# Question concearning notation in Mumford GIT

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

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

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.