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Lucas Piessevaux: The Balmer spectrum of finite spectra and prime fields in stable homotopy theory

Time: Fri 2022-10-07 13.15 - 14.15

Location: Alabano, Seminar room Kovalevsky

Participating: Lucas Piessevaux (Stockholms Universitet)

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In this talk, I will introduce work of Devinatz, Hopkins, and Smith on the classification of thick subcategories of finite spectra.   First, I will describe spectra as the basic objects of stable homotopy theory, and how they generalise abelian groups. Inspired by the decomposition of finite abelian groups into their p-torsion for any prime p, we attempt to construct a similar classification of finite spectra. The decomposition of finite abelian groups along primes and the construction of prime fields Z/pZ turns out to have an analogue in finite spectra, but with far richer structure, in particular a second direction of decomposition called chromatic height. This description is formalised in the field of tt-geometry and the construction of Balmer spectra.