A reduction approach to silting objects for derived categories of hereditary categories

Wei Dai, Changjian Fu Colloquium Mathematicum MSC: Primary 16E35; Secondary 18G80. DOI: 10.4064/cm8480-11-2021 Published online: 23 May 2022

Abstract

Let $\mathcal H$ be a hereditary abelian category over a field $k$ with finite-dimensional Hom and Ext spaces. It is proved that the bounded derived category $\mathcal D^b(\mathcal H)$ has a silting object iff $\mathcal H$ has a tilting object iff $\mathcal D^b(\mathcal H)$ has a simple-minded collection with acyclic Ext-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $\mathcal D^b(\mathcal H)$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $\mathcal R$ of $\mathcal D^b(\mathcal H)$ can be completed to a simple-minded collection iff the Ext-quiver of $\mathcal R$ is acyclic.

Authors

  • Wei DaiDepartment of Mathematics
    SiChuan University
    610064 Chengdu, P.R. China
    e-mail
  • Changjian FuDepartment of Mathematics
    SiChuan University
    610064 Chengdu, P.R. China
    e-mail

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