# Difference between revisions of "Projective scheme"

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Every projective scheme is complete (compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an [[Ample sheaf|ample]], [[Invertible sheaf|invertible]] sheaf on it. There are also other criteria of projectivity. | Every projective scheme is complete (compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an [[Ample sheaf|ample]], [[Invertible sheaf|invertible]] sheaf on it. There are also other criteria of projectivity. | ||

− | A generalization of the concept of a projective scheme is a ''projective morphism''. A morphism $ f: X \to Y $ of schemes is called ''projective'' (and $ X $ is called a scheme | + | A generalization of the concept of a projective scheme is a '''projective morphism'''. A morphism $ f: X \to Y $ of schemes is called '''projective''' (and $ X $ is called a '''projective scheme''' over $ Y $) if and only if $ X $ is a closed sub-scheme of the projective fiber bundle $ {\mathbf{P}_{Y}}(\mathcal{E}) $, where $ \mathcal{E} $ is a locally free $ \mathcal{O}_{Y} $-module. A composition of projective morphisms is projective. The projectivity of a morphism is also preserved under a [[Base change|base change]]; in particular, the fibers of a projective morphism are projective schemes (but not conversely). If a scheme $ X $ is projective and $ X \to Z $ is a finite surjective morphism, then $ Z $ is also projective. |

Any projective scheme (over $ Y $) can be obtained using the construction of the [[Projective spectrum of a ring|projective spectrum]]. Restricting to the case of an affine base, $ Y = \operatorname{Spec}(R) $, suppose that $ \displaystyle A = \bigoplus_{i \geq 0} A_{i} $ is a graded $ R $-algebra with the $ R $-module $ A_{1} $ being of finite type and generating the algebra $ A $, and suppose that $ \operatorname{Proj}(A) $ is the set of homogeneous prime ideals $ \mathfrak{p} \subseteq A $ not containing $ A_{1} $. Equipped with the natural topology and a structure sheaf, the set $ \operatorname{Proj}(A) $ is a projective $ Y $-scheme; moreover, any projective $ Y $-scheme has such a form. | Any projective scheme (over $ Y $) can be obtained using the construction of the [[Projective spectrum of a ring|projective spectrum]]. Restricting to the case of an affine base, $ Y = \operatorname{Spec}(R) $, suppose that $ \displaystyle A = \bigoplus_{i \geq 0} A_{i} $ is a graded $ R $-algebra with the $ R $-module $ A_{1} $ being of finite type and generating the algebra $ A $, and suppose that $ \operatorname{Proj}(A) $ is the set of homogeneous prime ideals $ \mathfrak{p} \subseteq A $ not containing $ A_{1} $. Equipped with the natural topology and a structure sheaf, the set $ \operatorname{Proj}(A) $ is a projective $ Y $-scheme; moreover, any projective $ Y $-scheme has such a form. | ||

====References==== | ====References==== | ||

+ | |||

<table> | <table> | ||

− | <TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, “Algebraic geometry”, '''1. Complex projective varieties''', Springer (1976). {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR> | + | <TR><TD valign="top">[1]</TD><TD valign="top"> |

+ | D. Mumford, “Algebraic geometry”, '''1. Complex projective varieties''', Springer (1976). {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR> | ||

</table> | </table> | ||

====Comments==== | ====Comments==== | ||

− | Given a vector bundle $ E $ over $ Y $ (or, equivalently, a locally free $ \mathcal{O}_{Y} $-module $ \mathcal{E} $), the associated projective bundle, or projective fiber bundle, has as fiber over $ y \in Y $ the projective space $ \mathbf{P}(E_{Y}) $ of all | + | |

+ | Given a vector bundle $ E $ over $ Y $ (or, equivalently, a locally free $ \mathcal{O}_{Y} $-module $ \mathcal{E} $), the associated projective bundle, or projective fiber bundle, has as fiber over $ y \in Y $ the projective space $ \mathbf{P}(E_{Y}) $ of all $ 1 $-dimensional sub-spaces of the vector space $ E_{Y} $. | ||

====References==== | ====References==== | ||

+ | |||

<table> | <table> | ||

− | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR> | + | <TR><TD valign="top">[a1]</TD><TD valign="top"> |

+ | R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR> | ||

</table> | </table> |

## Latest revision as of 08:05, 14 December 2016

A closed sub-scheme of a projective space $ \mathbf{P}_{\mathbb{k}}^{n} $. In homogeneous coordinates $ x_{0},\ldots,x_{n} $ on $ \mathbf{P}_{\mathbb{k}}^{n} $, a projective scheme is given by a system of homogeneous algebraic equations: $$ {f_{1}}(x_{0},\ldots,x_{n}) = 0, \quad \ldots \quad {f_{r}}(x_{0},\ldots,x_{n}) = 0. $$

Every projective scheme is complete (compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an ample, invertible sheaf on it. There are also other criteria of projectivity.

A generalization of the concept of a projective scheme is a **projective morphism**. A morphism $ f: X \to Y $ of schemes is called **projective** (and $ X $ is called a **projective scheme** over $ Y $) if and only if $ X $ is a closed sub-scheme of the projective fiber bundle $ {\mathbf{P}_{Y}}(\mathcal{E}) $, where $ \mathcal{E} $ is a locally free $ \mathcal{O}_{Y} $-module. A composition of projective morphisms is projective. The projectivity of a morphism is also preserved under a base change; in particular, the fibers of a projective morphism are projective schemes (but not conversely). If a scheme $ X $ is projective and $ X \to Z $ is a finite surjective morphism, then $ Z $ is also projective.

Any projective scheme (over $ Y $) can be obtained using the construction of the projective spectrum. Restricting to the case of an affine base, $ Y = \operatorname{Spec}(R) $, suppose that $ \displaystyle A = \bigoplus_{i \geq 0} A_{i} $ is a graded $ R $-algebra with the $ R $-module $ A_{1} $ being of finite type and generating the algebra $ A $, and suppose that $ \operatorname{Proj}(A) $ is the set of homogeneous prime ideals $ \mathfrak{p} \subseteq A $ not containing $ A_{1} $. Equipped with the natural topology and a structure sheaf, the set $ \operatorname{Proj}(A) $ is a projective $ Y $-scheme; moreover, any projective $ Y $-scheme has such a form.

#### References

[1] |
D. Mumford, “Algebraic geometry”, 1. Complex projective varieties, Springer (1976). MR0453732 Zbl 0356.14002 |

#### Comments

Given a vector bundle $ E $ over $ Y $ (or, equivalently, a locally free $ \mathcal{O}_{Y} $-module $ \mathcal{E} $), the associated projective bundle, or projective fiber bundle, has as fiber over $ y \in Y $ the projective space $ \mathbf{P}(E_{Y}) $ of all $ 1 $-dimensional sub-spaces of the vector space $ E_{Y} $.

#### References

[a1] | R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. MR0463157 Zbl 0367.14001 |

**How to Cite This Entry:**

Projective scheme.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Projective_scheme&oldid=38803