# zbMATH — the first resource for mathematics

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)