Stably thick subcategories of modules over Hopf algebras

Abstract

Abstract. We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional cocommutative Hopf algebra B. Our method is based on that of , who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F 2 . Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

Classifying subcategories of modules

“…The category of finite-dimensional rational vector spaces, as a subcategory of the category of abelian groups, is an example of a wide subcategory that is not a Serre class. The thick subcategories studied in [HP99,HP00] are, on the other hand, more general than wide subcategories.…”

Galois Theory of Thick Subcategories in Modular Representation Theory

“…The results in [HP99] give a classification of the Bousfield classes, for certain Hopf algebras B defined over algebraically closed fields: they are in bijection with arbitrary subsets of Proj Ext * B K K . We do not know how to prove an analogue of Theorem 1.7 for Bousfield classes, though, or for localizing subcategories.…”

Krull–Schmidt decompositions for thick subcategories

“…For each thick ideal A of finite objects in C, there is a finite localisation functor L A (also denoted as L f A ) on C whose finite acyclics are precisely the objects of A; see [12,Theorem 2.3]. For each bihomogeneous prime ideal p in π * (S), there are finite localisation functors L p and L <p whose finite acyclics are {X finite | X q = 0 ∀ q ⊆ p} and {X finite | X q = 0 ∀ q p} respectively.…”

“…We are now ready to state the thick subcategory theorem for C. 12]). Let C be a Noetherian stable homotopy category satisfying the tensor product property.…”

“…Example 5.7 ( [12,11]). Let B denote a finite dimensional graded co-commutative Hopf algebra over a field k (char(k) > 0).…”