A strong finiteness condition for smashing localisations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909778392907776 |
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| author | Longbottom, Isabel |
| author_facet | Longbottom, Isabel |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A strong finiteness condition for smashing localisations Longbottom, Isabel Algebraic Topology 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. |
| title | A strong finiteness condition for smashing localisations |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2509.07344 |