The algebraic $K$-theory of Green functors
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915452125446144 |
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| author | Chan, David Wisdom, Noah |
| author_facet | Chan, David Wisdom, Noah |
| contents | In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group. From the spectral sequence we deduce a complete calculation of the algebraic $K$-theory of the constant $C_2$-Green functor associated to the field with two elements, and a calculation of the $p$-completion of the algebraic $K$-theory of the constant $G$-Green functor associated to the integers when $G$ is a cyclic $p$-group. Additionally, we introduce the notion of a Green meadow to abstract the Green functor structure underlying clarified Tambara fields, and show, under mild conditions, that every finitely generated projective module over a $G$-Green meadow is free when $G$ is a cyclic $p$-group. This gives a computation of $K_0$ for such Green functors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The algebraic $K$-theory of Green functors Chan, David Wisdom, Noah K-Theory and Homology Algebraic Topology 19D50 (Primary) 19A22, 55P91, 16D40 (Secondary) In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group. From the spectral sequence we deduce a complete calculation of the algebraic $K$-theory of the constant $C_2$-Green functor associated to the field with two elements, and a calculation of the $p$-completion of the algebraic $K$-theory of the constant $G$-Green functor associated to the integers when $G$ is a cyclic $p$-group. Additionally, we introduce the notion of a Green meadow to abstract the Green functor structure underlying clarified Tambara fields, and show, under mild conditions, that every finitely generated projective module over a $G$-Green meadow is free when $G$ is a cyclic $p$-group. This gives a computation of $K_0$ for such Green functors. |
| title | The algebraic $K$-theory of Green functors |
| topic | K-Theory and Homology Algebraic Topology 19D50 (Primary) 19A22, 55P91, 16D40 (Secondary) |
| url | https://arxiv.org/abs/2508.14207 |