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

Simson, Daniel

doi:10.1090/S0002-9947-00-02575-7

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

Trans. Amer. Math. Soc. 352 (2000), 4843-4875

DOI: https://doi.org/10.1090/S0002-9947-00-02575-7

Published electronically: June 8, 2000

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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.

References

David M. Arnold, Representations of partially ordered sets and abelian groups, Abelian group theory (Perth, 1987) Contemp. Math., vol. 87, Amer. Math. Soc., Providence, RI, 1989, pp. 91–109. MR 995268, DOI 10.1090/conm/087/995268
D. Arnold and M. Dugas, Block rigid almost completely decomposable groups and lattices over multiple pullback rings, J. Pure Appl. Algebra 87 (1993), no. 2, 105–121. MR 1224215, DOI 10.1016/0022-4049(93)90119-E
D. Arnold, F. Richman, and C. Vinsonhaler, Representations of finite posets and valuated groups, J. Algebra 155 (1993), no. 1, 110–126. MR 1206624, DOI 10.1006/jabr.1993.1033
Klaus Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), no. 1, 1–12. MR 756773, DOI 10.1007/BF01455993
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders; Reprint of the 1981 original; A Wiley-Interscience Publication. MR 1038525
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.
Piotr Dowbor and Andrzej Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), no. 6, 522–529. MR 797444, DOI 10.1007/BF01193992
Piotr Dowbor and Andrzej Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), no. 2, 311–337. MR 896100, DOI 10.1007/BF02564450
Yu. A. Drozd and G.-M. Greuel, Tame-wild dichotomy for Cohen-Macaulay modules, Math. Ann. 294 (1992), no. 3, 387–394. MR 1188126, DOI 10.1007/BF01934330
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).
E. L. Green and I. Reiner, Integral representations and diagrams, Michigan Math. J. 25 (1978), no. 1, 53–84. MR 497882, DOI 10.1307/mmj/1029002006
Jeremy Haefner and Lee Klingler, Special quasi-triads and integral group rings of finite representation type. I, II, J. Algebra 158 (1993), no. 2, 279–322, 323–374. MR 1226795, DOI 10.1006/jabr.1993.1135
Hans-Joachim von Höhne and Daniel Simson, Bipartite posets of finite prinjective type, J. Algebra 201 (1998), no. 1, 86–114. MR 1608695, DOI 10.1006/jabr.1997.7261
Stanisław Kasjan, Minimal bipartite algebras of infinite prinjective type with prin-preprojective component, Colloq. Math. 76 (1998), no. 2, 295–317. MR 1618720, DOI 10.4064/cm-76-2-295-317
Stanisław Kasjan and Daniel Simson, Varieties of poset representations, tameness and minimal posets of wild prinjective type, Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992) Carleton-Ottawa Math. Lecture Note Ser., vol. 14, Carleton Univ., Ottawa, ON, 1992, pp. 45. MR 1206948
Stanisław Kasjan and Daniel Simson, Fully wild prinjective type of posets and their quadratic forms, J. Algebra 172 (1995), no. 2, 506–529. MR 1322415, DOI 10.1016/S0021-8693(05)80013-4
Stanisław Kasjan and Daniel Simson, Tame prinjective type and Tits form of two-peak posets $I$, J. Pure Appl. Algebra 106 (1996), no. 3, 307–330. MR 1375827, DOI 10.1016/0022-4049(94)00070-0
Stanisław Kasjan and Daniel Simson, Tame prinjective type and Tits form of two-peak posets. II, J. Algebra 187 (1997), no. 1, 71–96. MR 1425560, DOI 10.1006/jabr.1997.6751
Stanisław Kasjan and Daniel Simson, A subbimodule reduction, a peak reduction functor and tame prinjective type, Bull. Polish Acad. Sci. Math. 45 (1997), no. 1, 89–107. MR 1444674
Justyna Kosakowska and Daniel Simson, On Tits form and prinjective representations of posets of finite prinjective type, Comm. Algebra 26 (1998), no. 5, 1613–1623. MR 1622434, DOI 10.1080/00927879808826225
M. V. Milovanov, Determinability of solvable Lie groups from uniform discrete subgroups, Dokl. Akad. Nauk BSSR 21 (1977), no. 2, 108–111, 187 (Russian). MR 0437680
L. A. Nazarova and A. V. Roĭter, Representations of bipartite completed posets, Comment. Math. Helv. 63 (1988), no. 3, 498–526. MR 960771, DOI 10.1007/BF02566776
