Vol. 6, No. 3, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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

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.

graded modules, Betti numbers, multiplicity conjecture
Mathematical Subject Classification 2000
Primary: 13D02
Secondary: 13A02
Received: 2 July 2010
Revised: 24 January 2011
Accepted: 23 May 2011
Published: 5 July 2012
Mats Boij
Department of Mathematics
SE-100 44 Stockholm
Jonas Söderberg
Department of Mathematics
SE-100 44 Stockholm