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 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.

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