The representation invariants of 2-term silting complexes

Abstract

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.