Advertisement

Collectanea Mathematica

, Volume 63, Issue 2, pp 165–180 | Cite as

Presentations of rings with non-trivial semidualizing modules

  • David A. JorgensenEmail author
  • Graham J. Leuschke
  • Sean Sather-Wagstaff
Article

Abstract

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and \({{\rm Hom}_R(C,C)\cong R}\) . We prove that a Cohen–Macaulay ring R with dualizing module D admits a semidualizing module C satisfying \({R\ncong C \ncong D}\) if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that R admits a dualizing module if and only if R is Cohen–Macaulay and a homomorphic image of a local Gorenstein ring.

Keywords

Gorenstein rings Semidualizing modules Self-orthogonal modules Tor-independence Tate Tor Tate Ext 

Mathematics Subject Classification (2000)

13C05 13D07 13H10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Auslander, M.: Anneaux de Gorenstein, et torsion en algèbre commutative. Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, vol. 1966/67, Secrétariat mathématique, Paris. MR 37 #1435 (1967)Google Scholar
  2. 2.
    Auslander, M., Bridger, M.: Stable module theory. In: Memoirs of the American Mathematical Society, No. 94. American Mathematical Society, Providence (1969). MR 42 #4580Google Scholar
  3. 3.
    Avramov L.L., Foxby H.-B.: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc. (3) 75(2), 241–270 (1997) MR 98d:13014MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Avramov L.L., Martsinkovsky A.: Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. Lond. Math. Soc. (3) 85, 393–440 (2002) MR 2003g:16009MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bruns, W., Herzog, J.: Cohen-Macaulay rings, revised edn. In: Studies in Advanced Mathematics, vol. 39. University Press, Cambridge (1998). MR 1251956 (95h:13020)Google Scholar
  6. 6.
    Christensen, L.W.: Gorenstein dimensions. In: Lecture Notes in Mathematics, vol. 1747. Springer-Verlag, Berlin (2000). MR 2002e:13032Google Scholar
  7. 7.
    Christensen L.W.: Semi-dualizing complexes and their Auslander categories. Trans. Am. Math. Soc. 353(5), 1839–1883 (2001) MR 2002a:13017zbMATHCrossRefGoogle Scholar
  8. 8.
    Dorroh J.L.: Concerning adjunctions to algebras. Bull. Am. Math. Soc. 38(2), 85–88 (1932) MR 1562332MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ferrand D., Raynaud M.: Fibres formelles d’un anneau local noethérien. Ann. Sci. École Norm. Sup. (4) 3, 295–311 (1970) MR 0272779 (42 #7660)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Foxby H.-B.: Gorenstein modules and related modules. Math. Scand. 31(1972), 267–284 (1973) MR 48 #6094MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gerko A.A.: On the structure of the set of semidualizing complexes. Illinois J. Math. 48(3), 965–976 (2004) MR 2114263MathSciNetzbMATHGoogle Scholar
  12. 12.
    Golod E.S.: G-dimension and generalized perfect ideals. Trudy Mat. Inst. Steklov. 165, 62–66 (1984) Algebraic geometry and its applications. MR 85m:13011MathSciNetzbMATHGoogle Scholar
  13. 13.
    Grothendieck, A.: Théorèmes de dualité pour les faisceaux algébriques cohérents. Séminaire Bourbaki, vol. 4. Soc. Math. France, Paris, pp. Exp. No. 149, 169–193 (1995). MR 1610898Google Scholar
  14. 14.
    Hartshorne, R.: Local Cohomology. A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961. Springer-Verlag, Berlin (1967). MR 0224620 (37 #219)Google Scholar
  15. 15.
    Hochschild G.: On the cohomology groups of an associative algebra. Ann. Math. (2) 46, 58–67 (1945) MR 0011076 (6,114f)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Holm H., Jørgensen P.: Semi-dualizing modules and related Gorenstein homological dimensions. J. Pure Appl. Algebra 205(2), 423–445 (2006) MR 2203625MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Reiten I.: The converse to a theorem of Sharp on Gorenstein modules. Proc. Am. Math. Soc. 32, 417–420 (1972) MR 0296067 (45 #5128)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Sather-Wagstaff, S.: Bass numbers and semidualizing complexes. In: Commutative Algebra and its Appliciations, pp. 349–381. Walter de Gruyter, Berlin (2009). MR 2640315Google Scholar
  19. 19.
    Sather-Wagstaff, S., Sharif, T., White, D.: Tate cohomology with respect to semidualizing modules. Preprint (2009). arXiv:math.AC/0907.4969v1Google Scholar
  20. 20.
    Sharp R.Y.: Gorenstein modules. Math. Z. 115, 117–139 (1970) MR 0263801 (41 #8401)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sharp R.Y.: On Gorenstein modules over a complete Cohen-Macaulay local ring. Quart. J. Math. Oxford Ser. (2) 22, 425–434 (1971) MR 0289504 (44 #6693)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Vasconcelos, W.V.: Divisor theory in module categories. In: North-Holland Mathematics Studies, No. 14, Notas de Matemática No. 53. [Notes on Mathematics, No. 53]. North-Holland, Amsterdam (1974). MR 0498530 (58 #16637)Google Scholar
  23. 23.
    Wakamatsu T.: On modules with trivial self-extensions. J. Algebra 114(1), 106–114 (1988) MR 931903 (89b:16020)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Universitat de Barcelona 2010

Authors and Affiliations

  • David A. Jorgensen
    • 1
    Email author
  • Graham J. Leuschke
    • 2
  • Sean Sather-Wagstaff
    • 3
  1. 1.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA
  2. 2.Mathematics DepartmentSyracuse UniversitySyracuseUSA
  3. 3.Department of MathematicsNDSU Dept # 2750FargoUSA

Personalised recommendations