## EVENT DETAILS AND ABSTRACT

**Topology Seminar**
**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.

**Date:** Monday, November 13, 2006

**Time:** 4:10pm

**Where:** Lunt 104

**Contact Person:** Prof. Paul Goerss

**Contact email:** pgoerss@math.northwestern.edu

**Contact Phone:** 847-491-8544

Copyright © 1997-2024
Department of Mathematics, Northwestern University.