---- > [!definition] Definition. ([[projective space]]) > Let $A$ be a [[ring]]. We define **projective $n$-space over $A$** to be the [[scheme]] $\mathbb{P}^{n}_{A}:=\text{Proj }A[X_{0},\dots X_{n}],$ > where $\text{Proj }$denotes the [[proj construction]] and the [[polynomial 4|polynomial ring]] $A[X_{0},\dots,X_{n}]$ is taken with its usual [[graded ring|grading]] (elements of $A$ have degree zero). > > [!specialization] > Let $k$ be a field. The **$k$-projective $n$-space of dimension** is $k\mathbb{P}^{n}:= (k^{n+1}- \{ 0 \}) / k^{*},$ where $k^{*}$ denotes the [[unit|group of units]] of $k$, whose [[group action|action]] on the set $k^{n+1} - \{ 0 \}$ we are [[quotient set|quotienting by]]. ^specialization - [ ] except i am uneasy with how these two concepts align, especially with regard to [[topological space|topology]] > [!basicproperties] > > - (*as a smooth manifold*) [[projective space as a smooth manifold]] > - (*topological properties*) [[projective plane|Fundamental group of projective plane]] - [ ] link to other relevant notes ---- #### ---- #### 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 > ```