The cyclic homology of an exact category
Algebraic \(K\)-theory is at once an invariant of a ring and of an exact category, that is, the category of finitely generated modules over a ring. Constructions of homomorphisms between algebraic \(K\)-theory and other invariants of rings such as the Dennis trace map and the generalized Chern character, usually proceed through the ring-theoretic definition of \(K\)-theory. In this paper the author obtains all of the relevant additional structure on the invariants of rings that are most closely related to algebraic \(K\)-theory, that is, the Hochschild homology and the (negative, periodic) cyclic homology, by starting with an exact category. This not only generalizes the definitions of these invariants but it provides a uniform foundation for the relevant additional structure and provides means for further study of the relations between these invariants through categorical constructions. Let \(k\) denote a commutative ring with unit. The ingredients for this synthesis are the \(S\)-construction of Waldhausen applied to a category with cofibrations and the additive cyclic nerve of a small \(k\)-linear category \({\mathcal C}\); this is a cyclic \(k\)-modulo which at level \(n\) is given by \[ \text{CN}_ n ({\mathcal C}) = \bigoplus \Hom_{\mathcal C} (C_ 1,C_ 0) \otimes_ k \Hom_{\mathcal C} (C_ 2,C_ 1) \otimes_ k \cdots \otimes_ k \Hom_{\mathcal C} (C_ 0,C_ n), \] where the sum is over \(n + 1\)-tuples of objects in \({\mathcal C}\). The face maps are given by composition, the degeneracy maps by inserting an identity map, and the twist map is given by applying the cyclic permutation to the tensor product. The split Hochschild homology groups of \({\mathcal C}\) are given by the homology of the additive cyclic nerve. The composition of Waldhausen's \(S\)-construction applied to an exact category \({\mathcal E}\) followed by the additive cyclic nerve gives a simplicial cyclic abelian group, and the Hochschild homology of the exact category is given by \[ HH_ n ({\mathcal E}) = H_{n + 1}\text{(CN}.S. {\mathcal E}). \] Following Waldhausen, the algebraic \(K\)-theory of a \(k\)- linear category \({\mathcal C}\) with fibrations is identified with the space \(\Omega | \text{obj} (S.{\mathcal C}) |\) and likewise we can identify the Hochschild homology with the space \(\Omega | \text{CN}.S. {\mathcal C} |.\) The Dennis trace map is induced by the inclusion of obj\((S.{\mathcal C})\) into CN\(_ 0 (S.{\mathcal C})\) followed by the inclusion into \(\text{CN.}(S.{\mathcal C})\). The Chern character is similarly defined by passing through the simplicial double complex for negative cyclic homology. By defining the appropriate product structures on the \(S\)-construction the author obtains the multiplicative structure on the Dennis trace and the generalized Chern character. The additive cyclic nerve possesses an \(S^ 1\)-action and the author shows that the Dennis trace maps into the homotopy inverse limit of the fixed points of each finite subgroup of \(S^ 1\). This recovers observations due to Bökstedt. Some of the author's constructions require maps of cyclic objects that do not preserve the degeneracies. There is a useful appendix in which the properties of such semi-cyclic maps are developed.
scientific article; zbMATH DE number 584667
scientific article