Polynomial growth of Betti sequences over local rings

This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, $R$. The subsequences $\{{\beta }_i^R(M){\}}_{i\ge 0}$ with even, respectively, odd $i$ are known to be eventually given by polynomials in i with equal leading terms. We show that these polynomials coincide if $I^{\mathrm{\square }}$, the ideal generated by the quadratic relations of the associated graded ring of $R$, satisfies $\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}{I}^{\mathrm{\square }}\ge \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}R-1$, and that the converse holds if $R$ is homogeneous or $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}R\le 4$. Subsequently Avramov, Packauskas, and Walker proved that the terms of degree $j>\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}R-\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}{I}^{\mathrm{\square }}$ of the even and odd Betti polynomials are equal. We give a new proof of that result, based on an intrinsic characterization of residue rings of c.i. local rings of minimal multiplicity obtained in this paper. We also show that that bound is optimal.