I want to know if the notion of completed tensor product in [Stacks Project Tag 0AMU](https://stacks.math.columbia.edu/tag/0AMU) is the one that yields
$$k[[x]] \widehat{\otimes} k[[y]]≅k[[x,y]].$$
Here I should be considering the inverse limit topology in the power series rings, and $R=k$ a field (with the trivial topology?). If it is not, then what is the right notion of $\widehat{\otimes}$ ?


I believe this is broadly used when talking of formal groups laws (eg defining its contravariant bialgebra).



Completed tensor product and power series rings
$k[[x]]$ and $k[[y]]$ are topological $k$-algebras, and their completed tensor product is indeed isomorphic to $k[[x,y]]$ (with the topology defined by the maximal ideal $(x,y)$). This is because the completed tensor product is the projective limit of the algebras $k[x]/(x^m) \otimes k[y]/(y^n) = k[x,y]/(x^m,y^n)$ over $m,n \geq 1$.


