Research Article
BibTex RIS Cite

ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS

Year 2021, , 168 - 202, 17.07.2021
https://doi.org/10.24330/ieja.969656

Abstract

Let $K$ be a field and $S=K[x_1,\dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals.
More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.

References

  • L. Amata and M. Crupi, Computation of graded ideals with given extremal Betti numbers in a polynomial ring, J. Symbolic Comput., 93 (2019), 120-132.
  • A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z., 228(2) (1998), 353-378.
  • A. Aramova, J. Herzog and T. Hibi, Shifting operations and graded Betti numbers, J. Algebraic Combin., 12(3) (2000), 207-222.
  • D. Bayer, H. Charalambous and S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra, 221(2) (1999), 497-512.
  • [5] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998.
  • S. M. Cooper and S. S. Wagsta , Connections Between Algebra, Combinatorics, and Geometry, Springer Proceedings in Mathematics & Statistics, 76, Springer-Verlag, 2014.
  • M. Crupi, Extremal Betti numbers of graded modules, J. Pure Appl. Algebra, 220(6) (2016), 2277-2288.
  • M. Crupi, A constructive method for standard Borel fixed submodules withgiven extremal Betti numbers, Mathematics, 56(5) (2017), 1-26.
  • M. Crupi, Computing general strongly stable modules with given extremal Betti numbers, J. Commut. Algebra, 12(1) (2020), 53-70.
  • M. Crupi and C. Ferro, Squarefree monomial modules and extremal Betti numbers, Algebra Colloq., 23(3) (2016), 519-530.
  • M. Crupi and R. Utano, Extremal Betti numbers of lexsegment ideals, Geometric and Combinatorial Aspects of Commutative Algebra, Lecture Notes in Pure and Appl. Math., Dekker, New York, 217 (2001), 159-164.
  • M. Crupi and R. Utano, Extremal Betti numbers of graded ideals, Results Math., 43 (2003), 235-244.
  • D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995.
  • S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra, 129(1) (1990), 1-25.
  • 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, Ltd., London, 2011.
  • J. Herzog, L. Sharifan and M. Varbaro, The possible extremal Betti numbers of a homogeneous ideal, Proc. Amer. Math. Soc., 142(6) (2014), 1875-1891.
  • E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.
Year 2021, , 168 - 202, 17.07.2021
https://doi.org/10.24330/ieja.969656

Abstract

References

  • L. Amata and M. Crupi, Computation of graded ideals with given extremal Betti numbers in a polynomial ring, J. Symbolic Comput., 93 (2019), 120-132.
  • A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z., 228(2) (1998), 353-378.
  • A. Aramova, J. Herzog and T. Hibi, Shifting operations and graded Betti numbers, J. Algebraic Combin., 12(3) (2000), 207-222.
  • D. Bayer, H. Charalambous and S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra, 221(2) (1999), 497-512.
  • [5] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998.
  • S. M. Cooper and S. S. Wagsta , Connections Between Algebra, Combinatorics, and Geometry, Springer Proceedings in Mathematics & Statistics, 76, Springer-Verlag, 2014.
  • M. Crupi, Extremal Betti numbers of graded modules, J. Pure Appl. Algebra, 220(6) (2016), 2277-2288.
  • M. Crupi, A constructive method for standard Borel fixed submodules withgiven extremal Betti numbers, Mathematics, 56(5) (2017), 1-26.
  • M. Crupi, Computing general strongly stable modules with given extremal Betti numbers, J. Commut. Algebra, 12(1) (2020), 53-70.
  • M. Crupi and C. Ferro, Squarefree monomial modules and extremal Betti numbers, Algebra Colloq., 23(3) (2016), 519-530.
  • M. Crupi and R. Utano, Extremal Betti numbers of lexsegment ideals, Geometric and Combinatorial Aspects of Commutative Algebra, Lecture Notes in Pure and Appl. Math., Dekker, New York, 217 (2001), 159-164.
  • M. Crupi and R. Utano, Extremal Betti numbers of graded ideals, Results Math., 43 (2003), 235-244.
  • D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995.
  • S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra, 129(1) (1990), 1-25.
  • 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, Ltd., London, 2011.
  • J. Herzog, L. Sharifan and M. Varbaro, The possible extremal Betti numbers of a homogeneous ideal, Proc. Amer. Math. Soc., 142(6) (2014), 1875-1891.
  • E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Luca Amata This is me

Marilena Crupı This is me

Publication Date July 17, 2021
Published in Issue Year 2021

Cite

APA Amata, L., & Crupı, M. (2021). ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS. International Electronic Journal of Algebra, 30(30), 168-202. https://doi.org/10.24330/ieja.969656
AMA Amata L, Crupı M. ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS. IEJA. July 2021;30(30):168-202. doi:10.24330/ieja.969656
Chicago Amata, Luca, and Marilena Crupı. “ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS”. International Electronic Journal of Algebra 30, no. 30 (July 2021): 168-202. https://doi.org/10.24330/ieja.969656.
EndNote Amata L, Crupı M (July 1, 2021) ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS. International Electronic Journal of Algebra 30 30 168–202.
IEEE L. Amata and M. Crupı, “ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS”, IEJA, vol. 30, no. 30, pp. 168–202, 2021, doi: 10.24330/ieja.969656.
ISNAD Amata, Luca - Crupı, Marilena. “ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS”. International Electronic Journal of Algebra 30/30 (July 2021), 168-202. https://doi.org/10.24330/ieja.969656.
JAMA Amata L, Crupı M. ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS. IEJA. 2021;30:168–202.
MLA Amata, Luca and Marilena Crupı. “ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS”. International Electronic Journal of Algebra, vol. 30, no. 30, 2021, pp. 168-02, doi:10.24330/ieja.969656.
Vancouver Amata L, Crupı M. ON THE EXTREMAL BETTI NUMBERS OF SQUAREFREE MONOMIAL IDEALS. IEJA. 2021;30(30):168-202.