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VOL. 86 | 2020 Notes on noncommutative Fitting invariants (with an appendix by Henri Johnston and Andreas Nickel)
Andreas Nickel, Henri Johnston

Editor(s) Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji

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.

Information

Published: 1 January 2020
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.2969/aspm/08610027

Subjects:
Primary: 16H05 , 16H10 , 16L30

Keywords: annihilator , Fitting ideal , Fitting invariant

Rights: Copyright © 2020 Mathematical Society of Japan

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