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Absolutely Koszul Algebras and the Backelin-Roos Property

Acta Mathematica Vietnamica volume 40, pages 353–374 (2015)Cite this 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.

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References

  1. Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Vol. I. Grundlehren Math. Wiss., vol. 267. Springer, Berlin (1985)

    Book  Google Scholar 

  2. Avramov, L.L.: Local rings over which all modules have rational Poincaré series. J. Pure Appl. Algebra 91, 29–48 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  3. Avramov, L.L.: Infinite free resolutions. In: Six lectures on commutative algebra (Bellaterra, 1996), 1–118, Progr. Math., vol. 166. Birkhäuser (1998)

  4. Avramov, L.L., Iyengar, S.B., Şega, L.M.: Free resolutions over short local rings. J. London Math. Soc. 78, 459–476 (2008)

    MATH  Article  Google Scholar 

  5. Avramov, L.L., Eisenbud, D.: Regularity of modules over a Koszul algebra. J. Algebra 153, 85–90 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  6. Avramov, L.L., Peeva, I.: Finite regularity and Koszul algebras. Am. J. Math. 123, 275–281 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  7. Avramov, L.L., Foxby, H.-B., Herzog, B.: Structure of local homomorphisms. J. Algebra 164, 124–145 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  8. Backelin, J., Fröberg, R.: Koszul algebras, Veronese subrings, and rings with linear resolutions. Rev. Roumaine Math. Pures Appl. 30, 85–97 (1985)

    MATH  MathSciNet  Google Scholar 

  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. Bruns, W., Herzog, J.: Cohen-Macaulay rings. Revised edition, Cambridge Studies Adv. Math, vol. 39. University Press, Cambridge (1998)

    Book  Google Scholar 

  11. Conca, A.: Gröbner bases for spaces of quadrics of low codimension. Adv. App. Math. 24, 111–124 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  12. Conca, A.: Universally Koszul algebras. Math. Ann. 317, 329–346 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  13. Conca, A.: Universally Koszul algebras defined by monomials. Rend. Sem. Mat. Univ. Padova 107, 95–99 (2002)

    MATH  MathSciNet  Google Scholar 

  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)

  15. Conca, A., Rossi, M.E., Valla, G.: Gröbner flags and Gorenstein algebras. Compos. Math. 129(1), 95–121 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  16. Conca, A., Trung, N.V., Valla, G.: Koszul property for points in projective space. Math. Scand. 89, 201–216 (2001)

    MATH  MathSciNet  Google Scholar 

  17. Dress, A., Krämer, H.: Bettireihen von Faserprodukten lokaler Ringe. Math. Ann. 215, 79–82 (1975)

    MATH  MathSciNet  Article  Google Scholar 

  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)

  19. Eisenbud, D., Fløystad, G., Schreyer, F.-O.: Sheaf cohomology and resolutions over the exterior algebra. Trans. Am. Math. Soc. 355, 4397–4426 (2003)

    MATH  Article  Google Scholar 

  20. Fröberg, R.: On Stanley-Reisner rings. In: Topics in Algebra, vol. 26 Part 2, pp. 57–70. Banach Center Publications (1990)

  21. Henriques, I.B., Şega, L.M.: Free resolutions over short Gorenstein local rings. Math. Z. 267, 645–663 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  22. Herzog, J.: Algebra retracts and Poincaré series. Manuscr. Math. 21, 307–314 (1977)

    MATH  Article  Google Scholar 

  23. Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya. Math. J. 153, 141–153 (1999)

    MATH  MathSciNet  Google Scholar 

  24. Herzog, J., Hibi, T., Restuccia, G.: Strongly Koszul algebras. Math. Scand. 86, 161–178 (2000)

    MATH  MathSciNet  Google Scholar 

  25. Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Herzog, J., Iyengar, S.: Koszul modules. J. Pure Appl. Algebra 201, 154–188 (2005)

    MATH  MathSciNet  Article  Google Scholar 

  27. Kempf, G.: Syzygies for points in projective space. J. Algebra 145, 219–223 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  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)

    MATH  MathSciNet  Article  Google Scholar 

  29. Iyengar, S.B., Römer, T.: Linearity defects of modules over commutative rings. J. Algebra 322, 3212–3237 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  30. Priddy, S.B.: Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970)

    MATH  MathSciNet  Article  Google Scholar 

  31. Roos, J.-E.: Good and bad Koszul algebras and their Hochschild homology. J. Pure Appl. Algebra 201(1–3), 295–327 (2005)

    MATH  MathSciNet  Article  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genova, Italy

    Aldo Conca & Hop D. Nguyen

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Srikanth B. Iyengar

  3. Institut für Mathematik, Universität Osnabrück, 49069, Osnabrück, Germany

    Tim Römer

Authors
  1. Aldo Conca
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  2. Srikanth B. Iyengar
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  3. Hop D. Nguyen
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  4. Tim Römer
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Correspondence to Aldo Conca.

Additional information

Dedicated to Ngo Viet Trung on the occasion of his sixtieth birthday

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Conca, A., Iyengar, S.B., Nguyen, H.D. et al. Absolutely Koszul Algebras and the Backelin-Roos Property. Acta Math Vietnam 40, 353–374 (2015). https://doi.org/10.1007/s40306-015-0125-0

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  • DOI: https://doi.org/10.1007/s40306-015-0125-0

Keywords

  • Koszul algebras
  • Free resolutions
  • Veronese algebras

Mathematics Subject Classification (2010)

  • 13D02