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ON THE CARTAN MAP FOR CROSSED PRODUCTS AND HOPFGALOIS EXTENSIONS
 

Summary: ON THE CARTAN MAP FOR CROSSED PRODUCTS AND
HOPF­GALOIS EXTENSIONS
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. Introduction
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

  

Source: Ardakov, Konstantin - School of Mathematical Sciences, University of Nottingham

 

Collections: Mathematics