Derived equivalences and delooping levels

TL;DR

This paper constructs a finite-dimensional algebra derived equivalent to a known example, showing that certain invariants like delooping levels can change from infinite to zero under derived equivalence, challenging previous assumptions.

Contribution

It provides a new example of a derived equivalent algebra with finite delooping levels, illustrating that these invariants are not preserved under derived equivalences.

Findings

The constructed algebra has delooping levels equal to 0.

The Kershaw--Rickard example has infinite invariants.

Finiteness of invariants is not preserved under derived equivalence.

Abstract

We construct a finite-dimensional algebra derived equivalent to the example of Kershaw--Rickard. For the Kershaw--Rickard example the delooping level and the sub-derived delooping level are both infinite, while for our algebra both invariants are $0$. Thus the finiteness of these invariants is not preserved under derived equivalences.