Gorenstein Projection, Invariant Properties Of Cohomology Of Injective And Flat Modules

In this paper, we mainly diacuss the homological invariants properties of Gorenstein projective, injective and flat modules. We define the invariants Gspli(R)=sup{Gorenstein projiective lengths of Gorenstein injective left R-mudule}, Gsilp(R)=sup{Goren-stein injective lengths of Gorenstein projective left R-mudule},and discussed some properties of them, if Gsilp(R)<oo and Gspli(R)<∞,then Gsilp(R)=Gspli(R). For a finite group, we have Gsilp(ZG)=Gspli(ZG)=1. If{Mλ}λ∈A is a family of strongly cotorsion R-mudule on Gorenstein ring, then we can get the finiteness of GpdR{ⅡMλ} by the finiteness of Gsilp(R). We introduce the invariants Gsfli(R)=sup{Gorenstein flat lengths of Gorenstein injective left R-mudule} to discuss some properties of the former invariants, when R has finite Gorenstein weak dimension,we have Gsilp(R)=Gsilf(R).For the invariants Gsfli(R) we have if Gsfli(R)<ooandGsfli(Rop)<∞,then Gsfli(R)=Gsfli(Rop). If R has finite Gorenstein weak dimension,we give two equivalent conditions of Gspli(R)=Gsilf(R)<∞. In part four, some properties of resolving of Gorenstein projective, injective and flat modules are discussed. In the last part, we study some properties of the above invariants on integral group ring.