Derived equivalences for hereditary Artin algebras

Abstract

We study the role of the Serre functor in the theory of derived equivalences. Let $\mathcal{A}$ be an abelian category and let $(\mathcal{U},\mathcal{V})$ be a $t$-structure on the bounded derived category $D^b\mathcal{A}$ with heart $\mathcal{H}$. We investigate when the natural embedding $\mathcal{H}\to D^b\mathcal{A}$ can be extended to a triangle equivalence $D^b\mathcal{H}\to D^b\mathcal{A}$. Our focus of study is the case where $\mathcal{A}$ is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the $t$-structure is bounded and the aisle $\mathcal{U}$ of the $t$-structure is closed under the Serre functor.

As accepted by Advances in Mathematics; some differences in label and page numbering may occur between this version and the published version

References in corpus (2)

• On triangulated orbit categories
• A characterization of torsion theories in the cluster category of Dynkin type A_{\infty}

Cited by in corpus (4)

• Frobenius-Perron Theory of Representations of Quivers
• Derived categories of (one-sided) exact categories and their localizations
• Some examples of $t$-structures for finite-dimensional algebras
• Bounded t-structures on the bounded derived category of coherent sheaves over a weighted projective line