Endomorphism algebras of 2-term silting complexes

TL;DR

This paper investigates the global dimension of endomorphism algebras of 2-term silting complexes, establishing upper bounds for algebras with low global dimension and demonstrating the possibility of infinite global dimension in certain cases.

Contribution

It provides bounds on the global dimension of endomorphism algebras of 2-term silting complexes and constructs examples with infinite global dimension.

Findings

For algebras with global dimension ≤ 2, the endomorphism algebra's global dimension is at most 7.

Existence of algebras with arbitrary global dimension n > 2 that admit 2-term silting complexes with infinite endomorphism algebra global dimension.

Abstract

We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra $A$ whose global dimension $\mathop{\rm gl. dim}\nolimits A\leq 2$ and any 2-term silting complex $\mathbf{P}$ in the bounded derived category ${D^b(A)}$ of $A$, the global dimension of $\mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ is at most 7. We also show that for each $n>2$, there is an algebra $A$ with $\mathop{\rm gl. dim}\nolimits A=n$ such that ${D^b(A)}$ admits a 2-term silting complex $\mathbf{P}$ with $\mathop{\rm gl. dim}\nolimits \mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ infinite.