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Invariance properties of coHochschild homology. (English) Zbl 1454.13018

For any commutative ring \(\Bbbk\), the classical definition of Hochschild homology of \(\Bbbk\)-algebras [J.-L. Loday, Cyclic homology. Berlin: Springer-Verlag (1992; Zbl 0780.18009)] admits a straightforward extension to differential graded \((dg)\) \(\Bbbk\)-algebras. In [J. Pure Appl. Algebra 93, No. 3, 251–296 (1994; Zbl 0807.19002)] R. McCarthy extended the definition of Hochschild homology in another direction, to \(\Bbbk\)-exact categories, seen as \(k\)-algebras with many objects. As B. Keller showed in [J. Pure Appl. Algebra 136, No. 1, 1–56 (1999; Zbl 0923.19004)], 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) and Morita invariance. The notion of Hochschild homology of a differential graded (\(dg\)) algebra admits a natural dualization, the coHochschild homologyof a \(dg\) coalgebra, which was introduced by K. Hess et al. [J. Pure Appl. Algebra 213, No. 4, 536–556 (2009; Zbl 1238.16005)] as a tool to study free loop spaces.
The authors 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.
Next, generalizing the \(dg\) case, the authors 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 by the authors [Adv. Math. 290, 1079–1137 (2016; Zbl 1331.19003)], it is proved that “agreement” holds for coTHH as well.

MSC:

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16T15 Coalgebras and comodules; corings
18G80 Derived categories, triangulated categories
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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