scheme

scheme

1

Introduction

In order to extend

algebraic geometry

to deal with fields that are not

algebraically closed

, it is necessary to generalize the notions of

affine variety

and

projective variety

. The most suitable

generalization

seems to be the notion of a scheme. A scheme in some sense captures the

equations

defining an

algebraic

object, so that the points of that object can be examined over many different fields. In fact, the points of such an object take a secondary role: this is necessary because, for example, over a

finite field

most curves have no points at all until you pass to a suitable

field extension

. The underlying machinery allowing this is the tensor product of rings: given a

k

-

algebra

A

and a field extension

K

of

k

,

A

⊗

k

K

is the

K

-algebra that is, in some sense, defined by the same equations as

A

.

Along with this ability to deal with varying

base fields

, schemes offer a great number of other possibilities. In particular, the tools for dealing with

individual

schemes can usually also be adapted to deal with families of schemes. These families can even span different

characteristics

, which can be a powerful tool for applying the tools of characteristic zero geometry to problems in positive characteristic.

2

Definitions

An

affine scheme

is a

locally ringed space

(

X

,

𝒪

X

)

with the property that there exists a ring

R

(

commutative

, with

identity

) whose

prime spectrum

Spec

⁡

(

R

)

is

isomorphic

to

X

as a locally ringed space.

A

scheme

is a locally ringed space

(

X

,

𝒪

X

)

which has an

open cover

{

U

α

}

α

∈

I

with the property that each

open set

U

α

, together with its

restriction

sheaf

𝒪

X

|

U

α

, is an affine scheme.

We define a

morphism of schemes

between two schemes

(

X

,

𝒪

X

)

and

(

Y

,

𝒪

Y

)

to be a morphism of locally ringed spaces

f

:

(

X

,

𝒪

X

)

⟶

(

Y

,

𝒪

Y

)

.

With these definitions, schemes of course form a

category

. However, frequently one wishes to work in a slightly different category, such as the category of “

complex

schemes”, that is, schemes obtained from complex algebras. However, these schemes will have strange

automorphisms

obtained from automorphisms of the

complex numbers

. In order to prevent this problem, we often look at another category.

Fix a scheme

Y

. Then a

scheme over

Y

is defined to be a scheme

X

together with a morphism of schemes

X

⟶

Y

, called the structure morphism of

X

. A

morphism

X

→

X

′

of schemes over

Y

is a morphism

X

→

X

′

which makes the following

diagram

commute:

\xymatrix

⁢

X

⁢

\ar

⁢

[

r

⁢

r

]

⁢

\ar

⁢

[

d

⁢

r

]

⁢

&

⁢

&

⁢

X

′

⁢

\ar

⁢

[

d

⁢

l

]

⁢

&

⁢

Y

⁢

&

The resulting category is called the category of schemes over

Y

, and is sometimes denoted

Sch

/

Y

. Frequently,

Y

will be the spectrum of a ring (or especially a field)

R

, and in this case we will also call this the category of schemes over

R

(rather than schemes over

Spec

⁡

R

).

Observe that this resolves the problem of automorphisms of the complex numbers leading to automorphisms of schemes over

ℂ

.

If

Y

is not simply the spectrum of a field (perhaps it is a curve or some more general scheme) then one should interpret this as saying that

X

is a family of schemes; for each point of

Y

, we have a scheme, namely the fiber of

X

over that point of

Y

.

Note:

Some authors, notably Mumford and Grothendieck, require that a scheme be

separated

as well (and use the term

prescheme

to describe a scheme that is not separated), but we will not impose this requirement.

For many problems, the points of the underlying

topological space

of a scheme do not

represent

what we want to work with. For example, if we take the

coordinate ring

R

of a

variety

over an algebraically closed field

k

,

Spec

⁡

R

(as a scheme over

k

) behaves very much like the variety in question, but there are “extra” points in the underlying topological space: for every

irreducible

closed subset

, we have a point whose

closure

is the whole subset. Such a point is called a

generic point

. As a result, we have a more elaborate definition for a point on a scheme.

Fix a scheme

S

. Then an

S

-point on a scheme

X

is a morphism

S

→

X

. If

X

is a scheme over

Y

, then this morphism is expected to be a

Y

-morphism. The

functor

S

↦

Hom

⁡

(

S

,

X

)

is called the

functor of points

on

X

. The set of

S

-points on

X

is denoted

X

⁢

(

S

)

. If

R

is a ring, then

X

⁢

(

Spec

⁡

R

)

is often written

X

⁢

(

R

)

. Thus, for example, the complex points on

X

might be written

X

⁢

(

ℂ

)

.

3

Examples

•

Every affine scheme is clearly a scheme as well. In particular,

Spec

⁡

(

R

)

is a scheme for any commutative ring

R

.

•

Every variety can be interpreted as a scheme. An affine variety corresponds to the prime spectrum of its coordinate ring, and a projective variety has an open cover by affine pieces each of which is an affine variety, and hence an affine scheme.

4

References

See the

bibliography for algebraic geometry

(

http://planetmath.org/BibliographyForAlgebraicGeometry

). Hartshorne’s book,

Algebraic Geometry

, is an excellent reference for these ideas.

Title

scheme

Canonical name

Scheme

Date of creation

2013-03-22 12:48:17

Last modified on

2013-03-22 12:48:17

Owner

djao (24)

Last modified by

djao (24)

Numerical id

16

Author

djao (24)

Entry type

Definition

Classification

msc 14A15

Related topic

PrimeSpectrum

Related topic

Variety

Related topic

AlgebraicGeometry

Related topic

LocallyRingedSpace

Defines

affine scheme

Defines

morphism of schemes

Defines

point on a scheme

Defines

functor of points

Defines

structure morphism

Defines

generic point

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