UNIVERSIT ´E NICE SOPHIA ANTIPOLIS Ann´ee 2015/2016
Master 2 Math´ematiques Introduction to algebraic geometry
Exercise sheet 1
Exercice 1. LetRbe a ring, and let I⊂R be an ideal.
a) Show thatI is a prime ideal if and only ifR/Iis an integral domain.
b) Show thatI is a maximal ideal if and only ifR/I is a field.
Exercice 2. Letkbe a field andna positive integer.
a) For everyt∈T letSt⊂kn be an affine set. Show thatT
t∈TStis an affine set.
b) LetJ1, J2⊂k[X1, . . . , Xn] be ideals. Show that
V(J1J2) =V(J1)∪V(J2).
In particular a finite union of affine sets is an affine set.
Exercice 3. Letkbe a field andna positive integer.
a) IfX ⊂kn is an affine set, then we haveV(I(X)) =X. b) LetM (N be affine sets inkn. Then we haveI(N)(I(M).
c) IfJ ⊂k[X1, . . . , Xn] is an ideal, then we haveJ ⊂I(V(J)).
Exercice 4. Let X be a non-empty topological space. Show that the following properties are equivalent :
a) IfX =F∪GwithF andGclosed, thenX =F orX =G.
b) IfU, V ⊂X are open sets such that U∩V =∅, then U =∅ orV =∅.
c) IfU ⊂X is a non-empty open set, thenU is dense.
Exercice 5. Is the set
{(x1, x2)∈R2 |x2= sinx1} an affine set ?
Exercice 6.Letkbe an infinite field. We want to determine the function ringAi :=k[X1, X2]/(Fi) for
F1=X2−X12 F2=X1X2−1
a) Show thatA1'k[T] andA2'k[T, T−1]. Show thatA1 andA2 are not isomorphic.
b) DetermineA3 for
F3=X12+X22−1.
IsA3 isomorphic toA1 orA2? The answer depends on the fieldk.
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