---
date: 2022-04-18 12:19
modification date: Monday 18th April 2022 12:19:19
title: "projective"
aliases: [quasiprojective scheme, projective scheme, quasiprojective, projective variety]
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Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
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# projective scheme

- A morphism $f$ is **projective** if it factors as a closed immersion into $\mathbb{P}_{Y}^{n}$, followed by the projection map $\mathbb{P}_{Y}^{n} \rightarrow Y$ for some $n$. 

- $f$ is **quasiprojective** if it factors as an [open immersion](open%20immersion) followed by a projective morphism.

# Results

- Every projective variety is [complete](Unsorted/complete.md).
- For a [smooth](Unsorted/smooth%20scheme.md) projective variety, the [canonical sheaf](Unsorted/canonical%20bundle.md) can be identified with the [dualizing sheaf](Unsorted/dualizing%20sheaf.md).