# zbMATH — the first resource for mathematics

Non-commutative resolutions and Grothendieck groups. (English) Zbl 1327.13053
M. Auslander initiated the use of noncommutative algebras of finite dimension by studying the representation theory of Cohen-Macaulay rings, that is, more generally orders. Two classes of noncommutative algebras were singled out, the Auslander algebras and the non-singular orders. The representation theory of finite-dimensional representations is encoded in the structure of their Auslander algebras. All Cohen-Macaulay modules are projective, so that the study of the non-singular orders are more basic.
The study of such algebras gives applications in algebraic geometry. A first example is Van den Bergh’s definition of Noncommutative Crepant Resolution, NCCR: Let $$R$$ be a commutative noetherian normal domain. Then a reflexive $$R$$-module $$M$$ is said to give a NCCR of $$\text{Spec}R$$ if $$\Lambda=\text{End}_R(M)$$ is a nonsingular $$R$$-order, which means that $$\Lambda_{\mathfrak p}=\text{End}_{R_{\mathfrak p}}(M_{\mathfrak p},M_{\mathfrak p})$$ is a maximal Cohen-Macaulay $$R_{\mathfrak p}$$-module for each $$\mathfrak p\in\text{Spec}R$$.
This article considers the slightly simpler concept of NCR, that is Noncommutative resolution: A finitely generated module $$M$$ over a commutative noetherian ring $$R$$ is called a NCR of $$\text{Spec}(R)$$ if $$M$$ is faithful and $$\text{End}_R(M)$$ has finite global dimension. NCRs exist when $$R$$ is artinian, or reduced and one-dimensional. This article treats the question on which rings $$R$$ that have an NCR. The conditions are given in the terms of the Grothendieck group of the category of finitely generated $$R$$-modules, and its subcategories. The existence of an NCR forces strong constraints on the singularities of $$R$$. The formulations of the conditions on $$R$$ in terms of the Grothendieck group leads to influence and use of results from algebraic K-theory. This leads to one of the main results stating that for surface singularities over an algebraically closed field, the existence of a NCR characterise rational singularities. Thus the rationality of a surface singularity can be tested on the existence of a NCR.
The main results of the article, more or less verbatim, is as follows:
Theorem 2.5. Let $$R$$ be a semilocal ring and assume that $$M$$ gives a NCR of $$R$$. Let $$\mathcal C_M$$ be the full subcategory of mod$$R$$ consisting of $$X$$ satisfying $$\text{supp}X\subset\text{NG}(M).$$ Then $$K_0(R)/\langle\mathcal C_M\rangle$$ is a finitely generated abelian group. ($$K_0(R)$$ denotes the Grothendieck group of $$R$$, $$\text{NG}(M)$$ is the nongenerating locus of $$M$$.)
Theorem 3.11. Let $$R$$ be a normal, Cohen-Macaulay standard graded algebra over a subfield $$k$$ of $$\mathbb C$$. Let $$\mathfrak m$$ be the irrelevant ideal of $$R$$. Suppose that $$\text{Spec}R\setminus\{\mathfrak m\}$$ has only rational singularities. Suppose moreover that there exists an $$R$$-module $$M$$ giving a NCR. Then $$\text{Spec}R$$ has only rational singularities.
A few relevant, explicit examples are given, and all in all this article gives nice results from the noncommutative algebraic geometry to the commutative. Also, a really nice historical survey is given in the introduction, and a good list of references ends the article.

