Equivariant algebraic K-theory of G-rings
| Autor: | Mona Merling |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
General Mathematics
010102 general mathematics Dimension of an algebraic variety Mathematics::Algebraic Topology 01 natural sciences Algebraic cycle Algebra Mathematics::K-Theory and Homology Algebraic group 0103 physical sciences Algebraic surface FOS: Mathematics Real algebraic geometry Algebraic Topology (math.AT) Equivariant cohomology Equivariant map A¹ homotopy theory Mathematics - Algebraic Topology 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | Mathematische Zeitschrift. 285:1205-1248 |
| ISSN: | 1432-1823 0025-5874 |
| DOI: | 10.1007/s00209-016-1745-3 |
| Popis: | A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological $K$-theory, Atiyah's Real $K$-theory, and existing statements about algebraic $K$-theory spectra with $G$-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory. Comment: Final version to appear in Mathematische Zeitschrift. The last section about Waldhausen G-categories has been removed from this paper |
| Databáze: | OpenAIRE |
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