On the computation of stabilized tensor functors and the relative algebraic \(K\)-theory of dual numbers (Q2706616)

In [\textit{R. McCarthy}, Fields Inst. Commun. 16, 199-220 (1997; Zbl 0886.18007)] the author defines the stabilization NEWLINE\[NEWLINEF^{\text{st}}(\mathcal A)=\lim_{n\to\infty}F(S^{(n)}\mathcal A)[-n]NEWLINE\]NEWLINE of a functor \(F\) from the category of exact categories to chain complexes. Here \(S^{(n)}\) is Waldhausen's construction of algebraic K-theory applied \(n\) times. One example is the functor \(F=\mathbb{Z}\) which sends \(\mathcal A\) to the free abelian group on the objects of \(\mathcal A\) modulo the zero object. Then \(H_*(\mathbb{Z}^{\text{st}}(\mathcal A))\) is the stable homology of the K-theory spectrum of \(\mathcal A\). Another example is given by \(F(\mathcal A)=\bigoplus_{a\in \text{ob}\mathcal A}\Hom_\mathcal A(a,a)\) whose stabilization is equivalent to the topological Hochschild homology of \(\mathcal A\). NEWLINENEWLINENEWLINEIn this paper the author applies this machinery on functors derived form the tensor product, giving a new proof of Goodwillie's rational equivalence between relative algebraic K-theory and cyclic homology of dual numbers [\textit{T. G. Goodwillie}, Ann. Math., II. Ser. 124, 347-402 (1986; Zbl 0627.18004)]. This uses the model for the relative algebraic K-theory of the dual numbers of \textit{B. I. Dundas} and \textit{R. McCarthy} [Ann. Math., II. Ser. 140, No. 3, 685-701 (1994; Zbl 0833.55007)], and an identification of its homotopy by means of the stabilization of certain tensor products and permutation of tensor factors. An analysis shows that the stabilization of the tensor functor can be interpreted in terms of the cohomology of a small category, which in the rational case reduces to ordinary Hochschild homology, giving Goodwillie's result. The important point is here to take care of the permutation action, which reduces to a cyclic action. NEWLINENEWLINENEWLINEIn the integral case, the stabilization of the tensor products are expressed in terms of Mac Lane homology using the results ob \textit{M. Jibladze} and \textit{T. Pirashvili} [J. Algebra 137, No. 2, 253-296 (1991; Zbl 0724.18005)] and \textit{T. Pirashvili} and \textit{F. Waldhausen} [J. Pure Appl. Algebra 82, No. 1, 81-98 (1992; Zbl 0767.55010)].