UNIVERSIT ´ E NICE SOPHIA ANTIPOLIS Ann´ ee 2015/2016
Master 2 MPA Introduction to algebraic geometry
Exercise sheet 5
Let k be an algebraically closed field.
Exercice 1. Let (X, O X ) be an algebraic set and Y ⊂ X a subset. For every open set U ⊂ X we define
I Y (U ) := {f ∈ O X (U ) | f (x) = 0 ∀ x ∈ U ∩ Y }.
Prove that I Y is a sheaf (which we call the ideal sheaf of Y in X ).
Exercice 2. Let X ⊂ k n (resp. Y ⊂ k m ) be an affine set endowed with its structure sheaf O X
(resp. O Y ). Let
ϕ : X → Y
be a morphism of affine sets. Show that ϕ defines a morphism between the ringed spaces (X, O X ) and (Y, O Y ).
Exercice 3. (difficult, but interesting) We consider the affine plane k 2 endowed with its structure sheaf O k
2.
a) Let f 1 , f 2 ∈ k[X 1 , X 2 ] be non-zero polynomials such that gcd(f 1 , f 2 ) = 1. Show that the restriction map
r : k[X 1 , X 2 ] = O k
2(k 2 ) → O k
2(k 2 \ (V (f 1 ) ∩ V (f 2 ))) is an isomorphism.
b) The set k 2 \ (0, 0) is an open subset of the algebraic set (k 2 , O k
2), so it has an induced structure as algebraic set. Show that k 2 \ (0, 0) is not isomorphic to an affine set.
Hint : use (and prove) that O k
2(k 2 \ (0, 0)) ' O k
2(k 2 ).
Exercice 4. Let X be a topological space. Let F → X be a presheaf of functions, so for every open set U ⊂ X we have a set of functions F(U ) which are defined on U and have values in a set K. We define for every open set U ⊂ X the set of functions
F + (U ) := {f : U → K | ∀x ∈ U ∃V ⊂ U open such that x ∈ V and f | V ∈ F(V )}.
a) Show that for every open set U ⊂ X we have a natural inclusion F(U ) ⊂ F + (U ).
b) Show that F + is a sheaf on X (which we call the sheaf associated to the presheaf F).
c) Let K be a set and K the presheaf of constant functions with values in K, i.e. for every open set U ⊂ X we have
K(U ) := {f : U → K is constant}.
Show that K + is the sheaf of locally constant functions with values in K.
1
Exercice 5. For this exercise you need some basic knowledge about complex analysis in one variable. We consider X := C endowed with the euclidean topology. For every open set U ⊂ X we define
O X (U) := {f : U → C is holomorphic}
and
O ∗ X (U ) := {f : U → C is holomorphic and f (x) 6= 0 ∀ x ∈ U }.
a) Check that O X and O ∗ X are sheaves.
We define a map exp : O X → O X ∗ by setting for every open set U ⊂ X exp(U) : O X (U ) → O X ∗ (U), f 7→ exp(f ).
b) Show that exp(U ) is not always surjective (for example if U = C \ 0).
c) Show that for every x ∈ X the map exp induces a map between the stalks exp x : O X,x → O ∗ X,x .
d) Show that exp x is surjective for every x ∈ X.
Exercice 6. In this exercise we define the structure sheaf on the projective line P 1 (the case of the projective space P n is similar). For simplicity of notation, we set R := k[X 0 , X 1 ]
a) Let f ∈ R be a homogeneous polynomial of degree d > 0, and let R f be the localisation of R in f . If f g
r∈ R f with g ∈ R a homogeneous polynomial of degree e > 0, we define
deg( g
f r ) := e − dr.
Let R (f) ⊂ R f be the set of elements of degree 0. Show that R (f) is a subring.
b) Note that the elements of R (f) define functions on P 1 \ V P (f ). Show that we can identify an element X g
r0
