Milnor squares of algebras, I: derived equivalences

TL;DR

This paper introduces a method to generate new derived equivalences of Artin algebras using Milnor squares, revealing how derived equivalences can alter Frobenius types while preserving certain properties.

Contribution

It presents novel operations for constructing derived equivalences via Milnor squares, expanding the understanding of how algebra properties can change under these transformations.

Findings

Derived equivalences can change Frobenius type of algebras.

Operations like gluing vertices, unifying arrows, and identifying socle elements produce new derived equivalences.

Both tilting and almost ν-stable derived equivalences preserve Frobenius type.

Abstract

Derived equivalences for Artin algebras (and almost $\nu$-stable derived equivalences for finite-dimensional algebras) are constructed from Milnor squares of algebras. Particularly, three operations of gluing vertices, unifying arrows and identifying socle elements on derived equivalent algebras are presented to produce new derived equivalences of the resulting algebras from the given ones. As a byproduct, we construct a series of derived equivalences, showing that derived equivalences may change Frobenius type of algebras in general, though both tilting procedure and almost $\nu$-stable derived equivalences do preserve Frobenius type of algebras.