Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations

Let kE denote the group algebra of an elementary abelian p-group of rank r over an algebraically closed field of characteristic p  . We investigate the functors FiFi from kE  -modules of constant Jordan type to vector bundles on Pr−1(k)Pr−1(k), constructed by Benson and Pevtsova. For a kE-module M   of constant Jordan type, we show that restricting the sheaf Fi(M)Fi(M) to a dimension s−1s−1 linear subvariety of Pr−1(k)Pr−1(k) is equivalent to restricting M along a corresponding rank s shifted subgroup of kE   and then applying FiFi.In the case r=2r=2, we examine the generic kernel filtration of M   in order to show that Fi(M)Fi(M) may be computed on certain subquotients of M whose Loewy lengths are bounded in terms of i. More precise information is obtained by applying similar techniques to the nth power generic kernel filtration of M. The latter approach also allows us to generalise our results to higher ranks r.