zbMATH — the first resource for mathematics

Noncommutative resolutions and rational singularities. (English) Zbl 1177.14026
In this article, it is important that \(k\) is a fixed algebraically closed field of characteristic \(0\). The authors prove that the center of a homologically homogeneous, finitely generated \(k\)-algebra has rational singularities. Assume \(X=\text{Spec} R\) for an affine, Gorenstein \(k\)-algebra \(R\). In this article, a commutative resolution of singularities is a crepant homomorphism \(f:Y\rightarrow X,\) i.e. \(f^\ast\omega_Y=\omega_X.\) Bondal and Orlov conjectured that two such resolutions are derived equivalent, and this was later proved by Bridgeland. The authors generalize this to a third noncommutative crepant resolution explaining Bridgeland’s proof. This observation leads to different approaches to the Bondal-Orlov conjecture and related topics. The question now is how the existence of a noncommutative crepant resolution affects the original commutative singularity. It is known that if a Gorenstein singularity has a crepant resolution then it has rational singularities. The authors asks wether this is true for a noncommutative crepant resolution. The article answers this affirmatively.
Let \(\Delta\) be a prime affine \(k\)-algebra that is finitely generated as a module over its center \(Z(\Delta).\) \(\Delta\) is called homologically homogeneous of dimension \(d\) if all simple \(\Delta\)-modules have the same projective dimension \(d.\) The properties of homologically homogeneous rings are close to commutative regular rings, and the idea is to use such a ring \(\Delta\) as a noncommutative analogue of a crepant resolution. Formally, a noncommutative crepant resolution of \(R\) is any homologically homogeneous ring of the form \(\Delta=\text{End}_R(M)\) where \(M\) is a reflexive and finitely generated \(R\)-module. The main result of the article is the following:
Theorem. Let \(\Delta\) be a homologically homogeneous \(k\)-algebra. Then the center \(Z(\Delta)\) has rational singularities. In particular, if a normal affine \(k\)-domain \(R\) has a noncommutative crepant resolution, then it has rational singularities.
Also, examples are given proving that this theorem may fail in positive characteristic.
The article starts with the properties of homologically homogeneous rings, based on tame orders: If \(\Delta\) is a prime ring with simple Artinian ring of fractions \(A\) (i.e. \(\Delta\) is a prime order in \(A\)), \(\Delta\) is called a tame \(R\)-order if it is a finitely generated and reflexive \(R\)-module such that \(\Delta_{\mathfrak p}\) is hereditary for all prime ideals \(\mathfrak p\) in \(R\) of height \(1\).
A homologically homogeneous ring \(\Delta\) of dimension \(d\) is Cohen Macaulay (CM) over its center \(Z(\Delta)\), both GK\(\dim\Delta\) and the global homological dimension gl\(\dim\Delta\) of \(\Delta\) equal \(d\), the center \(Z=Z(\Delta)\) is an affine CM normal domain, and finally, \(\Delta\) is a tame \(Z\)-order.
The rest of the article is then used to prove the main theorem. This involves reduction to the Calabi-Yau case for proving that \(Z\) has rational singularities by a generalization of the commutative method where one constructs a Gorenstein cover of a \(\mathbb Q\)-Gorenstein singularity.
The article ends with examples proving, among other things, that the main theorem may fail in the case where \(k\) has positive characteristic.
The article is precise, and illustrates noncommutative algebraic geometry in a concrete way. It also give useful criterions and ideas to be followed in other settings in noncommutative geometry.

14A22 Noncommutative algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
16S38 Rings arising from noncommutative algebraic geometry
18G20 Homological dimension (category-theoretic aspects)
Full Text: DOI arXiv
[1] M. Artin, Wildly ramified \(\mathbb Z/2\) actions in dimension two, Proc. Amer. Math. Soc. 52 (1975), 60–64. · Zbl 0315.14015 · doi:10.2307/2040100
[2] R. Bezrukavnikov, Noncommutative counterparts of the Springer resolution, Proceedings of the International Congress of Mathematicians, vol. II, pp. 1119–1144, European Mathematical Society, Zürich, 2006. · Zbl 1135.17011
[3] R. Bezrukavnikov and D. B. Kaledin, McKay equivalence for symplectic resolutions of quotient singularities, Tr. Mat. Inst. Steklova 246 (2004), 20–42. · Zbl 1137.14301
[4] A. I. Bondal and D. O. Orlov, Derived categories of coherent sheaves, Proceedings of the International Congress of Mathematicians (Beijing, 2002), vol. II, pp. 47–56, Higher Education Press, Beijing, 2002. · Zbl 0996.18007
[5] ——, Semi-orthogonal decompositions for algebraic varieties, preprint, math.AG/950601.
