POLYNOMIAL GROWTH OF BETTI SEQUENCES OVER LOCAL RINGS

Authors

LUCHEZAR L. AVRAMOV, University of Nebraska-Lincoln
Alexandra Seceleanu, University of Nebraska-LincolnFollow
ZHENG YANG, University of Nebraska-Lincoln

Date of this Version

8-9-2022

Citation

2020 Mathematics Subject Classification. Primary: 13D02, 13D40. Secondary: 14M10, 16E45.

Comments

Used by permission.

Abstract

We study sequences of Betti numbers (β^R_i (M)) of finite modules M over a complete intersection local ring, R. It is known that for every M the subsequence with even, respectively, odd indices i is eventually given by some polynomial in i. We prove that these polynomials agree for all R-modules if the ideal I^☐ generated by the quadratic relations of the associated graded ring of R satisfies height I^☐ ≥ codim R − 1, and that the converse holds when R is homogeneous and when codim R ≤ 4. Avramov, Packauskas, and Walker [10] subsequently proved that the degree of the difference of the even and odd Betti polynomials is always less than codim R − height I^☐ − 1. We give a different proof, based on an intrinsic characterization of the residue rings of complete intersection local rings of minimal multiplicity obtained in this paper.