Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture

By M. Boij, Jonas Söderberg. 2006

136 citations recorded by the source corpus.

We give conjectures on the possible graded Betti numbers of Cohen–Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non‐negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions is known, that is: for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for proposing the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/840c463e354e516d896c42fde45b08bd05fa2409

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Citations in the corpus

Introduction to twisted commutative algebras (2012, methodology): “ follows from Lemma (9.4.3). 10. Existing applications 10.1. Construction of pure resolutions The constructions in this section appeared in [EFW] and were motivated by the Boij–S¨oderberg conjectures [BS], which were solved in [ES1]. The speciﬁcs of the conjectures will not be discussed here, but we”
Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case (2008, methodology): “In this paper we generalize the construction given in [Boij and Söderberg 2008] to include Cohen–Macaulay pure Betti diagrams of various codimensions.”
Betti numbers of graded modules and cohomology of vector bundles (2007, methodology): “tive Betti Numbers The correctness of the following algorithm for decomposing a Betti table may clarify the meaning of Theorem 0.2, from which it follows easily. It was conjectured in Boij-S¨oderberg [2006]. Decomposition Algorithm Input: A graded Cohen-Macaulay S-module M of codimension c. Output: ”
Boij-Söderberg theory: Introduction and survey (2011, background): “s whose resolution becomes pure of this type when taking total degrees by the map Z2 →Z given by (d 1,d 2) →d 1 + d 2. Let P(p,q) be the positive rational cone generated by such modules. Theorem 6.8 ([6]). When p and q are relatively prime the extremal rays in the cone P(p,q) are parametrized by pai”
STABILITY PATTERNS IN REPRESENTATION THEORY (2013, background): “r the homogeneous coordinate ring of a smooth quadric hypersurface. This construction is known for polynomial rings [EFW, ES] and is the ﬁrst step in the proof [ES] of the Boij–S¨oderberg conjectures [BS], which describe the linear inequalities that deﬁne the cone of graded Betti numbers of ﬁnitely ”