Exercise sheet 2

UNIVERSIT ´E NICE SOPHIA ANTIPOLIS Ann´ee 2015/2016

Master 2 Math´ematiques Introduction to algebraic geometry

Exercise sheet 2

Letkbe an algebraically closed field.

Exercice 1. LetRbe a ring andJ ⊂R an ideal.

a) Show that the radical rac(J) is an ideal.

b) Show that rac(J) = J if and only if the quotient ring A/J does not have any nilpotent elements (i.e. non-zero elementsgsuch thatg^m= 0 for somem∈N).

Exercice 2. LetF ∈k[X₁, . . . , X_n] be a polynomial such that F =F₁^m1· · ·F_r^mr where F_i ∈ k[X₁, . . . , X_n] are irreducible polynomials andm_i ∈N. Suppose also that the polynomials are not associated i.e. we do not haveF_i=λF_j for someλ∈k^∗.

a) Show thatI(V(F)) = (F₁· · ·F_r).

b) Show thatV(F) =V(F₁)∪. . .∪V(F_r).

Exercice 3. LetX be a topological space. Show the following properties : a) IfX is irreducible andU ⊂X is an open set, thenU is irreducible.

b) IfY ⊂X andY is irreducible, then Y ⊂X is irreducible.

Exercice 4. Determine the idealsI(X) of the following algebraic sets : a)V(X1X₂³+X₁³X2−X₁²+X2)

b)V(X₁²X2,(X1−1)(X2+ 1)²).

Exercice 5. LetX andY be topological spaces. The product topology on the productX×Y is the topology where the open setsU ⊂X×Y are unions

U =[

i

U_i,X×U_i,Y

whereU_i,X ⊂X (resp.U_i,Y ⊂Y) are open sets inX (resp.Y).

ConsiderX=kandY =kendowed with the Zariski topology. Show that the product topology onX×Y =k²is not the Zariski topology of k².

Exercice 6.LetX andY be affine sets endowed with the Zariski topology, and letϕ:X →Y be a regular map. Show thatϕis continuous.

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