函子、(上)同调模和平衡函子等概念。对它们的研究丰富拓展了经典的同调理论。全文总共分为四章。第一章叙述了研究背景和主要结论,并给出了后文中用到的记号说明。第二章探讨了Gorenstein投射模何时成为覆盖类和预包络类的条件。我们证明了这和经典环论的情形是相似的：若Gorenstein投射模和Gorenstein平坦模等价,则Gorenstein投射模类构成一个覆盖类。我们还给出了Gorenstein投射模类构成一个预包络类时环的完全刻画。一些特殊条件下的Gorenstein模的覆盖和包络条件也被给出。在第三章中,我们比较了基于不同恰当分解的相对上同调理论,并给出这些差异消失的一些充分条件,得到了这些条件和广义Tate上同调模的关系。我们证明了,在一定条件下,这两种上同调函子的平衡性质是等价的。作为应用,我们重新得到了早先的一些已知结论。最后我们探讨了和Auslander类以及Bass类相关的导出函子。在第四章中,对于给定的一个特殊模类X,我们探讨了χ-Gorenstein投射模的维数和覆盖性质。
[Abstract]:(pre) coverage and (pre) envelope modes play an important role in relative homological algebra. They are regarded as generalized injective and projective classes, these different classes of modules based on, we can appropriate decomposition of arbitrary norm structure is different from injective and projective. By proving the existence of these broad class of modules. In 2000, the relative homological algebra Enochs and Jenda et al, Eilenberg and Moore began to lay a more solid foundation. The application of these proper decomposition, we can get a more general dimension, derived functor, (a) modules and balanced functor concept. Their research enriches and develops the theory of classical coherence. This text is divided into four chapters. The first chapter introduces the research background and main conclusions, and gives the mark used. The second chapter discusses the Gorenstein projective modules will be covered and pre envelope of . we show that the classical ring theory situation is similar: if the Gorenstein projective modules and Gorenstein flat modules, projective equivalence, Gorenstein class form a cover class. We give a complete characterization of Gorenstein projective preenvelope class a ring. Gorenstein mode under special conditions of the cover and envelope conditions are also given. In the third chapter, we compare the different appropriate decomposition of relative cohomology based on the theory, some sufficient conditions are given and these differences disappeared, these conditions were obtained and generalized Tate cohomology relations. We prove that, under certain conditions, these two kinds of balance the nature of the homology functor is proved. As applications, we get some known conclusions earlier. At last we discussed and Auslander and Bass related to the derived functor. In the fourth chapter, for As a special class of X, we discuss the dimension and the covering properties of the X -Gorenstein projective modules.

【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O189.22

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