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Algebras and Representation Theory

, Volume 16, Issue 6, pp 1809–1827 | Cite as

Almost Split Sequences and Approximations

  • Shiping LiuEmail author
  • Puiman Ng
  • Charles Paquette
Article

Abstract

Let \(\mathcal A\) be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in \(\mathcal{A}\), which yields some necessary and sufficient conditions for \(\mathcal A\) to have almost split sequences. Then, we study when an almost split sequence in \(\mathcal A\) induces an almost split sequence in an exact subcategory \(\mathcal C\) of \(\mathcal A\). In case \(\mathcal A\) has almost split sequences and \(\mathcal C\) is Ext-finite and Krull–Schmidt, we obtain a necessary and sufficient condition for \(\mathcal C\) to have almost split sequences. Finally, we show some applications of these results.

Keywords

Krull–Schmidt categories Exact categories Almost split sequences Approximations Functorially finite subcategories Non-degenerate bilinear forms 

Mathematics Subject Classifications (2010)

16G20 16G70 18E10 18E40 

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References

  1. 1.
    Anderson, F.W., Fuller, K.R.: Rings and categories of modules. In: Graduate Text Books 13. Springer, New York (1973)Google Scholar
  2. 2.
    Auslander, M.: Functors and morphisms determined by objects. Representation theory of algebras (Proc. Conf. Temple Univ., 1976). In: Lecture Notes in Pure Appl. Math., vol. 37, pp. 1–244. Dekker, New York (1978)Google Scholar
  3. 3.
    Auslander, M.: Almost split sequences and algebraic geometry. Representations of algebras (Durham, 1985). In: London Math. Soc. Lecture Note Series, vol. 116, pp. 165–179. Cambridge University Press, Cambridge (1986)Google Scholar
  4. 4.
    Auslander, M., Reiten, I.: Representation theory of artin algebras III. Commun. Algebra 3, 239–294 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Auslander, M., Reiten, I.: Representation theory of artin algebras IV. Commun. Alebra 5, 443–518 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Auslander, M., Smalø, S.O.: Preprojective modules over artin algebras. J. Algebra 66, 61–122 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69, 426–454 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bautista, R., Liu, S., Paquette, C.: Representation theory of strongly locally finite quivers. Proc. Lond. Math. Soc. (2012). doi: 10.1112/plms/pds039 Google Scholar
  10. 10.
    Dickson, S.E.: A torsion theory for abelian categories. Trans. Am. Math. Soc. 121, 223–235 (1966)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gabriel, P., Roiter, A.V.: Representations of Finite Dimensional Algebras. Algebra VIII, Encyclopedia Math. Sci., vol. 73. Springer, Berlin (1992)zbMATHGoogle Scholar
  12. 12.
    Hoshino, M.: On splitting torsion theories induced by tilting modules. Commun. Algebra 11, 427–439 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications, vol. 37. American Mathematical Society, Providence (1956)zbMATHGoogle Scholar
  14. 14.
    Jørgensen, P.: Calabi–Yau categories and Poincaré duality spaces. In: Trends in Representation Theory of Algebras and Related Topics, EMS Ser. Congr. Rep., pp. 399–431. European Mathematical Society Publishing House, Zürich (2008)CrossRefGoogle Scholar
  15. 15.
    Jørgensen, P.: Auslander–Reiten triangles in subcategories. J. K-Theory 3, 583–601 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kleiner, M.: Approximations and almost split sequences in homologically finite subcategories. J. Algebra 198, 135–163 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kleiner, M., Perez, E.: Computation of almost split sequences with applications to relatively projective and prinjective modules. Algebr. Represent. Theory 6, 251–284 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Krause, H.: A short proof for Auslander’s defect formula. Linear Algebra Appl. 365, 267–270 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krause, H., Saorí n, M.: On minimal approximations of modules. In: Trends in the representation theory of finite-dimensional algebras (Seattle, 1997). Contemp. Math., vol. 229, pp. 227–236. Amer. Math. Soc., Providence (1998)CrossRefGoogle Scholar
  20. 20.
    Lenzing, H., Zuazua, R.: Auslander–Reiten duality for abelian categories. Bol. Soc. Mat. Mex. 10, 169–177 (2004)MathSciNetGoogle Scholar
  21. 21.
    Ng, P.: Existence of Auslander–Reiten sequences in subcategories. J. Pure Appl. Algebra 215, 2378–2384 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de SherbrookeSherbrookeCanada
  2. 2.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada

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