Triangulated Categories and Singularities

This website provides a collection of material on triangulated categories and singularities. In Paderborn we are running a seminar on this subject and there will be a workshop in the last week of May 2009. The website reflects these activities and their preparations; so it is far from being complete at this stage. Please contribute and send comments to Henning Krause.

The Triangulated Category of Singularities

The subject: With any algebraic variety X one can naturally associate two triangulated categories: the bounded derived category of coherent sheaves and the triangulated subcategory of perfect complexes on X. If the variety X is smooth, then these two categories coincide. For singular varieties this is no longer true, and the quotient of the bounded derived category of coherent sheaves by the full subcategory of perfect complexes captures many properties of the singularities of X. In 2003, Orlov introduced this quotient as the "triangulated category of singularities". There are a number of variations and alternative descriptions of this category in the Gorenstein case, studied first by Buchweitz some 20 years ago under the name "stable derived category". For instance, one can identify the triangulated category of singularities with stable categories of Cohen-Macaulay modules or matrix factorizations.

Orlov's theorem: A theorem of Orlov relates the bounded derived category of a generalized projective space to its triangulated category of singularities.  Due to the existence of tilting objects, interesting applications arise from the categories of coherent sheaves on weighted projective lines.

Connection with cluster categories: Cluster categories provide a categorification of cluster algebras. A cluster category admits a triangulated structure and a theorem of Keller describes the cluster category as a triangulated category of singularities. Thus for a given cluster category C, there exists a dg algebra A such that the quotient D^b(A)/per(A) of the derived category of dg A-modules with bounded homology modulo the perfect derived category is equivalent to C as a triangulated category; see Theorem 7.1 in [B. Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551-581] and its correction.

Seminar on Triangulated Categories and Singularities

The seminar is meant as a preparation for the workshop in May 2009.

