Let $A$ be a finite dimensional algebra over a field $k$ and $\textbf{P}$ be a 2-term silting complex in $K^{b}(\text{proj}A)$. In this paper, we investigate the representation dimension of $\text{End}_{D^{b}(A)} (\textbf{P})$ by using the silting theory. We show that if $\textbf{P}$ is a separating silting complex with certain homological restriction, then rep.dim $A=$rep.dim $\text{End}_{D^{b}(A)}(\textbf{P})$. This gives a proper generalization of the classical compare theorem of representation dimensions showed by Chen and Hu. It is well-known that $\text{H}^{0}(\textbf{P})$ is a tilting $A/\text{ann}_{A} (\textbf{P})$-module. We also show that rep.dim $\text{End}_{A} (\text{H}^{0}(\textbf{P})) = $rep.dim $A/\text{ann}_{A} (\textbf{P})$ if $\textbf{P}$ is a separating and splitting silting complex.

中文翻译：

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2项淤积复合体的表示不变量

设 $A$ 是 $k$ 域上的有限维代数，$\textbf{P}$ 是 $K^{b}(\text{proj}A)$ 中的 2 项淤积复合体。在本文中，我们使用淤泥理论研究了$\text{End}_{D^{b}(A)} (\textbf{P})$ 的表示维度。我们证明如果 $\textbf{P}$ 是具有一定同源性限制的分离淤泥复合体，则 rep.dim $A=$rep.dim $\text{End}_{D^{b}(A)}( \textbf{P})$。这给出了陈和胡展示的表示维数的经典比较定理的适当推广。众所周知，$\text{H}^{0}(\textbf{P})$ 是一个倾斜的 $A/\text{ann}_{A} (\textbf{P})$-module。我们还证明了 rep.dim $\text{End}_{A} (\text{H}^{0}(\textbf{P})) = $rep.dim $A/\text{ann}_{ A} (\textbf{P})$ 如果 $\textbf{P}$ 是一个分离和分裂的淤泥复合体。