American Journal of Mathematics

We introduce the space P(G) of abelian p-points of a finite group scheme over an algebraically closed field of characteristic p > 0. We construct a homeomorphism Ψ_G: P(G) → Proj |G| from P(G) to the projectivization of the cohomology variety for any finite group G. For an elementary abelian p-group (respectively, an infinitesimal group scheme), P(G) can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups). For a finite dimensional G-module M, Ψ_G restricts to a homeomorphism P(G)_M → Proj |G|_M, thereby giving a representation-theoretic interpretation of the cohomological support variety.