Mathematische Zeitschrift

, Volume 277, Issue 3–4, pp 847–866 | Cite as

t-Structures and cotilting modules over commutative noetherian rings

Article

Abstract

For a commutative noetherian ring \(R\), we establish a bijection between the resolving subcategories consisting of finitely generated \(R\)-modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category \(\mathcal {D}(R)\) that contain \(R[1]\) in their heart. Under this bijection, the t-structures \((\mathcal U,\mathcal V)\) such that the aisle \(\mathcal U\) consists of objects with homology concentrated in degrees \(<n\) correspond to the \(n\)-cotilting classes in \({{\mathrm{Mod}\text {-}R}}\). As a consequence of these results, we prove that the little finitistic dimension findim\(R\) of \(R\) equals an integer \(n\) if and only if the direct sum \(\bigoplus _{k=0}^n E_k(R)\) of the first \(n+1\) terms in a minimal injective coresolution \(0\rightarrow R\rightarrow E_0(R)\rightarrow E_1(R)\rightarrow \cdots \) of \(R\) is an injective cogenerator of \({{\mathrm{Mod}\text {-}R}}\).

Keywords

t-Structure Tilting module Cotilting module Resolving subcategory Finitistic dimension  Gorenstein-injective Gorenstein-flat 

Mathematics Subject Classification (2010)

13D09 13D05 16E30 

References

  1. 1.
    Ajitabh, K., Smith, S.P., Zhang, J.J.: Auslander–Gorenstein rings. Commun. Algebra 26, 2159–2180 (1998)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alonso, L., Jeremías, A., Saorín, M.: Compactly generated t-structures on the derived category of a Noetherian ring. J. Algebra 324, 313–346 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Alonso, L., Jeremías, A., Souto, M.J.: Construction of t-structures and equivalences of derived categories. Trans. Am. Math. Soc. 355, 2523–2543 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Angeleri Hügel, L., Coelho, F.U.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13, 239–250 (2001)MATHMathSciNetGoogle Scholar
  5. 5.
    Angeleri Hügel, L., Herbera, D.: Herbera: Mittag–Leffler conditions on modules. Indiana Univ. Math. J. 57, 2459–2518 (2008)Google Scholar
  6. 6.
    Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Tilting modules and Gorenstein rings. Forum Math. 18(2), 211–229 (2006)MATHMathSciNetGoogle Scholar
  7. 7.
    Angeleri Hügel, L., Pospíšil, D., Š\(\check{{\rm t}}\)ovíček, J., Trlifaj, J.: Tilting, cotilting, and spectra of commutative noetherian rings. Trans. Am. Math. Soc., to appearGoogle Scholar
  8. 8.
    Auslander, M., Buchsbaum, D.: Homological dimension in regular local rings. Trans. AMS 85, 390–405 (1957)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bazzoni, S.: Cotilting modules are pure-injective. Proc. Am. Math. Soc. 131, 3665–3672 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bazzoni, S.: Cotilting and tilting modules over Pruefer domains. Forum Math. 19, 1005–1027 (2007)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Cartan, H., Eilenberg, S.: Homological Algebra, 7th edn. Oxford University Press, Oxford (1973)Google Scholar
  13. 13.
    Colpi, R., Trlifaj, J.: Tilting modules and tilting torsion theories. J. Algebra 178(2), 614–634 (1995)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Dao, H., Takahashi, R.: Classification of resolving subcategories and grade consistent functions. Int. Math. Res. Not. Available at http://arxiv.org/pdf/1202.5605v1.pdf (2012, to appear)
  15. 15.
    Enochs, E., Jenda, O.: Gorenstein injective and projective modules. Math. Zeitschr. 220, 611–633 (1995)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Enochs, E., Jenda, O., Torrecillas, B.: Gorenstein flat modules. Nanjing Daxue Xuebao Shuxue Bannian Kan 10, 1–9 (1993)MATHMathSciNetGoogle Scholar
  17. 17.
    Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)MATHMathSciNetGoogle Scholar
  18. 18.
    Goebel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. W. de Gruyter, Berlin (2006)CrossRefMATHGoogle Scholar
  19. 19.
    Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 575 (1996)Google Scholar
  20. 20.
    Huisgen-Zimmermann, B.: The finitistic dimension conjectures—a tale of 3.5 decades. In: Facchini, A., Menini, C., (eds.) Abelian Groups and Modules. Kluwer, Dordrecht, pp. 501–517 (1995)Google Scholar
  21. 21.
    Iwanaga, Y.: On rings with finite self-injective dimension. Commun. Algebra 7, 394–414 (1979)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Keller, B.: Deriving dg categories. Ann. Sci. Ecole. Norm. Sup. 27(1), 63–102 (1994)MATHMathSciNetGoogle Scholar
  23. 23.
    Kerner, O., Š\(\check{{\rm t}}\)ovíček, J., Trlifaj, J.: Tilting via torsion pairs and almost hereditary noetherian rings. J. Pure Appl. Algebra 215, 2072–2085 (2011)Google Scholar
  24. 24.
    Krause, H.: The spectrum of a module category. Mem. Am. Math. Soc. 707 (2001)Google Scholar
  25. 25.
    Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)MATHGoogle Scholar
  26. 26.
    Miyachi, J.: Duality for derived categories and cotilting bimodules. J. Algebra 185, 583–603 (1996)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Miyachi, J.: Injective resolutions of noetherian rings and cogenerators. Proc. Am. Math. Soc. 128, 2233–2242 (2000)Google Scholar
  28. 28.
    Pospíšil, D., Š\(\check{{\rm t}}\)ovíček, J.: On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings. Preprint (arXiv:1212.3122)
  29. 29.
    Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39, 436–456 (1989)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Š\(\check{{\rm t}}\)ovíček, J.: All \(n\)-cotilting modules are pure-injective. Proc. AMS 134, 1891–1897 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Università degli Studi di VeronaVeronaItaly
  2. 2.Departamento de MatemáticasUniversidad de MurciaEspinardoSpain

Personalised recommendations