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TTF Triples in Functor Categories

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Abstract

We characterize the hereditary torsion pairs of finite type in the functor category of a ring R that are associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a condition under which they give rise to TTF triples.

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References

  1. Angeleri Hügel, L.: A key module over pure-semisimple hereditary rings. J. Algebra 307, 361–376 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Angeleri Hügel, L., Bazzoni, S., Herbera, D.: A solution to the Baer splitting problem. Trans. Amer. Math. Soc. 360(5), 2409–2421 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Tilting modules and Gorenstein rings. Forum Math. 18, 211–229 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Angeleri Hügel, L., Valenta, H.: A duality result for almost split sequences. Colloq. Math. 80, 267–292 (1999)

    MATH  MathSciNet  Google Scholar 

  5. Auslander, M.: Coherent functors. In: Proceedings of the Conference on Categorical Algbera (La Jolla1965), pp. 189–231. Spinger, New York (1966)

    Google Scholar 

  6. Auslander, M.: Functors and morphisms determined by objects. Lect. Notes in Pure and Appl. Math. 37, 1–244 (1978)

    MathSciNet  Google Scholar 

  7. Auslander, M.: Isolated singularities and almost split sequences. In: Representation Theory II. LNM, vol. 117, pp. 194–242. Springer, New York (1986)

    Google Scholar 

  8. Bazzoni, S.: Cotilting modules are pure-injective. Proc. Amer. Math. Soc. 131, 3665–3672 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bazzoni, S.: When are definable classes tilting or cotiling classes? J. Algebra 320(12), 4281–4299 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bazzoni, S., Herbera, D.: One dimensional tilting modules are of finite type. Algebr. Represent. Theory 11(1), 43–61 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bazzoni, S., Herbera, D.: Cotorsion pairs generated by modules of bounded projective dimension. Israel J. Math (in press)

  12. Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceux Perverse. Astèrisque 100, 19820

  13. Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), 207 pages (2007)

    Google Scholar 

  14. Buan, A.B., Krause, H.: Cotilting modules over tame hereditary algebras. Pacific J. Math. 211(1), 41–59 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Colby, R.R., Fuller, K.R.: Tilting, cotilting and serially tilted rings. Comm. Algebra 25(10), 3225–3237 (1997)

    Article  MathSciNet  Google Scholar 

  16. Colpi, R., Trlifaj, J.: Tilting modules and tilting torsion theories. J. Algebra 178, 614–634 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Crawley-Boevey, W.W.: Locally finitely presented additive categories. Comm. Algebra 22, 1644–1674 (1994)

    Google Scholar 

  18. Crawley-Boevey, W.W.: Infinite-dimensional modules in the representation theory of finite-dimensional algebras. In: Algebras and Modules, I (Trondheim, 1996). CMS Conf. Proc. Amer. Math. Soc., vol. 23, pp. 29–54. American Mathematical Society, Providence (1998)

    Google Scholar 

  19. Dickson, S.E.: A torsion theory for abelian categories. Trans. Amer. Math. Soc. 121, 223–235 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  20. Facchini, A.: Divisible modules over integral domains. Ark. Mat. 26(1), 67–85 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Facchini, A.: A tilting module over commutative integral domains. Comm. Algebra 15(11), 2235–2250 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fuchs, L., Salce, L.: Modules over non-noetherian domains. In: Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001)

    Google Scholar 

  23. Garkusha, G.A.: Algebra i Analiz. St. Petersburg Math. J. 13(2), 149–200 (2002) (transl.)

    MATH  MathSciNet  Google Scholar 

  24. Gentle, R.: T.T.F. theories in abelian categories. Comm. Algebra 16, 877–908 (1996)

    Article  MathSciNet  Google Scholar 

  25. Goebel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. W. de Gruyter, Berlin (2006)

    Book  MATH  Google Scholar 

  26. Gruson, L., Jensen, C.U.: Dimension cohomologique reliées aux functeurs \(\varprojlim^{(i)}\). In: Séminair d’algèbre. LNM, vol. 867, pp. 234–294 (1981)

  27. Herzog, I.: The Ziegler spectrum of a locally coherent Grothendieck catgeory. Proc. London Math. Soc. 74, 503–558 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jans, J.P.: Some aspects of torsion. Pacific J. Math. 15, 1249–1259 (1965)

    MATH  MathSciNet  Google Scholar 

  29. Jensen, C.U., Lenzing, H.: Model-theoretic algebra with particular emphasis on fields, rings, modules. In: Algebra, Logic and Applications, vol. 2. Gordon and Breach Science, New York (1989)

    Google Scholar 

  30. Kaplansky, I.: A characterization of Prüfer domains. J. Indian Math. Soc. 24, 279–281 (1960)

    MathSciNet  Google Scholar 

  31. Kerner, O., Trlifaj, J.: Tilting classes over wild hereditary algebras. J. Algebra 290, 538–556 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Krause, H.: The spectrum of a module category. Mem. Amer. Math. Soc. 149(707), 125 pages (2001)

    Google Scholar 

  33. Krause, H.: A short proof for Auslander’s defect formula. Linear Algebra Appl. 365, 267–270 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lukas, F.: Infinite-dimensional modules over wild hereditary algebras. J. London Math. Soc. 44, 401–419 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lukas, F.: A class of infinite-rank modules over tame hereditary algebras. J. Algebra 158, 18–30 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  36. Modoi, C.: Equivalences induced by adjoint functors. Comm. Algebra 31(5), 2327–2355 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nicolás, P.: Sobre las ternas TTF. Ph.D. thesis, Murcia (2007)

  38. Reiten, I., Ringel, C.M.: Infinite dimensional representations of canonical algebras. Canad. J. Math. 58, 180–224 (2006)

    MATH  MathSciNet  Google Scholar 

  39. Ringel, C.M.: Infinite dimensional representations of finite dimensional hereditary algebras. Sympos. Math. 23, 321–412 (1979)

    MathSciNet  Google Scholar 

  40. Stenström, B.: Rings of quotients. In: Grundleheren der Math., vol. 217. Springer, New York (1975)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica - Settore di Matematica, Universita’ di Verona, Strada le Grazie 15 - Ca’ Vignal, 37134, Verona, Italy

    Lidia Angeleri Hügel

  2. Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121, Padova, Italy

    Silvana Bazzoni

Authors
  1. Lidia Angeleri Hügel
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  2. Silvana Bazzoni
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Corresponding author

Correspondence to Silvana Bazzoni.

Additional information

First named author partially supported by the DGI and the European Regional Development Fund, jointly, through Project MTM2005–00934, and by the Comissionat per Universitats i Recerca of the Generalitat de Catalunya, Project 2005SGR00206.

First and second named authors supported by MIUR, PRIN 2005, project “Perspectives in the theory of rings, Hopf algebras and categories of modules”.

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Angeleri Hügel, L., Bazzoni, S. TTF Triples in Functor Categories. Appl Categor Struct 18, 585–613 (2010). https://doi.org/10.1007/s10485-009-9188-1

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Keywords

Mathematics Subject Classifications (2000)