リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Simple-minded objects in triangulated categories and Cohen-Macaulay differential graded modules」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。
リケラボ

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Simple-minded objects in triangulated categories and Cohen-Macaulay differential graded modules

JIN, Haibo 晋, 海波 名古屋大学

2020.10.19

概要

Part 1. Derived categories and triangulated categories appear in many areas of mathematics, such as algebraic geometry, representation theory and algebraic topology. An important way to construct a new triangulated category is the Verdier quotient. But usually the morphisms in the quotient category are too complicated to understand. It is known in some nice cases, Verdier quotient can be realized as a reduction process, which is another way to construct a new triangulated category. The reduction can be realized as a certain sub (or subfactor) category of the given one and is easier to study.

The first part of this thesis is devoted to introducing a new reduction process of triangulated categories with respect to simple-minded collections (or SMC for short). Simple modules are one of the most basic and important objects in the representation theory of algebras. There have been many generalizations and studies of simple modules. Among them, SMC in derived categories and SMS (=simple-minded system) in singularity categories have particular importance. We show that SMC behaves very nice under reduction process and there is a bijection between SMCs in the new triangulated category and SMCs in the original one containing certain objects. A similar process called the SMS reduction has been studied by Coelho Simoes and Pauksztello.

Moreover, we consider the pair (𝒯, 𝒯 𝑝 ) of a triangulated category 𝒯 and its thick subcategory 𝒯 𝑝 satisfying certain assumptions, and also the singularity category 𝒯𝑠𝑔: = 𝒯/𝒯 𝑝 . We show the quotient map sends SMCs in 𝒯 to SMSs in 𝒯𝑠𝑔, which is highly non-trivial. Further we prove that the SMS reduction is the shadow of the SMC reduction, that is, the following diagram of operations commutes

This is parallel to a result obtained by Iyama and Yang that Calabi-Yau reduction is the shadow of silting reduction.

Part 2. The second part of this thesis is devoted to introducing Cohen-Macaulay (CM) differential graded (dg) modules and study their representation theory. The notion of CohenMacaulay (CM) modules is classical in commutative algebras, and has natural generalizations for non-commutative algebras. The category of CM modules has been studied by many researchers in representation theory. On the other hand, the derived categories of differential graded (dg) categories introduced by Bondal-Kapranov and Keller is an active subject appearing in various areas of mathematics. We are aimed to introduce CM dg modules to connect these two subjects.

We consider a nice class of dg algebras 𝐴 called Gorenstein and introduce the category 𝐂𝐌 𝐴 of CM dg A-modules. For the case of Gorenstein algebras, the category of CM modules is a Frobenius exact category, and Buchweitz found a triangle equivalence between the stable category of CM modules and certain Verdier quotient of the derived category called the singularity category by Orlov. However, 𝐂𝐌 𝐴 does not have a natural structure of Frobenius exact category in our setting. Surprisingly we find that it has a structure of Frobenius extriangulated category, which is introduced by Nakaoka and Palu as a common generalization of exact category and triangulated category. As an analogue of Buchweitz's equivalence, we show that the stable category 𝑪𝑴 𝐴 is triangle equivalent to the singularity category. We study 𝐂𝐌 𝐴 from a point of view of representation theory, and show that it admits Auslander-Reiten extensions. One of the most important problems in CM representation is to understand CMfinite Gorenstein (dg) algebras, in the sense that they have only finitely many indecomposable CM (dg) modules. We give a complete classification of Auslander-Reiten quivers of CM-finite dsymmetric dg algebras for 𝑑 ≥ 1, which is one of the most basic cases of CM-finite Gorenstein dg algebras.

Part 3. The third part is the continuation of the first two parts. Recently there is increasing interest in negative Calabi-Yau (CY) triangulated categories, including 𝑪𝑴 𝐴 for 𝑑 −symmetric dg algebras 𝐴 (which are (−𝑑) − 𝐶𝑌). These categories often contain d-SMS, which plays a key role in the study. We show that there is a bijection between d-SMSs in (−𝑑) −CY cluster category and maximal simplices in ∆ 𝑑 (Φ) without negative simple roots, where ∆ 𝑑 (Φ) is introduced by Fomin and Reading as a generalization of the combinatorial structure of cluster algebras of finite type.

