Cyclic Homology of Dedekind Domains

The purpose of this paper is to calculate the cyclic homology of rings of integers of global fields. We accomplish this by explicitly computing the homology of the simple complex associated to Tsygan’s double complex. To accomplish this, we first compute the cyclic homology of cyclic algebras, i.e., algebras of the form A = R[t]/(P (t)), where P is a monic polynomial with coefficients in R. More precisely, we give a complex (2.9) of finite free A-modules and R-linear maps, whose homology gives the cyclic homology HCr(A/R) when A is an integral domain. This complex enjoys a great advantage over the usual complex, (2.1), for cyclic homology, in that the rank of the modules grows linearly rather than exponentially in r. As a result, one can compute the homology groups explicitly in many interesting cases. If R is a Dedekind domain whose field of fractions K is a global field (i.e. a number field or the fraction field of a curve over a finite field), and S is the integral closure of R in a finite separable extension L/K, the cyclic homology of S/R can be obtained from the cyclic homology of certain cyclic algebras, by a kind of adèlization procedure (3.2). To establish the relation between our cyclic complex and the “standard” cyclic complex of Tsygan, Loday, and Quillen [12], we describe a subcomplex of the latter, the small Tsygan complex, which is quasi-isomorphic to the full complex and which admits a quasiisomorphism to our cyclic complex. The construction of the latter quasi-isomorphism is the key technical feature of our paper. It is motivated by our attempts to solve the extension problem arising from the spectral sequence which computes the homology of the Tsygan complex. The differentials of this spectral sequence are implicitly worked out in the proof of Th. 2.10. The solution of the extension problem follows from the technical results Prop. 2.6 and Prop. 2.8. The first section discusses the Hochschild theory for cyclic algebras and Dedekind domains. For cyclic algebras, Hochschild homology can be computed from a periodic complex (1.6). This complex seems to be part of the folklore; but it seems to have been discovered independently, in varying degrees of generality, by a number of people, including Zack [18]; Cortiñas, Guccione, and Villamayor [4]; Burghelea and ViguéPoirrier [3]; Goodwillie (unpublished); Masuda and Natsume [14]; Wolffhardt [17]; and perhaps others) By adèlizing, we obtain (Prop. 1.9) the periodicity of Hochschild homology for Dedekind domains. By constructing explicit quasi-isomorphisms between the Zack complex and the standard complex for Hochschild homology, we can study the maps HH∗(S/R) → HH∗(T/R) → HH∗(T/S), where R ⊂ S ⊂ T is a triple of Dedekind domains (Props. 1.11, 1.13), as well as the ring structure of HH∗(S/R) (1.16). We conclude with a construction of the small subcomplex of the standard Hochschild complex, which motivates the small subcomplex of the Tsygan complex.