Then (X, OX) is a ringed space

1. Scheme

A ringed space is a pair (X, O_X), where X is a topological space X and O_X is a sheaf of rings on X. We call X and OX the underlying topological space and the structure sheaf of (X, O_X) respectively.

Example 1.1. Let X be any topological space and Z be the ring of integers with the discrete topology. The constant sheaf¹ Z on X is a sheaf of rings on X. The pair (X, Z) is a ringed space.

Example 1.2. Let X be a complex manifold and O_X be the sheaf of holomorphic functions on X. Then (X, OX) is a ringed space.

Let f : X → Y be holomorphic map between complex manifolds. For each open set V ⊂ Y and each holomorphic function g ∈ OY(V ), we can define a holomorphic function on f⁻¹(V ) by g ◦ f |_f⁻¹_{(V )}. We define a map

f_V^∗ : OY(V ) → OX(f⁻¹(V ))

by f_V^∗(g) = g ◦ f |_f⁻¹_{(V )}. We obtain a morphism of sheaves of rings: f^∗: OY → f∗O_X on Y.

This motivates the definition of morphism of ringed spaces.

A morphism (X, OX) → (Y, OY) of ringed spaces is a pair (f, f^#), where f : X → Y is a continuous map and f^# : O_Y → f_∗O_X is a morphism of sheaves of rings on Y. Ringed spaces together with the morphisms defined here form a category.

Example 1.3. A complex manifold is a ringed space (X, O_X) with the property that there exists an open covering {Ui : i ∈ I} such that the ringed space (Ui, OX|_U

i) is isomorphic to the ringed space (Cⁿ, O_Cⁿ). Notice that the stalk O_X,x is a local ring for each x ∈ X.

Definition 1.1. A locally ringed space is a ringed space (X, OX) such that the stalk OX,x

is a local ring for each x ∈ X.

If (f, f^#) : (X, OX) → (Y, OY) is a morphism of ringed space, for each open set V of Y, we have a ring homomorphism

f_V^#: OY(V ) → f∗O_X(V ) = OX(f⁻¹(V )).

For each V ⊃ V⁰, we have a homomorphism

O_Y(V ) → O_Y(V⁰), O_X(f⁻¹(V )) → O_X(f⁻¹(V⁰)) and hence we have the following commuting diagram:

O_Y(V ) ^f

#

−−−−→ OV _X(f⁻¹(V ))



 y



 y O_Y(V⁰) ^f

#

−−−−→ OV 0 _X(f⁻¹(V⁰)).

These diagrams induce a homomorphism of rings lim−→

f (x)∈V

O_Y(V ) → lim−→

x∈f⁻¹(V )

O_X(f⁻¹(V )).

1Let A be a ring with discrete topology. The constant sheaf A is the sheaf defined as follows. For each open set U in X, we set A(U ) = C⁰(U, A) the ring of continuous functions from U to A.

1

2

Since lim−→x∈f⁻¹(V )O_X(f⁻¹(V )) is mapped into lim−→x∈UO_X(f⁻¹(U )), we obtain a map f_x^#: lim−→

f (x)∈V

O_Y(V ) → lim−→

x∈U

O_X(U ).

By definition, we know lim−→f (x)∈V O_Y(V ) = O_{Y,f (x)} and lim−→x∈UO_X(U ) = O_X,x. We obtain a ring homomorphism fx^#: O_{Y,f (x)}→ O_X,x.

A morphism (X, OX) → (Y, OY) of locally ringed spaces is a morphism of ringed spaces (f, f^#) such that the induced map fx^#: O_{Y,f (x)}→ O_X,x is a local homomorphism² of rings for all x ∈ X. Locally ringed spaces together with morphisms defined here form a category.

Remark. The category of locally ringed spaces is a subcategory of ringed spaces but not a full subcategory.

An affine scheme is a locally ringed space (X, OX) such that (X, OX) is isomorphic to (Spec A, O_{Spec A}) for some commutative ring A. A scheme is a locally ringed space (X, O_X) such that there exists an open covering {Ui}_i∈I of X such that (Ui, OX|_U

i) is an affine scheme for all i ∈ I, or equivalently, a scheme is a locally ringed space (X, O_X) with the property that every point has an open neighborhood U such that (U, O_X|_U) is an affine scheme. A morphism of schemes is a morphism of locally ringed spaces.

If (X, O_X) is a scheme and U is an open subset of X such that (U, O_X|_U) is isomorphic to an affine scheme (Spec A, OSpec A), we say that U is affine open.

Remark. We will abuse the use of the notation of X as ringed spaces, locally ringed spaces, or schemes.

2Let A and B be local rings with maximal ideals m_Aand mB respectively. A ring homomorphism ϕ : A → B is a local homomorphism if ϕ⁻¹(mB) = mA.