Coherence Of Complexes Of Modules

The importance of coherent rings comes from the fact that such rings possess many interesting properties from viewpoints of ring theory,module theory as well as homological algebra.The notion of coherent modules is a natural generalization of coherent rings.Note that many theories on modules can be extended to complexes of modules.Inspired by previous results on related topics,we generalize the notion of coherence(resp.,pseudo-coherence andτ-coherence)of modules to complexes of modules.In Chapter 2,(pseudo-)coherent complexes of modules are introduced and characterized.Some known results on(pseudo-)coherent modules are extended to the settings of complexes.For instance,it is proved that a ring R is right noetherian if and only if every complex C of right R-modules is pseudo-coherent if and only if every finitely generated complexes of right R-modules is coherent.We also investigate the endocoherence of a complex C=(Ci,di)i∈Z of right R-modules.Let S be the endomorphism ring of C.There is a natural action of S on each term Ci so that C=(Ci,di)i∈Z becomes a complex of left S-modules.It is proved that S is a left coherent ring if and only if every complex A E presC has an addC-preenvelope.Moreover,if C is finitely presented then it is coherent as a complex of left S-modules if and only if every finitely presented complex of right R-modules has an addC-preenvelope.In Chapter 3 the notion of τ-coherent modules is generalized to complexes,where τ=(T,F)is a torsion pair in the category of right R-modules.To this end,we first introduce τfinitely generated(resp.,τ-finitely presented and τ-C-flat)complexes.Then we define a complex to be τ-coherent in terms of τ-finitely presented complexes and characterize τ-coherent complexes via τ-C-flatness of complexes.Let S be the endomorphism ring of a finitely presented complex C of right R-modules.It is proved that C is τ-coherent as a complex of left S-modules if and only if all direct products of copies of C are τ-C-flat as complexes of right R-modules,if and only if the class of all τ-C-flat complexes of right R-modules is closed under arbitrary direct products.