  1. Introduction to Cohen-Macaulay modules (Dirk Kussin)
    Thursday, November 27, 2008
    Abstract: This talk provides an elementary introduction to the theory of Cohen-Macaulay modules. Basic reference is the book of Yuji Yoshino with title "Cohen-Macaulay modules over Cohen-Macaulay rings".
  2. Rational singularities and almost split sequences (Xiao-Wu Chen)
    Thursday, November 27, 2008
    Abstract: The paper "Rational singularities and almost split sequences" of Auslander relates almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup G of GL(2,C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules of the quotient singularity associated with G.
  3. Cohen-Macaulay rings, unmixedness and chain conditions (Karl-Heinz Kiyek)
    Thursday, December 4, 2008
    Abstract: In this short presentation, I provide—without proofs—the characterization of noetherian Cohen-Macaulay rings by the unmixedness property. In addition, I mention that CM-rings behave well with respect to chain conditions for prime ideals. For proofs, I refer to ChapterXII of some unfinished book project.
  4. Phenomenology of singularities (Helmut Lenzing)
    Thursday, December 4, 2008
    Abstract: The talk will be of an introductory nature and focuses on an analysis of surface singularities. The classes of simple, parabolic and unimodular singularities will be discussed in some detail, in particular their relationship to symmetry groups related to tilings of the spherical, Euclidean or hyperbolic plane. The talk will advocate the aspect that the singularities themselves are best understood through the properties of the resulting quotients of the actions of such symmetry groups. The arising quotients form compact 2-orbifolds, their Euler characteristic being a measure of the complexity for the singularity in question. We will show that taking the orbifolds as a starting point, will provide naturally the links between singularities, group actions, group representations, Cohen Macaulay modules over the singularity, vector bundles on the orbifold, etc. This will, in particular, be illustrated for the case of simple singularities, again `explaining' McKay correspondence. Throughout, emphasis will be given to explain and illustrate the links between the various concepts. There will be only few proofs.
  5. Matrix factorizations (Marcel Wiedemann)
    Thursday, January 8, 2009
    Abstract: In my talk I will discuss Eisenbud's matrix factorization. This plays a key role in the treatment of Cohen-Macaulay modules over hypersurface singularities. A reference is [Yoshino 1990].
  6. Knoerrer's periodicity (Marcel Wiedemann)
    Thursday, January 15, 2009
    Abstract: I shall discuss that the notions of simple singularity and finite representation type are equivalent for hypersurfaces. A reference is [Yoshino 1990].
  7. Introduction to weighted projective lines I: Construction from canonical algebras (Karsten Dietrich)
    Thursday, January 22, 2009
    Abstract: After recalling the definition of a canonical algebra and the structure of its module category we will construct the category of coherent sheaves on the associated weighted projective line (inside the bounded derived category of the canonical algebra) and discuss some properties.
  8. Introduction to weighted projective lines II: Vector bundles and Cohen-Macaulay modules (Xiao-Wu Chen)
    Thursday, January 22, 2009
    Abstract: In this talk, I will recall the definition of weighted projective lines and sketch the proof of Serre's theorem, and a refinement of Serre's theorem which relates vector bundles with graded maximal Cohen-Macaulay modules. If time permits, to go back to canonical algebras, we will also discuss Serre duality and the canonical tilting sheaves.
  9. Introduction to weighted projective lines III: Axiomatic approach (Claudia Köhler)
    Thursday, February 5, 2009
    Abstract: We show that each abelian category satisfying certain properties (noetherian, hereditary, no non-zero projectives, existence of a tilting complex) is derived equivalent to the module category of a so-called squid algebra and therefore equivalent to the category of coherent sheaves on the associated weighted projective line.
  10. Introduction to weighted projective lines IV: Representation type and Euler characteristic (Dirk Kussin)
    Thursday, February 5, 2009
    Abstract: We will explain how the Euler characteristic of a weighted projective line determines the representation type of the category of coherent sheaves and of the associated category of graded Cohen-Macaulay modules, respectively.
  11. The triangulated category of singularities (Daniel Murfet)
    Thursday, February 12, 2009
    Abstract: I will survey the main results of Buchweitz's unpublished manuscript on the subject of maximal Cohen-Macaulay modules and Tate cohomology. The centerpiece will be the construction, for a Gorenstein ring, of four equivalent triangulated categories. Taking morphism sets in these categories defines a cohomology theory, called Tate cohomology, which reflects stable homological features of the ring.
  12. Orlov's theorem I (Marcel Wiedemann)
    Tuesday, March 24, 2009
    Abstract: I shall present all the definitions and background results necessary to prove Orlov's Theorem, such as semiorthogonal decompositions, triangulated categories of singularities and quotient categories of graded modules.
  13. Orlov's theorem II (Xiao-Wu Chen)
    Tuesday, March 24, 2009
    Abstract: I shall discuss the proof of Orlov's Theorem and some of its corollaries.
  14. Examples, illustrating Orlov's theorem (Helmut Lenzing)
    Tuesday, March 24, 2009
    Abstract: We are going to illustrate the trichotomy expressed by Orlov's theorem by a couple of examples. The most impressing examples we are going to discuss arise from weighted projective lines, whose Euler characteristic plays a key role for the Orlov context.

Literature on Triangulated Categories and Singularities

This is a preliminary list and will be extended.

