Profinite and continuous Higher K-Theory of Exact categories, schemes, Orders and Group-rings.
by
Aderemi Kuku
International Centre for Theoretical physics, Trieste, Italy.

Let C be an exact category. In this lecture, we define and study the profinite Higher K-theory and some other associated K-theories of C and prove several l-completeness results in the general setting of exact categories. We then specialise C to the category P(X) of locally free sheaves of O_X modules (X a scheme) and C = P(A) the category of finitely generated projective modules over any ring A (not necessarily commutative) with identity as well as the category C = M(x) of coherent sheaves of O_X modules (X a Noetherian Schemme) and the category M(A) of finitely generated modules over a (not necessarily commutative) Noetherian ring A. For computational purposes, we specialise further to the categories P(A) and M(A) where the A are orders in semisimple algebras over number fields and p-adic fields. We also define and study continuous Higher K-theory of orders A in p-adic semisimple algebras and show some connection between the profinite and continuous K-theory of A.

Date received: June 18, 2001