Socle degrees of Frobenius powers

Socle Degrees, Resolutions, and Frobenius Powers

We first describe a situation in which every graded Betti number in the tail of the resolution of R J may be read from the socle degrees of R J . Then we apply the above result to the ideals J and J [q]; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of R/J [q] are equal to the graded Betti numbers in the tail of a shift of the resolution of R/J...

Computations with Frobenius Powers

It is an open question whether tight closure commutes with localization in quotients of a polynomial ring in finitely many variables over a field. Katzman [K] showed that tight closure of ideals in these rings commutes with localization at one element if for all ideals I and J in a polynomial ring there is a linear upper bound in q on the degree in the least variable of reduced Gröbner bases in...

Frobenius Powers of Non-complete Intersections

The purpose of this paper is to address a number of issues raised by Avramov and Miller in a recent paper [1]. Let (R,m, k) be a Noetherian local ring of characteristic p > 0 with residue field k, and let φ : R → R be the the Frobenius homomorphism defined by φ(a) = a. For r ≥ 1, we denote by φrR the R-module structure on R via φ. That is, for a ∈ R and b ∈ φ r R, a · b = a r b. When R is a reg...

GLn(Fq) in polynomials modulo Frobenius powers

Associated primes of local cohomology modules and of Frobenius powers

We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the loc...