N-copresented Complexes And CP-Projective Complexes

In this paper,we mainly study n-copresentations of complexes of left R-modules and left orthogonal complexes of the class of finitely copresented complexes,namely CP-projective complexes,and obtain several equivalent characterizations of left ncocoherent rings,left V-rings and left n-cohereditary rings.In the first chapter,we first introduce the concept of finitely cogenerated complexes from categorical version,and then give the definition of n-copresented complexes,obtain some equivalent characterizations of n-copresented complexes,and show that the class of n-copresented complexes is closed under extensions,kernels of epimorphisms,finite direct sums,direct summands and shifts.We give some equivalent characterizations of left n-cocoherent rings and left n-cohereditary rings by n-copresented complexes,and prove that the class of n-copresented complexes is thick for left n-cocoherent rings.We also obtain some equivalent characterizations of left V-rings by means of finitely cogenerated(finitely copresented)exact complexes.Dual to λ-dimensions of complexes,we introduce λc-dimensions of complexes,get a equation of λc-dimensions between bounded complexes and their modules,give some relations of λc-dimensions among complexes in a short exact sequence of complexes.Under ring extensions,we study the invariance of n-copresented complexes,prove that λc-dimensions of D(as a R-module complex),D(as a S-module complex)and HomR(S,D)(as a S-module complex)are equal under an almost excellent extension of rings.In particular,if one of them is n-copresented,so are the others.Under a Morita duality,we obtain n-presented complexes and n-copresented complexes form a duality pair,show that the λ dimension and the λc-dimension of complexes and dual complexes are equal.In the second chapter,we introduce left orthogonal complexes of the class of finitely copresented complexes,namely CP-projective complexes,give some equivalent characterizations of CP-projective complexes,prove that a complex D is a CPprojective complex if and only if Ext1(D,F)=0 for every finitely copresented complex F,obtain some closeness of the class of CP-projective complexes under extensions,direct sums and direct summands,and prove that the class of CP-projective complexes is projectively resolving.Furthermore,we discuss the relation between CP-projective complexes and CP-projective modules,show that a complex D is CP-projective if and only if Dn is CP-projective for each n∈Z,and Hom(D,F)is exact for every finitely presented complex F.In particular,a bounded complex D is a CP-projective complex if and only if each Dn is CP-projective.Thus bounded CP-projective complexes and bounded FP-injective complexes form a dual pair under Morita duality.We also prove that bounded complexes in the left orthogonal class of the class of exact complexes with each item a finitely copresented R-module is exactly all bounded CP-projective complexes,and show that every CP-projective cover with the unique mapping property of a bounded complex D can be made up of CP-projective covers with the unique mapping property of every Dm of complex D.Finally,we obtain some equivalent characterizations of left 1-cohereditary ring and left V-ring by CPprojective complexes.