Endomorphism Rings of Finite Global Dimension

Leuschke, Graham J.

Canadian journal of mathematics, Tome 59 (2007) no. 2, pp. 332-342

Résumé

For a commutative local ring $R$ , consider (noncommutative) $R$ -algebras $\Lambda$ of the form $\Lambda \,=\,\text{En}{{\text{d}}_{R}}\left( M \right)$ where $M$ is a reflexive $R$ -module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec $R$ . For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $\mathbb{C}$ -algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $\Lambda$ with finite global dimension and which is maximal Cohen-Macaulay over $R$ (a “noncommutative crepant resolution of singularities”). We produce algebras $\Lambda \,=\,\text{En}{{\text{d}}_{R}}\left( M \right) $ having finite global dimension in two contexts: when $R$ is a reduced one-dimensional complete local ring, or when $R$ is a Cohen-Macaulay local ring of finite Cohen–Macaulay type. If in the latter case $R$ is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

Détail

DOI : 10.4153/CJM-2007-014-1

Mots-clés : 16G60, 13H10, 16E99, representation dimension, noncommutative crepant resolution, maximal Cohen–Macaulay modules

Bibliographie

[1] [1] Auslander, M., Representation dimension of artin algebras. In: Selected Works of Maurice Auslander. Part 1, Amer. Math. Soc., Providence, RI, 1999. Google Scholar

[2] [2] Auslander, M., Rational singularities and almost split sequences. Trans. Amer. Math. Soc. 293(1986), no. 2, 511–531. Google Scholar

[3] [3] Auslander, M., Reiten, I., and Smalø, S., Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, 1995. Google Scholar

[4] [4] Bondal, A. and Orlov, D., Derived categories of coherent sheaves. In: Proceedings of the International Congress of Mathematicians, II, Higher Education Press, Beijing, 2002, pp. 47–56. Google Scholar

[5] [5] Buchweitz, R.-O., Greuel, G.-M., and Schreyer, F.-O., Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math. 88(1987), no. 1, 165–182. Google Scholar

[6] [6] Çimen, N., One-dimensional rings of finite Cohen–Macaulay type. Ph.D. thesis, University of Nebraska–Lincoln, Lincoln, NE, 1994. Google Scholar

[7] [7] de Jong, T., An algorithm for computing the integral closure. J. Symbolic Computation 26(1998), no. 3, 273–277. Google Scholar

[8] [8] Drozd, Ju. A. and Roĭter, A. V., Commutative rings with a finite number of indecomposable integral representations. Izv. Akad. Nauk. SSSR Ser. Mat. 31(1967), 783–798. Google Scholar

[9] [9] Esnault, H., Reflexive modules on quotient surface singularities. J. Reine Angew. Math. 362(1985), 63–71. Google Scholar

[10] [10] Grayson, D. and Stillman, M., Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/. Google Scholar

[11] [11] Green, E. L. and Reiner, I., Integral representations and diagrams. Michigan Math. J. 25(1978), no. 1, 53–84. Google Scholar

[12] [12] Herzog, J., Ringe mit nur endlich vielen Isomorphieklassen von maximalen unzerlegbaren Cohen-Macaulay-Moduln. Math. Ann. 233(1978), no. 1, 21–34. Google Scholar

[13] [13] Iyama, O., Rejective subcategories of artin algebras and order. Fields. Inst. Commun. 40(2004), 45–64. Google Scholar

[14] [14] Iyama, O., Representation dimension and Solomon zeta function. In: Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Comm. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 45–64. Google Scholar

[15] [15] Iyama, O., Finiteness of representation dimension. Proc. Amer. Math. Soc. 131(2003), no. 4, 1011–1014. Google Scholar

[16] [16] Kapranov, M. and Vasserot, E., Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(2000), no. 3, 565–576. Google Scholar

[17] [17] Knörrer, H., Cohen-Macaulay modules on hypersurface singularities. I. Invent. Math. 88(1987), no. 1, 153–164. Google Scholar

[18] [18] Matsumura, H., Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986. Google Scholar

[19] [19] Rouquier, R., Dimensions of triangulated categories. arXiv:math.CT/0310134. Google Scholar

[20] [20] Van den Bergh, M., Non-commutative crepant resolutions. In: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770. Google Scholar

[21] [21] Van den Bergh, M., Three-dimensional flops and non-commutative rings. Duke Math. J. 122(2004), no. 3, 423–455. Google Scholar

[22] [22] Wiegand, R., Noetherian rings of bounded representation type. In: Commutative Algebra, Math. Sci. Res. Inst. Publ. 15, Springer, New York, 1989, pp. 497–516. Google Scholar

[23] [23] Wiegand, R., One-dimensional local rings with finite Cohen-Macaulay type. In: Algebraic Geometry and its Applications. Springer, New York, 1994, pp. 381–389. Google Scholar

[24] [24] Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings. London Mathematical Society Lecture Notes Series 146, Cambridge University Press, Cambridge, 1990. Google Scholar