Categorification of a linear algebra identity and factorization of Serre functors

TL;DR

This paper offers a categorical interpretation of a linear algebra identity, revealing how Serre functors in finite dimensional algebras can be factored into reflection functors, enriching the understanding of their structure.

Contribution

It introduces a novel categorification of a classical linear algebra identity and demonstrates how Serre functors can be decomposed into reflection functors in derived categories.

Findings

Serre functors lift to products of reflection functors

Categorification links linear algebra identities to functor isomorphisms

Provides new insights into the structure of derived categories of algebras

Abstract

We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of A.