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Module structures on the K-theory of graded rings. (English) Zbl 0611.13012
Let \(R\) be a commutative ring; \(A=A_ 0+A_ 1+..\). a graded \(R\)- algebra, with R concentrated in degree \(0;\) and, \(A_+\) the graded ideal \(A_ 1+A_ 2+... \). Let \(W(R)\) be the ring of Witt vectors over \(R\). Then there is a continuous \(W(R)\)-module structure on each group \(K_ i(A,A_+)\). This structure is natural on the category of graded \(A\)- algebras. If \(R\) contains \({\mathbb{Q}}\), then each \(K_ i(A,A_+)\) has a natural \(R\)-module structure via the ring map \(\lambda: R\to W(R).\) The \(W(R)\)-module structure agrees with a pairing given by S. Bloch [J. Algebra 53, 304-326 (1978; Zbl 0432.14014)]. Examples include formulas for the \(R\)- and \(W(R)\)-module structures on \(K_ 2(A,A_+)\).
Reviewer: R.M.Najar

MSC:
13D15 Grothendieck groups, \(K\)-theory and commutative rings
13K05 Witt vectors and related rings (MSC2000)
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