A brief introduction to algebraic \(K\)-theory.

*(English)*Zbl 1196.19003
Ji, Lizhen (ed.) et al., Cohomology of groups and algebraic \(K\)-theory. Selected papers of the international summer school on cohomology of groups and algebraic \(K\)-theory, Hangzhou, China, July 1–3, 2007. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-144-5/pbk). Advanced Lectures in Mathematics (ALM) 12, 167-185 (2010).

Summary: We give a brief survey of higher algebraic \(K\)-theory and its connection to motivic cohomology. We start with limits and colimits, and then pass to the combinatorial construction of topological spaces by means of “systems of simplices”, usefully mediated by simplicial sets. Definitions of \(K\)-theory are offered, and the main theorems are stated. A definition of motivic cohomology is offered and its major properties are listed. Many references to the literature are provided, especially for the development of motivic cohomology.

For the entire collection see [Zbl 1185.20001].

For the entire collection see [Zbl 1185.20001].

##### MSC:

19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |

19D55 | \(K\)-theory and homology; cyclic homology and cohomology |

14F42 | Motivic cohomology; motivic homotopy theory |

##### Keywords:

higher algebraic \(K\)-theory; motivic cohomology; combinatorial construction of topological spaces
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\textit{D. R. Grayson}, in: Cohomology of groups and algebraic \(K\)-theory. Selected papers of the international summer school on cohomology of groups and algebraic \(K\)-theory, Hangzhou, China, July 1--3, 2007. Somerville, MA: International Press; Beijing: Higher Education Press. 167--185 (2010; Zbl 1196.19003)