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

   

 

A realization theorem for modules of constant Jordan type and vector bundles


Authors: Dave Benson and Julia Pevtsova
Journal: Trans. Amer. Math. Soc. 364 (2012), 6459-6478
MSC (2010): Primary 20C20, 14F05
Published electronically: June 28, 2012
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Abstract: Let $ E$ be an elementary abelian $ p$-group of rank $ r$ and let $ k$ be a field of characteristic $ p$. We introduce functors $ \mathcal {F}_i$ from finitely generated $ kE$-modules of constant Jordan type to vector bundles over projective space $ \mathbb{P}^{r-1}$. The fibers of the functors $ \mathcal {F}_i$ encode complete information about the Jordan type of the module.

We prove that given any vector bundle $ \mathcal {F}$ of rank $ s$ on $ \mathbb{P}^{r-1}$, there is a $ kE$-module $ M$ of stable constant Jordan type $ [1]^s$ such that $ \mathcal {F}_1(M)\cong \mathcal {F}$ if $ p=2$, and such that $ \mathcal {F}_1(M) \cong F^*(\mathcal {F})$ if $ p$ is odd. Here, $ F\colon \mathbb{P}^{r-1}\to \mathbb{P}^{r-1}$ is the Frobenius map. We prove that the theorem cannot be improved if $ p$ is odd, because if $ M$ is any module of stable constant Jordan type $ [1]^s$, then the Chern numbers $ c_1,\dots ,c_{p-2}$ of $ \mathcal {F}_1(M)$ are divisible by $ p$.


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Additional Information

Dave Benson
Affiliation: Department of Mathematics, University of Aberdeen, King’s College, Meston Building, Aberdeen AB24 3UE, Scotland

Julia Pevtsova
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05482-1
PII: S 0002-9947(2012)05482-1
Received by editor(s): July 28, 2010
Received by editor(s) in revised form: September 28, 2010, and March 4, 2011
Published electronically: June 28, 2012
Additional Notes: The second author was partially supported by the NSF award DMS-0800940 and DMS-0953011
Article copyright: © Copyright 2012 David Benson and Julia Pevtsova