##### MSC:
 13D15 Grothendieck groups, $$K$$-theory and commutative rings 14B05 Singularities in algebraic geometry 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 16G30 Representations of orders, lattices, algebras over commutative rings 18G20 Homological dimension (category-theoretic aspects)
Full Text:
##### References:
  M. Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), pp. 1-244. Lecture Notes in Pure Appl. Math., Vol. 37, Dekker, New York, 1978. · Zbl 0383.16015  M. Auslander, Isolated singularities and existence of almost split sequences, In Representation Theory II. Lecture Notes in Mathematics 1178, Springer-Verlag, Berlin- Heidelberg, 1986, 194-242. · Zbl 0633.13007  M. Auslander, O. Goldman, Maximal Orders, Trans. Amer. Math. Soc. 97 (1960), 1-24. · Zbl 0117.02506 · doi:10.2307/1993361  M. Artin, Some Numerical Criteria for Contractability of Curves on Algebraic Surfaces, Amer. J. Math. (3) 84 (1962), 485-496. · Zbl 0105.14404 · doi:10.2307/2372985  I. Burban, Private communication.  S. Bloch, V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. (5) 105 (1983), 1235-1253. · Zbl 0525.14003 · doi:10.2307/2374341  C.-Y. J. Chan, Filtrations of modules, the Chow group, and the Grothendieck group, J. Algebra 219 (1999), 330-344. · Zbl 0949.13008 · doi:10.1006/jabr.1999.7888  C. W. Curtis, I. Reiner, Methods of representation theory. Vol. I. With applications to finite groups and orders, reprint of the 1981 original, Wiley Classics Library. A Wiley- Interscience Publication. John Wiley & Sons, Inc., New York, 1990. · Zbl 0698.20001  H. Dao, Some observations on local and projective hypersurfaces, Math. Res. Lett. 15 (2008), 207-219. · Zbl 1229.13014 · doi:10.4310/MRL.2008.v15.n2.a1 · arxiv:math/0701881  R. M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. · Zbl 0256.13001  K. R. Fuller, W. A. Shutters, Projective modules over non-commutative semilocal rings, Tohoku Math. J. (2) 27 (1975), no. 3, 303-311. · Zbl 0316.16026 · doi:10.2748/tmj/1203529242  W. Fulton, Intersection Theory, Springer-Verlag, Berlin (1998). · Zbl 0885.14002  V. Gavran, Kahn’s correspondence and Cohen-Macaulay modules over abstract surface and curve singularities, Journal of Singularities 4 (2012), 68-73. · Zbl 1292.13003 · doi:10.5427/jsing.2012.4d  R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, Springer-Verlag, New York, (1977). · Zbl 0367.14001  A. Heller, Some exact sequences in algebraic K-theory, Topology 4 (1965), 389-408. · Zbl 0161.01507 · doi:10.1016/0040-9383(65)90004-2  C. Huneke, R. Wiegand, Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), 161-183. · Zbl 0908.13010 · eudml:167433  O. Iyama, Rejective subcategories of artin algebras and orders, · Zbl 1124.16017 · arxiv.org  O. Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22-50. · Zbl 1115.16005 · doi:10.1016/j.aim.2006.06.002 · arxiv:math/0407052  O. Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51-82. · Zbl 1115.16006 · doi:10.1016/j.aim.2006.06.003 · arxiv:math/0411631  O. Iyama, M. Wemyss, The classification of special Cohen-Macaulay modules, Math. Z. 265 (2010), no. 1, 41-83. · Zbl 1192.13012 · doi:10.1007/s00209-009-0501-3 · arxiv:0809.1958  O. Iyama, M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), no. 3, 521-586. · Zbl 1308.14007 · doi:10.1007/s00222-013-0491-y · arxiv:1007.1296  K. Kurano, A remark on the Riemann-Roch formula for affine scheme associated with Noetherian local ring, Tohoku Math J. 48 (1996), 121-138. · Zbl 0882.14002 · doi:10.2748/tmj/1178225414  K. Kurano, Numerical equivalence defined on Chow groups of Notherian local rings, Invent. Math. 157 (2004), 575-619. · Zbl 1070.14007 · doi:10.1007/s00222-004-0361-8 · arxiv:math/0304220  G. J. Leuschke, Non-commutative crepant resolutions: scenes from categorical geometry, Progress in commutative algebra 1, 293-361, de Gruyter, Berlin, 2012. · Zbl 1254.13001 · arxiv:1103.5380  J. Lipman, Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. I.H.E.S. 36 (1969), 195-279. · Zbl 0181.48903 · doi:10.1007/BF02684604 · numdam:PMIHES_1969__36__195_0 · eudml:103893  D. Mumford, Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto. Univ. 9 (1968), 195-204. · Zbl 0184.46603  N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3. Academic Press, London-New York, 1973. · Zbl 0271.18006  A. A. Roitman, Rational equivalence of 0-cycles, Math. USSR Sbornik, 18 (1972), 571-588. · Zbl 0273.14001 · doi:10.1070/SM1972v018n04ABEH001860  I. Reiten, M. Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Memoirs Amer. Math. Soc. 80 (1989), no. 408. · Zbl 0677.16002  P. Salmon, Su un problema posto da P. Samuel, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 40 (1966), 801-803. · Zbl 0147.01601  K. E. Smith, F-Rational Rings have Rational Singularities, Amer. J. Math. 119 (1997), 159-180. · Zbl 0910.13004 · doi:10.1353/ajm.1997.0007 · muse.jhu.edu  J. T. Stafford, M. Van den Bergh, Noncommutative resolutions and rational singularities, Mich. Math. J. 57 (2008), 659-674. · Zbl 1177.14026 · doi:10.1307/mmj/1220879430 · arxiv:math/0612032
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.