Summary: ON THE CARTAN MAP FOR CROSSED PRODUCTS AND
KONSTANTIN ARDAKOV AND SIMON WADSLEY
Abstract. We study certain aspects of the algebraic Ktheory of HopfGalois
extensions. We show that the Cartan map from Ktheory to Gtheory 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 Gtheory of a right regular ring B coincide [5, Corollary 2 to Theorem
3]. More precisely, the Cartan map K i (B) # G i (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