Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen–Macaulay case

Boij, Mats; Söderberg, Jonas

doi:10.2140/ant.2012.6.437

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Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen–Macaulay case

Mats Boij and Jonas Söderberg

Vol. 6 (2012), No. 3, 437–454

DOI: 10.2140/ant.2012.6.437

Abstract

We use results of Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination of Betti diagrams of modules with a pure resolution. This implies the multiplicity conjecture of Herzog, Huneke, and Srinivasan for modules that are not necessarily Cohen–Macaulay and also implies a generalized version of these inequalities. We also give a combinatorial proof of the convexity of the simplicial fan spanned by pure diagrams.

Keywords

graded modules, Betti numbers, multiplicity conjecture

Mathematical Subject Classification 2000

Primary: 13D02

Secondary: 13A02

Milestones

Received: 2 July 2010

Revised: 24 January 2011

Accepted: 23 May 2011

Published: 5 July 2012

Authors

Mats Boij
Department of Mathematics KTH SE-100 44 Stockholm Sweden

Jonas Söderberg
Department of Mathematics KTH SE-100 44 Stockholm Sweden

Vol. 6, No. 3, 2012