Date of this Version
2020 Mathematics Subject Classification. Primary: 13D02, 13D40. Secondary: 14M10, 16E45.
We study sequences of Betti numbers (βRi (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  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.