Affine varieties

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 16 May 2012

Affine varieties

Let k and 𝕂_ be fields with k𝕂_ and 𝕂_ algebraically closed.

Let Y=V(S) be an affine (k,𝕂_) variety.

HW: Show that φ: k[t1,...,tn] {regular functions on Y} f f|Y has kerφ =S.

HW: Show that, in [t], t = t2 and t t2 .

HW: Let A be a finitely generated k-algebra and let u1,...,un be generators of A. Let 𝔞=kerφ, where φ: k[t1,...,tn] A tj uj so that A k[t1,...,tn] 𝔞 .

HW: Show that { affine (k,𝕂_)varieties } { finitely generated k-algebras with no nilpotent elements } Y k[Y] V(𝔞) A= k[t1,...,tn] 𝔞 is a bijection.

HW: Let Y be an affine (k,𝕂_) variety. Show that Y { k-algebra homomorphisms γ:k[Y]𝕂_ } x evx ( γ(u1),...,γ(un) ) γ where u1,...,un are generators of A=k[Y], is a bijection.

HW: Let Y be an affine (k,𝕂) variety,

  1. k[Y] the coordinate ring of Y,
  2. X=Spec(k[Y]), and
  3. |X| =Maxspec(k[Y]) the set of closed points of X.
Define φ: Y X x ker(evx) where evx: k[Y] 𝕂_ f f(x).
  1. Show that if k=𝕂_ then φ is injective and imφ =|X|.
  2. Show that if 𝕂_ has infinite transcendence degree over k then φ is surjective (since, if 𝔭 is a prime ideal of A=k[Y] then the field of fractions of A𝔭 is a finitely generated field extension of k so that A𝔭 is an algebraic extension of a pure transcendental extension of k so that there exists a homomorphism γ:A𝕂_ with kerγ=𝔭).

HW: Show that if k= and 𝕂_= and Y=V(S) where S= {t2+1} then Y = {i,-i} and k[Y] = [t] t2+1 . In this case X=Spec()=pt and X=|X| and the map φ:YX is surjective but not injective.

HW: Show that if k= and 𝕂_= and Y=V(S) where S = {t2+1} then Y = {i,-i} and k[Y] = [t] t2+1 . In this case X=Spec(k[Y]) has two points x1=t+i and x2= t-i and X=|X|. Then the map φ: Y |X|=X ix2 -ix1 is a bijection.

Notes and References

This definition of affine (k,𝕂) 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 (k,𝕂) varieties with affine schemes is in [Mac, Ch.3, p.23]. The definition of the structure sheaf of Y is given in [Mac, Ch.1, p.10].

Should the following left overs from a previous page be put somewhere?


[AM] M. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp. MR0242802.

[Mac] I.G. Macdonald, Algebraic geometry. Introduction to Schemes, W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp. MR0238845.

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