## 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} : **A**^{n} -> **A**^{1},
defined by f_{i}((a_{1}, ... , 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.