The Frobenius morphism in invariant theory

Abstract

Let R be the homogeneous coordinate ring of the Grassmannian G = Gr(2, n) defined over an algebraically closed field k of characteristic p ≥ max{n − 2, 3}. In this paper we give a description of the decomposition of R, considered as graded R p r -module, for r ≥ 2. This is a companion paper to [16], where the case r = 1 was treated, and taken together, our results imply that R has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators D k (R) is simple, that G has global finite F-representation type (GFFRT) and that R provides a noncommutative resolution for R p r .

On Segre products, $F$-regularity, and finite Frobenius representation type

“…Key examples of rings with FFRT include those that are graded direct summands of polynomial rings; such rings are also F-regular, and hence Cohen-Macaulay. Recent work on the FFRT property includes that of Hara and Ohkawa [HO], where they study the property for 2-dimensional normal graded rings in terms of Q-divisors, and [RSV1,RSV2] where Raedschelders, Špenko, and Van den Bergh prove that over an algebraically closed field of characteristic p max{n − 2, 3}, the Plücker homogeneous coordinate ring of the Grassmannian G(2, n) has FFRT.…”

Non-commutative resolutions of toric varieties

“…It should be noted that in a very nice paper,Špenko and Van den Bergh independently found less constructive proofs for the finiteness of global dimension in settings (1) and (2), albeit without explicit bounds on the global dimension [ŠVdB17b, 1.3.6]; they also beat us to publication. See also [RŠVdB17].…”

“…We speculate that for a strongly F-regular ring R with finite F-representative type (a class of rings including toric rings in prime characteristic), perhaps End R (R 1/p e ) always has finite global dimension. This speculation encompasses the classes of rings arising in representation theory where this idea has been explored from a somewhat different point of view in [ŠVdB17b,RŠVdB17].…”