Relative singularity categories and Gorenstein-projective modules.(English)Zbl 1244.18014
The notion of maximal Cohen-Macaulay approximation was introduced by Auslander and Buchweitz in the mid-80s. While their published account deals with this concept in a categorical setting, prior work had dealt with specific examples in local algebra, and in particular with Gorenstein-projective modules. It is well-known that mCM-modules are left-orthogonal to modules of finite projective dimension, and that the intersection of the two classes consists precisely of free modules. In a more general setting, when the ring is Cohen-Macaulay with a dualizing module, the same results are true, with some obvious modifications. Namely, modules of finite projective dimension should be replaced by modules of finite injective dimension, and the intersection would then consist of direct sums of copies of the dualizing module. In either case, the modules in the intersection would be self-orthogonal.
This is the starting point of the present paper. The author starts with a self-orthogonal category and then constructs relative CM-modules. It is convenient to work with these constructs in the derived category. In this setting, the author gives generalizations of some results well-known in the classical setup.
This is the starting point of the present paper. The author starts with a self-orthogonal category and then constructs relative CM-modules. It is convenient to work with these constructs in the derived category. In this setting, the author gives generalizations of some results well-known in the classical setup.
Reviewer: Alex Martsinkovsky (Boston)
MSC:
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 18G35 | Chain complexes (category-theoretic aspects), dg categories |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16E35 | Derived categories and associative algebras |
Keywords:
relative singularity category;Gorenstein-projective modules;compact objects;mCM-modules;Gorenstein local rings;relative CM-modulesCite
References:
| [1] | Auslander, Representation Dimension of Artin Algebras, Lecture Notes (1971) ·Zbl 0331.16026 |
| [2] | Auslander, The homological theory of maximal Cohen-Macaulay approximations, Memoire de la S.M.F. 2e serie, tome 38 pp 5– (1989) ·Zbl 0697.13005 |
| [3] | Auslander, Applications of contravariantly finite subcategories, Adv. Math. 86 (1) pp 111– (1991) ·Zbl 0774.16006 ·doi:10.1016/0001-8708(91)90037-8 |
| [4] | Auslander, Representation Theory of Artin Algebras (1995) ·Zbl 0834.16001 ·doi:10.1017/CBO9780511623608 |
| [5] | Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co)stabilization, Comm. Algebra 28 pp 4547– (2000) ·Zbl 0964.18006 ·doi:10.1080/00927870008827105 |
| [6] | Beligiannis, Homotopy theory of modules and Gorenstein rings, Mathematica Scand. 89 pp 5– (2001) ·Zbl 1023.55009 ·doi:10.7146/math.scand.a-14329 |
| [7] | Beligiannis, Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras, J. Algebra 288 pp 137– (2005) ·Zbl 1119.16007 ·doi:10.1016/j.jalgebra.2005.02.022 |
| [8] | Beligiannis, Homological and Homotopical Aspects of Torsion Theories Memories of the American Mathematical Society (2007) ·Zbl 1124.18005 |
| [9] | R. O. Buchweitz Maximal Cohen-Macaulay Modules and Tate Cohomology over Gorenstein Rings 1987 https://tspace.library.utoronto.ca/handle/1807/16682 |
| [10] | Chen, Quotient triangulated categories, Manuscripta Math. 123 pp 167– (2007) ·Zbl 1129.16011 ·doi:10.1007/s00229-007-0090-6 |
| [11] | Enochs, Relative Homological Algebra, de Gruyter Expositions in Mathematics (2000) ·Zbl 0952.13001 ·doi:10.1515/9783110803662 |
| [12] | Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Mathematical Society Lecture Notes Series (1988) ·Zbl 0635.16017 ·doi:10.1017/CBO9780511629228 |
| [13] | Happel, On Gorenstein Algebras, Progress in Mathematics (1991) |
| [14] | Hartshorne, Residue and Duality, Lecture Notes in Mathematics (1966) ·Zbl 0212.26101 ·doi:10.1007/BFb0080482 |
| [15] | Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 pp 553– (2002) ·Zbl 1016.55010 ·doi:10.1007/s00209-002-0431-9 |
| [16] | Iyengar, Acyclicity versus totally acyclicity for complexes over noetherian rings, Documenta Math. 11 pp 207– (2006) ·Zbl 1119.13014 |
| [17] | Jørgensen, The homotopy category of complexes of projective modules, Adv. Math. 193 pp 223– (2005) ·Zbl 1068.18012 ·doi:10.1016/j.aim.2004.05.003 |
| [18] | Keller, Chain complexes and stable categories, Manuscripta Math. 67 pp 379– (1990) ·Zbl 0753.18005 ·doi:10.1007/BF02568439 |
| [19] | Keller, Derived categories and universal problems, Comm. Algebra 19 pp 699– (1991) ·Zbl 0722.18002 ·doi:10.1080/00927879108824166 |
| [20] | Keller, On the construction of triangle equivalences, in: Derived Equivalences for Group Rings, Lecture Notes in Mathematics (1998) |
| [21] | Keller, Sous les cateégories dérivées, C. R. Acad. Sci. Paris 305 pp 225– (1987) |
| [22] | Krause, The stable derived category of a noetherian scheme, Compositio Math. 141 pp 1128– (2005) ·Zbl 1090.18006 ·doi:10.1112/S0010437X05001375 |
| [23] | Neeman, The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. 25 pp 547– (1992) ·Zbl 0868.19001 |
| [24] | Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent. Math. 174 (2) pp 255– (2008) ·Zbl 1184.18008 ·doi:10.1007/s00222-008-0131-0 |
| [25] | Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 246 (3) pp 227– (2004) ·Zbl 1101.81093 |
| [26] | Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 pp 303– (1989) ·Zbl 0685.16016 ·doi:10.1016/0022-4049(89)90081-9 |
| [27] | Verdier, Catégories dérivées, etat 0, Springer Lecture Notes 569 pp 262– (1977) |
| [28] | C. C. Xi 2008 http://math.bnu.edu.cn/ccxi/Papers/Articles/rart.pdf |
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