論文の公開元へ

関連論文

関連論文をもっと見る

参考文献

[ABu] Maurice Auslander, Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). M´em. Soc. Math. France (N.S.) No. 38 (1989), 5–37.

[ABr] Maurice Auslander, Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94 American Mathematical Society, Providence, R.I. 1969.

[AI] Takuma Aihara, Osamu Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668.

[AKLY] Lidia Angeleri H¨ugel, Steffen Koenig, Qunhua Liu, Dong Yang, Ladders and simplicity of derived module categories, J. Algebra 472 (2017), 15–66.

[Al] Salah Al-Nofayee, Simple objects in the heart of a t-structure, J. Pure Appl. Algebra 213 (2009), no. 1, 54–59.

[Am] Claire Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Annales de l’institut Fourier. 2009, 59(6): 2525–2590.

[AMY] Takahide Adachi, Yuya Mizuno, Dong Yang, Discreteness of silting objects and t-structures in triangulated categories, Proc. Lond. Math. Soc. (3) 118 (2019), no. 1, 1–42.

[An] Marcus Andrei, Tilting complexes for group graded algebras, J. Group Theory 6 (2003), no. 2, 175–193.

[Ar] Drew Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), no. 949.

[ARS] Maurice Auslander, Idun Reiten, Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995. xiv+423 pp. ISBN: 0-521-41134-3.

[B] Ragnar-Olaf Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, unpublished manuscript.

[BBD] Aleksandr A. Beilinson, Joseph N. Bernstein, Pierre Deligne, Faisceaux pervers (French) [Perverse sheaves] Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Ast´erisque, 100, Soc. Math. France, Paris, 1982.

[BGS] Alexander Beilinson, Victor Ginzburg, Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.

[BK] Alexey Bondal, Mikhail Kapranov, Enhanced triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683; translation in Math. USSR-Sb. 70 (1991), no.1, 93–107.

[BH] Winfried Bruns, J¨urgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.

[BMRRT] Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, Gordana Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618.

[Bo] Mikhail V. Bondarko, Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), no. 3, 387–504.

[BRT1] Aslak Bakke Buan, Idun Reiten, Hugh Thomas, Three kinds of mutation, J. Algebra 339 (2011), 97–113.

[BRT2] Aslak Bakke Buan, Idun Reiten, Hugh Thomas, From m-clusters to m-noncrossing partitions via exceptional sequences, Math. Z. 271 (2012), no. 3-4, 1117–1139.

[BY] Thomas Br¨ustle, Dong Yang, Ordered exchange graphs, Advances in representation theory of algebras, 135– 193, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨urich, 2013.

[C1] Xiao-Wu Chen, On extensions of covariantly finite subcategories, arXiv:0809.3387.

[C2] Xiao-Wu Chen, Singularity categories, Schur functors and triangular matrix rings, Algebr. Represent. Theory 12 (2009), no. 2-5, 181–191.

[C3] Xiao-Wu Chen, Gorenstein Homological Algebra of Artin Algebras, arXiv:1712.04587.

[CR] Charles W. Curtis, Irving Reiner, Methods of representation theory. Vol. I. With applications to finite groups and orders, Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1981.

[CS1] Raquel Coelho Sim˜oes, Hom-configurations and noncrossing partitions, J. Algebraic Combin. 35 (2012), no. 2, 313–343.

[CS2] Raquel Coelho Sim˜oes, Hom-configurations in triangulated categories generated by spherical objects, J. Pure Appl. Algebra 219 (2015), no. 8, 3322–3336.

[CS3] Raquel Coelho Sim˜oes, Mutations of simple-minded systems in Calabi-Yau categories generated by a spherical object, Forum Math. 29 (2017), no. 5, 1065–1081.

[CSP] Raquel Coelho Sim˜oes, David Pauksztello, Simple-minded systems and reduction for negative Calabi-Yau triangulated categories, Trans. Amer. Math. Soc. 373 (2020), no. 4, 2463–2498.

[CSPP] Raquel Coelho Sim˜oes, David Pauksztello, David Ploog, Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions, arXiv:2004.00604.

[D] Alex Dugas, Torsion pairs and simple-minded systems in triangulated categories, Appl. Categ. Structures 23 (2015), no. 3, 507–526. SIMPLE-MINDED OBJECTS AND CM DG MODULES 69

[EJ] Edgar E. Enochs, Overtoun M. G. Jenda, Relative homological algebra, Volume 1. Second revised and extended edition. De Gruyter Expositions in Mathematics, 30. Walter de Gruyter GmbH & Co. KG, Berlin, 2011.

