Let $k$ be a field.
Consider the inverse limit
$\varprojlim k[x,y]/(y\cdot x^n)$.
I wonder if there is a nice description of this ring?
Geometrically, we look at the union of the line $y=0$ along with an infinitesimal neighborhood of the line $x=0$. But what happens in the limit? I think we get $k[y][[x]]$? What is the geometric interpretation of this?
Any help will be appreciated.