Algebraic geometry M 4 P 33

Exercise sheet 2

1. Show that if X ⊂ A_{k}^{n} and Y ⊂ A_{k}^{m} are closed subsets, then X × Y ⊂ A_{k}^{n+m} is a closed subset.

2. (a) Show that SL(n) ⊂ A_{k}^{n^{2}} is a hypersurface of degree n (it is not completely obvious why it is irreducible!).

(b) Show that O(n) is not irreducible. (Hint: what are the possible values of det(M)?)

3. Show that the quasi-affine variety X = A_{k}^{n} \ {(0, …, 0)}, n ≥ 1, is isomorphic to an affine variety if and only if n = 1.

Hint: Consider the tautological embedding X → A_{k}^{n}. If X is affine, then this map is a morphism of affine varieties, as it is defined by n regular functions on X. Now use the fact that a morphism of affine varieties is uniquely determined by the induced homomorphism of coordinate rings.

3. Let X be A_{k}^{n} with the coordinate hyperplanes removed, that is, X is the open subset given by x_{1} … x_{n} ≠ 0.

(a) Prove that X is affine (and not just quasi-affine).

(b) Prove that k[X] is the ring of Laurent polynomials

k[x_{1}, …, x_{n}, x_{1}^{−1}, …, x_{n}^{−1}].

(c) As usual, k[X]^{∗} is the group of invertible elements of the ring k[X]. Deduce from (b) the Rosenlicht lemma which says that k[X]^{∗}/k^{∗} is a free abelian group freely generated by the classes of functions x_{1}, …, x_{n}.

4. When a rational function in one variable defines a birational map A_{k}^{1} − − > A_{k}^{1}?

5. Let m_{1}, …, m_{n} be integers. Consider the map f : t ↦ (t^{m_{1}}, t^{m_{2}}, …, t^{m_{n}}).

(a) Show that f(A_{k}^{1}) ⊂ A_{k}^{n} is an affine variety.

(b) When is f an isomorphism of A_{k}^{1} with f(A_{k}^{1})?

(c) When is f a birational equivalence of A_{k}^{1} with f(A_{k}^{1})?

6. Prove that an isomorphism of affine varieties is a stronger notion than a bijective morphism. This may be due to several reasons.

(a) If k is algebraically closed of characteristic p, then the Frobenius map A_{k}^{n} → A_{k}^{n}, defined by raising all the coordinates to the power p, is a morphism which gives a bijection on k-points, but has no inverse.

(b) In any characteristic, the morphism f : A_{k}^{1} → A_{k}^{2} given by t ↦ (t^{2}, t^{3}) is a bijection but not an isomorphism.

Hint: Compute the induced action of f on the tangent space to A_{k}^{1} at 0.