Algebraic Geometry I - Problem Set 1
Please write up solutions as legibly and clearly as you can, preferably in LaTeX!
1. Prove that a ring R satisfies the ascending chain condition (every infinite ascending chain
of ideals stabilizes) if and only if every ideal in R is finitely generated.
2. Prove that the set {(t, t2 , t3 )|t ∈ k} ⊆ A3 is an algebraic set.
3. Prove that if k is an infinite field, then A2 \ {(0, 0)} is not the vanishing set of any ideal
in k[X, Y ].
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4. An ideal in a ring R is called a radical ideal if I = I. Prove the following:
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(a) If I is an ideal then I is a√radical√ideal (make sure you also prove that it is an ideal!)
(b) If I1 ⊆ I2 are ideals, then I1 ⊆ I√2 .
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(c) If I1 , I2 are ideals, then prove that I1 I2 = I1 ∩ I2 .
(Think of an example in which I1 I2 6= I1 ∩ I2 !)
(d) If I1 , I2 are radical ideals, then their intersection is also a radical ideal.
For every algebraic set X ⊆ An we define the ideal associated to X to be:
I(X) = {f ∈ k[X1 , . . . , Xn ] | f (p) = 0,
∀p ∈ X}.
5. Prove that I(X) is a radical ideal.
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6. Prove that if I is an ideal in k[X1 , . . . , Xn ] then V (I) = V ( I) and
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I ⊆ I(V (I)).
7. Find the ideal in k[X, Y, Z] associated to the union of the point (1, 1, 1) and the X-axis.
8. Assume k = C. Prove that there are three points a, b, c ∈ A2 such that
p
< X 2 − 2XY 4 + Y 6 , Y 3 − Y > = ma ∩ mb ∩ mc ,
where ma is the (maximal) ideal associated to the point a.
Hint: interpret both sides geometrically.
9. Let Y be an algebraic set in An and p a point not in Y . Prove that there is a polynomial
f ∈ k[X1 , . . . , Xn ] such that f (q) = 0 for all q ∈ Y , but f (p) = 1. Hint: I(Y ) 6= I(Y ∪ {p}).
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