ArticleOriginal scientific text

Title

Tame three-partite subamalgams of tiled orders of polynomial growth

Authors 1

Affiliations

  1. Faculty of Mathematics and Informaticsi, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Abstract

Assume that K is an algebraically closed field. Let D be a complete discrete valuation domain with a unique maximal ideal p and residue field D/p ≌ K. We also assume that D is an algebra over the field K . We study subamalgam D-suborders Λ (1.2) of tiled D-orders Λ (1.1). A simple criterion for a tame lattice type subamalgam D-order Λ to be of polynomial growth is given in Theorem 1.5. Tame lattice type subamalgam D-orders Λ of non-polynomial growth are completely described in Theorem 6.2 and Corollary 6.3.

Bibliography

  1. D. M. Arnold and M. Dugas, Block rigid almost completely decomposable groups and lattices over multiple pullback rings, J. Pure Appl. Algebra 87 (1993), 105-121.
  2. C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, Wiley Classics Library Edition, New York, 1990.
  3. P. Dowbor and S. Kasjan, Galois covering technique and tame non-simply connected posets of polynomial growth, J. Pure Appl. Algebra 1999, in press.
  4. J. A. Drozd and M. G. Greuel, Tame-wild dichotomy for Cohen-Macaulay modules, Math. Ann. 294 (1992), 387-394.
  5. Y. A. Drozd, Cohen-Macaulay modules and vector bundles, in: Interactions between Ring Theory and Representations of Algebras (Murcia, 1998), Lecture Notes in Pure and Appl. Math., Marcel Dekker, 1999, to appear.
  6. P. Gabriel and A. V. Roiter, Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopedia Math. Sci. 73, Springer, 1992.
  7. E. L. Green and I. Reiner, Integral representations and diagrams, Michigan Math. J. 25 (1978), 53-84.
  8. S. Kasjan, Adjustment functors and tame representation type, Comm. Algebra 22 (1994), 5587-5597.
  9. S. Kasjan, Minimal bipartite algebras of infinite prinjective type with prin-preprojective component, Colloq. Math. 76 (1998), 295-317.
  10. S. Kasjan, A criterion for polynomial growth of w_~n-free two-peak posets of tame prinjective type, preprint, Toruń, 1998.
  11. S. Kasjan and D. Simson, Tame prinjective type and Tits form of two-peak posets I, J. Pure Appl. Algebra 106 (1996), 307-330.
  12. S. Kasjan and D. Simson, Tame prinjective type and Tits form of two-peak posets II, J. Algebra 187 (1997), 71-96.
  13. S. Kasjan and D. Simson, A subbimodule reduction, a peak reduction functor and tame prinjective type, Bull. Polish Acad. Sci. Math. 45 (1997), 89-107.
  14. L. A. Nazarova and V. A. Roiter, Representations of completed posets, Comment. Math. Helv. 63 (1988), 498-526.
  15. J. A. de la Pe na and D. Simson, Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences, Trans. Amer. Math. Soc. 329 (1992), 733-753.
  16. C. M. Ringel and K. W. Roggenkamp, Diagrammatic methods in the representation theory of orders, J. Algebra 60 (1979), 11-42.
  17. D. Simson, A splitting theorem for multipeak path algebras, Fund. Math. 138 (1991), 113-137.
  18. D. Simson, Right peak algebras of two-separate stratified posets, their Galois coverings and socle projective modules, Comm. Algebra 20 (1992), 3541-3591.
  19. D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon & Breach, New York, 1992.
  20. D. Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl. Algebra 90 (1993), 77-103.
  21. D. Simson, On representation types of module subcategories and orders, Bull. Polish Acad. Sci. Math. 41 (1993), 77-93.
  22. D. Simson, A reduction functor, tameness and Tits form for a class of orders, J. Algebra 174 (1995), 430-452.
  23. D. Simson, Triangles of modules and non-polynomial growth, C. R. Acad. Sci. Paris Sér. I 321 (1995), 33-38.
  24. D. Simson, Representation embedding problems, categories of extensions and prinjective modules, in: Representation Theory of Algebras (Cocoyoc, 1994), CMS Conf. Proc. 18, Amer. Math. Soc., 1996, 601-639.
  25. D. Simson, Socle projective representations of partially ordered sets and Tits quadratic forms with application to lattices over orders, in: Abelian Groups and Modules (Colorado Springs, 1995), Lecture Notes in Pure and Appl. Math. 182, Marcel Dekker, 1996, 73-111.
  26. D. Simson, Prinjective modules, propartite modules, representations of bocses and lattices over orders, J. Math. Soc. Japan 49 (1997), 31-68.
  27. D. Simson, Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders, in: Trends in the Representation Theory of Finite Dimensional Algebras (Seattle, 1997), Contemp. Math. 229, Amer. Math. Soc., 1998, 307-342.
  28. D. Simson, Three-partite subamalgams of tiled orders of finite lattice type, J. Pure Appl. Algebra 138 (1999), 151-184.
  29. D. Simson, A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders, Trans. Amer. Math. Soc., in press.
  30. D. Simson, Cohen-Macaulay modules over classical orders, in: Interactions between Ring Theory and Representations of Algebras (Murcia, 1998), Lecture Notes in Pure and Appl. Math., Marcel Dekker, 1999, to appear.
  31. A. Skowroński, Group algebras of polynomial growth, Manuscripta Math. 59 (1987), 499-516.
  32. A. Skowroński, Criteria for polynomial growth of algebras, Bull. Polish Acad. Sci. Math. 42 (1994), 173-183.
  33. Y. Yoshino, Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan 39 (1987), 719-739.
  34. Y. Yoshino, Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, 1990.
  35. A. G. Zavadskij and V. V. Kirichenko, Torsion-free modules over prime rings, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 57 (1976), 100-116 (in Russian).
  36. A. G. Zavadskij and V. V. Kirichenko, Semimaximal rings of finite type, Mat. Sb. 103 (1977), 323-345 (in Russian).
Pages:
237-262
Main language of publication
English
Received
1999-02-15
Published
1999
Exact and natural sciences