Last update: 16 May 2012
Let and be fields with
An affine variety is an element of the image of
Let be an affine
- A regular function on is
- The coordinate ring of is
- The (Zariski) topology on has closed sets
- The structure sheaf of
Show that, in
Let be a finitely generated algebra and let
be generators of Let
is a bijection.
Let be an affine variety. Show that
where are generators of is a bijection.
Let be an affine variety,
- the coordinate ring of
the set of closed points of .
- Show that if
then is injective and .
- Show that if has infinite transcendence
degree over then is surjective (since,
if is a prime ideal of
then the field of
fractions of is a finitely
generated field extension of so that
is an algebraic extension
of a pure transcendental extension of so that there exists a homomorphism
Show that if and
In this case
and the map
is surjective but not injective.
In this case
has two points
Then the map
Notes and References
This definition of affine varieties follows [Mac, Ch.1]. An alternative treatment is in [AM, Ch.1, Ex.27]. The exercise [AM, Ch.1, Ex.28] says that the affine varieties form a category. The comparison of affine varieties with affine schemes is in [Mac, Ch.3, p.23].
The definition of the structure sheaf of is given in [Mac, Ch.1, p.10].
Should the following left overs from a previous page be put somewhere?
A projective variety is a variety that can be embedded in a projective space
- A variety is complete if it satisfies:
- If is a variety then
is a closed map (with respect to the Zariski topology).
M. Atiyah and I.G. Macdonald, Introduction to commutative algebra,
Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.
I.G. Macdonald, Algebraic geometry. Introduction to Schemes,
W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp.