Title:

Some results on modules of constant Jordan type for elementary abelianρgroup

Let E be an elementary abelian pgroup of rank r and k an algebraically closed field of characteristic p. We investigate finitely generated kEmodules of stable constant Jordan type [a][b] with 1 ≤ a, b ≤ p − 1 using the functors Fi from finitely generated kEmodules 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 kEmodules 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 Wmodules in rank two, we also calculate the vector bundles Fi(M) for all Wmodules M. For E of rank two, we derive a duality formula for kEmodules 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 kEmodule 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 kEmodule 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.
