# Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$

Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $$A$$, Eilenberg and Mac Lane define its cubical construction $$QA$$ to be a certain connective chain complex, whose homology is isomorphic to the stable homology of Eilenberg-Mac Lane spaces: $$H_i(QA)\cong H_{i+j}(K(A,j)) \text{ for } j \gg i.$$In fact, $$QA$$ is a dg-ring, and it comes with a ring map $$QA \to A$$ inducing an isomorphism on $$H_0$$. You can find the construction and basic properties of the cubical construction in [1] or [2, Chapter 13]. One then defines the Mac Lane homology of $$A$$ to be the Hochschild homology $$HML_i(A):= HH_i(QA,A)$$and similarly the Mac Lane cohomology is $$HML^i(A):=HH^i(QA,A)$$.

In 1992, Pirashvili and Waldhausen [3] proved that $$HML_i(A)\cong THH_i(A)$$, where the right-hand side is topological Hochschild homology. The proof goes by identifying both of them with a functor homology group. Subsequently in 1995, Fiedorowicz-Pirashvili-Schwaenzl-Vogt-Waldhausen [1] outlined a `brave new algebra' proof of this fact. In modern terms, they noticed that the Pirashvili-Waldhausen result would follow if one has a stable equivalence of $$H\mathbb{Z}$$-algebra spectra $$HQA \simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}.$$Here, $$H$$ denotes the Eilenberg-Mac Lane functor taking dgas to $$H\mathbb{Z}$$-algebra spectra, and $$\mathbb{S}$$ is the sphere spectrum. (For concreteness, my preferred model of spectra is symmetric spectra in simplicial sets.) Indeed, if one has such a stable equivalence, then it follows by base change results that $$HML(A)\simeq THH(A)$$, for both homology and cohomology.

My question: Is it known that there is a stable equivalence $$HQA \simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$$ of $$H\mathbb{Z}$$-algebra spectra?

This may be a fact known to experts in $$THH$$, but I was unable to find anything in the literature. One can almost get there: because the homology of $$QA$$ is the stable cohomology of Eilenberg-Mac Lane spaces, it follows that one has $$\pi_i(HQA)\cong \pi_i(HA \wedge_{\mathbb{S}} H\mathbb{Z}).$$Moreover, both are $$H\mathbb{Z}$$-module spectra, hence wedges of Eilenberg-Mac Lane spectra, and so an isomorphism on $$\pi_*$$ actually lifts to a stable equivalence $$HQA \simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$$ of $$H\mathbb{Z}$$-module spectra. This argument already appears in [1], but it wasn't clear to the authors then, and it's certainly not clear to me now, how to lift this to a stable equivalence of $$H\mathbb{Z}$$-algebra spectra. I guess because $$QA$$ is connective, in theory one could write down both sides of the equation and match up the algebra structure somehow, but I am not sure if this is very tractable.

References:

[1] Fiedorowicz, Z.; Pirashvili, T.; Schwänzl, R.; Vogt, R.; Waldhausen, F. Mac Lane homology and topological Hochschild homology. Math. Ann. 303 (1995)

[2] Jean-Louis Loday, Cyclic homology, Springer 1998

[3] Pirashvili, Teimuraz; Waldhausen, Friedhelm. Mac Lane homology and topological Hochschild homology. J. Pure Appl. Algebra 82 (1992)

• Great question, and great write-up of a question. I would give more than one +1 if I could. Aug 26 '20 at 21:07

The answer to your question is yes, these two $$H\mathbb{Z}$$-algebras are equivalent (and if $$A$$ is commutative, they are equivalent as commutative $$H\mathbb{Z}$$-algebras). I am in fact in the process of writing this up with Maxime Ramzi. Hopefully the paper will be finished in a week or 2.