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1999 | 80 | 2 | 267-292
Tytuł artykułu

A duality result for almost split sequences

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.
Słowa kluczowe
Rocznik
Tom
80
Numer
2
Strony
267-292
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-08-11
poprawiono
1998-12-30
Twórcy
  • Mathematisches Institut der Universität, Theresienstraße 39, D-80333 München, Germany
  • Mathematisches Institut der Universität, Theresienstraße 39, D-80333 München, Germany
Bibliografia
  • [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed., Springer, New York, 1992.
  • [2] L. Angeleri Hügel, $P_1$-hereditary artin algebras, J. Algebra 176 (1995), 460-479.
  • [3] L. Angeleri Hügel, Almost split sequences arising from the preprojective partition, ibid. 194 (1997), 1-13.
  • [4] L. Angeleri Hügel, Finitely cotilting modules, Comm. Algebra, to appear.
  • [5] L. Angeleri Hügel, On some precovers and preenvelopes, preprint, 1998.
  • [6] M. Auslander, Large modules over artin algebras, in: Algebra, Topology, Category Theory, Academic Press, 1976, 1-17.
  • [7] M. Auslander, Functors and morphisms determined by objects, in: Lecture Notes in Pure and Appl. Math. 37, Marcel Dekker, 1978, 1-244.
  • [8] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. 94 (1969).
  • [9] M. Auslander and I. Reiten, Representation theory of artin algebras III. Almost split sequences, Comm. Algebra 3 (1975), 239-294.
  • [10] M. Auslander, I. Reiten and S. O. Smalο, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, 1995.
  • [11] M. Auslander and S. O. Smalο, Preprojective modules over artin algebras, J. Algebra 66 (1980), 61-122.
  • [12] R. R. Colby and K. R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), 1585-1615.
  • [13] R. Colpi, Some remarks on equivalences between categories of modules, ibid. 18 (1990), 1935-1951.
  • [14] R. Colpi, Tilting modules and *-modules, ibid. 21 (1993), 1095-1102.
  • [15] R. Colpi, G. D'Este and A. Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), 461-494.
  • [16] N. V. Dung, Preinjective modules and finite representation type of artinian rings, Comm. Algebra, to appear.
  • [17] P. Gabriel and A. V. Roiter, Algebra VIII: Representations of Finite-Dimensional Algebras, Encyclopedia Math. Sci. 73, Springer, 1992.
  • [18] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443.
  • [19] I. Herzog, A test for finite representation type, J. Pure Appl. Algebra 95 (1994), 151-182.
  • [20] M. Hoshino, Tilting modules and torsion theories, Bull. London Math. Soc. 14 (1982), 334-336.
  • [21] M. Hoshino, On splitting torsion theories induced by tilting modules, Comm. Algebra 11 (1983), 427-439.
  • [22] R C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984.
  • [23] M. Schmidmeier, A dichotomy for finite length modules induced by local duality, Comm. Algebra 25 (1997), 1933-1944.
  • [24] M. Schmidmeier, The local duality for homomorphisms and an application to pure semisimple PI-rings, Colloq. Math. 77 (1998), 121-132.
  • [25] D. Simson, Pure semisimple categories and rings of finite representation type, J. Algebra 48 (1977), 290-296; Corrigendum, ibid. 67 (1980), 254-256.
  • [26] D. Simson, On pure-semisimple Grothendieck categories, I, Fund. Math. 100 (1978), 211-222.
  • [27] D. Simson, Partial Coxeter functors and right pure semisimple hereditary rings, J. Algebra 71 (1981), 195-218.
  • [28] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon and Breach, 1992.
  • [29] D. Simson, An Artin problem for division ring extensions and the pure semisimplicity conjecture I, Arch. Math. (Basel) 66 (1996), 114-122.
  • [30] D. Simson, A class of potential counter-examples to the pure semisimplicity conjecture, in: Adv. Algebra Model Theory 9, Gordon and Breach, 1997, 345-373.
  • [31] H. Valenta, Existence criteria and construction methods for certain classes of tilting modules, Comm. Algebra 22 (1994), 6047-6072.
  • [32] W. Zimmermann, Existenz von Auslander-Reiten-Folgen, Arch. Math. (Basel) 40 (1983), 40-49.
  • [33] B. Zimmermann-Huisgen, Strong preinjective partitions and representation type of artinian rings, Proc. Amer. Math. Soc. 109 (1990), 309-322.
  • [34] B. Zimmermann-Huisgen and W. Zimmermann, On the sparsity of representations of rings of pure global dimension zero, Trans. Amer. Math. Soc. 320 (1990), 695-711.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv80i2p267bwm
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