[FP] Elsa A. Fern´andez, Mar´ıa In´es Platzeck, Isomorphic trivial extensions of finite dimensional algebras, J. Pure Appl. Algebra 204 (2006), no. 1, 9–20.

[FR] Sergey Fomin, Nathan Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 2005, no. 44, 2709–2757.

[FZ1] Sergey Fomin, Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529.

[FZ2] Sergey Fomin, Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121.

[Ga] Peter Gabriel, Auslander-Reiten sequences and representation-finite algebras, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 1–71, Lecture Notes in Math., 831, Springer, Berlin, 1980.

[Gu] Lingyan Guo, Cluster tilting objects in generalized higher cluster categories, J. Pure Appl. Algebra 215 (2011), no. 9, 2055–2071.

[GR] Peter Gabriel, Christine Riedtmann, Group representations without groups, Comment. Math. Helv. 54 (1979), no. 2, 240–287.

[H1] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988.

[H2] Dieter Happel, On Gorenstein algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 389–404, Progr. Math., 95, Birkh¨auser, Basel, 1991.

[HJY] Thorsten Holm, Peter Jørgensen, Dong Yang, Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object, Bull. Lond. Math. Soc. 45 (2013), no. 1, 120–130.

[I1] Osamu Iyama, τ-categories I: Ladders, Algebr. Represent. Theory 8 (2005), no. 3, 297–321.

[I2] Osamu Iyama, Tilting Cohen-Macaulay representations, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. II. Invited lectures, 125–162, World Sci. Publ., Hackensack, NJ,2018.

[INP] Osamu Iyama, Hiroyuki Nakaoka, Yann Palu, Auslander–Reiten theory in extriangulated categories, arXiv:1805.03776.

[IW] Osamu Iyama, Michael Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), no. 3, 521–586.

[IYa1] Osamu Iyama, Dong Yang, Silting reduction and Calabi-Yau reduction of triangulated categories, Trans. Amer. Math. Soc. 370 (2018), no. 11, 7861–7898.

[IYa2] Osamu Iyama, Dong Yang, Quotients of triangulated categories and Equivalences of Buchweitz, Orlov and Amiot–Guo–Keller, to appear in Amer. J. Math. also arXiv:1702.04475.

[IYo] Osamu Iyama, Yuji Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168.

[Ji1] Haibo Jin, Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations, arXiv:1812.03737.

[Ji2] Haibo Jin, Reductions of triangulated categories and simple-minded collections, arXiv:1907.05114.

[Jo1] Peter Jørgensen, Auslander-Reiten theory over topological spaces, Comment. Math. Helv. 79 (2004), no. 1,160–182.

[Jo2] Peter Jørgensen, The Auslander-Reiten quiver of a Poincar´e duality space, Algebr. Represent. Theory 9 (2006), no. 4, 323–336.

[K1] Bernhard Keller, Deriving DG categories, Ann. Sci. Ecole Norm. Sup. (4) 27 (1994), no. 1, 63–102. ´

[K2] Bernhard Keller, On triangulated orbit categories. Doc. Math. 10 (2005), 551–581.

[K3] Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Z¨urich, 2006.

[K4] Bernhard Keller, Corrections to ‘On triangulated orbit categories’, https://webusers.imj-prg.fr/ ˜bernhard.keller/publ/corrTriaOrbit.pdf.

[KaY1] Martin Kalck, Dong Yang, Relative singularity categories I: Auslander resolutions, Adv. Math. 301 (2016), 973–1021.

[KaY2] Martin Kalck, Dong Yang, Derived categories of graded gentle one-cycle algebras, J. Pure Appl. Algebra 222 (2018), no. 10, 3005–3035.

[KL] Steffen Koenig, Yuming Liu, Simple-minded systems in stable module categories, Q. J. Math. 63 (2012), no. 3, 653–674.

[KMV] Bernhard Keller, Daniel Murfet, Michel Van den Bergh, On two examples by Iyama and Yoshino, Compos. Math. 147 (2011), no. 2, 591–612.

