On the generic kernel filtration for modules of constant Jordan type

Abstract

Let \({E \cong (\mathbb{Z}/p)^2}\) be an elementary abelian p-group of rank two k an algebraically closed field of characteristic p, and let J = J(kE). We investigate finitely generated kE-modules M of constant Jordan type and their generic kernels \({\mathfrak{K}(M)}\) . In particular, we answer a question posed by Carlson, Friedlander, and Suslin regarding whether or not the submodules \({J^{-i} \mathfrak{K}(M)}\) have constant Jordan type for all i ≥ 0. We show that this question has an affirmative answer whenever p = 3 or \({J^2 \mathfrak{K}(M) = 0}\) . We also show that this question has a negative answer in general by constructing a kE-module M of constant Jordan type for p ≥ 5 such that \({J^{-1} \mathfrak{K}(M)}\) does not have constant Jordan type.