Cartan-Eilenberg Gorenstein AC-Cohomology Dimension Of Complexes

Throughout this paper,R is a commutative ring with an identity,R-Mod denotes the category of R-modules.The main content of this paper is as follows:In the first chapter,some basic definitions related to this paper are given.In the second chapter,we show that a complex X is CE type FP∞ if and only if there is CE exact sequence→ P₂→P₁→P₀→X→0,by CE finitely generated pro-jective complexes P_i.Secondly,some equivalent characterizations of CE absolutely clean complex and CE level complex are introduced.In particular,we have Ext（?）（X,A）=0 for any CE type FP∞ complex X.Then,M+is a CE level complex for any CE absolutely clean complex M.Finally,the classes of CE absolutely clean complexes and CE level complexes are closed under direct products,direct limits,pure subcomplexes and pure quotients.In particular,the class of CE absolutely clean complexes is closed under CE pure subcomplexes and CE pure quotients.In the third chapter,some equivalent characterizations of CE Gorenstein AC-injective（projective）complexes are discussed.In particular,it is proved that a complex G is CE Gorenstein AC-injective（projective）if and only if G has a CE strongly complete injective（L complete projective）resolution.Then,the relations of L,M and N are givenin CE exact sequence 0→L→M→N→0.Further,CE Gorenstein AC-injective（projection）complexes are Gorenstein AC-injective（projection）complexes.Fi-nally,（CE（⊥（gI））,CE（9I））and（CE（gP）,CE（（gP）⊥））are hereditary and complete cotorsion pairs ralative to Ext（?）（-,-）.In particular,any complexes has CE Gorenstein AC-injective preenvelope（projective precover）.In the fourth chapter,introducing definitions of CE absolutely clean dimensions,CE level dimensions and CE Gorenstein AC-injective（projective）dimensions are given.And characterizations of the finiteness of CE Gorenstein AC-injective（projective）dimensions are given.Then,CE Gorenstein AC-injective（projective）dimensions of complex equal to supremum of CE Gorenstein AC-injective（projective）dimensions of the mth boundary and homology modules.Then,the relations of CE Gorenstein AC-injective（projective）dimensions L,M and N are given in CE exact sequence 0→L→M→N→0.Finally,the CE Gorenstein AC-injective（projective）dimensions of direct products（direct sums）of a family of complexes equal to supremum of CE Gorenstein AC-injective（pro-jective）dimensions of all complexes.