Picard Groups in Equivariant Algebra and Stable Homotopy Theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909968470376448 |
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| author | Keyes, Jesse Sawdy, Jordan |
| author_facet | Keyes, Jesse Sawdy, Jordan |
| contents | Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the equivariant stable homotopy category. Though there is a canonical map of abelian groups $\text{RO}(G) \rightarrow \text{Pic}(\text{Ho}(\text{Sp}^G))$ relating the two, this map is neither injective or surjective in general. Fausk, Lewis, and May give an algebraic expression of $\text{Pic}(\text{Ho}(\text{Sp}^G))$ in terms of the Picard group of the Burnside ring $A(G)$, and this work suggests a folklore isomorphism between $\text{Pic}(A(G))$ and $\text{Pic}(\text{Mack}_G)$. We prove the existence of this folklore isomorphism in the setting of finite groups, then leverage our analysis to prove a classification of invertible Mackey functors in the setting of finite abelian groups. As a consequence, we furnish a classification of invertible $A(G)$-modules again for $G$ a finite abelian group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Picard Groups in Equivariant Algebra and Stable Homotopy Theory Keyes, Jesse Sawdy, Jordan Algebraic Topology Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the equivariant stable homotopy category. Though there is a canonical map of abelian groups $\text{RO}(G) \rightarrow \text{Pic}(\text{Ho}(\text{Sp}^G))$ relating the two, this map is neither injective or surjective in general. Fausk, Lewis, and May give an algebraic expression of $\text{Pic}(\text{Ho}(\text{Sp}^G))$ in terms of the Picard group of the Burnside ring $A(G)$, and this work suggests a folklore isomorphism between $\text{Pic}(A(G))$ and $\text{Pic}(\text{Mack}_G)$. We prove the existence of this folklore isomorphism in the setting of finite groups, then leverage our analysis to prove a classification of invertible Mackey functors in the setting of finite abelian groups. As a consequence, we furnish a classification of invertible $A(G)$-modules again for $G$ a finite abelian group. |
| title | Picard Groups in Equivariant Algebra and Stable Homotopy Theory |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2512.16002 |