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A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders


Author: Daniel Simson
Journal: Trans. Amer. Math. Soc. 352 (2000), 4843-4875
MSC (2000): Primary 16G30, 16G50, 15A21; Secondary 15A63, 16G60
DOI: https://doi.org/10.1090/S0002-9947-00-02575-7
Published electronically: June 8, 2000
MathSciNet review: 1695036
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Abstract:

Let $D$ be a complete discrete valuation domain with the unique maximal ideal ${\mathfrak{p}}$. We suppose that $D$ is an algebra over an algebraically closed field $K $ and $D/{\mathfrak{p}} \cong K$. Subamalgam $D$-suborders $\Lambda ^{\bullet }$ of a tiled $D$-order $\Lambda $ are studied in the paper by means of the integral Tits quadratic form $q_{\Lambda ^{\bullet }}: {\mathbb{Z} }^{n_{1}+2n_{3}+2 } \,\,\longrightarrow {\mathbb{Z} }$. A criterion for a subamalgam $D$-order $\Lambda ^{\bullet }$ to be of tame lattice type is given in terms of the Tits quadratic form $q_{{\Lambda ^{\bullet }}}$ and a forbidden list $\Omega _{1},\ldots ,\Omega _{17}$ of minor $D$-suborders of $\Lambda ^{\bullet }$presented in the tables.


