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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.07344 |
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Table of Contents:
- We define a class of smashing localisations which we call compactly central, and classify compactly central localisations of $Sp_{(p)}$ and of $Sp$. Our main result is that $L_n^f$ is a compactly central localisation. A map $α: 1 \to A$ in a presentably symmetric monoidal $\infty$-category $\mathscr{C}$ is central if there exists a homotopy $α\otimes id_A \simeq id_A \otimes α: A \to A \otimes A$. A central map $α$ can be used to produce a smashing localisation $L_α$ of $\mathscr{C}$, because the free $\mathbb{E}_1$ algebra on the $\mathbb{E}_0$ algebra $α$ is an idempotent commutative algebra. When both the monoidal unit and $A$ are compact, we call $L_α$ compactly central. We show that when $\mathscr{C}$ is (compactly generated) rigid, all compactly central localisations are finite in the sense of Miller. Not all finite localisations of $Sp$ are compactly central. To exhibit $L_n^f$ as compactly central, we determine properties of the $K(n)$-homology of a map between $p$-local finite spectra which ensure that some tensor power of the map is central.