Failure of F-purity and F-regularity in Certain Rings of Invariants

Let Fq be a finite field of characteristic p, K a field containing it, and R = K[X1, . . . , Xn] a polynomial ring in n variables. The general linear group GLn(Fq) has natural action on R by degree preserving ring automorphisms. L. E. Dickson showed that the subring of elements which are fixed by this group action is a polynomial ring, [Di], though for an arbitrary subgroupG of GLn(Fq), the structure of the ring of invariants R may be rather mysterious. If the order of the group |G| is relatively prime to the characteristic p of the field, there is anR–linear retraction ρ : R → R, the Reynolds operator . This retraction makes R a direct summand of R as an R–module, and so R is F–regular. However when the characteristic p divides |G|, this method no longer applies, and the ring of invariants R need not even be Cohen–Macaulay. M.–J. Bertin showed that when R is a polynomial ring in four variables and G is the cyclic group with four elements which acts by permuting the variables in cyclic order, then the ring of invariants R is a unique factorization domain which is not Cohen–Macaulay, providing the first example of such a ring, [Be]. Related work and bounds on the depth of R can be found in the work of R. M. Fossum and P. A. Griffith, see [FG]. More recently D. Glassbrenner studied the invariant subrings of the action of the alternating group An on a polynomial ring in n variables over a field of characteristic p, constructing examples of F–pure rings which are not F–regular, [G1, G2]. Both these families of examples study rings of invariants of K[X1, . . . , Xn] under the action of a subgroup G of the symmetric group on n elements, i.e., an action which permutes the variables, and Glassbrenner shows that for such a group the ring of invariants is F–pure, see [G1, Proposition 0.6.7].