Coalgebraic $K$-theory
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915947933073408 |
|---|---|
| author | Gerhardt, Teena Péroux, Maximilien Soré, W. Hermann B. |
| author_facet | Gerhardt, Teena Péroux, Maximilien Soré, W. Hermann B. |
| contents | We establish comparison maps between the classical algebraic $K$-theory of algebras over a field and its analogue $K^c$, an algebraic $K$-theory for coalgebras over a field. The comparison maps are compatible with the Hattori--Stallings (co)traces. We identify conditions on the algebras or coalgebras under which the comparison maps are equivalences. Notably, the algebraic $K$-theory of the power series ring is equivalent to the $K^c$-theory of the divided power coalgebra. We also establish comparison maps between the $G$-theory of finite dimensional representations of an algebra and its analogue $G^c$ for coalgebras. In particular, we show that the Swan theory of a group is equivalent to the $G^c$-theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coalgebraic $K$-theory Gerhardt, Teena Péroux, Maximilien Soré, W. Hermann B. K-Theory and Homology Algebraic Topology Primary: 16T15, 19D50. Secondary: 18M05, 55U30 We establish comparison maps between the classical algebraic $K$-theory of algebras over a field and its analogue $K^c$, an algebraic $K$-theory for coalgebras over a field. The comparison maps are compatible with the Hattori--Stallings (co)traces. We identify conditions on the algebras or coalgebras under which the comparison maps are equivalences. Notably, the algebraic $K$-theory of the power series ring is equivalent to the $K^c$-theory of the divided power coalgebra. We also establish comparison maps between the $G$-theory of finite dimensional representations of an algebra and its analogue $G^c$ for coalgebras. In particular, we show that the Swan theory of a group is equivalent to the $G^c$-theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology. |
| title | Coalgebraic $K$-theory |
| topic | K-Theory and Homology Algebraic Topology Primary: 16T15, 19D50. Secondary: 18M05, 55U30 |
| url | https://arxiv.org/abs/2503.04897 |