Zygotop seminar
We are an informal pedagogical seminar concerning algebraic topology and homotopy theory, localized at Harvard and MIT. Unless otherwise specified, talks will proceed on Wednesdays at 4:30pm ET, located in SC 310, and following a topic chosen by the speaker, organized according to these principles.
This seminar is organized by Keita Allen, who you should email with questions or to be added to the signal group chat. Click here to add this seminar to your google calendar.
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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, we will 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 extremely finite since in most settings it is a stronger condition than the usual notion of finiteness. We will classify all extremely finite localisations of the category of spectra, and give a simple criterion in terms of the Balmer spectrum for when a finite localisation is extremely finite. Using these results, one can conjecture more generally about the situation in \(D(X)\) for a qcqs (Noetherian) scheme \(X\).
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