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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: David Eisenbud and Jerzy Weyman
Journal: Trans. Amer. Math. Soc. 355 (2003), 4451-4473
MSC (2000): Primary 13C99, 13C05, 13D02, 16D70, 17B70
Published electronically: June 10, 2003
MathSciNet review: 1990758
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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.


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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: de@msri.org

Jerzy Weyman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: j.weyman@neu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03198-2
PII: S 0002-9947(03)03198-2
Received by editor(s): March 20, 2002
Received by editor(s) in revised form: May 29, 2002
Published electronically: June 10, 2003
Additional Notes: The second named author is grateful to the Mathematical Sciences Research Institute for support in the period this work was completed. Both authors are grateful for the partial support of the National Science Foundation.
Article copyright: © Copyright 2003 American Mathematical Society