Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

hello everybody! Let $X$ a smooth variety and $I$ an ideal sheaf of $\mathcal{O}_X$ . I have read that if $dim(X)\geq 2$ an ideal cannot be written as product of prime ideals, but the codimension 1 primes can be separeted from the rest. That is, there is a unique largest effective divisor $Div(I)$ such that $I\subset\mathcal{O}_X(-Div(I))$. but i don't understand what are the codimensions primes. can someone could show me an example? thanks

share|improve this question

2 Answers 2

Codimension=height.

To find that divisor and the corresponding ideal look at the ideal of the codimension $1$ part of the subscheme $\big(\mathrm{supp}(\mathscr O_X/I), \mathscr O_X/I\big)$.

share|improve this answer
    
thanks! but what do you mean by the support of $\mathcal{O}_X /I$? –  oxydo Mar 29 '11 at 7:54

Suppose that $X$ is affine: $X=\mathrm{Spec}(A)$. Then $I$ is an ideal of the noetherian, integrally closed ring $A$ (this property in fact is all you need, smoothness is not required). The ring $A$ can be written as

$A=\bigcap\limits_{p\in\mathrm{Spec} (A):\mathrm{height}(p)=1}A_p$.

The divisorial hull of $I$ is defined as

$\widehat{I}=\bigcap\limits_{p\in\mathrm{Spec} (A):\mathrm{height}(p)=1}IA_p$.

The ideal $\widehat{I}$ then defines a Weil divisor $D$ on $X$ in the usual way: namely

$v_p(D):=v_p(f_p) $ for $p\in\mathrm{Spec} (A):\mathrm{height}(p)=1$,

where $v_p$ is the discrete valuation associated to the valuation ring $A_p$ and $\widehat{I}A_p=f_pA_p$. This divisor is the one you are searching for.

share|improve this answer

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.