A strong finiteness condition for smashing localisations

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 $\alpha: 1 \to A$ in a presentably symmetric monoidal $\infty$-category $\mathscr{C}$ is central if there exists a homotopy $\alpha \otimes id_A \simeq id_A \otimes \alpha: A \to A \otimes A$. A central map $\alpha$ can be used to produce a smashing localisation $L_\alpha$ of $\mathscr{C}$, because the free $\mathbb{E}_1$ algebra on the $\mathbb{E}_0$ algebra $\alpha$ is an idempotent commutative algebra. When both the monoidal unit and $A$ are compact, we call $L_\alpha$ 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.