Low dimensional orders of finite representation type

Abstract

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups \(G \subset {{{\,\mathrm{GL}\,}}_2}\), explicitly computing \(H^2(G,k^*)\), and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let \(B = k_{\zeta } \llbracket x,y \rrbracket \) be the skew power series ring where \(\zeta \) is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form \(A = B/(f)\) where \(f \in Z(B)\) which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Auslander, M., Reiten, I.: McKay quivers and extended Dynkin diagrams. Trans. Am. Math. Soc. 293(1), 293–301 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Arnol’d, V.I.: Normal forms of functions near degenerate critical points, the Weyl groups \(A_{k}, D_{k}, E_{k}\) and Lagrangian singularities. Funkc. Anal. Priložen. 6(4), 3–25 (1972)

    Google Scholar 

  3. 3.

    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, Volume 36 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997). (Corrected reprint of the 1995 original)

    Google Scholar 

  4. 4.

    Artin, M.: Maximal orders of global dimension and Krull dimension two. Invent. Math. 84(1), 195–222 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Artin, M.: Two-dimensional orders of finite representation type. Manuscr. Math. 58(4), 445–471 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Auslander, M.: Isolated singularities and existence of almost split sequences. Representation Theory, II (Ottawa, Ont., 1984), Volume 1178 of Lecture Notes in Mathematics, pp. 194–242. Springer, Berlin (1986)

    Google Scholar 

  7. 7.

    Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15(1), 27–75 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bannai, E.: Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension \(2\). J. Math. Soc. Jpn. 28(3), 447–454 (1976)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29(2), 178–218 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Buchweitz, R.-O., Faber, E., Ingalls, C.: Noncommutative resolutions of discriminants (2017)

  11. 11.

    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, Volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2nd edn. Springer, Berlin (2004)

    Google Scholar 

  12. 12.

    Chan, D., Chan, K.: Rational curves and ruled orders on surfaces. J. Algebra 435, 52–87 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Chan, D.: Lectures on orders. https://web.maths.unsw.edu.au/~danielch/Lect_Orders.pdf (2011)

  14. 14.

    Chan, D., Hacking, P., Ingalls, C.: Canonical singularities of orders over surfaces. Proc. Lond. Math. Soc. (3) 98(1), 83–115 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Chan, D., Ingalls, C.: The minimal model program for orders over surfaces. Invent. Math. 161(2), 427–452 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Coxeter, H.S.M.: Regular Complex Polytopes, 2nd edn. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  17. 17.

    Curtis, C.W., Reiner, I.: Methods of Representation Theory, Vol. I. With Applications to Finite Groups and Orders. Wiley, New York (1990). (Reprint of the 1981 original)

    Google Scholar 

  18. 18.

    Drozd, Y.A., Kiričenko, V.V.: Primary orders with a finite number of indecomposable representations. Izv. Akad. Nauk SSSR Ser. Mat. 37, 715–736 (1973)

    MathSciNet  Google Scholar 

  19. 19.

    Drozd, Y.A., Roĭter, A.V.: Commutative rings with a finite number of indecomposable integral representations. Izv. Akad. Nauk SSSR Ser. Mat. 31, 783–798 (1967)

    MathSciNet  Google Scholar 

  20. 20.

    Drozd, Y.A., Roggenkamp, K.W.: Cohen–Macaulay rings of global dimension two and Krull dimension two. Commun. Algebra 22(9), 3297–3329 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Val, P.D.: Homographies, Quaternions and Rotations. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964)

    Google Scholar 

  22. 22.

    Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260(1), 35–64 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Grieve, N., Ingalls, C.: On the kodaira dimension of maximal orders. arXiv:1611.10278 (2016)

  24. 24.

    Greuel, G.-M., Knörrer, H.: Einfache Kurvensingularitäten und torsionsfreie Moduln. Math. Ann. 270(3), 417–425 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Goursat, E.: Sur les substitutions orthogonales et les divisions régulières de l’espace. Ann. Sci. École Norm. Sup. 3(6), 9–102 (1889)

    MATH  Article  Google Scholar 

  26. 26.

    Hijikata, H., Nishida, K.: Primary orders of finite representation type. J. Algebra 192(2), 592–640 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Isaacs, I.: Martin: Character Theory of Finite Groups. AMS Chelsea Publishing, Providence (2006). Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]

    Google Scholar 

  28. 28.

    Iyama, O., Wemyss, M.: Maximal modifications and Auslander–Reiten duality for non-isolated singularities. Invent. Math. 197(3), 521–586 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Jacobinski, H.: Sur les ordres commutatifs avec un nombre fini de réseaux indécomposables. Acta Math. 118, 1–31 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original)

    Google Scholar 

  31. 31.

    Knörrer, H.: Cohen–Macaulay modules on hypersurface singularities. I. Invent. Math. 88(1), 153–164 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Le Bruyn, L., Van den Bergh, M., Van Oystaeyen, F.: Graded Orders. Birkhäuser Boston Inc, Boston (1988)

    Google Scholar 

  33. 33.

    Luo, X.: 0-Calabi-Yau configurations and finite Auslander–Reiten quivers of Gorenstein orders. J. Pure Appl. Algebra 219(12), 5590–5630 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Ringel, C.M., Roggenkamp, K.W.: Diagrammatic methods in the representation theory of orders. J. Algebra 60(1), 11–42 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Reiten, I., Riedtmann, C.: Skew group algebras in the representation theory of Artin algebras. J. Algebra 92(1), 224–282 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Reiten, I., Van den Bergh, M.: Two-dimensional tame and maximal orders of finite representation type. Mem. Am. Math. Soc. 80(408), viii+72 (1989)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Simson, D.: Linear Representations of Partially Ordered Sets and Vector Space Categories, Volume 4 of Algebra, Logic and Applications. Gordon and Breach Science Publishers, Montreux (1992)

    Google Scholar 

  38. 38.

    Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Wiedemann, A.: Die Auslander–Reiten Köcher der gitterendlichen Gorensteinordnungen. Bayreuth. Math. Schr. 23, 1–134 (1987)

    MATH  Google Scholar 

  40. 40.

    Yoshino, Y.: Cohen–Macaulay Modules Over Cohen–Macaulay Rings, Volume 146 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  41. 41.

    Zavadskiĭ, A.G., Kiričenko, V.V.: Semimaximal rings of finite type. Mat. Sb. (N.S.) 103(145), 323–345, 463 (1977)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Anthony Henderson for his help regarding automorphisms of Dynkin quivers. The second author would like to thank Osamu Iyama for useful conversations. We would also like to thank an anonymous referee for many helpful suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Colin Ingalls.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Daniel Chan was supported by the Australian Research Council, Discovery Project Grant DP0880143. Colin Ingalls was supported by an NSERC Discovery Grant.

Appendix

Appendix

We present some of the McKay graphs of finite subgroups of \({{\,\mathrm{GL}\,}}_2\).

figurea
figureb
figurec
figured
figuree

In these tables we use the following abbreviations: we abbreviate \({{\,\mathrm{{\mathbb {Z}}}\,}}/a\) as simply a, similarly we write \(a \oplus b\) for the group \({\mathbb {Z}}/a \oplus {{\,\mathrm{{\mathbb {Z}}}\,}}/b\), and as before \(a \vee b\) for the greatest common divisor of a and b. Below is a list of the cohomology groups \(H^1(G,k^*), H^2(G,k^*)\) for all subgroups G of \({{\,\mathrm{GL}\,}}_2\) as computed in Sects. 5 and 6.

