A seminar in the academic year 2020/21 (both semesters) as a branch of the MSTR Elective Seminar (NMAG475). It is intended for Master students of the program Mathematical Structures. The aim is to learn about modern homological algebra and its applications in commutative algebra and representation theory.

Basic information

The semanar has already finished.

The seminar runs on Zoom and is currently scheduled on Wednesday at 2-3.30pm.

This is a branch of the MSTR Elective Seminar (NMAG475) and is intended for Master students of the program Mathematical Structures.

Our aim is to learn about derived categories and derived equivalences. In the ideal case, one obtains sufficient background to apprechiate the results explained in [I18], which use derived equivalences and representation theory of finite dimensional algebras in order to understand interesting modules in commutative algebra.

In case of interest, please send me an e-mail.


A brief overview of what has been told in the seminar will be updated below.

November 25, 2020
Recollections of abelian categories, Yoneda Ext functors (see [ML63], Sec. XII.5) and a glimpse of derived categories.
[PDF - notes from the seminar]
December 2, 2020
Recollections on pullbacks and pushouts in abelian categories, their properties and relation to Yoneda Ext, complexes and their cohomologies, quasi-isomorphisms and construction of the derived category.
[PDF - notes from the seminar]
December 9, 2020
Cohomological ∂-functors and their universality (after [Ro09, Sec. 6.2.2] and [ML63, Sec. XII.7-XII.9]), balancedness of Ext.
[PDF - notes from the seminar]
December 16, 2020
Localization of categories at a class of morphisms (following [GZ67, Ch. I]) and the homotopy category of complexes over an abelian category.
[PDF - notes from the seminar]
January 6, 2021
The homotopy category of an additive category is the localization of the category of complexes at the class of homotopy isomorphisms (see [GM03, III.4.2-3]). The calculus of left and right fractions. The class of quasi-isomorphisms in the homotopy category of an abelian category admits such a calculus ([Kr10, Section 3] and [GZ67, Ch. I]).
[PDF - notes from the seminar]
January 18, 2021
Triangulated categories and their basic properties. The standard triangulated structure on the derived category of an abelian category. A glimpse of derived equivalences and tilting complexes (see [Ke07]).
[PDF - notes from the seminar]
February 8, 2021
Tilting modules, tilting complexes, constructing derived equivalences (after [Ha87], [Ri89], [Ke94]).
[PDF - notes from the seminar]
February 15, 2021
Frobenius exact categories (see [Bu10]) and algebraic triangulated categories (see [Ha88, Sec. I.2] for the construction). Basic interaction of commutative algebra with homological algebra and the hierarchy of commutative noetherian local rings based on how nicely they behave homologically (regular => Gorenstein => Cohen-Macaulay). An outline of what how we can use triangle equivalences to understand the category of MCM modules over Kleinian singularities.
[PDF - notes from the seminar]
March 10, 2021
Differential graded (=DG) algebras and DG-modules over them, the homotopy and derived categories of DG-modules. The Hom-complex of a pair of DG-modules and the interpretation of its cocycles, coboundaries and cohomologies. The endomorphism DG-algebra of a DG-module. Literature: [Ke94] and [Av13].
March 17, 2021
The endomorphism DG-algebra of a DG-module continued. The internal Hom of complexes as a functor. A detailed computed example of the endomorphism DG-algebra of a complex of finitely generated modules over a finite dimensional algebra.
March 31, 2021
Introduction to spectral sequences - filtered complexes, bicomplexes. Proof of the Universal Coefficient Theorem as an application. The Leray-Serre spectral sequence (converging to the homology of the total space of a Serre fibration). [MP4 - recorded talk] [PDF - presentation] [PDF - handwritten notes]
April 7, 2021
Cohomological spectral sequences and the multiplicative structure, the Serre cohomological spectral sequence, a computed example: the cohomology ring of ℂℙ. [MP4 - recorded talk] [PDF - presentation] [PDF - handwritten notes]
April 14, 2021
An introduction to the Auslander-Reiten theory - motivation, the radical of a preadditive category [Ass06, App. A.3], the Auslander-Reiten formulas [Ass06, Sec. IV.2] and a sketch of the relation to the radical of the module category. [MP4 - recorded talk] [PDF - notes from the seminar]
May 5, 2021
Various aspects of the theory of triangulated categories following the presentation in [Ne01]. [MP4 - recorded talk]
May 12, 2021
Axioms of triangulated categories, what are algebraic triangulated categories. [MP4 - recorded talk] [PDF - notes from the seminar]
May 19, 2021
Grothendieck spectral sequence for the composition of right derived functors, along with preliminaries on derived functors via adapted subcategories and subsets, and basic facts about sheaves of modules over sheaves of rings in order to approach examples. [MP4 - recorded talk] [PDF - presentation]
June 2, 2021
Well generated triangulated categories and the Dual Brown Representability Theorem for them, following [Ne01]. [MP4 - recorded talk] [PDF - notes from the seminar]
June 16, 2021
Iwanaga-Gorenstein rings, basic facts about them, and the stable category of Cohen-Macaulay modules. [PDF - presentation]


