EVENT DETAILS AND ABSTRACT

Title: Galois Representations and Descent in Algebraic K-theory
Speaker: Grace Lyo
Speaker Info: University of California, Berkeley
Brief Description:
Special Note:
Abstract:

We will discuss a conjectural model for the completed $K$-theory spectrum of a field in terms of the semilinear representation theory of its absolute Galois group $G_F$. More specifically G. Carlsson has conjectured that if $F$ is a field with a separable closure $\bar{F}$, and $k$ is an algebraically closed subfield of $\bar{F}$, then there is a weak equivalence of completed $K$-theory spectra, $\dcomplete{KF}{p} \to \dcomplete{Kk}{p}$. Here, $p$ is a prime different from the characteristic of $F$, the functor $\dcomplete{-}{p}$ is the derived completion at the prime $p$, and $k$ is the twisted group ring, which is a generalization of the ordinary group ring that incorporates the action by $G_F$ on $k$. We will start by discussing the derived completion, which is a construction on ring spectra. This completion is modeled after the completion in algebra and, in addition to possessing many naturality properties, agrees with the algebraic notion of completion on Eilenberg-MacLane spectra in many cases. Next, fixing primes $p eq l$, we will sketch a proof that Carlsson's conjecture holds when $F$ is the extension of the field $\mathbb{F}_l((x))$ whose tame Galois group is a semidirect product of the $p$-adic integers with itself, and $k$ is a separable closure of a finite field.