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  • [1] D. M. Arnold, Representations of partially ordered sets and abelian groups, Contemporary Math. 87 (1989), 91-109. MR 90j:20118
  • [2] D. M. Arnold and M. Dugas, Block rigid almost completely decomposable groups and lattices over multiple pullback rings, J. Pure Appl. Alg. 87 (1993), 105-121. MR 94f:20102
  • [3] D. M. Arnold, F. Richman and C. Vinsonhaler, Representations of finite posets and valuated groups, J. Algebra 155 (1993), 110-126. MR 94d:20064
  • [4] K. Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), 1-12. MR 86k:16023
  • [5] Ch. W. Curtis and I. Reiner, ``Methods of Representation Theory'', Vol. I,, Wiley Classics Library Edition, New York, 1990. MR 90k:20001
  • [6] P. Dowbor and S. Kasjan, Galois covering technique and tame non-simply connected posets of polynomial growth, J. Pure Appl. Algebra 147 (2000), 1-24.
  • [7] P. Dowbor and A. Skowronski, On Galois coverings of tame algebras, Arch. Math. 44 (1985), 522-529. MR 87a:16035
  • [8] P. Dowbor and A. Skowronski, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337. MR 88m:16020
  • [9] J. A. Drozd and M. G. Greuel, Tame-wild dichotomy for Cohen-Macaulay modules, Math. Ann. 294 (1992), 387-394. MR 93h:16023
  • [10] Y. A. Drozd, Cohen-Macaulay modules and vector bundles, Proc. Euroconference ``Interactions between Ring Theory and Representations of Algebras'', Murcia, 12-17 January 1998, Lecture Notes in Pure and Appl. Math. (to appear).
  • [11] E. L. Green and I. Reiner, Integral representations and diagrams, Michigan Math. J. 25 (1978), 53-84. MR 80g:16032
  • [12] J. Haefner and L. Klingler, Special quasi-triads and integral group rings of finite representation type, I and II, J. Algebra 158 (1993), 279-374. MR 94k:16028
  • [13] H.-J. von Höhne and D. Simson, Bipartite posets of finite prinjective type, J. Algebra 201 (1998), 86-114. MR 99h:16027
  • [14] S. Kasjan, Minimal bipartite algebras of infinite prinjective type with prin-preprojective component, Colloquium Math. 76 (1998), 295-317. MR 99b:16024
  • [15] S. Kasjan and D. Simson, Varieties of poset representations and minimal posets of wild prinjective type, in Proceedings of the Sixth International Conference on Representations of Algebras, vol. 14, Canadian Mathematical Society Conference Proceedings, 1993, pp. 245-284. MR 94e:16025
  • [16] S. Kasjan and D. Simson, Fully wild prinjective type of posets and their quadratic forms, J. Algebra 172 (1995), 506-529. MR 96m:16020
  • [17] S. Kasjan and D. Simson, Tame prinjective type and Tits form of two-peak posets I, J. Pure Appl. Algebra 106 (1996), 307-330. MR 97d:16018
  • [18] S. Kasjan and D. Simson, Tame prinjective type and Tits form of two-peak posets II, J. Algebra 187 (1997), 71-96. MR 98h:16020
  • [19] S. Kasjan and D. Simson, A subbimodule reduction, a peak reduction functor and tame prinjective type, Bull. Pol. Acad. Sci. Math. 45 (1997), 89-107. MR 99a:16009
  • [20] J. Kosakowska and D. Simson, On Tits form and prinjective representations of posets of finite prinjective type, Comm. Algebra 26 (1998), 1613-1623. MR 99d:16013
  • [21] L. A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR 39 (1975), 963-991; English transl., Math. USSR Izv. 9 (1975), 911-938. MR 55:10604
  • [22] L. A. Nazarova and V. A. Roiter, Representations of bipartite completed posets, Comment. Math. Helv. 63 (1988), 498-526. MR 89m:06003
  • [23] J. A. de la Peña and D. Simson, Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences, Trans. Amer. Math. Soc. 329 (1992), 733-753. MR 92e:16005
  • [24] I. Reiner, ``Maximal Orders'', Academic Press, London, 1975. MR 52:13910
  • [25] K. W. Roggenkamp, Auslander-Reiten species of Bäckstrom orders, J. Algebra 85 (1983), 449-476. MR 85g:16005
  • [26] K. W. Roggenkamp, Lattices over subhereditary orders and socle-projective modules, J. Algebra 121 (1989), 40-67. MR 90h:16016
  • [27] C. M. Ringel and K. W. Roggenkamp, Diagrammatic methods in the representation theory of orders, J. Algebra 60 (1979), 11-42. MR 81b:16008
  • [28] D. Simson, Socle reductions and socle projective modules, J. Algebra 103 (1986), 18-68. MR 88a:16058
  • [29] D. Simson, Representations of bounded stratified posets, coverings and socle projective modules, in ``Topics in Algebra, Part I: Rings and Representations of Algebras'', Banach Center Publications, vol. 26, PWN, Warszawa, 1990, pp. 499-533. MR 93g:16019
  • [30] D. Simson, A splitting theorem for multipeak path algebras, Fund. Math. 138 (1991), 113-137. MR 93a:16007
  • [31] D. Simson, Right peak algebras of two-separate stratified posets, their Galois coverings and socle projective modules, Comm. Algebra 20 (1992), 3541-3591. MR 94f:16031
  • [32] D. Simson, ``Linear Representations of Partially Ordered Sets and Vector Space Categories'', Algebra, Logic and Applications, Vol. 4, Gordon & Breach Science Publishers, New York, 1992. MR 95g:16013
  • [33] D. Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl. Algebra 90 (1993), 77-103. MR 95b:16011
  • [34] D. Simson, On representation types of module subcategories and orders, Bull. Pol. Acad. Sci., Math. 41 (1993), 77-93. MR 97g:16024
  • [35] D. Simson, A reduction functor, tameness and Tits form for a class of orders, J. Algebra 174 (1995), 430-452. MR 96d:16022
  • [36] D. Simson, Triangles of modules and non-polynomial growth, C. R. Acad. Sci. Paris, Série I 321 (1995), 33-38. MR 96g:16015
  • [37] D. Simson, Representation embedding problems, categories of extensions and prinjective modules, in Proceedings of the Seventh International Conference on Representations of Algebras, Canadian Mathematical Society Conference Proceedings, Vol. 18, 1996, 601-639. MR 98g:16011
  • [38] D. Simson, Socle projective representations of partially ordered sets and Tits quadratic forms with application to lattices over orders, in Proceedings of the Conference on Abelian Groups and Modules, Colorado Springs, August 1995, Lecture Notes in Pure and Appl. Math., Vol. 182, 1996, pp. 73-111. MR 97j:16024
  • [39] D. Simson, Prinjective modules, propartite modules, representations of bocses and lattices over orders, J. Math. Soc. Japan 49 (1997), 31-68. MR 98e:16015
  • [40] D. Simson, Three-partite subamalgams of tiled orders of finite lattice type, J. Pure Appl. Algebra 138 (1999), 151-184. CMP 99:12
  • [41] D. Simson, Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders, in Proc. AMS-IMS-SIAM Summer Research Conference ``Trends in the Representation Theory of Finite Dimensional Algebras'', The University of Washington, July 20-24, 1997, Contemporary Math. 229 (1998), 307-342. CMP 99:09
  • [42] D. Simson, Tame three-partite subamalgams of tiled orders of polynomial growth, Colloq. Math. 81 (1999), 237-262.
  • [43] D. Simson, Cohen-Macaulay modules over classical orders, Proc. Euroconference ``Interactions between Ring Theory and Representations of Algebras'', Murcia, 12-17 January 1998, Lecture Notes in Pure and Appl. Math. (to appear).
  • [44] Y. Yoshino, ``Cohen-Macaulay Modules over Cohen-Macaulay Rings'', London Math. Soc. Lecture Notes Series, Vol. 146, Cambridge University Press, 1990. MR 92b:13016
  • [45] A. G. Zavadskij and V. V. Kirichenko, Torsion-free modules over prime rings, Zap. Naychn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 57 (1976), 100-116; English tranl., J. Soviet Math. 11 (1979), 598-612. MR 58:11026
  • [46] A. G. Zavadskij and V. V. Kirichenko, Semimaximal rings of finite type, Mat. Sbornik 103 (1977), 323-345; English transl., Math. USSR Sb. 32 (1997), 273-291. MR 56:15706
  • [47] A. G. Zavadskij, Representations of partially ordered sets of finite growth, Kievskij Ordena Trudovovo Krasnovo Znameni Inzinerno-Stroitelnyi Institut (KISI), Dep. Ukr. NIINTI, No. 413-Yk-D83, Kiev, 1983, pp. 1-76 (Russian).
  • [48] A. G. Zavadskij, Differentiation algorithm and classification of representations, Izv. Akad. Nauk SSSR, Ser. Mat. 55 (1991), 1007-1048 (in Russian); English transl., Math. USSR Izvestia 39 (1992), 975-1012.

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Additional Information

Daniel Simson
Affiliation: Faculty of Mathematics and Informatics, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Email: simson@mat.uni.torun.pl

DOI: https://doi.org/10.1090/S0002-9947-00-02575-7
Received by editor(s): November 12, 1997
Published electronically: June 8, 2000
Additional Notes: Partially supported by Polish KBN Grant 2 P0 3A 012 16.
Dedicated: Dedicated to Klaus Roggenkamp on the occasion of his 60th birthday
Article copyright: © Copyright 2000 American Mathematical Society

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