Osaka Journal of Mathematics

Generic initial ideals and fibre products

Aldo Conca and Tim Römer

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Abstract

We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute explicitly the generic initial ideal of a fibre product in a special case. We also prove that the fibre product of two graded ideals is componentwise linear if and only if both ideals have this property.

Article information

Source
Osaka J. Math., Volume 47, Number 1 (2010), 17-32.

Dates
First available in Project Euclid: 19 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1266586783

Mathematical Reviews number (MathSciNet)
MR2666122

Zentralblatt MATH identifier
1189.13011

Subjects
Primary: 13C99: None of the above, but in this section 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Secondary: 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10]

Citation

Conca, Aldo; Römer, Tim. Generic initial ideals and fibre products. Osaka J. Math. 47 (2010), no. 1, 17--32. https://projecteuclid.org/euclid.ojm/1266586783


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References

  • CoCoATeam: CoCoA: A System for Doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
  • A. Conca: Koszul homology and extremal properties of Gin and Lex, Trans. Amer. Math. Soc. 356 (2004), 2945--2961.
  • A. Conca: Symmetric ladders, Nagoya Math. J. 136 (1994), 35--56.
  • A. Conca and J. Herzog: Castelnuovo-Mumford regularity of products of ideals, Collect. Math. 54 (2003), 137--152.
  • A. Conca, J. Herzog and G. Valla: Sagbi bases with applications to blow-up algebras, J. Reine Angew. Math. 474 (1996), 113--138.
  • D. Eisenbud: Commutative Algebra, With a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer, New York, 1995.
  • M.L. Green: Generic initial ideals; in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math. 166, Birkhäuser, Basel, 1998, 119--186.
  • J. Herzog: Generic initial ideals and graded Betti numbers; in Computational Commutative Algebra and Combinatorics (Osaka, 1999), Adv. Stud. Pure Math. 33, Math. Soc. Japan, Tokyo, 2002, 75--120.
  • J. Herzog and T. Hibi: Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141--153.
  • J. Herzog, V. Reiner and V. Welker: Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (1999), 211--223.
  • G. Kalai: Algebraic shifting; in Computational Commutative Algebra and Combinatorics (Osaka, 1999), Adv. Stud. Pure Math. 33, Math. Soc. Japan, Tokyo, 2002, 121--163.
  • D.R. Grayson and M.E. Stillman: Macaulay 2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
  • E. Nevo: Algebraic shifting and basic constructions on simplicial complexes, J. Algebraic Combin. 22 (2005), 411--433.
  • E. Nevo: Algebraic shifting and f-vector theory. Ph. D. thesis, Hebrew University of Jerusalem (2007).
  • G.-M. Greuel, G. Pfister and H. Schönemann: Singular 3.0. A Computer Algebra System for Polynomial Computations, available at http://www.singular.uni-kl.de/.
  • B. Sturmfels: Gröbner Bases and Convex Polytopes, University Lecture Series 8, Amer. Math. Soc., Providence, RI, 1996.