Download Algebraic Geometry I - Problem Set 2

Algebraic Geometry I - Problem Set 2
You may pick any six of the following eight problems. Please write up solutions as legibly
and clearly as you can, preferably in LaTeX!
1. Prove that a subset Y of a topological space X (considered with the induced topology) is
irreducible if and only if its closure in X is irreducible. In particular, any non-empty open
subset of an irreducible topological space is irreducible.
2. Let X be a Noetherian topological space. Prove that:
(a) If an irreducible closed subset Y is contained in a union ∪Xi of finitely many closed
subsets Xi , then Y ⊆ Xi for some i.
(b) X is a finite union of of irreducible closed subsets X1 , . . . Xl . Prove that this decomposition is unique (up to relabelling) if we assume there are no redundancies (i.e.,
Xi 6⊆ Xj for i 6= j).
3. Let X be a topological space. Prove that X is Noetherian if and only if every open set is
quasi-compact (i.e., any open cover has a finite open subcover).
4. (a) Let X = V (xy − 1) in Ak2 . Prove that X is an irreducible curve. Prove that A(X) is
not isomorphic to k[t].
(b) Let X be the algebraic set given by y 2 = x(x − 1)(x − λ) (where λ ∈ k). Prove that X
is irreducible.
5. Let X = V (xy − zw) ⊆ A4 and U = D(y) ∪ D(w) ⊂ X. Define a regular function
f : U → k by f = x/w on D(w) and f = z/y on D(y). Prove that there are no polynomial
functions p, q ∈ A(X) such that q(a) 6= 0 and f (a) = p(a)/q(a) for all a ∈ U .
6. Let X be an algebraic set such that the coordinate ring A(X) is a unique factorization
domain. Let U ⊂ X be an open set. Prove that if f : U → k is any regular function, then
there exist p, q ∈ A(X) such that q(x) 6= 0 and f (x) = p(x)/q(x) for all x ∈ U .
7. Prove that if X is an algebraic set and f ∈ A(X), then O(D(f )) = A(X)f . Recall:
g
| g ∈ A(X), n ≥ 0 }.
A(X)f = {
fn
(Hint: follow the proof we did in class for proving that O(X) = A(X)f .)
8. Let X = {(t, t3 , t5 )| t ∈ k } ⊂ A3 . Prove that X is an irreducible curve. Compute
I(X). (Hint: this is similar to the twisted cubic example we did in class. One thing I used:
If R is a ring, and r1 , r2 are elements of R, then R[X, Y ]/ < X − r1 , Y − r2 > is isomorphic
to R. Why?)
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