[6] J. F. Boutot, Singularités rationelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65–68. · Zbl 0619.14029 · doi:10.1007/BF01405091 · eudml:143444
[7] A. Braun, On symmetric, smooth and Calabi–Yau algebras, J. Algebra 317 (2007), 519–533. · Zbl 1172.16306 · doi:10.1016/j.jalgebra.2007.08.021
[8] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613–632. · Zbl 1085.14017 · doi:10.1007/s002220100185
[9] K. A. Brown and C. R. Hajarnavis, Homologically homogeneous rings, Trans. Amer. Math. Soc. 281 (1984), 197–208. JSTOR: · Zbl 0531.16019 · doi:10.2307/1999529 · links.jstor.org
[10] K. A. Brown, C. R. Hajarnavis, and A. B. MacEacharn, Rings of finite global dimension integral over their centres, Comm. Algebra 11 (1983), 67–93. · Zbl 0502.16020 · doi:10.1080/00927878308822837
[11] R.-O. Buchweitz, G. Leuschke, and M. Van den Bergh, Noncommutative desingularization of the generic determinant, in preparation. · Zbl 1204.14003
[12] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ, 1956. · Zbl 0075.24305
[13] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom. 61 (2002), 227–261. · Zbl 1090.14003
[14] C. W. Curtis and I. Reiner, Methods of representation theory I. With applications to finite groups and orders, Wiley, New York, 1981. · Zbl 0469.20001
[15] K. De Naeghel and M. Van den Bergh, Ideal classes of three dimensional Artin–Schelter regular algebras, J. Algebra 283 (2005), 399–429. · Zbl 1069.16033 · doi:10.1016/j.jalgebra.2004.06.011
[16] R. M. Fossum, The Noetherian different of projective orders, Bull. Amer. Math. Soc. 72 (1966), 898–900. · Zbl 0156.04201 · doi:10.1090/S0002-9904-1966-11607-5
[17] V. Ginzburg, Calabi–Yau algebras, preprint, math.AG/0612139.
[18] O. Iyama and I. Reiten, Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras, preprint, math.RT/0605136. · Zbl 1162.16007 · doi:10.1353/ajm.0.0011
[19] D. Kaledin, On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.) 9 (2003), 529–555. · Zbl 1066.14003 · doi:10.1007/s00029-003-0308-8
[20] ——, Derived equivalences by quantization, preprint, math.AG/0504584.
[21] Y. Kawamata, \(D\) -equivalence and \(K\) -equivalence, J. Differential Geom. 61 (2002), 147–171. · Zbl 1056.14021
[22] D. S. Keeler, D. Rogalski, and J. T. Stafford, Naï ve noncommutative blowing up, Duke Math. J. 126 (2005), 491–546. · Zbl 1082.14003 · doi:10.1215/S0012-7094-04-12633-8
[23] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings I, Lecture Notes in Math., 339, Springer-Verlag, Berlin, 1973. · Zbl 0271.14017
[24] F. Knop, Der kanonische Modul eines Invariantenrings, J. Algebra 127 (1989), 40–54. · Zbl 0716.20021 · doi:10.1016/0021-8693(89)90271-8
[25] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998.
[26] L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded orders, Birkhäuser, Boston, 1988.
[27] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, Chichester, 1987. · Zbl 0644.16008
[28] E. Nauwelaerts and F. Van Oystaeyen, Finite generalized crossed products over tame and maximal orders, J. Algebra 101 (1986), 61–68. · Zbl 0588.16002 · doi:10.1016/0021-8693(86)90096-7
[29] J. Rainwater, Global dimension of fully bounded noetherian rings, Comm. Algebra 15 (1987), 2143–2156. · Zbl 0628.16010 · doi:10.1080/00927878708823527
[30] I. Reiner, Maximal orders, London Math. Soc. Monogr. (N.S.), 5, Academic Press, London, 1975.
[31] L. Silver, Tame orders, tame ramification and Galois cohomology, Illinois J. Math. 12 (1968), 7–34. · Zbl 0162.05602
[32] J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian PI rings, J. Algebra 168 (1994), 988–1026. · Zbl 0812.16046 · doi:10.1006/jabr.1994.1267
[33] M. Van den Bergh, Existence theorems for dualizing complexes over noncommutative graded and filtered rings, J. Algebra 195 (1997), 662–679. · Zbl 0894.16020 · doi:10.1006/jabr.1997.7052
[34] ——, Noncommutative crepant resolutions, The legacy of Niels Henrik Abel, pp. 749–770, Springer-Verlag, Berlin, 2004. · Zbl 1047.00019 · doi:10.1007/978-3-642-18908-1
[35] ——, Three dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), 423–455. · Zbl 1074.14013 · doi:10.1215/S0012-7094-04-12231-6
[36] A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), 41–84. · Zbl 0790.18005 · doi:10.1016/0021-8693(92)90148-F
[37] ——, Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. (2) 60 (1999), 723–746. · Zbl 0954.16006 · doi:10.1112/S0024610799008108
[38] A. Yekutieli and J. J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), 1–51. · Zbl 0948.16006 · doi:10.1006/jabr.1998.7657
[39] ——, Residue complexes over noncommutative rings, J. Algebra 259 (2003), 451–493. · Zbl 1035.16004 · doi:10.1016/S0021-8693(02)00579-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.