Algebra & Number Theory

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

Mats Boij and Jonas Söderberg

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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.

Article information

Algebra Number Theory, Volume 6, Number 3 (2012), 437-454.

Received: 2 July 2010
Revised: 24 January 2011
Accepted: 23 May 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 13A02: Graded rings [See also 16W50]

graded modules Betti numbers multiplicity conjecture


Boij, Mats; Söderberg, Jonas. Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen–Macaulay case. Algebra Number Theory 6 (2012), no. 3, 437--454. doi:10.2140/ant.2012.6.437.

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