image in formal completion ring

Question

Let $X$ be a scheme and $C\subset X$ be an integral subscheme and $I$ be a sheaf of ideals over $X$.

Denote $mult_C I:=Sup\{n\in \Bbb N: I\mathcal O_{C,\xi_C} \subset \mathfrak m_{\xi_C}^n\}$ the multiplicity of I where $\xi_C$ is the generic point of $C$ and $\mathfrak m_{\xi_C}$ the maximal ideal of $\mathcal O_{C,\xi_C}$ and $\kappa$ the corresponding residue field.

Suppose that $C$ is defined by $x_1=x_2=...=x_d=0$ where $(x_1,x_2,...,x_n)$ is a system of coordinates at a point $P\in X$.

Prove that $mult_C I \ge r \Leftrightarrow \forall f\in I,$ the image of $f$ in $\kappa[[x_1,x_2,...,x_n]]$ have all its terms of degree $\ge r$ with $x_1,x_2,...,x_d$ coordinates.

It seems obvious exercise but I don't understand:

1. What is the image of $f$ in $\kappa[[x_1,x_2,...,x_n]]$? Is it the completion map?
2. Does $f\in I$ mean that f can be written as the sum of products of $x_i, i>d$?
3. Is $\mathcal O_{C,\xi_C} \cong \kappa[x_1,x_2,...,x_n]/I$?
4. Is $I\mathcal O_{C,\xi_C}=(x_1,x_2,...,x_d)$, i.e. the ideal generated by $x_1,x_2,...,x_d$?
5. Does this exercise make sense?

Thank you for your help.

Edit

What I tried: Take $X=\operatorname{Spec} k[X,Y]/(Y-X^2) \cong \operatorname{Spec} k[X]$ The curve C is defined by the ideal $(Y-X^2)$, but what is the corresponding equation in terms of local coordinate system? I guess it is easier to take the example of the x-axis in $\Bbb A^2$?

I tried to compute $\xi_C$ and $\mathcal O_{C,\xi_C}$. I know that generic point of $k[X]$ is $(X)$ but I don't see why?

if $\xi_C = (X)$ then $\mathcal O_{C,\xi_C} = k(X)$ so $\mathfrak m_{\xi_C}=(0)$.

If $mult_C I \ge r$ then we always have $I=0$ which doesn't make sense. I guess this contradicts 2).

If I step back and take $C$ as the x-axis, I end up with same computation and defining equation $X=0$ for $C$. Hence $d=1$ in this case which contradicts 4) as $I=0$.

I guess then that 2) 4) are false statements. 3) also maybe false unless I is a maximal ideal of $k[X,Y]$, but even in this case I am not sure.

Edit Main question

Now I try to write down things for the main question:Prove that $mult_C I \ge r \Leftrightarrow \forall f\in I,$ the image of $f$ in $\kappa[[x_1,x_2,...,x_n]]$ have all its terms of degree $\ge r$ with $x_1,x_2,...,x_d$ coordinates.

I assume that $X$ is of finite type over a field $k$, so that $\mathcal O_{X,\xi_C}$ is artinian as it is of finite type over $k$, which means that powers of its maximal ideal (for simplicity noted $\mathfrak m$ here) stagnate, let's say at the power $s$, hence $ \widehat{\mathcal O_{X,\xi_C}}= \prod_{i=1...s} \mathcal O_{X,\xi_C}/ \mathfrak m^i$. I don't know if making this assumption is necessary?

Let $f=\sum_{j=0}^\infty b_i$ with $b_i\in I$ a monomial of degree $i$ which is a product of $i$ elements of the coordinate system.

Suppose $mult_C I \ge r$. For $i > r$, we have $I \not \subset \mathfrak m^i$, can we deduce from this that $b_i \not \in \mathfrak m^i$? But in that case $b_i$ doesn't vanish on $C$ and it contradicts the fact that $b_i$ should be product of some $x_j$ with $1\le j \le d$.

Thanks for you help.

