Invariance properties of coHochschild homology

Abstract

The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in [23] as a tool to study free loop spaces. In this article we prove “agreement” for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra C is isomorphic to the Hochschild homology of the dg category of appropriately compact C-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space X is equivalent to the suspension spectrum of the free loop space on X, as long as X is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established in [24], we prove that “agreement” holds for coTHH as well.

Introduction

For any commutative ring $\mathbb{k}$, the classical definition of Hochschild homology of $\mathbb{k}$-algebras [33] admits a straightforward extension to differential graded (dg) $\mathbb{k}$-algebras. In [37] McCarthy extended the definition of Hochschild homology in another direction, to $\mathbb{k}$-exact categories, seen as $\mathbb{k}$-algebras with many objects. As Keller showed in [29], there is a common refinement of these two extended definitions to dg categories, seen as dg algebras with many objects. This invariant of dg categories satisfies many useful properties, including “agreement” (the Hochschild homology of a dg algebra is isomorphic to that of the dg category of compact modules) [29, 2.4] and Morita invariance (a functor in the homotopy category of dg categories that induces an isomorphism between the subcategories of compact objects also induces an isomorphism on Hochschild homology) [47, 4.4].

The notion of Hochschild homology of a differential graded (dg) algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, which was introduced by Hess, Parent, and Scott in [23], generalizing the non-differential notion of [13]. They showed in particular that the coHochschild homology of the chain coalgebra on a simply connected space X is isomorphic to the homology of the free loop space on X and that the coHochschild homology of a connected dg coalgebra C is isomorphic to the Hochschild homology of ΩC, the cobar construction on C.

In this article we establish further properties of coHochschild homology, analogous to the invariance properties of Hochschild homology recalled above. We first prove a sort of categorification of the relation between coHochschild homology of a connected dg coalgebra C and the Hochschild homology of ΩC, showing that there is a dg Quillen equivalence between the categories of C-comodules and of ΩC-modules (Proposition 2.8). We can then establish an “agreement”-type result, stating that the coHochschild homology of a dg coalgebra C is isomorphic to the Hochschild homology of the dg category spanned by certain compact C-comodules (Proposition 2.12). Thanks to this agreement result, we can show as well that coHochschild homology is a Morita invariant (Proposition 2.23), using the notion of Morita equivalence of dg coalgebras formulated in [4], which extends that of Takeuchi [46] and which we recall here. Proving these results required us to provide criteria under which a dg Quillen equivalence of dg model categories induces a quasi-equivalence of dg subcategories (Lemma 2.13); this technical result, which we were unable to find in the literature, may also be useful in other contexts.

The natural analogue of Hochschild homology for spectra, called topological Hochschild homology (THH), has proven to be an important and useful invariant of ring spectra, particularly because of its connection to K-theory via the Dennis trace. Blumberg and Mandell proved moreover that THH satisfies both “agreement,” in the sense that THH of a ring spectrum is equivalent to THH of the spectral category of appropriately compact R-modules, and Morita invariance [5]

We define here an analogue of coHochschild homology for spectra, which we call topological coHochschild homology (coTHH). We show that coTHH is homotopy invariant, as well as independent of the particular model category of spectra in which one works. We prove moreover that coTHH of the suspension spectrum ${\mathrm{\Sigma }}_{+}^{\infty }X$ of a connected Kan complex X is equivalent to ${\mathrm{\Sigma }}_{+}^{\infty }\mathcal{L}X$, the suspension spectrum of the free loop space on X, whenever X is EMSS-good, i.e., whenever ${\pi }_{1}X$ acts nilpotently on the integral homology of the based loop space on X (Theorem 3.7).

This equivalence was already known for simply connected spaces X, by work of Kuhn [31] and Malkiewicz [34], though they did not use the term coTHH. The extension of the equivalence to EMSS-good spaces is based on new results concerning total complexes of cosimplicial suspension spectra, such as the fact that $\overline{\mathrm{Tot}}({\mathrm{\Sigma }}^{\infty }{Y}^{•})\simeq {\mathrm{\Sigma }}^{\infty }\overline{\mathrm{Tot}}{Y}^{•}$ whenever the homology spectral sequence for a cosimplicial space $Y^{•}$ with coefficients in $\mathbb{Z}$ strongly converges (Corollary A.3). We also show that if X is an EMSS-good space, then the Anderson spectral sequence for homology with coefficients in $\mathbb{Z}$ for the cosimplicial space $\mathrm{Map}(S_{•}^1,X)$ strongly converges to $H_{⁎}(\mathcal{L}X;\mathbb{Z})$ (Proposition A.4).

In [7], Bökstedt and Waldhausen proved that $\mathrm{THH}({\mathrm{\Sigma }}_{+}^{\infty }\mathrm{\Omega }X)\simeq {\mathrm{\Sigma }}_{+}^{\infty }\mathcal{L}X$ for simply connected X. It follows thus from Theorem 3.7 that if X is simply connected, then $\mathrm{THH}({\mathrm{\Sigma }}_{+}^{\infty }\mathrm{\Omega }X)\simeq \mathrm{coTHH}({\mathrm{\Sigma }}_{+}^{\infty }X)$, analogous to the result for dg coalgebras established in [23]. Combining this result with the spectral Quillen equivalence between categories of ${\mathrm{\Sigma }}_{+}^{\infty }\mathrm{\Omega }X$-modules and of ${\mathrm{\Sigma }}_{+}^{\infty }X$-comodules established in [24] and with THH-agreement [5], we obtain coTHH-agreement for simply connected Kan complexes X: $\mathrm{coTHH}({\mathrm{\Sigma }}_{+}^{\infty }X)$ is equivalent to THH of the spectral category of appropriately compact ${\mathrm{\Sigma }}_{+}^{\infty }\mathrm{\Omega }X$-modules (Corollary 3.11).

We do not consider Morita invariance for coalgebra spectra in this article, as the duality requirement of the framework in [4] is too strict to allow for interesting spectral examples. We expect that a meaningful formulation should be possible in the ∞-category context.

In parallel with writing this article, the second author collaborated with Bohmann, Gerhardt, Høgenhaven, and Ziegenhagen on developing computational tools for coHochschild homology, in particular an analogue of the Bökstedt spectral sequence for topological Hochschild homology constructed by Angeltveit and Rognes [2]. For C a coalgebra spectrum, the $E_2$-page of this spectral sequence is the associated graded of the classical coHochschild homology of the homology of C with coefficients in a field $\mathbb{k}$, and the spectral sequence abuts to the $\mathbb{k}$-homology of $\mathrm{coTHH}(C)$. If C is connected and cocommutative, then this is a spectral sequence of coalgebras. In [6] the authors also proved a Hochschild-Kostant-Rosenberg-style theorem for coHochschild homology of cofree cocommutative differential graded coalgebras.

In future work we will construct and study an analogue of the Dennis trace map, with source the K-theory of a dg or spectral coalgebra C and with target its (topological) coHochschild homology.

Acknowledgments

Work on this article began while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during Spring 2014, partially supported by the National Science Foundation under Grant No. 0932078000. The second author was also supported by NSF grants DMS-1104396, DMS-1406468, and DMS-1811278 during this work. We would also like to thank the University of Illinois at Chicago, the EPFL, and the University of Chicago for their hospitality during research visits enabling us to complete the research presented in this article. The authors would also like to thank The Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the Fall 2018 program “Homotopy harnessing higher structures” where this paper was finished. This work was supported by EPSRC grant No. EP/K032208/1.