Summary: ON THE CARTAN MAP FOR CROSSED PRODUCTS AND
KONSTANTIN ARDAKOV AND SIMON WADSLEY
Abstract. We study certain aspects of the algebraic K-theory of Hopf-Galois
extensions. We show that the Cartan map from K-theory to G-theory of such
an extension is a rational isomorphism, provided the ring of coinvariants is
regular, the Hopf algebra is finite dimensional and its Cartan map is injective
in degree zero. This covers the case of a crossed product of a regular ring with
a finite group and has an application to the study of Iwasawa modules.
1.1. The Cartan map. Recall that a ring is said to be right regular if it is right
Noetherian and every finitely generated right module has finite projective dimen-
sion. So any Noetherian ring of finite global dimension is necessarily regular.
One consequence of Quillen's celebrated Resolution Theorem is that the K-
theory and the G-theory of a right regular ring B coincide [5, Corollary 2 to Theorem
3]. More precisely, the Cartan map Ki(B) Gi(B) is an isomorphism for all i 0.
Now if G is a finite group and A = B G is a crossed product then A need not
be regular, so the Resolution Theorem does not apply. This is evident even in the
simplest case when B = k is a field of characteristic p > 0, p divides the order of
G and A = kG is the group algebra of G -- in fact, in this case the Cartan map