Total Betti numbers of modules of finite projective dimension

Mark E. Walker¹

¹ University of Nebraska, Lincoln, NE

Annals of mathematics, Tome 186 (2017) no. 2, pp. 641-646.

Résumé

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the $i^{\mathrm{t}\mathrm{h}}$ Betti number ${\beta }_i(M)$ of a nonzero module $M$ of finite length and finite projective dimension over a local ring $R$ of dimension $d$ should be at least $(\genfrac{}{}{0ex}{}{d}{i})$. It would follow from the validity of this conjecture that $\sum _i{\beta }_i(M)\ge 2^d$. We prove the latter inequality holds in a large number of cases and that, when $R$ is a complete intersection in which $2$ is invertible, equality holds if and only if $M$ is isomorphic to the quotient of $R$ by a regular sequence of elements.

Détail

DOI : 10.4007/annals.2017.186.2.6