Hartshorne's Algebraic Geometry uses the following facts on page 35 without proof:
The completion of $(k[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is $k[[x,y]]/(y^2-x^2-x^3)$ and that of $(k[x,y]/(xy))_{(x,y)}$ is $k[[x,y]]/(xy)$.
I studied Atiyah-Macdonald's chapter on completions, but I don't know why the above is true. It might be related to the isomorphism $\hat{M}\cong M\otimes_A \hat{A}$ where $A$ is Noetherian and $M$ is a finitely generated $A$-module, but I'm not sure. I don't have much experience with power series.
Would you please tell me why this is true? Thank you.
If $(R,m)$ is Noetherian local, then $m$-adic completion is exact. Moreover $(f)\widehat{R} \cong \widehat{(f)}$ for any $f \in m$. Thus the exact sequence
$$0 \to (f) \to R \to R/(f) \to 0$$ induces an exact sequence
$$0 \to f\widehat{R} \to \widehat{R} \to \widehat{R/(f)} \to 0$$