Topological Hochschild homology is a generalization of Hochschild homology from rings to $E\_\infty$-ring spectra. On the other hand, there is a natural way to extend the notion of Hochschild homology to dg rings (either by explicitly writing down the bar complex, which is now a double complex, or by defining it as a derived tensor product and using semi-free resolutions).


On the other hand, if $k$ is a $\mathbb{Q}$-algebra, then $E\_\infty$-ring spectra over $Hk$ and dg $k$-algebras are equivalent. Under this equivalence, does topological Hochschild homology coincide with dg-Hochschild homology? Is this written down anywhere?



Topological Hochschild homology and Hochschild homology of dg algebras
The notion of Hochschild homology can be defined abstractly in any suitable homotopical context in which a tensor product exist (say, in any presentably symmetric monoidal $\infty$-category). Given an associative algebra object $A$ in such a context, its Hochschild homology is the (suitably defined) tensor product of $A$ with itself over $A^{\rm op} \otimes A$. For example, dg-Hochschild homology is the one obtained by working in the $\infty$-category of chain-complexes (and their tensor product), while topological Hochschild homology is the one associated to spectra (and their smash product). Sometimes one comes across objects which might a-priori be interpreted as belonging to two different $\infty$-categories. For example, if we have an ordinary ring, then we may think of it as an object in either chain-complexes or spectra (via the associated Eilenberg-MacLane spectrum), and consequently define both its Hochschild homology and its topological Hochschild homology, which may be different. 


When $A$ is a $\mathbb{Q}$-algebra its associated Eilenberg-MacLane spectrum is rational (i.e., the map $HA \to H\mathbb{Q} \wedge HA$ is an equivalence). The $\infty$-category of rational spectra is equivalent to the $\infty$-category of chain-complexes over $\mathbb{Q}$. Moreover, this equivalence is symmetric monoidal and identifies the smash product on the spectra side with the tensor product on the chain-complex side. As a result, the topological Hochschild homology of such an Eilenberg-MacLane spectrum (defined using smash product of spectra) will coincide with its Hochschild homology when considered as a dg-algebra over $\mathbb{Q}$. You can also decide that you add an action of some $\mathbb{Q}$-algebra $k$ on both sides, but this will not matter much: as soon as the two interpretations yield $\infty$-categories which are symmetrically monoidal equivalent, they will produce the same Hochschild homology, essentially by definition.


