Derivative category
The derived category of an Abelian category is an important object in modern homological algebra . It was introduced by Grothendieck's student Verdier .
Quasi-isomorphism
First one forms the Abelian category of all chain complexes in . A chain homomorphism in is called a quasi-isomorphism if it becomes an isomorphism under homology, that is, if an isomorphism is for every integer .
Homotopy category
Analogous to the conventional homotopy category , the homotopy category is formed by identifying chain homotopic morphisms with one another. is a triangulated category .
Derivative category
Analogous to the localization makes it the derived category of by declaring all Quasiisomorphismen for inverted.
Set theoretical problem
Be two objects . In the totality of all morphisms from to is not always a set. The most important papers consider this problem to be insignificant.
literature
- Sergei I. Gelfand, Yuri I. Manin : Methods of Homological Algebra . 2nd Edition. Springer, Berlin 2003, ISBN 3-642-07813-3 .
- Wolfgang Soergel : Derived Categories and Functors. (PDF) Mathematical Institute, University of Freiburg, April 7, 2017, accessed on April 8, 2017 (lecture notes).
- Charles A. Weibel : An introduction to homological algebra (= Cambridge Studies in Advanced Mathematics . No. 38 ). Cambridge University Press, 1994, ISBN 0-521-43500-5 , chapter 10 .
Individual evidence
- ^ RP Thomas: Derived Categories for the Working Mathematician . In: Cumrun Vafa , S.-T. Yau (Ed.): Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999) (= AMS / IP Studies in Advanced Mathematics . No. 23 ). American Mathematical Society , Providence (Rhode Island) 2001, ISBN 0-8218-2159-8 , pp. 349–361 , arxiv : math / 0001045 : "... the creators of derived categories (principally Verdier, or as he is traditionally known in this context, Grothendieck's student Verdier)"
- ↑ For an example from Freyd , see Carles Casacuberta, Amnon Neeman: Brown representability does not come for free . In: Mathematical Research Letters . tape 16 , no. 1 . International Press, 2009, ISSN 1073-2780 , pp. 1-5 , arxiv : 0807.1872 .
- ^ Charles A. Weibel : An introduction to homological algebra (= Cambridge Studies in Advanced Mathematics . No. 38 ). Cambridge University Press, 1994, ISBN 0-521-43500-5 , Set-Theoretic Remark 10.3.3 : "The standard references […] all ignore these set-theoretic problems."