 
Summary: Problem Set 9
For this entire problem set, R = k[x1, x2, . . . , xn] with k a eld.
Problem 1. Dene a map : A1
A4
by t (t, t2
, t3
, t4
). This induces a map
: k[A, B, C, D] k[t].
a) Find ker().
b) Compute J = ker() k[A, B, C].
c) How does this compare with the ideal of the twisted cubic?
d) Compute I = ker() k[B, C].
e) How does this compare with the ideal in Problem 3 on Set 8?
f) Is I = J k[B, C]?
Problem 2. Let I = (x2
 y2
, xy  1) be an ideal in k[x, y].
a) Compute a Gröbner basis for I with respect to the lex order.
b) Find a reduced Gröbner basis for I with respect to the lex order.