$$\begin{aligned} \begin{array}{l|l|l} \hbox {type} &{} H^1 &{} H^2 \\ \hline B_n^m &{} \begin{array}{ll} 4m &{} n \text{ odd } , m \text{ even } \\ 2 \oplus 2m &{} \text{ else } \\ \end{array} &{} (m+1) \vee 2 \\ \hline D_n^m &{} \begin{array}{ll} 2 \oplus 2 \oplus m &{} n \text{ even } \\ 4 \oplus m &{} n \text{ odd } , m \text{ odd } \\ 2 \oplus 2m &{} n \text{ odd } , m \text{ even } \end{array} &{} (m \vee 2) \oplus (m \vee n \vee 2) \\ \hline E_6^m &{} 3 \oplus m &{} (m \vee 3) \\ \hline E_7^m &{} 2 \oplus m &{} (m \vee 2)\\ \hline E_8^m &{} m &{} 1 \\ \hline CD_n^m &{} 2 \oplus 2m &{} (m+n-1) \vee 2 \\ \hline F_{41}^m &{} 2m &{} 1 \\ \hline G_{21}^m &{} 3m &{} 1 \\ \hline BT_n^m &{} 2(2m-1) &{} 1 \end{array} \end{aligned}$$

The next table lists the information about the pseudo-reflection subgroup of \(G = G(Z_1,Z_2;G_1,G_2)\) as mentioned in Theorem 8.2. In the cases where there are two parameters mn we let \({\overline{m}} = m/(m\vee n), {\overline{n}} = n/(m\vee n)\). The notation \(\frac{1}{a}(b,c)\) denotes the cyclic group \( \left\langle \left( {\begin{matrix} \zeta _a^b &{} 0 \\ 0 &{} \zeta _a^c \end{matrix}}\right) \right\rangle \).

