Stable \(K\)-theory and topological Hochschild homology. (English) Zbl 0833.55007

Waldhausen’s algebraic \(K\)-theory of spaces is the extension of classical algebraic \(K\)-theory of rings, viewed as \(\mathbb{Z}\)-algebras, to rings up to coherent homotopies, viewed as algebras over \(QS^0= \text{codim}_n \Omega^n S^n\). In a similar way, topological Hochschild homology THH is the translation of Hochschild homology over the ground ring \(\mathbb{Z}\) to the new “ground ring” \(QS^0\). Admitting topological spaces as inputs allows stabilization processes; stable \(K\)-theory \(K^s\) is obtained from Waldhausen \(K\)-theory this way.
Right from the start of this new algebra Goodwillie conjectured that stable \(K\)-theory agrees with topological Hochschild homology. The present paper contains an elegant proof of this conjecture in the case of simplicial rings \(R_\bullet\) and simplicial \(R_\bullet\)-bimodules \(M_\bullet\).
For a ring \(R\) and an \(R\)-bimodule \(M\) let \(K(R; M)\) be the algebraic \(K\)-theory of the exact category with objects \((P, \alpha)\), \(P\) a finitely generated projective \(R\)-module and \(\alpha: P\to\) \(P\otimes M\) an \(R\)-linear map. Extend this definition to simplicial bimodules degreewise, and define \(K(R; M; X)= K(R; M[M]/ M[*])\) for simplicial sets \(X\). The authors show that \(\text{THH} (R; M)\) is weakly homotopy equivalent to the underlying space of the Goodwillie derivative of \(K(R; M; -)\). Since \(K^S (R, M)\) is the Goodwillie derivative of the functor \(X\mapsto \Omega K(R\oplus M[X]/ M[*])\) the result is obtained by comparing \(K(R; -)\) with \(k(R \oplus -)\). Here \(R\oplus M\) is the ring with multiplication \((r, m)\cdot (r', m')= (r\cdot r', rm'+ mr')\), and \(R\oplus M[X]/ M[*]\) is the simplicial ring using this multiplication degreewise.
There is an alternative proof of Goodwillie’s conjecture due to Schwänzl, Staffeldt and Waldhausen. It is based on the analysis of Nil- terms in the algebraic \(K\)-theory of free products of rings up to homotopy and includes the general case of rings up to homotopy (to appear). Recently, the first author has also announced a proof of the general case. It consists of a reduction to the case of simplicial rings and bimodules using a line of arguments suggested by Goodwillie.


55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55N15 Topological \(K\)-theory
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