Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have been trying to read Fulton's Intersection Theory, and the following puzzles me.

All schemes below are algebraic over some field $k$ in the sense that they come together with a morphism of finte type to $Spec k$.

Let $X$ be a variety (reduced irreducible scheme), and let $Y$ be a codimension $1$ subvariety, and let $A$ be its local ring (in particular a $1$-dimensional Noetherian local domain). Let $K(X)$ be the ring of rational functions on $X$ (the local ring at the generic point of X). Let $K(X)^*$ be the abelian group (under multiplication) of units of $K(X)$, and $A^*$ -- the group of units of $A$.

On the one hand, for any $r\in K(X)^*$ define the order of vanishing of $r$ at $Y$ to be $ord_Y(r)=l_A(A/(a))-l_A(A/(b))$ where $r=a/b$ for $a$ and $b$, and $l_A(M)$ is the length of an $A$-module $M$. On the other hand, it turns out that $Y$ is in the support of the principal Cartier divisor $div(r)$ if and only if $r\not\in A^*\subset K(X)^*$.

It is obvious that $Y$ not in the support of $div(r)$ implies that $ord_r(Y)=0$, since the former claims that $r\in A^*$ from which it follows that $ord_Y(b)=ord_Y(rb)=ord_Y(a)$ since obviously $ord_Y(r)=0$ for any unit. The contrapositive states that $ord_r(Y)\neq0$ implies $Y$ is in the support of $div(r)$. Because the latter can be shown to be a proper closed set and thus containing finitely many codim. $1$ subvarieties, which shows that the associated Weyl divisor $[D]=\sum_Y ord_Y(r)[Y]$ is well-defined.

What is not clear to me is whether or not the converse is true, i.e. whether $Y$ in the support of $div(r)$ implies that $ord_Y(r)\neq0$. I find myself doubting since if I am not mistaken, this is equivalent to the statement $l_A(A/(a))=l_A(A/(b))$ if and only if $(a)=(b)$, where $A$ is any $1$-dimensional local Noetherian domain (maybe even a $k$-algebra) which seems much too strong. Any (geometric) examples to give me an idea of what is going would be much appreciated.

share|improve this question
    
Are you also assuming X locally factorial? I only ask because you bring up Weil divisors. Second, I think there is a typo in the last paragraph, "Y in the support of $div(r)$ implies that $ord_Y(r)=0$" should probably be $ord_Y(r)\neq 0$? This should probably just follow from looking at generic points. –  Matt Mar 2 '12 at 5:42
    
@Matt, thanks for catching the typo. No, I am not assuming that $X$ is locally factorial. I think I am only assuming that $X$ is a Noetherian variety at this stage (the assumption that $X$ is a $k$-scheme seems to come into play only later to give the line bundle<->Cartier divisor relationship). –  Vladimir Sotirov Mar 3 '12 at 18:40
add comment

2 Answers 2

up vote 1 down vote accepted

Let's test your hypothesis with an explicit example. Since I bet everything works out nicely for regular schemes, let's take a simple singular one, with $k$ a field of characteristic not 2.

  • $X = \mathop{\text{Spec}} k[x,y] / (y^2 - x^3 - x^2)$
  • $Y = \mathop{\text{Spec}} k[x,y] / (x,y)$

$Y$ is the singular point of $X$. The functions defining the two tangent lines at $Y$ are likely to be the ones that give us problems. They are:

  • $a = x-y$
  • $b = x+y$

Then, we have

  • $A/(a) \cong k[x]/(x^3)$ has length 3.
  • $A/(b) \cong k[x]/(x^3)$ has length 3

But $(a) \neq (b)$, because $A/(a,b) \cong k$

share|improve this answer
add comment

The support of a Cartier divisor $D$ on $X$ is the union of all closed subvarieties $Z\subset X$ such that the local equation of $D$ at the generic point $z$ of $Z$ is not a unit of the local ring $O_{X,z}$. Note that $Z$ can be of codimension $>1$. However, let $f_Z$ be the local equation of $D$ and let $p\in\mathrm{Spec}(O_{X,z})$ such that $f_ZO_{X,z}\subseteq p$ and $p$ is minimal with that property. Then by the Principal Ideal Theorem $p$ is of height $1$ and $f_Z$ is not a unit in the localization $(O_{X,z})_p$. The latter is the local ring of the codimenions $1$ subvariety $Y$ having $p$ as its generic point. This shows that if $Z$ is in the support of $D$, then every codimension $1$ subvariety $Y\subset X$ such that $Z\subseteq Y$ is in the support of $D$. Does this solve your problem?

share|improve this answer
    
Sorry: I don't see how to translate the above into a statement about the order of $D$ at codim. $1$ subvarieties in the support of $D$. Could you say a little bit more? –  Vladimir Sotirov Mar 3 '12 at 18:53
add comment

Your Answer

 
discard

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.