Covering Classes And Direct Limits

Relative homological algebra has gained momentum over the past two decades by the discovery of a number of new precovering and preenvelopping classes by Enochs et al.[40,51].It is known that if a precovering class F is closed under direct limits,then it is covering class[40,Corollary 5.2.7],[51,Theorem 5.31].As a corollary,we get that if a cotorsion pair C =(A,B)is complete and closed,then C is perfect[51,Corollary 5.32].In the late 1990s,Enochs posed a question:is any covering class F of left R-modules is closed under direct limits?(see[51,Open problems 5.4],[2,§5]).In particular,is any perfect cotorsion pair closed?These problems are still open in general.In this thesis,we prove that a ring R is left coherent if and only if the class Abs of absolutely pure left R-modules is a covering class,answering the open ques-tion in[63,Remark 6.8](or[66,Remark 2.8]).It is a particular instance of Enochs’question.We also give some characterizations of tree modules,and make use of these characterizations to give an equivalent condition of Enochs,question.The thesis consists of four chapters.In Chapter 1,we give the research background and list the main results of this paper.In Chapter 2,some known results and terminologies on set theory and relative homology algebra are introduced.In Chapter 3,we prove that if A is a covering class of modules closed under pure submodules,extensions and direct products,then A is closed under direct limits.As a corollary,we get that a ring R is left coherent if and only if Abs is a covering class.It is also shown that(Abs,Abs⊥)is a cotorsion pair if and only if R is left coherent and absolutely pure as a left R-module.In Chapter 4,we give a brief construction of a tree module,and then some char-acterizations of tree modules are given.We make use of these characterizations to give an equivalent condition of the Enochs’ question that whether any covering class of left R-modules is closed under direct limits.