Algebra/Topology Seminar by Lukas Brantner

Algebra/Topology Seminar by Lukas Brantner (Harvard)

Title: The Lubin-Tate Theory of K(n)-local Lie Algebras 

AbstractL Quillen models the rational homotopy type of spaces by d.g. Lie algebras over Q. Recent work of Behrens-Rezk and Heuts considers K(n)-local Lie algebras as a modular generalisation. The homotopy groups of these Lie algebras are out of computational reach. 

We study a more accessible variant, namely Lie algebras in complete O_D^x-equivariant module spectra over Lubin-Tate space. These also appear as the basic building blocks in the chromatic Goodwillie spectral sequence.
 

We compute the operations which act on the homotopy groups of said Lie algebras. For this, we apply a new general technique for intertwining unstable power operations with operadic Koszul duality and use discrete Morse theory to study the equivariant topology of the partition poset. This also yields a short and purely combinatorial proof of an old theorem of Goerss on the cotangent complex in derived algebraic geometry.