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Irakli Patchkoria: Franke’s conjecture and derived infinity categories

Time: Thu 2022-03-17 14.15 - 16.00

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: Irakli Patchkoria (University of Bonn, online)

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Abstract: For a “nice enough” homology theory on a stable infinity category, we introduce a derived infinity category which encodes the Adams spectral sequence associated to the homology theory. By running the Goerss–Hopkins obstruction theory in the later, we prove Franke’s algebraicity conjecture from 1996 which asserts the following: Suppose we are given a “nice enough” homology theory on a stable infinity category C with values in an abelian category A and assume that A has finite cohomological dimension and is sufficiently sparse. Then the homotopy category of C is equivalent to the homotopy category of differential complexes in A. In fact it turns out that up to some fixed level, higher homotopy categories are equivalent too. This gives algebraic models in the following special cases: Modules over sufficiently sparse ring spectra, chromatic stable homotopy category for large primes, diagram categories such as filtered objects or towers, and chromatic spectral Mackey functors in the coprime case for large primes. This is all joint with Piotr Pstrągowski.