Linear Resolutions of Powers and Products

  • Winfried BrunsEmail author
  • Aldo Conca


The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see, this condition is strongly correlated to good primary decompositions of the products and good homological and arithmetical properties of the associated multi-Rees algebras. The following families will be discussed in detail: polymatroidal ideals, ideals generated by linear forms, and Borel-fixed ideals of maximal minors. The main tools are Gröbner bases and Sagbi deformation.


Determinantal ideal Gröbner basis Ideal of linear forms Koszul algebra Linear resolution Polymatroidal ideal Primary decomposition Rees algebra Regularity Toric deformation 

AMS Subject classes (2010)

13A30 13D02 13C40 13F20 14M12 13P10 


  1. 1.
    Abbott, J., Bigatti, A.M., Lagorio, G.: CoCoA-5: a system for doing computations in ccommutative algebra. Available at
  2. 2.
    Aramova, A., Crona, K., De Negri, E.: Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions. J. Pure Appl. Algebra 150, 215–235 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blum, S.: Subalgebras of bigraded Koszul algebras. J. Algebra 242, 795–809 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brion, M.: Multiplicity-free subvarieties of flag varieties. Contemp. Math. 331, 13–23 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bruns, W., Conca, A.: KRS and determinantal rings. In: Herzog, J., Restuccia, G. (eds.) Geometric and Combinatorial Aspects of Commutative Algebra. Lecture Notes in Pure and Applied Mathematics, vol. 217, pp. 67–87. Dekker, New York (2001)Google Scholar
  6. 6.
    Bruns, W., Conca, A.: Gröbner bases and determinantal ideals. In: Herzog, J., Vuletescu, V. (eds.) Commutative Algebra, Singularities and Computer Algebra, pp. 9–66. Kluwer, Dordrecht (2003)CrossRefGoogle Scholar
  7. 7.
    Bruns, W., Conca, A.: Products of Borel fixed ideals of maximal minors. Preprint (2016). arXiv:1601.03987 [math.AC]Google Scholar
  8. 8.
    Bruns, W., Herzog, J.: Cohen-Macaulay Rings, Revised edition. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1998)Google Scholar
  9. 9.
    Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)Google Scholar
  10. 10.
    Bruns, W., Conca, A., Varbaro, M.: Maximal minors and linear powers. J. Reine Angew. Math. 702, 41–53 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bruns, W., Ichim, B., Sieg, R., Römer, T., Söger, C.: Normaliz. Algorithms for rational cones and affine monoids. Available at
  12. 12.
    Cartwright, D., Sturmfels, B.: The Hilbert scheme of the diagonal in a product of projective spaces. Int. Math. Res. Not. 9, 1741–1771 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Conca, A.: Regularity jumps for powers of ideals. In: Corso, A. (ed.) Commutative Algebra. Lecture Notes in Pure and Applied Mathematics, vol. 244, pp. 21–32. Chapman & Hall/CRC, Boca Raton, FL (2006)Google Scholar
  14. 14.
    Conca, A., Herzog, J.: Castelnuovo–Mumford regularity of products of ideals. Collect. Math. 54, 137–152 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Conca, A., Herzog, J., Valla, G.: Sagbi bases with applications to blow–up algebras. J. Reine Angew. Math. 474, 113–138 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases for maximal minors. Int. Math. Res. Not. 11, 3245–3262 (2015)zbMATHGoogle Scholar
  17. 17.
    Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases and Cartwright-Sturmfels ideals. Preprint (2016)Google Scholar
  18. 18.
    De Negri, E.: Toric rings generated by special stable sets of monomials. Math. Nachr. 203, 31–45 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-0-2 — a computer algebra system for polynomial computations. Available at
  20. 20.
    Derksen, H., Sidman, J.: Castelnuovo-Mumford regularity by approximation. Adv. Math. 188, 10–123 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88, 89–133 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. Available at
  23. 23.
    Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, 2nd extended edn. Springer, Berlin (2007)Google Scholar
  24. 24.
    Herzog, J., Hibi, T.: Discrete polymatroids. J. Algebr. Comb. 16, 239–268 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, Berlin (2010)Google Scholar
  26. 26.
    Herzog, J., Vasconcelos, W.: On the divisor class group of Rees-Algebras. J. Algebra 93, 182–188 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Herzog, J., Hibi, T., Vladoiu, M.: Ideals of fiber type and polymatroids. Osaka J. Math. 42, 807–829 (2005)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Herzog, J., Rauf, A., Vladoiu, M.: The stable set of associated prime ideals of a polymatroidal ideal. J. Algebraic Comb. 37, 289–312 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lasoń, M., Michałek, M.: On the toric ideal of a matroid. Adv. Math. 259, 1–12 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Robbiano, L., Sweedler, M.: Subalgebra bases. In: Bruns, W., Simis, A. (eds.) Commutative Algebra. Proceedings Salvador 1988. Lecture Notes in Mathematics, vol. 1430, pp. 61–87. Springer, Berlin (1990)Google Scholar
  32. 32.
    Römer, T.: Homological properties of bigraded algebras. Ill. J. Math. 45, 1361–1376 (2001)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence, RI (1996)Google Scholar
  34. 34.
    West, E.: Primes associated to multigraded modules. J. Algebra 271, 427–453 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für MathematikUniversität OsnabrückOsnabrückGermany
  2. 2.Dipartimento di MatematicaUniversità degli Studi di GenovaGenovaItaly

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