Socle degrees, Resolutions, and Frobenius powers

TL;DR

This paper explores the relationship between socle degrees and graded Betti numbers in resolutions, particularly focusing on Frobenius powers of ideals, revealing conditions where these invariants are directly related.

Contribution

It introduces a new method to read Betti numbers from socle degrees and applies it to Frobenius powers, establishing when their resolutions are related by shifts.

Findings

Betti numbers in the tail of resolutions can be derived from socle degrees

Resolutions of Frobenius powers relate to shifted resolutions of original ideals

Conditions identified for equality of Betti numbers in tails of different resolutions

Abstract

We first describe a situation in which every graded Betti number in the tail of the resolution of $\frac RJ$ may be read from the socle degrees of $\frac RJ$. 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 $R/J$.