UNIVERSIT ´E NICE SOPHIA ANTIPOLIS Ann´ee 2015/2016
Master 2 Math´ematiques Introduction to algebraic geometry
Exercise sheet 3
Letkbe an algebraically closed field.
Exercice 1. LetX⊂kn andY ⊂kmbe affine sets.
a) Show thatX×Y ⊂kn+mis an affine set.
b) Letϕ:X →Y be a regular map, and let Γ⊂X×Y be the graph ofϕ. Show that Γ is an affine set. Show that the projectionp:X×Y →X induces an isomorphism Γ'X.
Exercice 2. Letpbe a prime number, and letkbe the algebraic closure of the finite field Fp. We call
F:k→k, x 7→ xp the Frobenius map.
a) Show thatF is a bijective morphism.
b) IsF an isomorphism of affine sets ?
Exercice 3. We consider the regular map
f :k→k3, t 7→ (t, t2, t3), and denote byC:=f(k) its image.
a) Show thatC is an affine set.
b) Compute the idealI(C) and show that Γ(C)'k[T].
Exercice 4. LetX⊂kn andY ⊂kmbe two affine sets. Let f : Γ(Y)→Γ(X) be a morphism ofk-algebras.
a) Show that ifm⊂Γ(X) is a maximal ideal, then f−1(m) is a maximal ideal in Γ(Y).
b) Show that there exists a regular map
ϕ:X→Y such thatf =ϕ∗.
Exercice 5. (more difficult) Ink4we consider the codimension two subspaces X =V(X1, X2), Y =V(X3, X4).
The unionX∪Y is an affine set. Show that the idealI(X∪Y) is not generated by two elements.
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