Research Article
Year 2021,
Volume: 30 Issue: 30, 168 - 202, 17.07.2021
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,
Volume: 30 Issue: 30, 168 - 202, 17.07.2021
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 | |
| Publication Date | July 17, 2021 |
| Published in Issue | Year 2021 Volume: 30 Issue: 30 |