HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH

Year 2020, Volume: 27 Issue: 27, 271 - 287, 07.01.2020

Luca Amata Marilena Crupi

https://doi.org/10.24330/ieja.663094

Cited By: 2

Abstract

Let $K$ be a field, $E$ the exterior algebra of a finite dimensional $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\deg g_1 \le \deg g_2 \le \cdots \le \deg g_r$. Given the Hilbert function of a graded $E$--module of the type $F/M$, with $M$ graded submodule of $F$, the existence of the unique lexicographic submodule of $F$ with the same Hilbert function as $M$ is proved by a new algorithmic approach.
Such an approach allows us to establish a criterion for determining if a sequence of nonnegative integers defines the Hilbert function of a quotient of a free $E$--module only via the combinatorial Kruskal--Katona's theorem.

Keywords

Exterior algebra, Hilbert function, monomial submodule, lexicographic submodule

References

• J. Abbott, A. M. Bigatti and G. Lagorio, CoCoA-5: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it.
• L. Amata and M. Crupi, ExteriorIdeals: A package for computing monomial ideals in an exterior algebra, J. Softw. Algebra Geom., 8(1) (2018), 71-79.
• L. Amata and M. Crupi, Bounds for the Betti numbers of graded modules with given Hilbert function in an exterior algebra via lexicographic modules, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), Tome 61(109) No. 3 (2018), 237-253.
• L. Amata and M. Crupi, A generalization of Kruskal{Katona's Theorem, An. Stiint. Univ. \Ovidius" Constanta Ser. Mat., to appear.
• A. Aramova, J. Herzog and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra, 191(1) (1997), 174-211.
• W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
• M. Crupi and C. Ferro, Squarefree monomial modules and extremal Betti numbers, Algebra Colloq., 23 (3) (2016), 519-530.
• W. Decker, G. M. Greuel, G. Pfister and H. Schonemann, Singular 4-1-0 | A computer algebra system for polynomial computations, (2016), available at http://www.singular.uni-kl.de.
• D. Eisenbud, Commutative Algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
• D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2.
• J. Herzog and T. Hibi, Monomial Ideals, Graduate texts in Mathematics, 260, Springer-Verlag, London, 2011.
• G. Katona, A theorem of finite sets, in Theory of graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, (1968), 187-207.
• J. B. Kruskal, The number of simplices in a complex, in Mathematical optimization techniques (R. Bellman, ed.), University of California Press, Berkeley, (1963), 251-278.
• F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc., 26 (1927), 531-555.
• R. P. Stanley, Cohen-Macaulay rings and constructible polytopes, Bull. Amer. Math. Soc., 81 (1975), 133-135.