Harvard-MIT Algebraic Geometry Seminar
		      April 9, 2002 at 3:00 p.m.
			    MIT Room 4-163



	   K-theory of a henselian discrete valuation field
		    with non-perfect residue field



			Lars Hesselholt (MIT)



Abstract:

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 $F_p$, 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)  -->  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.