----- > [!proposition] Proposition. ([[universal line bundle on projective space]]) > Let $k$ be a [[field]] and $\mathbb{P}^{n}$ denote [[projective space|projective]] $n$-[[projective space|space]] over $k$, that is, $\mathbb{P}^{n}$ denotes the [[scheme]] $\mathbb{P}^{n}:=\text{Proj }\overbrace{ k[X_{0},\dots X_{n}] }^{ =: S },$ > where $\text{Proj }$denotes the [[proj construction]] and the [[polynomial 4|polynomial ring]] $k[X_{0},\dots,X_{n}]$ is taken with its usual [[graded ring|grading]] (elements of $A$ have degree zero). > > There is a 'universal [[locally free sheaf|line bundle]]' on $\mathbb{P}^{n}$, denoted $\mathcal{O}_{\mathbb{P}^{n}}(1)$, [[lines bundles and transition functions|constructed by]]: > - Taking as trivializing [[cover]] the set $\mathscr{U}:=\{ D_{+} (X_{i})\}_{i=0}^{n}$.[^1] Recall that $D_{+}(X_{i}) \cong \text{Spec }S_{(X_{i})}$. > - Specifying [[lines bundles and transition functions|transition functions]] $g_{ij} \in \mathcal{O}_{\mathbb{P}^{n}}^{*}\big(\underbrace{ D_{+}(X_{i} ) \cap D_{+}(X_{j}) }_{ D_{+}(X_{i}X_{j}) } \big) \cong S_{(X_{i}X_{j})}$ as $g_{ij}:=\frac{X_{i}}{X_{j}}$.[^2] > > > Called 'universal' because of role as a universal object in [[contravariant functor represented by projective space]] and, in turn, its status as generator of the [[Picard group]] $\text{Pic }\mathbb{P}^{n}$. ----- #### [^1]: Recall that $D_{+}(X_{i}):= \{ \mathfrak{p} \in \text{Proj }S: \mathfrak{p} \not \ni X_{i} \}$, $i=0,\dots,n$. This collection of $n+1$ sets covers $\mathbb{P}^{n}$ because if we had a [[homogeneous ideal|homogeneous]] [[prime ideal]] $\mathfrak{p} \in \text{Spec }A$ such that $\mathfrak{p} \ni X_{i}$ for all $i=0,\dots,n$, then $\mathfrak{p}$ contains the [[irrelevant ideal]] which is outlawed in the [[proj construction|proj]] definition. Intuitively, should be reminded of the standard [[real projective space|charting of]] $\mathbb{R}P^{n}$. [^2]: I think specifically we do $\frac{X_{i}}{X_{j}}=\frac{X_{i}^{2}}{X_{i}X_{j}} \in S_{(X_{i}X_{j})}$ ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```