A duality result for almost split sequences

Lidia Hügel; Helmut Valenta

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 2, page 267-292
  • ISSN: 0010-1354

Abstract

top
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.

How to cite

top

Hügel, Lidia, and Valenta, Helmut. "A duality result for almost split sequences." Colloquium Mathematicae 80.2 (1999): 267-292. <http://eudml.org/doc/210718>.

@article{Hügel1999,
abstract = {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.},
author = {Hügel, Lidia, Valenta, Helmut},
journal = {Colloquium Mathematicae},
keywords = {hereditary Artinian rings; endolengths; almost split sequences; tilting; torsion theories; preinjective modules; finite representation type; right pure semisimple rings; categories of finitely generated modules; preprojective modules},
language = {eng},
number = {2},
pages = {267-292},
title = {A duality result for almost split sequences},
url = {http://eudml.org/doc/210718},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Hügel, Lidia
AU - Valenta, Helmut
TI - A duality result for almost split sequences
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 2
SP - 267
EP - 292
AB - 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.
LA - eng
KW - hereditary Artinian rings; endolengths; almost split sequences; tilting; torsion theories; preinjective modules; finite representation type; right pure semisimple rings; categories of finitely generated modules; preprojective modules
UR - http://eudml.org/doc/210718
ER -

References

top
  1. [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed., Springer, New York, 1992. Zbl0765.16001
  2. [2] L. Angeleri Hügel, P 1 -hereditary artin algebras, J. Algebra 176 (1995), 460-479. 
  3. [3] L. Angeleri Hügel, Almost split sequences arising from the preprojective partition, ibid. 194 (1997), 1-13. 
  4. [4] L. Angeleri Hügel, Finitely cotilting modules, Comm. Algebra, to appear. 
  5. [5] L. Angeleri Hügel, On some precovers and preenvelopes, preprint, 1998. 
  6. [6] M. Auslander, Large modules over artin algebras, in: Algebra, Topology, Category Theory, Academic Press, 1976, 1-17. 
  7. [7] M. Auslander, Functors and morphisms determined by objects, in: Lecture Notes in Pure and Appl. Math. 37, Marcel Dekker, 1978, 1-244. 
  8. [8] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. 94 (1969). Zbl0204.36402
  9. [9] M. Auslander and I. Reiten, Representation theory of artin algebras III. Almost split sequences, Comm. Algebra 3 (1975), 239-294. Zbl0331.16027
  10. [10] M. Auslander, I. Reiten and S. O. Smalο, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, 1995. 
  11. [11] M. Auslander and S. O. Smalο, Preprojective modules over artin algebras, J. Algebra 66 (1980), 61-122. Zbl0477.16013
  12. [12] R. R. Colby and K. R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), 1585-1615. Zbl0703.16013
  13. [13] R. Colpi, Some remarks on equivalences between categories of modules, ibid. 18 (1990), 1935-1951. Zbl0708.16002
  14. [14] R. Colpi, Tilting modules and *-modules, ibid. 21 (1993), 1095-1102. Zbl0795.16004
  15. [15] R. Colpi, G. D'Este and A. Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), 461-494. Zbl0876.16004
  16. [16] N. V. Dung, Preinjective modules and finite representation type of artinian rings, Comm. Algebra, to appear. Zbl0953.16011
  17. [17] P. Gabriel and A. V. Roiter, Algebra VIII: Representations of Finite-Dimensional Algebras, Encyclopedia Math. Sci. 73, Springer, 1992. Zbl0839.16001
  18. [18] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443. Zbl0503.16024
  19. [19] I. Herzog, A test for finite representation type, J. Pure Appl. Algebra 95 (1994), 151-182. Zbl0814.16011
  20. [20] M. Hoshino, Tilting modules and torsion theories, Bull. London Math. Soc. 14 (1982), 334-336. Zbl0486.16019
  21. [21] M. Hoshino, On splitting torsion theories induced by tilting modules, Comm. Algebra 11 (1983), 427-439. Zbl0506.16018
  22. [22] R C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984. 
  23. [23] M. Schmidmeier, A dichotomy for finite length modules induced by local duality, Comm. Algebra 25 (1997), 1933-1944. Zbl0885.16010
  24. [24] M. Schmidmeier, The local duality for homomorphisms and an application to pure semisimple PI-rings, Colloq. Math. 77 (1998), 121-132. Zbl0915.16001
  25. [25] D. Simson, Pure semisimple categories and rings of finite representation type, J. Algebra 48 (1977), 290-296; Corrigendum, ibid. 67 (1980), 254-256. Zbl0409.16030
  26. [26] D. Simson, On pure-semisimple Grothendieck categories, I, Fund. Math. 100 (1978), 211-222. Zbl0392.18012
  27. [27] D. Simson, Partial Coxeter functors and right pure semisimple hereditary rings, J. Algebra 71 (1981), 195-218. Zbl0477.16014
  28. [28] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon and Breach, 1992. Zbl0818.16009
  29. [29] D. Simson, An Artin problem for division ring extensions and the pure semisimplicity conjecture I, Arch. Math. (Basel) 66 (1996), 114-122. Zbl0873.16010
  30. [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. Zbl0936.16010
  31. [31] H. Valenta, Existence criteria and construction methods for certain classes of tilting modules, Comm. Algebra 22 (1994), 6047-6072. Zbl0827.16005
  32. [32] W. Zimmermann, Existenz von Auslander-Reiten-Folgen, Arch. Math. (Basel) 40 (1983), 40-49. Zbl0513.16019
  33. [33] B. Zimmermann-Huisgen, Strong preinjective partitions and representation type of artinian rings, Proc. Amer. Math. Soc. 109 (1990), 309-322. 
  34. [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. Zbl0699.16019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.