[KN] Bernhard Keller, Pedro Nicol´as, Weight structures and simple dg modules for positive dg algebras, Int. Math. Res. Not. IMRN 2013, no. 5, 1028-1078.

[KoY] Steffen Koenig, Dong Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438.

[KR] Bernhard Keller, Idun Reiten, Acyclic Calabi-Yau categories, Compos. Math. 144 (2008), no. 5, 1332–1348. 70 HAIBO JIN

[KYZ] Bernhard Keller, Dong Yang, Guodong Zhou, The Hall algebra of a spherical object, J. Lond. Math. Soc. (2) 80 (2009), no. 3, 771–784.

[L] Xueyu Luo, 0-Calabi-Yau configurations and finite Auslander-Reiten quivers of Gorenstein orders, J. Pure Appl. Algebra 219 (2015), no. 12, 5590–5630.

[LW] Graham J. Leuschke, Roger Wiegand, Cohen-Macaulay representations, Mathematical Surveys and Monographs, 181. American Mathematical Society, Providence, RI, 2012.

[Ma] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge,1989.

[Min] Hiroyuki Minamoto, Resolutions of DG-modules and their applications to commutative DG-algebras, arXiv:1802.01994.

[Miy] Jun-ichi Miyachi, Localization of triangulated categories and derived categories, J. Algebra 141 (1991), no. 2,463–483.

[Ne] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.

[NP] Hiroyuki Nakaoka, Yann Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. G´eom. Diff´er. Cat´eg. 60 (2019), no. 2, 117–193.

[O] Dmitri Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 2004, no. 3(246), 227–248.

[P] David Pauksztello, Compact corigid objects in triangulated categories and co-t-structures, Cent. Eur. J. Math. 6 (2008), no. 1, 25–42.

[PV] Chrysostomos Psaroudakis, Jorge Vit´oria, Recollements of module categories Appl. Categ. Structures 22 (2014), no. 4, 579–593.

[Ric1] Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317.

[Ric2] Jeremy Rickard, Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002), no. 2, 460–481.

[Rie1] Christine Riedtmann, Algebren, Darstellungsk¨ocher, ¨uberlagerungen und zur¨uck, Comment. Math. Helv. 55 (1980), no. 2, 199–224.

[Rie2] Christine Riedtmann, Representation-finite self-injective algebras of class An, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), pp. 449–520, Lecture Notes in Math., 832, Springer, Berlin, 1980.

[Rie3] Christine Riedtmann, Representation-finite self-injective algebras of class Dn. Compositio Math. 49 (1983), no. 2, 231–282.

[Rie4] Christine Riedtmann, Configurations of ZDn, J. Algebra 82 (1983), no. 2, 309–327.

[RV] Idun Reiten, Michel Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366.

[Schm] Karsten Schmidt, Families of Auslander-Reiten components for simply connected differential graded algebras, Math. Z. 264 (2010), no. 1, 43–62.

[Schr] Sibylle Schroll, Trivial extensions of gentle algebras and Brauer graph algebras, J. Algebra 444 (2015), 183–200.

[Si] Daniel Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications, 4. Gordon and Breach Science Publishers, Montreux, 1992.

[STW] Christian Stump, Hugh Thomas, Nathan Williams, Cataland: Why the Fuss?, arXiv:1503.00710.

[SY] Hao Su, Dong Yang, From simple-minded collections to silting objects via Koszul duality, Algebr. Represent. Theory 22 (2019), no. 1, 219–238.

[T] Bertrand To¨en, The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167 (2007), no. 3, 615–667.

[W] Alfred Wiedemann, Die Auslander-Reiten K¨ocher der gitterendlichen Gorensteinordnungen, Bayreuth. Math. Schr. Volume 23, Pages 1–134.

[Ya] Kunio Yamagata, On Artinian rings of finite representation type, J. Algebra 50 (1978), no. 2, 276–283.

[Ye1] Amnon Yekutieli, Derived Categories, arXiv:1610.09640.

[Ye2] Amnon Yekutieli, Duality and Tilting for Commutative DG Rings, arXiv:1312.6411.

[Yo] Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.

[Z] Bin, Zhu, Generalized cluster complexes via quiver representations, J. Algebraic Combin. 27 (2008), no. 1, 35–54.

参考文献をもっと見る

全国の大学の
卒論・修論・学位論文
を
一発検索！

この論文の関連論文を見る