Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|cite|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.