0
$\begingroup$

Let $F_0,\ldots,F_m\in\mathbb{Z}[x_0,\ldots,x_n]$ be polynomials with integer coefficients and let $p$ be a prime integer. Consider the two ideals: $$I_0:=(F_0,\ldots,F_m)\subset \mathbb{Q}[x_0,\ldots,x_n]$$ and $$I_p:=(F_0\,\mathrm{mod}\,p,\ldots,F_m\,\mathrm{mod}\,p)\subset \mathbb{Z}/(p)[x_0,\ldots,x_n].$$

Is it true that $\mathrm{codim}(I_p) \leq \mathrm{codim}(I_0) $?

Thanks in advance.

$\endgroup$
4
$\begingroup$

That depends on what you mean by "codimension", but I think most interpretations give an answer of "no". For instance, let $m=n$ be $2$, let $F_0$ be $p(px_0-1)$, let $F_1$ be $(px_0 - 1)x_1$, and let $F_2$ be $(px_0-1)x_2$. Then the codimension of $\langle F_0,F_1,F_2 \rangle$ in $\mathbb{Q}[x_0,x_1,x_2]$ equals $1$; indeed, the ideal is just the principal prime ideal $\langle px_0 - 1 \rangle$. However, the codimension of $\langle F_0,F_1,F_2 \rangle$ in $(\mathbb{Z}/p\mathbb{Z})[x_0,x_1,x_2]$ equals $2$; indeed, the ideal is just the prime ideal $\langle x_1,x_2 \rangle$.

Edit. I should make clear that, in the above, by "codimension" of an ideal, I mean what is usually called "height", i.e., the minimal height of a prime ideal that contains the ideal (assuming the ideal is not the entire ring).

$\endgroup$
  • $\begingroup$ Is this a properness over $Spec\ \mathbb Z$ issue? $\endgroup$ – Allen Knutson Dec 13 '14 at 14:04
  • $\begingroup$ @AllenKnutson: "Is this a properness over $\text{Spec} \mathbb{Z}$ issue?" Yes, precisely. The example above only works because the induced morphism from the zero scheme of $F_0$, $F_1$, $F_2$ to $\text{Spec} \mathbb{Z}$ is not proper. $\endgroup$ – Jason Starr Dec 13 '14 at 14:08
  • $\begingroup$ With "codimension" I just mean "height". Thank you. Is it possible to find a counterexample when the $F_i$'s are homogeneous polynomials? $\endgroup$ – gio Dec 13 '14 at 14:31
  • 1
    $\begingroup$ @gio: If the polynomials $F_i$ are homogeneous, then the semicontinuity theorem holds. This was also Allen Knutson's point. I suggest you look up in a textbook "semicontinuity" (for dimension of fibers) and "properness". Since semicontinuity is a result on the domain of the morphism, and you want a result on the target, you need to use properness. $\endgroup$ – Jason Starr Dec 14 '14 at 12:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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