DubTop Seminar: Galois descent in algebraic K-theory

Jay Reiter, UW
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CMU 230

The Lichtenbaum--Quillen conjecture concerns the convergence of a motivic spectral sequence that begins with the étale cohomology of a variety and abuts to its algebraic \$\mathrm{K}\$-theory. In the 1980's, Thomason showed that (under appropriate circumstances) after either rationalization or \$\mathrm{K}(1)\$-localization, algebraic \$\mathrm{K}\$-theory is a sheaf for the étale topology, so in these cases the motivic spectral sequence is nothing more than an étale descent spectral sequence. For a while, it was known that the obstruction to \$\mathrm{K}\$ being an étale sheaf is its failure to satisfy Galois descent. Using a general theory of descendability and "deformations of descendable objects," Clausen--Mathew--Naumann--Noel extend Thomason's result to show that \$\mathrm{K}\$ satisfies Galois descent after localization at the mapping telescope \$\mathrm{T}(n)\$ of a type-\$n\$ spectrum for any \$n\$. In this talk, we'll give an overview of their techniques, and (time permitting) outline how Yuan uses this result to give a formal proof of chromatic redshift for Morava \$\mathrm{E}\$-theory

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