Completed stalk of intersection

Question

Let $Y_{i}$ be infinitely many reduced closed subschemes of a smooth scheme $X$ over an algebraically closed field. Suppose that they have a point $y$ in common and $y$ is closed in $X$. Let $Z$ be the intersection of $Y_{i}$'s in $X$. What is the relation between $\operatorname{Spf}(\hat{Z}_{y})$ and $\operatorname{Spf}(\hat{\left(Y_{i}\right)}_{y})$? Here, $\hat{A}_{a}$ refers to the completed stalk and $\operatorname{Spf}$ denotes the formal spectrum.

Answer 1

Topologically, the formal spectrum of a completed local ring is just a point, since an open prime ideal of $\hat{R}$ is a prime ideal of $\hat{R}/\mathbb m$, which is a field, so it has just one prime ideal.

Thus, the only question is the relationship between the rings $\hat{Z}_y$ and $\left(\hat Y_i\right)_y$. They are quotients of $\hat{X}_y$ by the images of the defining ideals of $Z$ and $Y_i$ respectively. Since the defining ideal of $Z$ is the sum of the defining ideals of $Y_i$, $\hat{Z}_y$ is the biggest quotient of $\hat{X}_y$ that factors through the $\left(\hat Y_i\right)_y$.

Proof: Since everything is local we can take $X$ to be affine. Then the $Y_i$ correspond to ideals in the coordinate ring of $X$ and $Z$ is their sum. If $m$ is the ideal of $y$, then $I(Y_i) \subset I(Z) \subset m$ and $\hat{Z}_y= \lim_{n\to \infty} R/(I(Z),m^n)$ and we have similar descriptions for everything else that let us just check this explicitly.