On support varieties for Lie superalgebras and finite supergroup schemes

Abstract

We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras we obtain results in positive characteristic that are strikingly similar to results of Duflo and Serganova in characteristic zero. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting.

Graded analogues of one-parameter subgroups and applications to the cohomology of GL|()

“…This verifies a conjecture we made in [7]. Note that for n = 0, one gets GL m|n(r) = GL m(r) , the r-th Frobenius kernel of GL m , and V r (GL m|n ) = V r (GL m ) is the variety of r-tuples of commuting p-nilpotent matrices.…”

“…Given v ∈ V , we will write v (r) to denote the element v ⊗ ϕ r 1 ∈ V (r) . We generally follow the notation, terminology, and conventions of [4,5,7]. In particular, we assume that the reader is familiar with the sign conventions of 'super' linear algebra.…”

“…A fundamental first step in developing this theory is to provide a concrete description of the spectrum. The authors' previous paper [7] provides such a description in several natural settings. Relevant to the present work is the case of the first Frobenius kernel GL m|n (1) of GL m|n .…”

“…Set g = gl(m|n). In [7], we proved that there is a finite morphism Φ : Max (H • (G 1 , k)) → gl(m|n)…”

“…, α r−1 , β) ∈(g 0 ) ×r × g 1 : [α i , α j ] = 0, [α i , β] = 0 for all i, j, α [p] i = 0 for 0 ≤ i ≤ r − 2, and α 1 In the preprint version of [7], the last condition appears instead as α [p] = 1 2 [β, β]. This is because of an implicit sign error in the argument in the first paragraph of the proof of [7,Proposition 5.4.2]. We correct the issue here in Remark 2.3.1 and in the discussion of Section 6.…”