Stably thick subcategories of modules over Hopf algebras (Q2725000)

Cohomological or support varieties were originally introduced (about thirty years ago) to study finite-dimensional modules over the group algebra \(kG\) of a finite group \(G\) over a field \(k\) of prime characteristic. Over the years, these varieties have been considered for modules over various finite-dimensional Hopf algebras. \textit{D. J. Benson, J. F. Carlson}, and \textit{J. Rickard} [in Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)], extended the definition of varieties for finite groups to arbitrary (i.e. possibly infinite-dimensional) modules. And they used these varieties [in loc. cit.] to classify thick subcategories of the stable module category. The goal of this paper is to develop these ideas for an arbitrary finite-dimensional cocommutative Hopf algebra.NEWLINENEWLINENEWLINEThe authors' perspective is one of axiomatic stable homotopy theory and they rely on previous results of the authors and \textit{N. P. Strickland} [Axiomatic stable homotopy theory, Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. They show that one can classify thick subcategories (among other nice properties) if the Hopf algebra satisfies the so-called tensor product property. For finite dimensional modules, the tensor product property says that the variety of a tensor product of two modules is the intersection of the varieties of the modules. For graded connected Hopf algebras, they show that the tensor product property holds if it holds for all quasi-elementary sub-Hopf algebras. For these subalgebras, one would like to use so-called rank varieties to show that the tensor product property holds. Unfortunately, quasi-elementary subalgebras are difficult to identify in general. One case of interest to topologists where this can be done is for a finite-dimensional subalgebra of the mod 2 Steenrod algebra.NEWLINENEWLINENEWLINEFinally, we refer the interested reader to further work of the authors [J. Algebra 230, No. 2, 713-729 (2000; Zbl 0962.20006)] in which they remove the neccessity that \(k\) be algebraically closed (used here) for the classification of thick subcategories.