## Journal of Commutative Algebra

### Non-simplicial decompositions of Betti diagrams of complete intersections

#### Abstract

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-S\"oderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression of the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from \cite{ES1}. In higher codimension, obstructions arise that inspire our work on an alternative algorithm.

#### Article information

Source
J. Commut. Algebra Volume 7, Number 2 (2015), 189-206.

Dates
First available in Project Euclid: 14 July 2015

https://projecteuclid.org/euclid.jca/1436909531

Digital Object Identifier
doi:10.1216/JCA-2015-7-2-189

Mathematical Reviews number (MathSciNet)
MR3370483

Zentralblatt MATH identifier
1327.13048

Subjects
Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 13C99: None of the above, but in this section

#### Citation

Gibbons, Courtney; Jeffries, Jack; Mayes, Sarah; Raicu, Claudiu; Stone, Branden; White, Bryan. Non-simplicial decompositions of Betti diagrams of complete intersections. J. Commut. Algebra 7 (2015), no. 2, 189--206. doi:10.1216/JCA-2015-7-2-189. https://projecteuclid.org/euclid.jca/1436909531

#### References

• Mats Boij and Jonas Söderberg, Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, J. Lond. Math. Soc. 78 (2008), 85–106.
• ––––, Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case, Alg. Num. Theor. 6 (2012), 437–454.
• David Eisenbud and Frank-Olaf Schreyer, Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), 859–888.
• Sabine El Khoury, Manoj Kummini and Hema Srinivasan, Bounds for the multiplicity of gorenstein algebras, arXiv (1211.1316), 2012.
• Daniel R. Grayson and Michael E. Stillman, Macaulay$2$, A software system for research in algebraic geometry, available at http:// www.math.uiuc.edu/Macaulay2/.
• Jason McCullough, A polynomial bound on the regularity of an ideal in terms of half of the syzygies, Math. Res. Lett. 19 (2012), 555–565.
• Uwe Nagel and Stephen Sturgeon, Combinatorial interpretations of some Boij-Söderberg decompositions, J. Alg. 381 (2013), 54–72.
• Fanya Wyrick-Flax, Algebraic relations and Boij–Söderberg theory, Undergraduate thesis, Bard College, Annandale-On-Hudson, NY, 2013.