Let $A$ be a $\mathbb{C}[[h]]$ algebra (not necessarily commutative). The Hochschild homology is then defined via a bar construction and that $HH\_0(A)=A/[A,A]$. Note that each $HH\_i(A)$ is a $\mathbb{C}[[h]]$-module. We can define $\overline{HH\_0}(A):=A/\overline{[A,A]}$, where the overline means taking the $h$-adic completion. I want to ask if there is a version of the bar resolution that produces the 'completed' version of $HH$ and how $\overline{HH\_i}$ and $HH\_i$ are related and when they are equal.


Edit: More assumptions on $A$ might be needed to make the question 'nice'.



Completed Hochschild (co)homology
The book:


Hübl, Reinhold:
*Traces of differential forms and Hochschild homology*.
Lecture Notes in Mathematics, **1368**. Springer-Verlag, Berlin, 1989.


treats the case of Hochschild co/homology for topological algebras. In the first chapter it gives an introduction to the basic constructions in the topological case.