J. A. de la Peña and D. Simson, Prinjective modules, reflection functors, quadratic forms, and Auslander-Reiten sequences, Trans. Amer. Math. Soc. 329 (1992), no. 2, 733–753. MR 1025753, DOI 10.1090/S0002-9947-1992-1025753-3
I. Reiner, Maximal orders, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0393100
K. W. Roggenkamp, Auslander-Reiten species of Bäckström orders, J. Algebra 85 (1983), no. 2, 449–476. MR 725095, DOI 10.1016/0021-8693(83)90107-2
K. W. Roggenkamp, Lattices over subhereditary orders and socle-projective modules, J. Algebra 121 (1989), no. 1, 40–67. MR 992315, DOI 10.1016/0021-8693(89)90084-7
Claus Michael Ringel and Klaus W. Roggenkamp, Diagrammatic methods in the representation theory of orders, J. Algebra 60 (1979), no. 1, 11–42. MR 549096, DOI 10.1016/0021-8693(79)90106-6
Daniel Simson, Socle reductions and socle projective modules, J. Algebra 103 (1986), no. 1, 18–68. MR 860687, DOI 10.1016/0021-8693(86)90167-5
Daniel Simson, Representations of bounded stratified posets, coverings and socle projective modules, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 499–533. MR 1171251
Daniel Simson, A splitting theorem for multipeak path algebras, Fund. Math. 138 (1991), no. 2, 113–137. MR 1124540, DOI 10.4064/fm-138-2-113-137
Daniel Simson, Right peak algebras of two-separate stratified posets, their Galois coverings and socle projective modules, Comm. Algebra 20 (1992), no. 12, 3541–3591. MR 1191967, DOI 10.1080/00927879208824529
Daniel Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications, vol. 4, Gordon and Breach Science Publishers, Montreux, 1992. MR 1241646
Daniel Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl. Algebra 90 (1993), no. 1, 77–103. MR 1246276, DOI 10.1016/0022-4049(93)90138-J
Daniel Simson, On representation types of module subcategories and orders, Bull. Polish Acad. Sci. Math. 41 (1993), no. 2, 77–93 (1994). MR 1414754
Daniel Simson, A reduction functor, tameness, and Tits form for a class of orders, J. Algebra 174 (1995), no. 2, 430–452. MR 1334218, DOI 10.1006/jabr.1995.1133
Daniel Simson, Triangles of modules and non-polynomial growth, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 1, 33–38 (English, with English and French summaries). MR 1340078
Daniel Simson, Representation embedding problems, categories of extensions and prinjective modules, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 601–639. MR 1388077
Daniel Simson, Socle projective representations of partially ordered sets and Tits quadratic forms with applications to lattices over orders, Abelian groups and modules (Colorado Springs, CO, 1995) Lecture Notes in Pure and Appl. Math., vol. 182, Dekker, New York, 1996, pp. 73–111. MR 1415624
Daniel Simson, Prinjective modules, propartite modules, representations of bocses and lattices over orders, J. Math. Soc. Japan 49 (1997), no. 1, 31–68. MR 1419438, DOI 10.2969/jmsj/04910031
D. Simson, Three-partite subamalgams of tiled orders of finite lattice type, J. Pure Appl. Algebra 138 (1999), 151–184.
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.
D. Simson, Tame three-partite subamalgams of tiled orders of polynomial growth, Colloq. Math. 81 (1999), 237–262.
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).
Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. MR 1079937, DOI 10.1017/CBO9780511600685
A. G. Zavadskiĭ and V. V. Kiričenko, Torsion-free modules over primary rings, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 57 (1976), 100–116, 177 (Russian). Modules in representation theory and algebraic geometry. MR 0491829
A. G. Zavadskiĭ and V. V. Kiričenko, Semimaximal rings of finite type, Mat. Sb. (N.S.) 103(145) (1977), no. 3, 323–345, 463 (Russian). MR 0457501
A. G. Zavadskij, Representations of partially ordered sets of finite growth, Kievskij Ordena Trudovovo Krasnovo Znameni Inžinerno-Stroitelnyi Institut (KISI), Dep. Ukr. NIINTI, No. 413–Yk–D83, Kiev, 1983, pp. 1–76 (Russian).
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.