Compactly central localisations in spectra and beyond

A research talk I gave for zygotop in Fall 2025, as a sequel to a previous talk. In a symmetric monoidal \(\infty\)-category \(\mathscr{C}\) we are often interested in studying smashing localisations, i.e. those localisations compatible with the symmetric monoidal structure. Finite localisations are an important family of smashing localisations, and in some simple cases we are able to classify all smashing localisations just by understanding the finite ones. In general, one is often able to classify the finite localisations in terms of Thomason-closed subsets of the Balmer spectrum. In this talk, I explained how to reimagine smashing localisations in terms of \(\mathbb{E}_1\)-algebras over central maps. From this perspective a natural notion of finiteness presents itself, which we call being compactly central. We classified all compactly central localisations of the category of spectra, and gave a simple criterion in terms of the Balmer spectrum for when a finite localisation is compactly central. I also discussed some more general conjectures about the situation in \(D(X)\) for a qcqs (Noetherian) scheme \(X\).