Seminar

Franke’s conjecture and derived infinity categories

Higher algebraic structures in algebra, topology and geometry

17 March 14:15 - 16:00

Irakli Patchkoria (Online) - University of Aberdeen

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.

 

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Organizers
Gregory Arone
Stockholm University
Tilman Bauer
KTH Royal Institute of Technology
Alexander Berglund
Stockholm University
Søren Galatius
University of Copenhagen
Jesper Grodal,
University of Copenhagen
Thomas Kragh
Uppsala University

Program
Contact

Alexander Berglund

alexb@math.su.se

Søren Galatius

galatius@math.ku.dk

Thomas Kragh

thomas.kragh@math.uu.se

Other
information

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