三角形矩阵环上的Gorenstein与对偶对
Gorenstein and duality pair over triangular matrix rings
摘要：
设$A$，$B$是两个环，$T=\left(\begin{smallmatrix} A & M \\ 0 & B \\\end{smallmatrix}\right)$与$M$是$A$ - $B$ -双模.首先利用$A$ -Mod和$B$ -Mod中的对偶对分别构造了$T$ -模的半完全对偶对$\mathcal{D}_{T}$.然后我们刻画了左$T$ -模是Gorenstein $D_{T}$ -投射的，Gorenstein $D_{T}$ -内射的或Gorenstein $D_{T}$ -平坦的.这三类$T$ -模将诱导出$T$ -模上的模型结构。最后我们证明了上述模型结构的同伦范畴相对于相应的稳定范畴存在一个等价范畴。我们的结果为这一方向的早期结果提供了新的特征。
英文摘要：
Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M \\ 0 & B \\\end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. We first construct a semi-complete duality pair $\mathcal{D}_{T}$ of $T$-modules using duality pairs in $A$-Mod and $B$-Mod respectively. Then we characterize when a left $T$-module is Gorenstein $D_{T}$-projective, Gorenstein $D_{T}$-injective or Gorenstein $D_{T}$-flat. These three class of $T$-modules will induce model structures on $T$-Mod. Finally we show that the homotopy category of each of model structures above admits a recollement relative to corresponding stable categories. Our results give new characterizations to earlier results in this direction.
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