<i>n</i>阶形式下三角矩阵环上的<i>m</i>-Gorenstein投射模

n阶形式下三角矩阵环上的m-Gorenstein投射模

摘要: 研究了 n 阶形式下三角矩阵环 T 上的 m -Gorenstein投射模，证明了：设 U_ij(1\leq j < i\leq n) 是 \left(A_i,A_j\right) -双模，并且作为左 A_i -模 U_ij 是投射模，作为右 A_j -模 U_ij 平坦维数有限. 若 M 是 m -Gorenstein投射左 T -模，则 M_1 是 \left(m-1\right) -Gorenstein投射左 A_1 -模，并且对任意整数 1\leq i\leq n-1,\varphi _i+1,i^M 是单同态，Coker \varphi _i+1,i^M 是 \left(m-1\right) -Gorenstein投射左 A_i+1 -模；反之，若 M_1 是 m -Gorenstein投射左 A_1 -模， \varphi _i+1,i^M 是单同态，Coker \varphi _i+1,i^M 是 m -Gorenstein投射左 A_i+1 -模，则 M 是 m -Gorenstein投射左 T -模.

Abstract: m -Gorenstein projective modules are investigated over formal lower triangular matrix ring T of order n . Let U_ij(1\leq j < i\leq n) be a \left(A_i,A_j\right) -bimodule, U_A_j has a finite flat dimension and A_iU is projective. It is proved that if M is a m -Gorenstein projective left T -module, then M_1 is a \left(m-1\right) -Gorenstein projective left A_1 -module, \varphi _i+1,i^M is a monomorphism for any 1\leq i\leq n-1 , and \textCoker\varphi _i+1,i^M is a \left(m-1\right) -Gorenstein projective left A_i+1 -module; Conversely, if M_1 is a m -Gorenstein projective left A_1 - module, \varphi _i+1,i^M is a monomorphism, and Coker \varphi _i+1,i^M is a m -Gorenstein projective left A_i+1 -module, then M is a m -Gorenstein projective left T -module.