Frobenius powers of complete intersections

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...

Powers of Complete Intersections: Graded Betti Numbers and Applications

Abstract. Let I = (F1, . . . , Fr) be a homogeneous ideal of the ring R = k[x0, . . . , xn] generated by a regular sequence of type (d1, . . . , dr). We give an elementary proof for an explicit description of the graded Betti numbers of Is for any s ≥ 1. These numbers depend only upon the type and s. We then use this description to: (1) write HR/Is , the Hilbert function of R/Is, in terms of HR...

Almost complete intersections and the Lex-Plus-Powers Conjecture

We prove the almost complete intersection case of the Lex-Plus-Powers Conjecture on graded Betti numbers. We show that the resolution of a lex-plus-powers almost complete intersection provides an upper bound for the graded Betti numbers of any other ideal with regular sequence in the same degrees and the same Hilbert function. A key ingredient is finding an explicit comparison map between two K...

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