Advertisement

Acta Mathematica Vietnamica

, Volume 40, Issue 3, pp 353–374 | Cite as

Absolutely Koszul Algebras and the Backelin-Roos Property

  • Aldo ConcaEmail author
  • Srikanth B. Iyengar
  • Hop D. Nguyen
  • Tim Römer
Article

Abstract

We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos property and their behavior under standard algebraic operations. In particular, we identify some Veronese subrings of polynomial rings that have the Backelin-Roos property and conjecture that the list is indeed complete. Among other things, we prove that every universally Koszul ring defined by monomials has the Backelin-Roos property.

Keywords

Koszul algebras Free resolutions Veronese algebras 

Mathematics Subject Classification (2010)

13D02 

References

  1. 1.
    Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Vol. I. Grundlehren Math. Wiss., vol. 267. Springer, Berlin (1985)CrossRefGoogle Scholar
  2. 2.
    Avramov, L.L.: Local rings over which all modules have rational Poincaré series. J. Pure Appl. Algebra 91, 29–48 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Avramov, L.L.: Infinite free resolutions. In: Six lectures on commutative algebra (Bellaterra, 1996), 1–118, Progr. Math., vol. 166. Birkhäuser (1998)Google Scholar
  4. 4.
    Avramov, L.L., Iyengar, S.B., Şega, L.M.: Free resolutions over short local rings. J. London Math. Soc. 78, 459–476 (2008)zbMATHCrossRefGoogle Scholar
  5. 5.
    Avramov, L.L., Eisenbud, D.: Regularity of modules over a Koszul algebra. J. Algebra 153, 85–90 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Avramov, L.L., Peeva, I.: Finite regularity and Koszul algebras. Am. J. Math. 123, 275–281 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Avramov, L.L., Foxby, H.-B., Herzog, B.: Structure of local homomorphisms. J. Algebra 164, 124–145 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Backelin, J., Fröberg, R.: Koszul algebras, Veronese subrings, and rings with linear resolutions. Rev. Roumaine Math. Pures Appl. 30, 85–97 (1985)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Bruns, W., Conca, A.: Gröbner bases and determinantal ideals. Commutative algebra, singularities and computer algebra (Sinaia, 2002), 9–66, NATO Sci. Ser. II Math. Phys, vol. 115. Kluwer Acad. Publ., Dordrecht (2003)Google Scholar
  10. 10.
    Bruns, W., Herzog, J.: Cohen-Macaulay rings. Revised edition, Cambridge Studies Adv. Math, vol. 39. University Press, Cambridge (1998)CrossRefGoogle Scholar
  11. 11.
    Conca, A.: Gröbner bases for spaces of quadrics of low codimension. Adv. App. Math. 24, 111–124 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Conca, A.: Universally Koszul algebras. Math. Ann. 317, 329–346 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Conca, A.: Universally Koszul algebras defined by monomials. Rend. Sem. Mat. Univ. Padova 107, 95–99 (2002)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Conca, A., De Negri, E., Rossi, M.E.: Koszul algebra and regularity. In: Peeva, I. (ed.) Commutative Algebra: Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday, pp 285–315. Springer, New York (2013)Google Scholar
  15. 15.
    Conca, A., Rossi, M.E., Valla, G.: Gröbner flags and Gorenstein algebras. Compos. Math. 129(1), 95–121 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Conca, A., Trung, N.V., Valla, G.: Koszul property for points in projective space. Math. Scand. 89, 201–216 (2001)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Dress, A., Krämer, H.: Bettireihen von Faserprodukten lokaler Ringe. Math. Ann. 215, 79–82 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Eisenbud, D.: Green’s conjecture: an orientation for algebraists. In: Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance, UT, 1990), vol. 2, pp. 51–78, Res. Notes Math. Jones and Bartlett, Boston (1992)Google Scholar
  19. 19.
    Eisenbud, D., Fløystad, G., Schreyer, F.-O.: Sheaf cohomology and resolutions over the exterior algebra. Trans. Am. Math. Soc. 355, 4397–4426 (2003)zbMATHCrossRefGoogle Scholar
  20. 20.
    Fröberg, R.: On Stanley-Reisner rings. In: Topics in Algebra, vol. 26 Part 2, pp. 57–70. Banach Center Publications (1990)Google Scholar
  21. 21.
    Henriques, I.B., Şega, L.M.: Free resolutions over short Gorenstein local rings. Math. Z. 267, 645–663 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Herzog, J.: Algebra retracts and Poincaré series. Manuscr. Math. 21, 307–314 (1977)zbMATHCrossRefGoogle Scholar
  23. 23.
    Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya. Math. J. 153, 141–153 (1999)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Herzog, J., Hibi, T., Restuccia, G.: Strongly Koszul algebras. Math. Scand. 86, 161–178 (2000)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Herzog, J., Iyengar, S.: Koszul modules. J. Pure Appl. Algebra 201, 154–188 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Kempf, G.: Syzygies for points in projective space. J. Algebra 145, 219–223 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Kustin, A., Palmer Slattery, S.: The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function. J. Pure Appl. Algebra 95, 271–295 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Iyengar, S.B., Römer, T.: Linearity defects of modules over commutative rings. J. Algebra 322, 3212–3237 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Priddy, S.B.: Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Roos, J.-E.: Good and bad Koszul algebras and their Hochschild homology. J. Pure Appl. Algebra 201(1–3), 295–327 (2005)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  • Aldo Conca
    • 1
    Email author
  • Srikanth B. Iyengar
    • 2
  • Hop D. Nguyen
    • 1
  • Tim Römer
    • 3
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenovaItaly
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  3. 3.Institut für MathematikUniversität OsnabrückOsnabrückGermany

Personalised recommendations