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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 \(H^{ML}_ *(R)\).

MSC:

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.)

Citations:

Zbl 0724.18005
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References:

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