The representation invariants of 2-term silting complexes

Abstract

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

18pages

References in corpus (1)

• On the representation dimension of artin algebras