Abstract
We prove that the multi-Rees algebra \({\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)\) of a collection of strongly stable ideals \(I_1, \ldots , I_r\) is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of \({\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)\) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of \({\mathcal {R}}(I_1 \oplus I_2)\) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.
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References
Bayer, D., Stillman, M.: A theorem on refining division orders by the reverse lexicographical order. Duke Math. J. 55, 321–328 (1987)
Blasiak, J.: The toric ideal of a graphic matroid is generated by quadrics. Combinatorica 28(3), 283–297 (2008)
Blum, S.: Subalgebras of bigraded Koszul algebras. J. Algebra 242, 795–808 (2001)
Bruns, W., Conca, A.: Linear resolutions of powers and products. In: Singularities and Computer Algebra, pp. 47–69. Springer, Cham (2017)
Chen, F., Wang, W., Liu, Y.: Computing singular points of plane rational curves. J. Symb. Comput. 43, 92–117 (2008)
Conca, A.: Symmetric ladders. Nagoya Math. J. 136, 35–36 (1994)
Conca, A., De Negri, E., Rossi, M.: Koszul algebras and regularity. In: Commutative Algebra, pp. 285–315. Springer, New York (2013)
Cox, D.: Applications of polynomial systems. In: CBMS Regional Conference Series in Mathematics, vol. 134. American Mathematical Society (2020)
Cox, D., Lin, K.-N., Sosa, G.: Multi-Rees algebras and toric dynamical systems. Proc. Am. Math. Soc. 147(11), 4605–4616 (2019)
De Negri, E.: Toric rings generated by special stable sets of monomials. Math. Nachr. 203(1), 31–45 (1999)
DiPasquale, M., Francisco, C., Mermin, J., Schweig, J., Sosa, G.: The Rees algebra of a two-Borel ideal is Koszul. Proc. Am. Math. Soc. 147(2), 467–479 (2019)
DiPasquale, M., Jabbar Nezhad, B.: Koszul multi-Rees algebras of principal \(L\)-Borel ideals. J. Algebra 581, 353–385 (2021)
Ene, V., Herzog, J.: Gröbner Bases in Commutative Algebra. American Mathematical Society, Providence (2011)
Fröberg, R.: Koszul algebras. In: Advances in Commutative Ring Theory, Lecture Notes in Pure and Applied Math, vol. 205, pp. 337–350. Dekker, New York (1999)
Galligo, A.: À propos du théorème de-préparation de Weierstrass. In: Fonctions de Plusieurs Variables Complexes, Lecture Notes in Math, vol. 409, pp. 543–579. Springer, Berlin (1974)
Grayson, Daniel R., Stillman, Michael E.: Macaulay2, a Software System for Research in Algebraic Geometry. Available at http://www.math.uiuc.edu/Macaulay2/
Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics. Springer-Verlag, London (2011)
Herzog, J., Hibi, T., Vladoiu, M.: Ideals of fiber type and polymatroids. Osaka J. Math. 42(4), 807–829 (2005)
Jabarnejad, B.: Equations defining the multi-Rees algebras of powers of an ideal. J. Pure Appl. Algebra 222, 1906–1910 (2018)
Lin, K.-N., Polini, C.: Rees algebras of truncations of complete intersections. J. Algebra 410(9), 36–52 (2014)
Schweig, J.: Toric ideals of lattice path matroids and polymatroids. J. Pure Appl. Algebra 215(11), 2660–2665 (2011)
Sosa, G.: On the Koszulness of Multi-Rees Algebras of Certain Strongly Stable Ideals. arXiv preprint arXiv:1406.2188, (2014)
Sturmfels, B.: Gröbner bases and convex polytopes. University Lecture Series. American Mathematical Society, Providence (1996)
Vasconcelos, W.: Arithmetic of Blowup Algebras, London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1994)
Acknowledgements
The authors sincerely thank the reviewers for their helpful and constructive suggestions which improved the presentation of the manuscript immensely. The first named author was partially supported by the University of South Alabama Arts and Sciences Support and Development Award. Thanks to this award, part of this work was done while the second author visited the University of South Alabama. The second author thanks the Department of Mathematics and Statistics for their hospitality. Many of the computations related to this paper was done using Macaulay2 [16].
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Kara, S., Lin, KN. & Castillo, G.S. Multi-Rees algebras of strongly stable ideals. Collect. Math. 75, 213–246 (2024). https://doi.org/10.1007/s13348-022-00385-2
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DOI: https://doi.org/10.1007/s13348-022-00385-2
Keywords
- Rees algebra
- Special fiber ring
- Toric ring
- Koszul algebra
- Gröbner basis
- Strongly stable ideals
- Fiber graph