Cyclic homology of Dedekind domains
In this paper the cyclic homology of \(A:=R[t]/P(t)\) is computed for an integral domain \(R\) and a monic irreducible polynomial \(P\). This is then applied to the ring of algebraic integers in an algebraic number field. In section 1 a local global principle is derived for the Hochschild homology of the integral closure \(S\) of a Dedekind domain \(R\) in a separable extension field \(L\) of the field of fractions \(K\) of \(R\). In case of perfect residue fields the homology groups are computed. Moreover, for \(a\in A\) dividing \(P'(t)\) the ring structure of the Hochschild homology with coefficients in \(A/a\cdot A\) is determined. -- Next the authors give a complex, the ranks of whose modules grow only linearly, and the homology of which is the cyclic homology of \(A\). For a Dedekind domain \(R\) with global field of fractions \(K\) and \(S\) the integral closure in a separable extension \(L\) of \(K\) the computation of the cyclic homology is reduced to that of certain cyclic algebras. The results are then applied to give a formula for cyclic homology of tamely ramified extensions and for cyclotomic extensions, by employing a graded structure. Also the cyclic homology of integral group rings of cyclic \(p\)-groups is computed. [Reviewer's remark: The results here for cyclic \(p\)-groups should generalize to blocks with normal cyclic defect groups, and may be -- via derived equivalence (or Green correspondence) -- to arbitrary blocks with cyclic defect].
scientific article; zbMATH DE number 149775