\documentclass[12pt]{article}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}

\topmargin=0.0in
\textwidth=6.5in
\textheight=8.5in
\oddsidemargin=0.0in
\parindent=0.0in
\font\bigboldfont=cmbx10 scaled \magstep4

\begin{document}

\begin{titlepage}

\begin{center}
{\bigboldfont
Harvard-M.I.T. Algebraic Geometry Seminar}

\vspace{0.5in}

{\large \bf
%%% title goes here in all caps
K-THEORY OF A HENSELIAN DISCRETE VALUATION FIELD WITH NON-PERFECT RESIDUE FIELD
}

\vspace{0.5in}

\begin{large}
%%% Speaker in caps, school in lower case
{\bf 
LARS HESSELHOLT
}

Massachusetts Institute of Technology

\end{large}

\vspace{0.5in}
%%% Abstract goes here; comment out otherwise
{\bf \textsc{Abstract:} }

\end{center}

\newcommand{\Fp}{\mathbb{F}_p}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\xto}{\xrightarrow}

Twenty-five years ago, Bloch introduced the complex of $p$-typical
curves on Quillen's algebraic $K$-groups for the purpose of
calculating crystalline cohomology. This lead Deligne and Illusie to
define the de~Rham-Witt complex, which gives crystalline cohomology. I
have showed that for a regular scheme over $\Fp$, the two complexes, in
fact, are isomorphic. Both complexes can be defined also for schemes
over a discrete valuation ring $V$ of mixed characteristic. And in this
case, they are not the same. In this talk, I will explain the
structure of the two complexes for a smooth $V$-scheme $X$. Let me
just mention here that the Frobenius fixed set of the de~Rham-Witt
complex (modulo $p$) is isomorphic to the sheaf of $p$-adic vanishing
cycles.

The calculation of Bloch's $p$-typical curves in the mixed
characteristic setting has the following consequence for algebraic
$K$-theory (whence the title): Let $K$ be the quotient field of the
henselian local ring of $X$ at the generic point of the special
fiber. Then, assuming that $\mu_p\subset K$, there is a canonical
isomorphism
$$K_*^M(K)\otimes_{\Z}S_{\Z/p}(\mu_p) \xto{\sim}
K_*(K,\Z/p),$$
which to $\zeta\in\mu_p$ assigns the corresponding Bott element
$b_{\zeta}\in K_2(K,\Z/p)$. This is the value of the $K$-groups
predicted by the Beilinson-Licthenbaum conjectures.

\vfill

%%% Date goes here, and location (Harvard Rm 507 or MIT Room 4--163)
\begin{center}
{\large
April 9, 2002

3:00 p.m.

%Harvard Room 507
MIT Room 4-163
}
\end{center}

\vspace{0.5in}

\begin{center}
%%% Seminar website goes here
\verb+http://www-math.mit.edu/~abuch/seminar/+
\end{center}

\end{titlepage}

\end{document}