Tame three-partite subamalgams of tiled orders of polynomial growth

Daniel Simson

Colloquium Mathematicae (1999)

• Volume: 81, Issue: 2, page 237-262
• ISSN: 0010-1354

Access Full Article

top

 Access to full text

 Full (PDF)

Abstract

top

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 ${\Lambda }^{•}$ (1.2) of tiled D-orders Λ (1.1). A simple criterion for a tame lattice type subamalgam D-order ${\Lambda }^{•}$ to be of polynomial growth is given in Theorem 1.5. Tame lattice type subamalgam D-orders ${\Lambda }^{•}$ of non-polynomial growth are completely described in Theorem 6.2 and Corollary 6.3.

How to cite

top

• MLA
• BibTeX
• RIS

Simson, Daniel. "Tame three-partite subamalgams of tiled orders of polynomial growth." Colloquium Mathematicae 81.2 (1999): 237-262. <http://eudml.org/doc/210737>.

@article{Simson1999,
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.},
author = {Simson, Daniel},
journal = {Colloquium Mathematicae},
keywords = {forbidden substructures; tame orders of polynomial growth; tame three-partite subamalgams; matrix algebras; amalgamations; integral quadratic forms; hypercritical two-peak posets; two-peak garlands},
language = {eng},
number = {2},
pages = {237-262},
title = {Tame three-partite subamalgams of tiled orders of polynomial growth},
url = {http://eudml.org/doc/210737},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Simson, Daniel
TI - Tame three-partite subamalgams of tiled orders of polynomial growth
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 2
SP - 237
EP - 262
AB - 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.
LA - eng
KW - forbidden substructures; tame orders of polynomial growth; tame three-partite subamalgams; matrix algebras; amalgamations; integral quadratic forms; hypercritical two-peak posets; two-peak garlands
UR - http://eudml.org/doc/210737
ER -

References

top

1. [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. Zbl0818.20065
2. [2] C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, Wiley Classics Library Edition, New York, 1990. 
3. [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. Zbl0998.16010
4. [4] J. A. Drozd and M. G. Greuel, Tame-wild dichotomy for Cohen-Macaulay modules, Math. Ann. 294 (1992), 387-394. Zbl0760.16005
5. [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. [6] P. Gabriel and A. V. Roiter, Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopedia Math. Sci. 73, Springer, 1992. 
7. [7] E. L. Green and I. Reiner, Integral representations and diagrams, Michigan Math. J. 25 (1978), 53-84. Zbl0365.16015
8. [8] S. Kasjan, Adjustment functors and tame representation type, Comm. Algebra 22 (1994), 5587-5597. Zbl0823.16010
9. [9] S. Kasjan, Minimal bipartite algebras of infinite prinjective type with prin-preprojective component, Colloq. Math. 76 (1998), 295-317. Zbl0912.16007
10. [10] S. Kasjan, A criterion for polynomial growth of $\tilde{}n$-free two-peak posets of tame prinjective type, preprint, Toruń, 1998. 
11. [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. Zbl0856.16007
12. [12] S. Kasjan and D. Simson, Tame prinjective type and Tits form of two-peak posets II, J. Algebra 187 (1997), 71-96. Zbl0944.16013
13. [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. Zbl0959.16012
14. [14] L. A. Nazarova and V. A. Roiter, Representations of completed posets, Comment. Math. Helv. 63 (1988), 498-526. Zbl0671.18002
15. [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. Zbl0789.16010
16. [16] C. M. Ringel and K. W. Roggenkamp, Diagrammatic methods in the representation theory of orders, J. Algebra 60 (1979), 11-42. Zbl0438.16021
17. [17] D. Simson, A splitting theorem for multipeak path algebras, Fund. Math. 138 (1991), 113-137. Zbl0780.16010
18. [18] D. Simson, Right peak algebras of two-separate stratified posets, their Galois coverings and socle projective modules, Comm. Algebra 20 (1992), 3541-3591. Zbl0791.16011
19. [19] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon & Breach, New York, 1992. Zbl0818.16009
20. [20] D. Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl. Algebra 90 (1993), 77-103. Zbl0815.16006
21. [21] D. Simson, On representation types of module subcategories and orders, Bull. Polish Acad. Sci. Math. 41 (1993), 77-93. Zbl0805.16011
22. [22] D. Simson, A reduction functor, tameness and Tits form for a class of orders, J. Algebra 174 (1995), 430-452. Zbl0831.16011
23. [23] D. Simson, Triangles of modules and non-polynomial growth, C. R. Acad. Sci. Paris Sér. I 321 (1995), 33-38. Zbl0842.16010
24. [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. Zbl0929.16014
25. [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. Zbl0869.16008
26. [26] D. Simson, Prinjective modules, propartite modules, representations of bocses and lattices over orders, J. Math. Soc. Japan 49 (1997), 31-68. Zbl0937.16019
27. [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. Zbl0921.16007
28. [28] D. Simson, Three-partite subamalgams of tiled orders of finite lattice type, J. Pure Appl. Algebra 138 (1999), 151-184. Zbl0928.16014
29. [29] D. Simson, A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders, Trans. Amer. Math. Soc., in press. Zbl0953.16016
30. [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. [31] A. Skowroński, Group algebras of polynomial growth, Manuscripta Math. 59 (1987), 499-516. Zbl0627.16007
32. [32] A. Skowroński, Criteria for polynomial growth of algebras, Bull. Polish Acad. Sci. Math. 42 (1994), 173-183. Zbl0865.16011
33. [33] Y. Yoshino, Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan 39 (1987), 719-739. Zbl0615.13008
34. [34] Y. Yoshino, Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, 1990. Zbl0745.13003
35. [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. [36] A. G. Zavadskij and V. V. Kirichenko, Semimaximal rings of finite type, Mat. Sb. 103 (1977), 323-345 (in Russian). Zbl0396.16002

Citations in EuDML Documents

top

1. Stanisław Kasjan, Simply connected right multipeak algebras and the separation property

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.

Language to use for this widget. български (bg) čeština (cs) Deutsch (de) ελληνικά (el) English (en) español (es) français (fr) italiano (it) lietuvių kalba (lt) polski (pl) português (pt) română (ro) русский (ru) slovenčina (sk)

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

Number of notes per page 5 10 15 20

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.

OK