Fitting’s Lemma for $\mathbb {Z}/2$-graded modules

Abstract:

Let $\phi :\; R^{m}\to R^{d}$ be a map of free modules over a commutative ring $R$. Fitting’s Lemma shows that the “Fitting ideal,” the ideal of $d\times d$ minors of $\phi$, annihilates the cokernel of $\phi$ and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a $\mathbb {Z}/2$-graded skew-commutative algebra and prove corresponding theorems about the annihilator; for example, the Fitting ideal and the annihilator of the cokernel are equal in the generic case. Our results generalize the classical Fitting Lemma in the commutative case and extend a key result of Green (1999) in the exterior algebra case. They depend on the Berele-Regev theory of representations of general linear Lie superalgebras. In the purely even and purely odd cases we also offer a standard basis approach to the module $\operatorname {coker}\phi$ when $\phi$ is a generic matrix.