Basic information
The aim of the course is to introduce derived categories and their basic properties. For an outline of the course and general information see the description in the Student Information System.
Literature
It was not easy to find a single book or lecture notes covering the whole course, so different parts are taken from different sources:
- The Yoneda definition of Ext^{1} is covered in Sections 1 and 2 of Chapter III in [McL]. The relation to the definition using projective resolutions is given in Theorem III.6.4.
- A direct proof for existence of the long exact sequence of Ext groups using the Yoneda definition can be found in Chapter XII, Section 5 ("Ext without Projectives") in [McL].
- Basic properties of pullbacks and pushouts in module categories are discussed in [Wis], Section 10 ("Pullback und Pushout").
- General aspects of localization of categories (i.e. adding formal inverses to some morphisms) are introduced in [Kr], Sections 2.1 and 2.2. Details are to be found in [GZ].
- Fundamental diagram lemmas (Five Lemma, Snake Lemma, 3x3 Lemma, existence of the long exact sequence of homologies coming from a short exact sequence of complexes) are proved in [McL], Sections I.3, II.4 and II.5.
- Basics on abelian categories are included for instance in [Po], Section 2. Mitchell Embedding Theorem is presented in [Po] as Theorem 4.11.6, page 229.
- Good references for exact categories are [Bu] or Appendix A in [Kel]. A brief introduction to Frobenius exact categories, their stable categories and the relation to categories of complexes can be found in Chapter I of [Ha].
- Basic properties of triangulated categories are discussed in detail in Chapter I of [Nee]. The triangulated structure on the homotopy category of complexes over an additive category is briefly described in Chapter I of [Ha].
- The presentation of the calculus of fractions and multiplicative systems has been taken from [Kr], Sections 3.1 and 4.3. Details can be found in [GZ] and [Ver].
- The construction of the derived category of an abelian category is described in [Wei], Chapter 10.
References
[Bu] | T. Bühler, Exact categories, Expo. Math. 28 (2010), 1-69. [Fulltext] |
[GZ] | P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag New York, Inc., New York, 1967. |
[Ha] | D. Happel, Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, LMS Lecture Note Series 119, Cambridge Univ. Press, 1988. |
[Kel] | B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379--417. |
[Kr] | H. Krause, Localization theory for triangulated categories, in Triangulated Categories, LMS Lecture Note Series 375, Cambridge Univ. Press, 2010. [Preprint] |
[McL] | S. Mac Lane, Homology, Academic Press, Inc., Publishers, New York, 1963. |
[Nee] | A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton, NJ, 2001. |
[Po] | N. Popescu, Abelian Categories with Applications to Rings and Modules, LMS Monographs 3, Academic Press, London-New York, 1973. |
[Ver] | J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque No. 239 (1996). |
[Wei] | C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge, 1994. |
[Wis] | R. Wisbauer, Grundlagen der Modul- und Ringtheorie, Verlag Reinhard Fischer, Munich, 1988. |