M953 Homework 4

M953 Homework 4 (Click here for solutions)

Due Friday, February 22, 2002

[1] Let f : V -> A^m be a morphism of algebraic sets. Let W be an algebraic set in A^m. Show that f^-1(W) is an algebraic subset of V. [Aside: this is analogous to the fact that the inverse image of a closed subset under a continuous map is closed.]

[2] Again let f : V -> A^m be a morphism of algebraic sets.

(a) If V is irreducible, show that V(I(f(V))) is an irreducible algebraic subset of A^m.
(b) In particular, if f : V -> W is a surjective morphism of algebraic sets with V irreducible, conclude that W is irreducible, too.

[3] Show that projection f_i : Aⁿ -> A¹, defined by f_i((a₁, ... , a_n)) = a_i, is a morphism, and determine the corresponding homomorphism of coordinate rings.

[4] Show that the image of an algebraic set need not be an algebraic set. [Hint: look at the projection of the hyperbola xy = 1.]

[5] Do Problem 2-8(b) on p. 39; i.e., show that X = V(ideal(x*z-y^2, y*z-x^3, z^2-x^2*y)) is irreducible [N.B.: the book gives a hint]. Also, check to see what Macaulay2 has to say about the irreducible components of X.