$$\begin{aligned} \begin{array}{l|l|l} \text{ type } &{} RG &{} G/RG \\ \hline A^c_{m,n} &{} m \oplus n\vee c &{} n/n\vee c \\ B^m_n &{} \begin{array}{ll} G(2n,2{\overline{n}},2) &{} m \vee 2 = 1 \\ \mu ^2_{m \vee n} &{} m \vee 2 = 2 \\ &{} \end{array} &{} \begin{array}{ll} \frac{1}{{\overline{m}}}({\overline{n}},1) &{} m \vee 2 = 1 \\ (2{\overline{m}},2{\overline{m}};D_{{\overline{n}}+2},D_{{\overline{n}}+2}) &{} m \vee 2 = 2, {\overline{m}} \vee 2 = 1 \\ (4{\overline{m}},2{\overline{m}};D_{{\overline{n}}+2},A_{2{\overline{n}}}) &{} m \vee 2 = 2, {\overline{m}} \vee 2 = 2\\ \end{array} \\ \hline D_{n+2}^m &{} \begin{array}{ll} G(2n,2{\overline{n}},2) &{} m \vee 2 = 2 \\ \mu ^2_{m \vee n} &{} m \vee 2 = 1 \end{array} &{} \begin{array}{ll} \frac{1}{{\overline{m}}}({\overline{n}},1) &{} m \vee 2 = 2 \\ (2{\overline{m}},2{\overline{m}};D_{{\overline{n}}+2},D_{{\overline{n}}+2}) &{} m \vee 2 = 1 \end{array} \\ \hline E_6^m &{} \begin{array}{ll} 1 &{} m \vee 6=1 \\ D_4^2 &{} m \vee 6=2 \\ E_6^{m \vee 6} &{} \text{ else } \\ \end{array} &{} \begin{array}{ll} E_6^m &{} m \vee 6=1 \\ 3m/2 &{} m \vee 6=2 \\ m/m \vee 6 &{} \text{ else } \\ \end{array} \\ \hline E_7^m &{} \begin{array}{ll} 1 &{} m \vee 12=1 \\ E_6^3 &{} m \vee 12=3 \\ E_7^{m \vee 12} &{}\text{ else } \\ \end{array} &{} \begin{array}{ll} E_7^m &{} m \vee 12=1 \\ 2m/3 &{} m \vee 12=3 \\ m/m \vee 12 &{} \text{ else } \\ \end{array} \\ \hline E_8^m &{} \begin{array}{ll} 1 &{} m \vee 30=1 \\ E_8^{m\vee 30} &{} m \vee 30 \ne 1 \end{array} &{} \begin{array}{ll} E_8^m &{} m \vee 30=1 \\ m/m \vee 30 &{} m \vee 30 \ne 1 \end{array} \\ \hline CD^m_{n+1} &{} \begin{array}{ll} G(2n,2{\overline{n}},2) &{} m \vee 2 = 2, ({\overline{m}}{\overline{n}}) \vee 2 = 2\\ G(2n,{\overline{n}},2) &{} m \vee 2 = 2, ({\overline{m}}{\overline{n}}) \vee 2 = 1 \\ G(2n,2{\overline{n}},2) &{} m \vee 2 = 1, ({\overline{m}}{\overline{n}}) \vee 2 = 2 \\ G(2n,{\overline{n}},2) &{} m \vee 2 = 1, ({\overline{m}}{\overline{n}}) \vee 2 = 1 \end{array} &{} \begin{array}{ll} \frac{1}{2{\overline{m}}}({\overline{m}} + {\overline{n}},1)&{} m \vee 2 = 2, ({\overline{m}}{\overline{n}}) \vee 2 = 2 \\ \frac{1}{{\overline{m}}}(\frac{{\overline{m}} + {\overline{n}}}{2},1)&{} m \vee 2 = 2, ({\overline{m}}{\overline{n}}) \vee 2 = 1 \\ \frac{1}{2{\overline{m}}}({\overline{m}} + {\overline{n}},1) &{} m \vee 2 = 1, ({\overline{m}}{\overline{n}}) \vee 2 = 2 \\ \frac{1}{{\overline{m}}}(\frac{{\overline{m}} + {\overline{n}}}{2},1)&{} m \vee 2 = 1, ({\overline{m}}{\overline{n}}) \vee 2 = 1 \end{array} \\ \hline F^m_{41} &{} \begin{array}{ll} D_4^2 &{} m \vee 12=4 \\ E_6^6 &{} m \vee 12=12 \\ F_{41}^{m\vee 12} &{}\text{ else } \\ \end{array} &{} \begin{array}{ll} 3m &{} m \vee 12=4 \\ m/3 &{} m \vee 12=12 \\ m/m \vee 12 &{} \text{ else } \\ \end{array} \\ \hline G^m_{21} &{} \begin{array}{ll} 1 &{} m \vee 6=3 \\ D_4^2 &{} m \vee 6=6 \\ G_{21}^{m \vee 6} &{} \text{ else } \\ \end{array} &{} \begin{array}{ll} G_{21}^m &{} m \vee 6=3 \\ 3m/2 &{} m \vee 6=6 \\ m/m \vee 6 &{} \text{ else } \\ \end{array} \\ \hline BT^{m}_{\frac{n-1}{2}} &{} \begin{array}{ll} G(n,{\overline{n}},2) \\ \end{array} &{} \begin{array}{ll} \frac{1}{{\overline{m}}}(\frac{3m+1}{4}{\overline{n}}, \frac{1+{\overline{m}}}{2}) &{} m \equiv 1 \mod 4 \\ \frac{1}{{\overline{m}}}(\frac{m+1}{4}{\overline{n}}, \frac{1+{\overline{m}}}{2}) &{} m \equiv 3 \mod 4 \end{array} \\ \end{array} \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chan, D., Ingalls, C. Low dimensional orders of finite representation type. Math. Z. (2020). https://doi.org/10.1007/s00209-020-02552-2

Download citation

Keywords

  • Maximal Cohen–Macaulay modules
  • Matrix factorizations

Mathematics Subject Classification

  • 14E16