Cohen-Macaulay modules
  1. M. Auslander, Rational singularities and almost split sequences. Trans. Amer. Math. Soc.  293 (1986), 511-531.
  2. R. O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. Unpublished manuscript (1987) 155 pp.
  3. I. Burban and Y. Drozd, Maximal Cohen-Macaulay modules over surface singularities. Trends in Representations of Algebras and Related Topics. EMS Publishing House, 2008, 101-166, arXiv:0803.0117v1.
  4. H. Kajiura, K. Saito and A. Takahashi, Matrix factorization and representations of quivers. II. Type ADE case, Adv. Math. 211 (2007), 327-362.
  5. H. Lenzing and J. A. de la Pena, Spectral analysis of finite dimensional algebras and singularities. Trends in Representations of Algebras and Related Topics. EMS Publishing House, 2008, 541-588, arXiv:0805.1018v1.
  6. G. Leuschke, The McKay correspondence. Unpublished notes (2006).
  7. D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246 (2004), 227-248.
  8. D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities. arXiv:math/0503632v2.
  9. H. Schoutens, Projective dimension and the singular locus, Comm. Algebra 31 (2003), 217-239.
  10. Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings. London Mathematical Society Lecture Note Series, 146. Cambridge University Press, Cambridge, 1990. viii+177 pp.
Singularities (compiled by Helmut Lenzing)
  1. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko: Singularities of Differentiable Maps, Volumes I,II, Birkhäuser 1985.
  2. W. Ebeling: Funktionentheorie, Differentialtopologie und Singularitäten. Vieweg+Teubner, 2001.
  3. A. Dimca: Singularities and Topology of Hypersurfaces. Springer, 1992. (Appendix A).
  4. J. Milnor: The 3-dimensional Brieskorn manifolds M(p,q,r). In: "Knots, Groups and 3-Manifolds, L.P. Neuwirth, ed.". Princeton University Press, 1975.
  5. J. M. Montesinos: Classical Tesselations and 3-Manifolds. Springer, 1987.
Comments: [1] is a standard reference. [2] is a highly recommended introduction to the various aspects of the subject. Appendix A of [3] gives a quick review of the start of the hierarchy of singularities. The main theme of [4] and [5] is 3-manifolds, not singularities; nevertheless they provide the background (actions of triangle groups and 2-orbifolds)  for a quick analysis of surface singularities.

Weighted projective lines (compiled by Dirk Kussin)

  1. M. Artin and J. J. Zhang, Noncommutative projective schemes. Adv. Math. 109 (1994), 228–287.
  2. W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras. In: Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987.
  3. W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves. J. Algebra 144 (1991), 273–343.
  4. H. Lenzing and J. A. de la Pe ̃na, Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. (3) 78 (1999), 513–540.
  5. H. Lenzing, Hereditary noetherian categories with a tilting complex. Proc. Amer. Math. Soc. 125 (1997), 1893–1901.
  6. H. Lenzing, Hereditary categories. In: Handbook of Tilting Theory, 105–146, London Math. Soc. Lecture Note Ser., 332, Cambridge University Press, Cambridge, 2007.
  7. C. M. Ringel, Tame algebras and integral quadratic forms. Lecture Notes in Mathematics 1099, Springer-Verlag, Berlin, 1984.

Comments: See here.

Unbounded Singularity Categories

This material is compiled by Daniel Murfet.

The subject: In the study of triangulated categories of singularities, it is sometimes convenient to pass to larger categories where some version of colimits (in this case, homotopy colimits) exist. In these "unbounded singularity categories", which are typically compactly generated triangulated categories, homotopy-theoretic techniques such as Brown Representability can be used. In the Gorenstein case singularity categories come in various guises, such as stable categories of Cohen-Macaulay modules or homotopy categories of acyclic complexes of finitely generated projectives; the corresponding unbounded singularity categories are given, respectively, by the stable category of Gorenstein projective modules [2,9] and the homotopy category of acyclic complexes of projective modules [2,3,6].

In the non-Gorenstein and/or non-affine (i.e. scheme-theoretic) situations, constructing the unbounded singularity category is more difficult. The general construction, via acyclic complexes of injective sheaves, is given in [5]. Another version of the construction, via acyclic complexes of flat sheaves, can be found in [13,14]. The relationship between acyclic complexes of injectives and flats is given by Grothendieck duality [6,14].

A related construction is given by taking the quotient of the bounded derived category of (possibly infinite) modules, by the full subcategory of bounded complexes of projective modules. This quotient is studied in [2,9] and the scheme-theoretic analogue plays an important role in [4].

The introduction of homotopy-theoretic techniques has been particularly influential in the modular representation theory of finite groups [1]. Recent developments include [10,11,12].

