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

Year 2020, , 271 - 287, 07.01.2020
https://doi.org/10.24330/ieja.663094

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.

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.
Year 2020, , 271 - 287, 07.01.2020
https://doi.org/10.24330/ieja.663094

Abstract

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.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Luca Amata This is me

Marilena Crupi This is me

Publication Date January 7, 2020
Published in Issue Year 2020

Cite

APA Amata, L., & Crupi, M. (2020). HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH. International Electronic Journal of Algebra, 27(27), 271-287. https://doi.org/10.24330/ieja.663094
AMA Amata L, Crupi M. HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH. IEJA. January 2020;27(27):271-287. doi:10.24330/ieja.663094
Chicago Amata, Luca, and Marilena Crupi. “HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 271-87. https://doi.org/10.24330/ieja.663094.
EndNote Amata L, Crupi M (January 1, 2020) HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH. International Electronic Journal of Algebra 27 27 271–287.
IEEE L. Amata and M. Crupi, “HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH”, IEJA, vol. 27, no. 27, pp. 271–287, 2020, doi: 10.24330/ieja.663094.
ISNAD Amata, Luca - Crupi, Marilena. “HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH”. International Electronic Journal of Algebra 27/27 (January 2020), 271-287. https://doi.org/10.24330/ieja.663094.
JAMA Amata L, Crupi M. HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH. IEJA. 2020;27:271–287.
MLA Amata, Luca and Marilena Crupi. “HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 271-87, doi:10.24330/ieja.663094.
Vancouver Amata L, Crupi M. HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH. IEJA. 2020;27(27):271-87.

Cited By

A generalization of Kruskal–Katona’s theorem
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https://doi.org/10.2478/auom-2020-0018