The second Brauer-Thrall conjecture for isolated singularities of excellent hypersurfaces. (English) Zbl 0743.13014
Let \(R\) be an excellent local Henselian \(k\)-algebra (with \(k\) an algebraically closed field of arbitrary characteristic) whose completion \(R^{}\) is isomorphic to \(k[[x_ 0,\ldots,x_ n]]/(f)\), where \(f\) has an isolated singularity at the origin. The aim of this paper is to extend and to give new proofs of certain results of E. Dieterich [Comment. Math. Helv. 62, 654-676 (1987; Zbl 0654.14002) and in Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht/Pfalz 1985, Lect. Notes Math. 1273, 244-264 (1987; Zbl 0632.14004)]. As a sample we mention the following.
Theorem. If \(R\) is not of finite representation type, there is a strictly increasing sequence \(\{n_ i\}_{i\geq 1}\) of positive integers such that for all \(i\) there are infinitely many isomorphism classes of indecomposable maximal Cohen-Macaulay \(R\)-modules of rank \(n_ i\). Dieterich proved this result in the case \(R=k[[x_ 0,\ldots,x_ n]]/(f)\) and \(\hbox{char}(k)\neq 2\). The authors are able to prove it in arbitrary characteristic and for \(R\) Henselian as above.
Theorem. If \(R\) is not of finite representation type, there is a strictly increasing sequence \(\{n_ i\}_{i\geq 1}\) of positive integers such that for all \(i\) there are infinitely many isomorphism classes of indecomposable maximal Cohen-Macaulay \(R\)-modules of rank \(n_ i\). Dieterich proved this result in the case \(R=k[[x_ 0,\ldots,x_ n]]/(f)\) and \(\hbox{char}(k)\neq 2\). The authors are able to prove it in arbitrary characteristic and for \(R\) Henselian as above.
Reviewer: L.Bădescu (Bucureşti)
MSC:
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13J15 | Henselian rings |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
Software:
classifyCeq.libReferences:
| [1] | [Au 1] Auslander, M., Functors and morphisms determined by objects, Proc. Conf. Representation Theory, Philadelphia, 1976, New York, Basel 1978, 1–244 |
| [2] | [Au 2] Auslander, M., Isolated singularities and existence of almost split sequences, Proc. ICRA IV, Springer Lecture Notes in Mathematics No 1178 (1986), 194–241 |
| [3] | [BGS] Buchweitz, R.O., Greuel, G.M., Schreyer, F.O., Cohen–Macaulay modules on hypersurface singularities II, Inventiones math. 88 (1987), 165–183 · Zbl 0617.14034 · doi:10.1007/BF01405096 |
| [4] | [Di 1] Dieterich, E., Reduction of isolated singularities, Comment. Math. Helvetici 62 (1987), 654–604 · Zbl 0654.14002 · doi:10.1007/BF02564469 |
| [5] | [Di 2] Dieterich, E., The Auslander-Reiten quiver of an isolated singularity, Springer Lecture Notes in Mathematics 1273 (1987), 244–264 · Zbl 0632.14004 · doi:10.1007/BFb0078848 |
| [6] | [EG] Evans, E.G., Griffith, P., Syzygies, London Mathematical Society Lecture Note Series, Cambridge 1985 |
| [7] | [Ei] Eisenbud, D. Homological algebra on a complete intersection with an application to group representations, Trans. AMS 260,1 (1980), 35–64 · Zbl 0444.13006 · doi:10.1090/S0002-9947-1980-0570778-7 |
| [8] | [GK] Greuel, G.M., Kröning, H., Simple singularities in positive characteristic, Math. Z. 203 (1990), 339–354 · Zbl 0715.14001 · doi:10.1007/BF02570742 |
| [9] | [HPR] Happel, D., Preiser, U., Ringel, C.M., Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules, Proceedings of the third International Conference on Representations of algebras, Carleton University, Ottawa 1979, Springer Lecture Notes in Mathematics 832, 280–294 · Zbl 0446.16032 |
| [10] | [Kn] Knörrer, H., Cohen Macaulay modules on hypersurface singularities I, Inventiones math. 88 (1987), 153–164 · Zbl 0617.14033 · doi:10.1007/BF01405095 |
| [11] | [Ma] Matsumura, H., Commutative ring theory, Cambridge 1986 · Zbl 0603.13001 |
| [12] | [Po] Popescu, D., Indecomposable Cohen Macaulay modules and their multiplicities, Trans. AMS 320 (1991) · Zbl 0725.13009 |
| [13] | [PR] Popescu, D., Roczen, M., Indecomposable Cohen Macaulay modules and irreducible maps, Compositio Mathematica 76 (1990), 277–294 · Zbl 0715.13010 |
| [14] | [Re] Reiten, L, Finite dimensional algebras and singularities, Springer Lecture Notes in Mathematics 1273 (1987), 35–57 · Zbl 0646.90092 · doi:10.1007/BFb0078837 |
| [15] | [Ri] Ringel, C.M., Tame algebras and integral quadratic forms, Springer Lecture Notes in Mathematics 1099 (1984) · Zbl 0546.16013 |
| [16] | [Sc] Schreyer, F. O., Finite and countable CM-representation type, Springer Lecture Notes in Mathematics No 1273 (1987), 9–34 · doi:10.1007/BFb0078836 |
| [17] | [So] Solberg, Ø., Hypersurface singularities of finite Cohen Macaulay type, Proc. London Math. Soc., III. Ser. 58 (1989), 258–280 · Zbl 0631.13019 · doi:10.1112/plms/s3-58.2.258 |
| [18] | [Yo] Yoshino, Y., Brauer– Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan 39, 4 (1984), 719–739 · Zbl 0615.13008 · doi:10.2969/jmsj/03940719 |
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