0
$\begingroup$

Hey folks,

I found some interesting piece of notation in Mumfords GIT and I'm not sure, what it means:

For example on page 104 you can find $S \times \mathbb{P}_n \cong \mathbb{P} [H^{0}(\mathcal{O} (1)) \otimes \mathcal{O}_S]$ or $\mathbb{P} (\varepsilon \vert U_i) \cong U_i \times \mathbb{P}_n$. What does $\mathbb{P} ( ??? )$ stand for? I searched through his book but can't find any explanation.

Greetings from Austria and thanks in advance

Xaver

$\endgroup$
2
  • $\begingroup$ That notation means the associated projective bundle of a locally free coherent sheaf. $\endgroup$ Apr 27, 2012 at 11:27
  • $\begingroup$ Thanks Jason, that's what I thought, but I wasn't sure :-) $\endgroup$ Apr 27, 2012 at 12:24

1 Answer 1

2
$\begingroup$

If $\mathcal{E}$ is some quasi-coherent $\mathcal{O}_S$-module, the functor $(\mathrm{Sch}/S)^{\text{op}} \to \mathrm{Set}$ which sends $f : T \to S$ to the set of invertible quotients of $f^* \mathcal{E}$, is representable by a (separated) scheme $\mathbb{P}(\mathcal{E})$ over $S$. If $\mathcal{E}$ is locally free, this is called the projective space bundle associated to $\mathcal{E}$. The intuition is: Replace fiberwise $\mathbb{A}^n$ by $\mathbb{P}^n$. You can find this functorial approach, more generally for Grassmannians, in EGA I (1970), §9.

As for the notation $H^0(\mathcal{F}) \otimes \mathcal{O}_S$: This is an external tensor product. If $M$ is some $R$-module and $\mathcal{E}$ is some quasi-coherent $\mathcal{O}_S$-module, where $S$ is an $R$-scheme, then $M \otimes_R \mathcal{E}$ is defined either locally in the obvious way, or by the obvious universal property, or by the rule $R^n \otimes_R \mathcal{E} = \mathcal{E}^n$ and cocontinuous extension.

$\endgroup$

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.