M953 Homework 7 (Click
here for solutions)
Due Monday, April 15, 2002
 Let X and Y be nonempty subsets of projective space, Pn.
- (a) If C(X) = C(Y), then show that X = Y.
- (b) Show that X is a projective algebraic set if and only if C(X) is an affine
- (c) Say X is a projective algebraic set. Show that
X is irreducible if and only if C(X) is. (You may assume that X is irreducible if
and only if Ip(X) is prime.)
 Let X = Vp(x2y - z3) in P2.
Show that k(X) is isomorphic to k(A1).
 Let X be an algebraic subset of An, which
we regard as the subset U0 of Pn, and let
Y be a projective algebraic subset of Pn.
Take k[Pn] to be k[x0, ..., xn].
- (a) Show that the union of X and Vp(x0) is a projective algebraic set.
- (b) Show that X is contained in X*, which is contained in the union
of X and Vp(x0) which equals the union
of X* and Vp(x0).
- (c) Show that Y** is contained in Y.