Homology of powers of ideals: Artin--Rees numbers of syzygies and the Golod property

TL;DR

This paper investigates the asymptotic behavior of syzygies and Betti numbers of powers of ideals in regular local rings, establishing the constancy of Artin-Rees numbers and the Golod property of quotient rings for large powers.

Contribution

It extends the understanding of syzygy modules and Betti numbers from graded to local rings, introducing the Artin-Rees number as a key invariant and proving Golodness for large powers.

Findings

Artin-Rees number stabilizes for large powers of ideals.

R/I^k is Golod for sufficiently large k.

Betti numbers of R/I^k are polynomial in k.

Abstract

For an ideal I in a regular local ring (R,m)$ with residue class field K = R/m or a standard graded K-algebra R we show that for k >> 0 --> the Artin--Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby present the Artin-Rees number as a proper replacement of regularity in the local situation, --> the ring R/I^k is Golod, its Poincer{\'e}-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of work of Kodiyalam and Cutkosky, Herzog & Trung on the regularity of I^k for k >> 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of work by Kodiyalam on the behavior of Betti numbers of the minimal free resolution of R/I^k over R.