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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Angeleri Hügel, L.: A key module over pure-semisimple hereditary rings. J. Algebra 307, 361–376 (2007)
Angeleri Hügel, L., Bazzoni, S., Herbera, D.: A solution to the Baer splitting problem. Trans. Amer. Math. Soc. 360(5), 2409–2421 (2008)
Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Tilting modules and Gorenstein rings. Forum Math. 18, 211–229 (2006)
Angeleri Hügel, L., Valenta, H.: A duality result for almost split sequences. Colloq. Math. 80, 267–292 (1999)
Auslander, M.: Coherent functors. In: Proceedings of the Conference on Categorical Algbera (La Jolla1965), pp. 189–231. Spinger, New York (1966)
Auslander, M.: Functors and morphisms determined by objects. Lect. Notes in Pure and Appl. Math. 37, 1–244 (1978)
Auslander, M.: Isolated singularities and almost split sequences. In: Representation Theory II. LNM, vol. 117, pp. 194–242. Springer, New York (1986)
Bazzoni, S.: Cotilting modules are pure-injective. Proc. Amer. Math. Soc. 131, 3665–3672 (2003)
Bazzoni, S.: When are definable classes tilting or cotiling classes? J. Algebra 320(12), 4281–4299 (2008)
Bazzoni, S., Herbera, D.: One dimensional tilting modules are of finite type. Algebr. Represent. Theory 11(1), 43–61 (2008)
Bazzoni, S., Herbera, D.: Cotorsion pairs generated by modules of bounded projective dimension. Israel J. Math (in press)
Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceux Perverse. Astèrisque 100, 19820
Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), 207 pages (2007)
Buan, A.B., Krause, H.: Cotilting modules over tame hereditary algebras. Pacific J. Math. 211(1), 41–59 (2003)
Colby, R.R., Fuller, K.R.: Tilting, cotilting and serially tilted rings. Comm. Algebra 25(10), 3225–3237 (1997)
Colpi, R., Trlifaj, J.: Tilting modules and tilting torsion theories. J. Algebra 178, 614–634 (1995)
Crawley-Boevey, W.W.: Locally finitely presented additive categories. Comm. Algebra 22, 1644–1674 (1994)
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)
Dickson, S.E.: A torsion theory for abelian categories. Trans. Amer. Math. Soc. 121, 223–235 (1966)
Facchini, A.: Divisible modules over integral domains. Ark. Mat. 26(1), 67–85 (1988)
Facchini, A.: A tilting module over commutative integral domains. Comm. Algebra 15(11), 2235–2250 (1987)
Fuchs, L., Salce, L.: Modules over non-noetherian domains. In: Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001)
Garkusha, G.A.: Algebra i Analiz. St. Petersburg Math. J. 13(2), 149–200 (2002) (transl.)
Gentle, R.: T.T.F. theories in abelian categories. Comm. Algebra 16, 877–908 (1996)
Goebel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. W. de Gruyter, Berlin (2006)
Gruson, L., Jensen, C.U.: Dimension cohomologique reliées aux functeurs \(\varprojlim^{(i)}\). In: Séminair d’algèbre. LNM, vol. 867, pp. 234–294 (1981)
Herzog, I.: The Ziegler spectrum of a locally coherent Grothendieck catgeory. Proc. London Math. Soc. 74, 503–558 (1997)
Jans, J.P.: Some aspects of torsion. Pacific J. Math. 15, 1249–1259 (1965)
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)
Kaplansky, I.: A characterization of Prüfer domains. J. Indian Math. Soc. 24, 279–281 (1960)
Kerner, O., Trlifaj, J.: Tilting classes over wild hereditary algebras. J. Algebra 290, 538–556 (2005)
Krause, H.: The spectrum of a module category. Mem. Amer. Math. Soc. 149(707), 125 pages (2001)
Krause, H.: A short proof for Auslander’s defect formula. Linear Algebra Appl. 365, 267–270 (2003)
Lukas, F.: Infinite-dimensional modules over wild hereditary algebras. J. London Math. Soc. 44, 401–419 (1991)
Lukas, F.: A class of infinite-rank modules over tame hereditary algebras. J. Algebra 158, 18–30 (1993)
Modoi, C.: Equivalences induced by adjoint functors. Comm. Algebra 31(5), 2327–2355 (2003)
Nicolás, P.: Sobre las ternas TTF. Ph.D. thesis, Murcia (2007)
Reiten, I., Ringel, C.M.: Infinite dimensional representations of canonical algebras. Canad. J. Math. 58, 180–224 (2006)
Ringel, C.M.: Infinite dimensional representations of finite dimensional hereditary algebras. Sympos. Math. 23, 321–412 (1979)
Stenström, B.: Rings of quotients. In: Grundleheren der Math., vol. 217. Springer, New York (1975)
Author information
Authors and Affiliations
Corresponding author
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”.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-009-9188-1