张量环上Gorenstein投射模的构造

Construction of Gorenstein projective modules over tensor rings

设R为任意环,M为一个幂零R-双模,TR(M)为相应的张量环。假设对任意i≥0,Ext1R(G,M⊗Ri⊗RP)=0=TorR1(M,M⊗Ri⊗RG),其中P为投射R-模,G为Gorenstein投射R-模。证明一个TR(M)-模(X,u)如果满足u是单同态并且u的余核是Gorenstein投射R-模,则(X,u)是Gorenstein投射TR(M)-模。

Let R be any ring,M a nilpotent R-bimodule,TR(M)is the associated tensor ring.Suppose that Ext1R(G,M⊗Ri⊗RP)=0=TorR1(M,M⊗Ri⊗RG),where P is a projective R-module,G is a Gorenstein projective R-module,and i≥0.It is proved that a TR(M)-module:(X,u)is a Gorenstein projective TR(M)-module if u is a monomorphism and the cokernel of u is a Gorenstein projective R-module.