# Derived categories, resolutions, and Brown representability

@article{Krause2005DerivedCR, title={Derived categories, resolutions, and Brown representability}, author={Henning Krause}, journal={arXiv: K-Theory and Homology}, year={2005} }

These notes are based on a series of five lectures given during the summer school ``Interactions between Homotopy Theory and Algebra'' held at the University of Chicago in 2004.

## 98 Citations

### On the centre of a triangulated category

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2011

Abstract We discuss some basic properties of the graded centre of a triangulated category and compute examples arising in representation theory of finite-dimensional algebras.

### Derived categories and their applications

- Mathematics
- 2007

In this notes we start with the basic definitions of derived cate- gories, derived functors, tilting complexes and stable equivalences of Morita type. Our aim is to show via several examples that…

### From triangulated categories to Lie algebras: A theorem of Peng and Xiao

- Mathematics
- 2005

We present a simplified and more intuitive proof of a theorem of Peng and Xiao, which constructs a Lie algebra from any 2-periodic triangulated k-category (satisfying some finiteness assumptions).

### Spherical Subcategories

- Mathematics
- 2012

To an arbitrary object of a triangulated category with twodimensional graded endomorphism algebra, we canonically associate a triangulated subcategory, in which the object becomes spherical. We study…

### Homotopy cartesian diagrams in $n$-angulated categories

- MathematicsHomology, Homotopy and Applications
- 2019

It has been proved by Bergh and Thaule that the higher mapping cone axiom is equivalent to the higher octahedral axiom for n-angulated categories. In this note, we use homotopy cartesian diagrams to…

### DERIVED HALL ALGEBRAS FOR STABLE HOMOTOPY THEORIES

- Mathematics
- 2009

JULIA E. BERGNERAbstract. In this paper we extend To¨en’s derived Hall algebra construction,in which he obtains unital associative algebras from certain stable model cat-egories, to one in which such…

### Spherical subcategories in algebraic geometry

- Mathematics
- 2016

We study objects in triangulated categories which have a two‐dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which…

### How to categorify the ring of integers localized at two

- MathematicsQuantum Topology
- 2019

We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring, naturally isomorphic to the ring of integers localized at two.

### Composition of Roofs in Derived Category

- Mathematics
- 2015

In that paper, we prove that the composition of two roofs is another roof by using mapping cone of a morphism of cochain complexes.

### The Ungraded Derived Category

- Mathematics
- 2012

By means of the ungraded derived category we prove that the orbit category of the bounded derived category of an iterated tilted algebra with respect to translation is triangulated in such a way that…

## References

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Considering homological algebra, this text is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory are…

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© Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions…

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