Graded Betti numbers of good filtrations

TL;DR

This paper extends the understanding of the asymptotic behavior of graded Betti numbers from polynomial rings to more general Z-graded algebras over Noetherian local rings, focusing on filtrations related to Rees algebras.

Contribution

It introduces a generalization of the quasi polynomial behavior of graded Betti numbers to Z-graded algebras and analyzes Betti tables of filtrations over Rees algebras.

Findings

Extended quasi polynomial behavior to Z-graded algebras over Noetherian local rings.

Analyzed Betti tables of filtrations finite or integral over Rees algebras.

Provided new insights into the asymptotic properties of Betti numbers in broader algebraic contexts.

Abstract

The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi polynomial behavior of graded Betti numbers of powers of homogenous ideals to Z-graded algebra over Notherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.