  1. J. Rickard, Idempotent modules in the stable category. J. London Math. Soc. (2) 56 (1997), no.1, 149-170.
  2. A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stabilization. Comm. Algebra 28 (2000), no.10, 4547-4596.
  3. P. Jorgensen, Spectra of modules. J. Algebra 244 (2001), no.2, 744-784.
  4. D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246 (2004), 227-248.
  5. H. Krause, The stable derived category of a Noetherian scheme. Compos. Math. 141 (2005), no.5, 1128-1162.
  6. S. Iyengar, H. Krause, Acyclicity versus total acyciclity for complexes over Noetherian rings. Doc. Math. 11 (2006), 207-240.
  7. H. Krause and J. Le, The Auslander-Reiten formula for complexes of modules. Adv. Math. 207 (2006), no.1, 133-148.
  8. P. Jorgensen, Existence of Gorenstein projective resolutions and Tate cohomology. J. Eur. Math. Soc. 9 (2007), no.1, 59-76.
  9. X. W. Chen, Relative singularity categories and Gorenstein-Projective modules, arxiv:0709.1762v1.
  10. D. Benson, S. Iyengar and H. Krause, Local cohomology and support for triangulated categories, arxiv:math/0702610v3.
  11. D. Benson and H. Krause, Complexes of injective $kG$-modules. Algebra Number Theory 2 (2008), no.1, 1-30.
  12. D. Benson, S. Iyengar and H. Krause, Stratifying modular representations of finite groups, arxiv:0810.1339v1.
  13. A. Neeman, The homotopy category of flat modules, and Grothendieck duality. Invent. Math. 174 (2008), no.2, 255-308.
  14. D. Murfet, The mock homotopy category of projectives and Grothendieck duality, PhD thesis (2007), available online.
  15. D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, arxiv:0901.1859v1.

Well Generated Triangulated Categories

The subject: Well generated triangulated categories were introduced by Neeman in his book published in 2001. They form a reasonable class of triangulated categories which admit arbitrary coproducts and a suitable set of generators. This class is closed under forming various basic operations. For instance, a localizing subcategory generated by a set of objects and the corresponding Verdier quotient are again well generated.

Relation to abelian categories: The concept of a well generated triangulated category may be viewed as a triangulated analogue of the concept of an abelian Grothendieck category. Recall that an abelian category is a Grothendieck category if it admits arbitrary coproducts, a set of generators, and satisfies the AB5 condition. Recent work shows that this analogy between abelian and triangulated categories can be made precise. In fact, there is an interesting interplay between both worlds. For example,  the derived category of any abelian Grothendieck category is a well generated triangulated category.

Publications: Here is a list of publications on this subject. It is not complete and will be extended.

  1. A. Heider, Two results from Morita theory of stable model categories, arXiv:0707.0707v1.
  2. H. Krause, On Neeman's well generated triangulated categories, Doc. Math. 6 (2001), 121 - 126 (electronic).
  3. H. Krause, Localization for triangulated categories, arXiv:0806.1324v1.
  4. A. Neeman, Triangulated categories, Ann. of Math. Stud., 148, Princeton Univ. Press, Princeton, NJ, 2001.
  5. M. Porta, The Popescu-Gabriel theorem for triangulated categories, arXiv:0706.4458v1.
  6. J. Rosicky, Generalized Brown representability in homotopy categories, Theory Appl. Categ. 14 (2005), 451 - 479 (electronic).
  7. J. Stovicek, Locally well generated homotopy categories of complexe, arXiv:0810.5684v1.
  8. G. Tabuada, Homotopy theory of well-generated algebraic triangulated categories, arXiv:math/0703172v1.

Workshop on Triangulated Categories and Singularities

Organisers: R.-O. Buchweitz, H. Krause, D. Kussin, H. Lenzing, A. Neeman

Dates: May 26 - 30, 2009. Talks start Tuesday morning and finish Saturday at noon.

The workshop will take place at the University of Paderborn.The first half will be devoted to the theory of triangulated categories (with some focus on well generated categories); the second half will be devoted to representations of algebras and singularities (with some focus on triangulated categories of singularities).

For more information, see here.

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