Polynomial growth of Betti sequences over local rings

arXiv2208.04770MaRDI QIDQ6506392

Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang

Abstract: We study sequences of Betti numbers $$ 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 $I^{s}quare$ generated by the quadratic relations of the associated graded ring of $R$ satisfies $mheight{I}^{s}quaregemcodimR-1$, and that the converse holds when $R$ is homogeneous and when $mcodimRle4$. Avramov, Packauskas, and Walker subsequently proved that the degree of the difference of the even and odd Betti polynomials is always less than $mcodimR-mheight{I}^{s}quare-1$. 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.

Mathematics Subject Classification ID

Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series (13D40) Complete intersections (14M10) Syzygies, resolutions, complexes and commutative rings (13D02) Differential graded algebras and applications (associative algebraic aspects) (16E45)

This page was built for publication: Polynomial growth of Betti sequences over local rings

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6506392)