A Fitting Lemma for Z/2-graded modules

TL;DR

This paper generalizes the classical Fitting Lemma to Z/2-graded modules over graded skew-commutative algebras, using Lie superalgebra representation theory to analyze annihilators and Fitting ideals.

Contribution

It introduces a new Fitting lemma for Z/2-graded modules, extending classical results and connecting to Lie superalgebra representations.

Findings

Annihilator of cokernel described by Fitting ideal in generic case

Relations between Fitting ideal and annihilator established

Results extend Green's 1999 key theorem

Abstract

We study the annihilator of the cokernel of a map of free Z/2-graded modules over a Z/2-graded skew-commutative algebra in characteristic 0 and define analogues of its Fitting ideals. We show that in the ``generic'' case the annihilator is given by a Fitting ideal, and explain relations between the Fitting ideal and the annihilator that hold in general. Our results generalize the classical Fitting Lemma, and extend the key result of Green [1999]. They depend on the Berele-Regev theory of representations of general linear Lie super-algebras.