Notes on noncommutative Fitting invariants

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.

Survey article that contains the material of the first part of my Preparatory Lecture Series on `Non-abelian Stark-type conjectures and noncommutative Iwasawa theory' at the Iwasawa 2017 conference. You may consider arXiv:1707.04432 for the second part. 24 pages, with an appendix by Henri Johnston and Andreas Nickel; several minor changes since v1

References in corpus (2)

• Noncommutative Fitting invariants and improved annihilation results
• On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture

Cited by in corpus (1)

• Fitting invariants in equivariant Iwasawa theory