K₀ of Invariant Rings and Nonabelian H¹

Abstract:

This article is concerned with the relationship between  K₀(S)  and  K₀(R), where  S  is a
commutative ring and  R=S^G  denotes the subring of invariants under the action of a finite
group  G  on  S.

Specifically, working under the assumption that the trace map  S  --> R  is surjective,
we study the kernel of the induction map  Ind: K₀(R) --> K₀(S)  that is associated with the
inclusion of  R  into  S.  We describe an embedding of   Ker(Ind)  into the cohomology set
H¹(G,GL(S)) . Moreover, we endow  H¹(G,GL(S))  with a natural commutative monoid
structure, essentially coming from the ``block diagonal" maps  GL_n x GL_m --> GL_n+m ,
such that our embedding identifies  Ker(Ind)   with the group of units  U(H¹(G,GL(S))) .
We further describe this unit group using certain restriction-reduction maps.

As applications, we present a version of Hilbert's  Theorem 90 for Galois actions on
commutative rings and quickly derive the  (known) structure of the Picard groups of linear
and multiplicative invariants. Some open problems are also discussed.
 
 

Electronic Preprint:
postscipt file (208K)
dvi file (91K)
 
 

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