Completions can also be used to analyze the local structure of singularities of a scheme. For example, the affine schemes associated to ${\displaystyle \mathbb {C} [x,y]/(xy)}$ and the nodal cubic plane curve ${\displaystyle \mathbb {C} [x,y]/(y^{2}-x^{2}(1+x))}$ have similar looking singularities at the origin when viewing their graphs (both look like a plus sign). Notice that in the second case, any Zariski neighborhood of the origin is still an irreducible curve. If we use completions, then we are looking at a "small enough" neighborhood where the node has two components. Taking the localizations of these rings along the ideal ${\displaystyle (x,y)}$ and completing gives ${\displaystyle \mathbb {C} [[x,y]]/(xy)}$ and ${\displaystyle \mathbb {C} [[x,y]]/((y+u)(y-u))}$ respectively, where ${\displaystyle u}$ is the formal square root of ${\displaystyle x^{2}(1+x)}$ in ${\displaystyle \mathbb {C} [[x,y]].}$ More explicitly, the power series:


Since both rings are given by the intersection of two ideals generated by a homogeneous degree 1 polynomial, we can see algebraically that the singularities "look" the same. This is because such a scheme is the union of two non-equal linear subspaces of the affine plane.