Answer 1

First, let's get question 1 out of the way: yes, your interpretation is correct. They mean the image of $f$ in $k[[x_1,\cdots,x_n]]$ under the completion map.

Now let's look at your example. Fix $P$ to be the parabola cut out in $\Bbb A^2_k$ by $y-x^2$. A local coordinate system of a variety $X$ at a point $x$ is a choice for some elements $f_1,\cdots,f_n\in \mathfrak{m}_x\subset \mathcal{O}_{X,x}$ so that $f_1,\cdots,f_n$ generate $\mathfrak{m}_x$. For us, working with the parabola, we should pick the coordinate system $\{y-x^2,x-a\}$ at a closed point $(a,a^2)$.

The generic point $\xi_P \in \Bbb A^2$ is the prime ideal $(y-x^2)$. This has local ring $\mathcal{O}_{\Bbb A^2,\xi_P}= k[x,y]_{(y-x^2)}$, the localization. The local ring of $\xi_P$ inside $P$ is $\mathcal{O}_{P,\xi_P}$ is $\operatorname{Frac}(k[x,y]/(y-x^2))$. In order for the multiplicity invariant you define to be useful, we should be considering $\mathcal{O}_{X,\xi_C}$ instead of what you have written. In our case, $\mathcal{O}_{X,\xi_C}$ is $k[x,y]_{(y-x^2)}$.

(Your calculation of $(X)$ as the generic point of $k[X]$ is not correct - $(X)$ is maximal as $k[X]/(X)=k$ and thus a closed point, so it is not the unique point who's closure is $\operatorname{Spec} k[X]$. You want this to be $(0)$ - in general, for an irreducible affine scheme $\operatorname{Spec} A$, the generic point is the minimal prime of $A$. For non-irreducible affine schemes, minimal primes are in bijection with the irreducible components.)

Once we make the fix of considering $\mathcal{O}_{X,\xi_C}$ in the definition of multiplicity, things will get better, but 2,3, and 4 are all still wrong. Let's explain:

For question 2, recall that the problem assumes $C$ is given by $x_1=\cdots=x_d=0$. But if $f\in I$ is to have positive multiplicity, then we should have that it's a sum of monomials which are all divisible by some $x_i$ for $1\leq i\leq d$, so that $f$ vanishes along $C$. On the other hand, this doesn't rule out being divisible by any specific $x_i$: consider the product $x_1x_2\cdots x_n$. For instance, in our example, at the origin, the function $xy-x^3$ vanishes to order 1 along the parabola but is also divisible by $x$ which doesn't vanish to any order along the parabola.

For question 3, we've already discussed this a little, but one reason this can't be right in general is that it's not generally a local ring. For a point $x\in X$, we always have that $\mathcal{O}_{X,x}$ is a local ring. We've already discussed what this should be in our example further up in the post.

For question 4, this is also not correct - we could always choose $I=(x_1^k,x_2^k,\cdots,x_d^k)$ for any integer $k\geq 0$ for instance.

Lastly, for question 5, I think this question does make sense, except that we need to talk about $\mathcal{O}_{X,\xi_C}$ to get something meaningful out of the multiplicity definition.

---

Now for your attempt on the question:

You have major problems. Finite type rings are not necessarily artinian - $k[x]$ is finite type over $k$ but not artinian, for instance, so your conclusion about $\mathcal{O}_{X,\xi_C}$ is incorrect.

Instead, what you should be doing here is looking at what happens to the ideal $I\mathcal{O}_{X,\xi_C}$ in the natural map to the completion. The ideal $\mathfrak{m}_{X,\xi_C}^p$ is generated by monomials of degree $p$ in $x_1,\cdots,x_d$ and it's image in $k[[x_1,\cdots,x_n]]$ is again the submodule generated by monomials of degree $p$ in $x_1,\cdots,x_d$. This should be enough to finish the problem. (If you have further difficulties, I might suggest asking a new question - you've already accepted an answer here, which will mean you'll get fewer people looking at this to help you with your issues when you edit this versus asking a new question.)