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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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