Let E be an elementary abelian p-group of rank r and k an algebraically closed field of characteristic p. We investigate finitely generated kE-modules of stable constant Jordan type [a][b] with 1 ≤ a, b ≤ p − 1 using the functors Fi from finitely generated kE-modules to vector bundles on the projective space Pr−1 constructed by Benson and Pevtsova. In particular, we study relations on the Chern numbers of the trivial bundle M to obtain restrictions on a and b for sufficiently large ranks and primes. We then study kE-modules with the constant image property and define the constant image layers of a module with respect to its maximal submodule having the constant image property. We prove that almost all such subquotients are semisimple. Focusing on the class of W-modules in rank two, we also calculate the vector bundles Fi(M) for all W-modules M. For E of rank two, we derive a duality formula for kE-modules M of constant Jordan type and their generic kernels K(M). We use this to answer a question of Carlson, Friedlander and Suslin regarding whether or not the submodules J−iK(M) also have constant Jordan type for all i ≥ 0. We show that this question has an affirmative answer whenever p = 3 or J2K(M) = 0. We also show that it has a negative answer in general by constructing a kE-module M of constant Jordan type for p ≥ 5 such that J−1K(M) does not have constant Jordan type. Finally, we use ideas from a theorem of Benson to show that if M is a kE-module of constant Jordan type containing no Jordan blocks of length one, then there always exist submodules of J−1K(M)/J2K(M) having a particularly nice structure.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:569614 |
| Date | January 2012 |
| Creators | Baland, Shawn |
| Publisher | University of Aberdeen |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=192251 |
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