3阶三角矩阵环上的Gorenstein投射模及其维数

Gorenstein projective modules and dimensions over triangular matrix ring of order 3

设T=(A_(1)U_(21)U_(31)0 A_(2)U_(32)00A_(3))是3阶三角矩阵环,其中A_(i)是环(i=1,2,3),U_(ij)是(A_(i),A_(j))-双模(1≤j<i≤3).探讨了环T上的模是Gorenstein投射模的等价条件.设M=(M_(1)M_(2)M_(3))φ^(M)是左T-模.若U_(ij)(1≤j<i≤3)作为左A_(i)-模时投射维数有限,U_(32)作为右A_(2)-模时平坦,以及U_(il)(i=2,3)作为右A_(1)-模时平坦维数有限,则M是Gorenstein投射左T-模当且仅当M_(1)是Gorenstein投射左A_(1)-模,以及对每个i=1,2,φ_(i)^(M)是单同态,且cokerφ_(i)^(M)是Gorenstein投射左A_(i+1)-模.同时刻画了左T-模M的Gorenstein投射维数.

LetT=(A_(1)U_(21)U_(31)0 A_(2)U_(32)00A_(3))be triangular matrix ring of order 3,where A_(i)are rings(i=1,2,3)and U_(ij)are(A_(i),A_(j))-bimodule(1≤j<i≤3).We obtain equivalent condition that module over ring T is Gorenstein projective module.Suppose M=(M_(1)M_(2)M_(3))φ^(M)is left-T module.If U_(ij)(1≤j<i≤3)as left A-module,they have finite projective dimensions,U_(32)as right A_(2)-module,it is flat,and U_(il)(i=2,3)as rightA_(1)-module,they have finite flat dimensions,then N is Gorenstein projective left T-module if and only if Mis Gorenstein projective lef T-module,and eachi=1,2,φ_(i)^(M)U_(i+1)i■A_(i)is monomorphism,andφ_(i)^(M)is Gorenstein projective left A_(i+1)-module.At the same time,we also characterize Gorenstein projective dimensions of left T-module M.