Notes on noncommutative Fitting invariants

TL;DR

This paper surveys the extension of Fitting invariants from commutative rings to certain noncommutative rings, discussing their properties, behavior under direct sums, and introducing a new Morita equivalence approach.

Contribution

It provides a comprehensive overview of noncommutative Fitting invariants, including recent results, open problems, and a novel perspective using Morita equivalence.

Findings

Fitting invariants are contained in annihilators and easier to compute.

Behavior of Fitting invariants under direct sums is analyzed.

A new approach via Morita equivalence is introduced.

Abstract

To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fitting ideal of $M$ over $R$. This is of interest because it is always contained in the $R$-annihilator $\mathrm{Ann}_{R}(M)$ of $M$, but is often much easier to compute. This notion has recently been generalised to that of so-called `Fitting invariants' over certain noncommutative rings; the present author considered the case in which $R$ is an $\mathfrak{o}$-order $\Lambda$ in a finite dimensional separable algebra, where $\mathfrak{o}$ is an integrally closed commutative noetherian complete local domain. This article is a survey of known results and open problems in this context. In particular, we investigate the behaviour of Fitting invariants under direct sums. In the appendix, we present a new approach to Fitting invariants via Morita equivalence.