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$$K$$-theory of endomorphism rings and of rings of invariants. (English) Zbl 0878.16007
Let $$T$$ be a ring, $$P$$ a finitely generated $$T$$-module, and $$I$$ the trace ideal of $$P$$ in $$T$$. The main general result of this interesting article asserts that $$K_0(\text{End }P_T)$$ is isomorphic to a subgroup of $$K_0(T,I)$$. This fact is then applied to the situation where a finite group $$G$$ acts on a ring $$S$$, with $$|G|^{-1}\in S$$. Namely, letting $$T=S*G$$ denote the skew group ring associated with this action and taking $$P=S_T$$, one has $$\text{End }P_T\cong S^G$$, the ring of $$G$$-invariants in $$S$$. Thus, as a corollary, the author is able to prove triviality of $$K_0(S^G)$$, that is, $$K_0(S^G)=\langle[S^G]\rangle$$, for various rings of invariants $$S^G$$, commutative and noncommutative.
Most notably, this is established for $$G$$-actions on the symmetric algebra $$S=S(V)$$ of a finite-dimensional $$G$$-representation $$V$$ over an algebraically closed field of characteristic 0 under the assumption that $$V$$ decomposes into fixed point free and trivial constituents only. This provides a positive answer to a special case of an open question, posed by H. Kraft [in: CMS Conf. Proc. 10, 111-123 (1989; Zbl 0703.14009)] as to whether $$K_0(S(V)^G)$$ is always trivial, for any rational representation $$V$$ of a reductive algebraic group $$G$$.

##### MSC:
 16E20 Grothendieck groups, $$K$$-theory, etc. 16W20 Automorphisms and endomorphisms 16S50 Endomorphism rings; matrix rings 19A49 $$K_0$$ of other rings 14L30 Group actions on varieties or schemes (quotients) 20G05 Representation theory for linear algebraic groups
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