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 . ${\displaystyle D({\mathcal {A}})}$ ${\displaystyle {\mathcal {A}}}$

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 . ${\displaystyle \mathop {Ch} ({\mathcal {A}})}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle f_{*}\colon C_{*}\rightarrow D_{*}}$ ${\displaystyle \mathop {Ch} ({\mathcal {A}})}$ ${\displaystyle H_{n}(f)\colon H_{n}(C)\rightarrow H_{n}(D)}$ ${\displaystyle n}$

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 . ${\displaystyle K({\mathcal {A}})}$ ${\displaystyle \mathop {Ch} ({\mathcal {A}})}$ ${\displaystyle K({\mathcal {A}})}$

Derivative category

Analogous to the localization makes it the derived category of by declaring all Quasiisomorphismen for inverted. ${\displaystyle D({\mathcal {A}})}$ ${\displaystyle K({\mathcal {A}})}$

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. ${\displaystyle A,B}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle D({\mathcal {A}})}$ ${\displaystyle A}$ ${\displaystyle B}$

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

1. ^ 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)"
2. ↑ 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 .
3. ^ 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."