Sources for motivation

A motivating survey about the use of derived equivalences in commutative algebra and representation theory of finite dimensional algebra:

[I18] O. 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. [arXiv:1805.05318]

Many of these results are motivated and generalize in one or another way the McKay correspondence. This is shortly explained (with colorful pictures of the corresponding singularities of algebraic varieties) in the following presentation:

[Fa16] E. Faber, A McKay correspondence for reflections groups, presentation at the Maurice Auslader Conference in 2016. [slides in PDF]

Classical homological algebra

A few standard references for homological algebra:

[CE56] H. Cartan, S. Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956.
[ML63] S. Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg 1963.
[Ro09] J. J. Rotman, An introduction to homological algebra, Second edition. Universitext. Springer, New York, 2009.

Many useful more specialized results, which relate e.g. commutative and homological algebra, can be found in this monograph:

[EJ00] E. E. Enochs, O. M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, 30. Walter de Gruyter & Co., Berlin, 2000.

Background material on abelian categories can be found for example in:

[Fr64] P. Freyd, Abelian categories, an introduction to the theory of functors, Harper's Series in Modern Mathematics Harper & Row, Publishers, New York 1964.
[Mi65] B. Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII Academic Press, New York-London 1965.
[Po73] N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press, London-New York, 1973.

Quite often, one also needs the concept of exact category, which axiomatizes extension closed subcategories of abelian categories together with the collection of short exact sequences with all terms in the subcategory. A comprehensive source with a long list of references can be found here:

[Bu10] T. Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1-69. [arXiv:0811.1480]

As an aside, a fancy and rather general proof that the two ways to right derive the Hom functor coincide (i.e. balancedness of Ext) can be found here:

[EPZ20] S. Estrada, M. A. Pérez, H. Zhu, Balanced pairs, cotorsion triplets and quiver representations, Proc. Edinb. Math. Soc. (2) 63 (2020), no. 1, 67-90. [arXiv:1802.09989]

Gorenstein rings

The history of the interest in commutative Gorenstein rings is enlightened in an older paper by Bass. A widely used non-commutative homological generalization of the concept of Gorenstein rings (in the form of a strong symmetry between projective and injective dimensions) goes back to Iwanaga.

[Ba63] H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28.
[Iw80] Y. Iwanaga, On rings with finite self-injective dimension II, Tsukuba J. Math. 4 (1980), no. 1, 107-113.

From our point of view, there is an interesting class of modules over Iwanaga-Gorenstein rings, which are called the (maximal) Cohen-Macaulay (aka Gorenstein projective) modules and which form a Frobenius exact category. One can find more about their homological properties in the papers

[AB89] M. Auslander, R.-O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. France (N.S.) No. 38 (1989), 5-37.
[Bu86] R.-O. Buchweitz, Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings preprint. [link to PDF]

Derived categories

A list of sources (alphabetically ordered and quite diverse in style) which one can use to learn about derived categories:

[GM03] S. I. Gelfand, Yu. I. Manin, Methods of homological algebra, Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
[Ha88] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988.
[KS06] M. Kashiwara, P. Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer-Verlag, Berlin, 2006.
[Kr07] H. Krause, Derived categories, resolutions, and Brown representability, Interactions between homotopy theory and algebra, 101-139, Contemp. Math., 436, Amer. Math. Soc., Providence, RI, 2007. [arXiv:math/0511047]
[We94] Ch. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.
[Ye20] A. Yekutieli, Derived categories, Cambridge Studies in Advanced Mathematics 183, Cambridge University Press, Cambridge, 2020.
[Zi14] A. Zimmermann, Representation theory, a homological algebra point of view, Algebra and Applications 19, Springer, Cham, 2014.

