MacLane homology and topological Hochschild homology. (English) Zbl 0767.55010

The authors prove that the topological Hochschild homology \(THH_ *(R)\) of a discrete ring \(R\) as defined by Bökstedt in 1985 is isomorphic to the MacLane homology \(H^{ML}_ *(R)\) of \(R\) as defined by MacLane in 1957. Moreover they construct an Atiyah-Hirzebruch type of spectral sequence whose \(E^ 2\)-term is Hochschild homology of \(R\) with coefficients in MacLane homology of \(\mathbb{Z}\) converging to MacLane homology of \(R\). In fact, they prove a stronger version of both results by allowing coefficients in functors from the category \({\mathcal P}({\mathcal R})\) of finitely generated projective \(R\)-modules to the category of all \(R\)-modules. The main result is proved by showing that \(THH_ *(R)\) satisfies a set of axioms characterizing the homology of \({\mathcal P}({\mathcal R})\) with coefficients in the Hom-functor. Previously, M. Dzhibladze and T. Pirashvili [J. Algebra 137, 253-296 (1991; Zbl 0724.18005)] proved the same for \(H^{ML}_ *(R)\).


55P99 Homotopy theory
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)


Zbl 0724.18005
Full Text: DOI Link


[1] Baues, H. J.; Wirsching, G. J., Cohomology of small categories, J. Pure Appl. Algebra, 38, 187-211 (1985) · Zbl 0587.18006
[2] M. Bökstedt, Topological Hochschild homology, Preprint.; M. Bökstedt, Topological Hochschild homology, Preprint.
[3] M. Bökstedt, The topological Hochschild homology of \(ZZp\); M. Bökstedt, The topological Hochschild homology of \(ZZp\)
[4] Cartan, A.; Eilenberg, S., Homological Algebra (1956), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0075.24305
[5] Dold, A.; Puppe, D., Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier, 11, 201-312 (1961) · Zbl 0098.36005
[6] Eilenberg, S.; MacLane, S., On the groups \(H(π,n)\), Ann. of Math., 60, 49-139 (1954), II · Zbl 0055.41704
[7] Goodwillie, T. G., On the general linear group and Hochschild homology, Ann. of Math., 121, 383-407 (1985) · Zbl 0566.20021
[8] Jibladze, M.; Pirashvili, T., Cohomology of algebraic theories, J. Algebra, 137, 253-296 (1991) · Zbl 0724.18005
[9] Kassel, C., \(La K\)-théorie stable, Bull. Soc. Math., 110, 381-416 (1982), France · Zbl 0507.18003
[10] MacLane, S., Homologie des anneaux et des modules, Coll. Topologie Algébrique, 55-80 (1956), Louvain · Zbl 0084.26703
[11] Pirashvili, T., Higher additivizations, Proc. Math. Inst. Tbilisi, 91, 44-54 (1988) · Zbl 0705.18008
[12] Pirashvili, T., New homology and cohomology of rings, Bull. Acad. Sci. Georgian SSR, 133, 477-480 (1989) · Zbl 0676.16024
[13] R. Schwänzl, R. Staffeldt and F. Waldhausen (in preparation).; R. Schwänzl, R. Staffeldt and F. Waldhausen (in preparation).
[14] Shukla, M. U., Cohomologie des algébres associatives, Ann. Sci. École Norm. Sup., 78, 163-209 (1961) · Zbl 0228.18005
[15] Waldhausen, F., Algebraic \(K\)-theory of topological spaces II, (Lecture Notes in Mathematics, 763 (1979), Springer: Springer Berlin), 356-394 · Zbl 0431.57004
[16] Zisman, M., Suite spectrale d’homotopie et ensembles bisimpliciaux (1975), Université Scientifique et Medicale de Grenoble
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.