Computations with Frobenius Powers

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...

GLn(Fq) in polynomials modulo 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...

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...

Infinitely many associated primes of Frobenius powers and local cohomology

A modification of Katzman’s example is given to produce a two-generated ideal in a two-dimensional Noetherian integral domain for which the set of associated primes of all the Frobenius powers is infinite. A further modification yields a four-dimensional Noetherian integral domain and a five-dimensional Noetherian local integral domain for which an explicit second local cohomology module has in...