example of functor of points of a scheme

example of functor of points of a scheme

Let

X

be an

affine scheme

of

finite type

over a field

k

. Then we must have

X

=

Spec

⁡

k

⁢

[

X

1

,

…

,

X

n

]

/

⟨

f

1

,

…

,

f

m

⟩

,

with the

structure morphism

X

→

Spec

⁡

k

induced

from the natural embedding

k

→

k

⁢

[

X

1

,

…

,

X

n

]

.

Let

k

′

be some field extension of

k

. What are the

k

′

-points of

X

? Recall that a

k

′

-point of

X

is by definition a morphism

Spec

⁡

k

′

→

X

(observe that since we have an

embedding

k

→

k

′

we have a morphism

Spec

⁡

k

′

→

Spec

⁡

k

, so

Spec

⁡

k

′

is natuarlly a

k

-scheme). Since

X

is affine, this must come from a

ring homomorphism

k

⁢

[

X

1

,

…

,

X

n

]

/

⟨

f

1

,

…

,

f

m

⟩

→

k

′

which takes elements of

k

to themselves inside

k

′

. Such a

homomorphism

is completely specified by specifying the images of

X

1

,

…

,

X

n

; for it to be a homomorphism, these images must

satisfy

f

1

,

…

,

f

m

. In other words, a

k

′

-point on

X

is identified with an element of

(

k

′

)

n

satisfying all the

polynomials

f

i

.

If

k

′

is an algebraically closed field, a point on

X

corresponds uniquely to a point on an

affine variety

defined by the same

equations

as

X

. If

k

′

is just any

extension

of

k

, then we have simply found which new points belong on

X

when we extend the base field. T

For an example of why schemes contain much more information than the list of points over their base field, take

X

=

Spec

⁡

ℝ

⁢

[

X

]

/

⟨

X

2

+

1

⟩

. Then

X

has

no

points over

ℝ

, its natural base field. Over

ℂ

, it has two points, corresponding to

i

and

-

i

.

This suggests that schemes may be the appropriate adaptation of

varieties

to deal with non-algebraically closed fields.

Observe that we never used the fact that

k

′

(or in fact

k

) was a field. One often chooses

k

′

as something other than a field in order to solve a problem. For example, one can take

k

′

=

k

⁢

[

ϵ

]

/

⟨

ϵ

2

⟩

. Then specifying a

k

′

-point on

X

amounts to choosing an image

κ

i

+

λ

i

⁢

ϵ

for each

X

i

. It is clear that the

κ

i

must satisfy the

f

j

. But upon reflection, we see that the

λ

i

must specify a tangent vector to

X

at the point specified by the

κ

i

. So the

k

⁢

[

ϵ

]

/

⟨

ϵ

2

⟩

-points tell us about the tangent bundle to

X

. Observe that we made no

assumption

about the field

k

— we can extract these “tangent vectors” in positive characteristic or over a non-complete field.

The ring

k

⁢

[

ϵ

]

/

⟨

ϵ

2

⟩

and rings like it (often any Artinian ring) can be used to define and study infinitesimal deformations of schemes, as a simple case of the study of families of schemes.

Title

example of functor of points of a scheme

Canonical name

ExampleOfFunctorOfPointsOfAScheme

Date of creation

2013-03-22 14:11:07

Last modified on

2013-03-22 14:11:07

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Example

Classification

msc 14A15

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