Tate Homology And The Depth Formula Via Gorenstein AC Flat Dimension

Tate homology is of great significance in the homological algebra.We construct the complete AC resolution of complexes of finite Gorenstein AC flat dimension,and on this basis,we study Tate homology of modules of finite Gorenstein AC flat dimension,and give a condition under which the derived depth formula holds.Firstly,we introduce Gorenstein AC flat dimension for complexes and give some characterizations of GFac-dim M≤n for any complex M,and prove that the complex M has finite Gorenstein AC flat dimension if and only if M admits a complete AC resolution(?).Based on the complete AC resolution of module of finite Gorenstein AC flat dimension,we investigate Tate homology and its induced long sequence lemmas,the dimensions transfer formula of Tate homology,as well as the long exact sequences associated with absolute homology,relative homology and Tate homology.As an application,we give a condition under which the derived depth formula holds.We prove that if R is a commutative Noetherian local ring,M is a complex of finite Goenstein AC flat dimension and N a bounded above complex,as well as one has ToriR(M,N)=0 for all i ∈Z,then the derived depth formula holds for M and N,that is,depthR(M(?)RLN)=depthRM+depthRN-depthR.