Trace methods for equivariant algebraic K-theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910947936829440 |
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| author | Chan, David Gerhardt, Teena Klang, Inbar |
| author_facet | Chan, David Gerhardt, Teena Klang, Inbar |
| contents | In the past decades, one of the most fruitful approaches to the study of algebraic $K$-theory has been trace methods, which construct and study trace maps from algebraic $K$-theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic $K$-theory have emerged, but thus far few tools are available for the study and computation of these theories. In this paper, we lay the foundations for a trace methods approach to equivariant algebraic $K$-theory. For $G$ a finite group, we construct a Dennis trace map from equivariant algebraic $K$-theory to a $G$-equivariant version of topological Hochschild homology; for $G$ the trivial group this recovers the ordinary Dennis trace map. We show that upon taking fixed points, this recovers the trace map of Adamyk--Gerhardt--Hess--Klang--Kong, and gives a trace map from the fixed points of coarse equivariant $A$-theory to the free loop space. We also establish important properties of equivariant topological Hochschild homology, such as Morita invariance, and explain why it can be considered as a multiplicative norm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_11327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trace methods for equivariant algebraic K-theory Chan, David Gerhardt, Teena Klang, Inbar Algebraic Topology K-Theory and Homology 55P91, 19D55, 16E40 In the past decades, one of the most fruitful approaches to the study of algebraic $K$-theory has been trace methods, which construct and study trace maps from algebraic $K$-theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic $K$-theory have emerged, but thus far few tools are available for the study and computation of these theories. In this paper, we lay the foundations for a trace methods approach to equivariant algebraic $K$-theory. For $G$ a finite group, we construct a Dennis trace map from equivariant algebraic $K$-theory to a $G$-equivariant version of topological Hochschild homology; for $G$ the trivial group this recovers the ordinary Dennis trace map. We show that upon taking fixed points, this recovers the trace map of Adamyk--Gerhardt--Hess--Klang--Kong, and gives a trace map from the fixed points of coarse equivariant $A$-theory to the free loop space. We also establish important properties of equivariant topological Hochschild homology, such as Morita invariance, and explain why it can be considered as a multiplicative norm. |
| title | Trace methods for equivariant algebraic K-theory |
| topic | Algebraic Topology K-Theory and Homology 55P91, 19D55, 16E40 |
| url | https://arxiv.org/abs/2505.11327 |