In fact, derived categories were introduced in Verdier's 1967 thesis which was published 30 years later:

[Ve97] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, with a preface by Luc Illusie, edited and with a note by Georges Maltsiniotis, Astérisque No. 239 (1996), xii+253 pp. (1997).

Useful facts about non-commutative localization of categories and the calculus of left and right fractions can be found in

[GZ67] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer-Verlag New York, Inc., New York, 1967.
[Kr10] H. Krause, Localization theory for triangulated categories, Triangulated categories, 161-235, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press, Cambridge, 2010. [arXiv:0806.1324]

An axiomatic approach to triangulated categories (which derived categories are examples of) is taken in these monographs:

[HPS97] M. Hovey, J. H. Palmieri, N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997).
[Ne01] A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, Princeton, NJ, 2001.

It is also worth to point out a very short, but also extremely dense paper which explains various phenomena related to derived categories:

[KV87] B. Keller, D. Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225-228.

Derived equivalences and tilting theory

A nice overview of derived equivalences was written up by Keller:

[Ke07] B. Keller, Derived categories and tilting, Handbook of tilting theory, 49–104, London Math. Soc. Lecture Note Ser., 332, Cambridge Univ. Press, Cambridge, 2007.

Ancient examples of derived equivalences go back to work of Beilinson and a systematic study of their existence via tilting modules was done by Happel. This was promoted to characterization of tilting complexes due to Rickard and conceptual theory was developed by Keller.

[Be78] A. A. Beilinson, Coherent sheaves on Pn and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12 (1978), no. 3, 68-69.
[Ha87] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389.
[Ke94] B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102.
[Ri89] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436-456.
[Ri91] J. Rickard, Derived equivalences as derived functors J. London Math. Soc. (2) 43 (1991), no. 1, 37-48.

As many find it difficult to learn about differential graded algebras and modules directly from [Ke94], one can also try alternative sources (in the first case, the calculus of dg-algebras and dg-modules is explained in the appendices):

[Av13] L. L. Avramov, (Contravariant) Koszul duality for DG algebras, Algebras, quivers and representations, 13-58, Abel Symp., 8, Springer, Heidelberg, 2013. [arXiv:1305.4230]
[Sa16] M. Saorín, Dg algebras with enough idempotents, their dg modules and their derived categories, Algebra Discrete Math. 23 (2017), no. 1, 62-137. [arXiv:1612.04719]

Auslander-Reiten theory

When it comes to a detailed description of a small abelian or triangulated category which is Hom-finite over a field, it is often done using Auslander-Reiten short exact sequences/triangles. They provide us with generators of the Jacobson radical of the category in question as well as with so-called mesh relations between these. The AR sequences/triangles are often construced using dualities in the category: the so-called Auslander-Reiten formula for module or derived categories over finite-dimensional algebras, the Serre duality in the context of coherent sheaves on a projective scheme, or Grothendieck's local duality in the context of maximal Cohen-Macaulay modules.

A chapter on the Auslader-Reiten theory for modules over finite dimensional algebras is contained in most textbooks on representation theory of such algebras, e.g. in:

[ASS06] I. Assem, D. Simson, A. Skowronski, Elements of the representation theory of associative algebras, Vol. 1., London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.
[ARS97] M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, corrected reprint of the 1995 original, Cambridge Studies in Advanced Mathematics 36. Cambridge University Press, Cambridge, 1997.

The derived categories of finite dimensional algebras were studied in [Ha87] and [Ha88] above, the maximal Cohen-Macaulay modules in

[Yo90] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series 146, Cambridge University Press, Cambridge, 1990.

The relation between a classical Serre duality from algebraic geometry and the Auslander-Reiten theory is described in Section I of

[RV02] I. Reiten, M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295-366.

A unified treatment of the triangulated version of the Auslander-Reiten theory using abstract notions in the theory of triangulated categories (the so-called Brown representability), followed by a discusion of bounded derived categories, is also given in

[KL06] H. Krause, J. Le, The Auslander-Reiten formula for complexes of modules, Adv. Math. 207 (2006), no. 1, 133-148. [arXiv:math/0412085]