Detection Theorem for Finite Group Schemes

The classical detection theorem for finite groups (due to D. Quillen [Q] and B. Venkov [Q-V]) tells that if π is a finite group and Λ is a Z/p[π]-algebra then a cohomology class z ∈ H(π,Λ) is nilpotent iff for every elementary abelian psubgroup i : π0 →֒ π the restriction i (z) ∈ H(π0,Λ) of z to π0 is nilpotent. This theorem is very useful for the identification of the support variety of the group π and for the identification of support varieties of π-modules. In [S-F-B] we proved a similar detection theorem for cohomology of infinitesimal group schemes G. The role of elementary abelian p-subgroups is played in this case by the so called one parameter subgroups of G, i.e. closed subgroup schemes i : Ga(r) →֒ G. The analogy between elementary abelian p-groups and one parameter subgroups is emphasized by the fact that the corresponding cocommutative Hopf algebras (which happen to be commutative as well in this case) are isomorphic as algebras: k[Ga(r)] # ∼ = k[(Z/p)] (but have quite different coproducts) and hence have isomorphic cohomology algebras. Note that above we use very similar notations k[Ga(r)] and k[(Z/p) ] for two very different objects: on the left we have the coordinate algebra of the group scheme Ga(r), which is a commutative Hopf algebra for any group scheme, while on the right we have a group ring of a finite group. In this paper we are dealing with algebro-geometric objects so we will try to avoid the notation k[π] for the group algebra. In particular if we want to consider the finite group π as a discrete group scheme over k then k[π] will stand for the coordinate algebra of this discrete group scheme (which is dual to the group algebra of π), i.e. k[π] = k. The main purpose of this paper is to prove the general detection theorem, which covers both the discrete and the infinitesimal cases and looks as follows.