Download 1. Scheme A ringed space is a pair (X,OX), where X is a topological

1. Scheme
A ringed space is a pair (X, OX ), where X is a topological space X and OX is a sheaf of
rings on X. We call X and OX the underlying topological space and the structure sheaf of
(X, OX ) 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 OX 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
fV∗ : OY (V ) → OX (f −1 (V ))
by fV∗ (g) = g ◦ f |f −1 (V ) . We obtain a morphism of sheaves of rings: f ∗ : OY → f∗ OX 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 # : OY → f∗ OX 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, OX ) with the property that there
exists an open covering {Ui : i ∈ I} such that the ringed space (Ui , OX |Ui ) is isomorphic to
the ringed space (Cn , OCn ). Notice that the stalk OX,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
fV# : OY (V ) → f∗ OX (V ) = OX (f −1 (V )).
For each V ⊃ V 0 , we have a homomorphism
OY (V ) → OY (V 0 ),
OX (f −1 (V )) → OX (f −1 (V 0 ))
and hence we have the following commuting diagram:
f#
OY (V ) −−−V−→ OX (f −1 (V ))




y
y
OY (V
0)
fV#0
−−−−→ OX (f −1 (V 0 )).
These diagrams induce a homomorphism of rings
lim OY (V ) →
−→
f (x)∈V
lim
−−1
→
x∈f
OX (f −1 (V )).
(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 ) = C 0 (U, A) the ring of continuous functions from U to A.
1
2
Since limx∈f −1 (V ) OX (f −1 (V )) is mapped into limx∈U OX (f −1 (U )), we obtain a map
−→
−→
fx# : lim OY (V ) → lim OX (U ).
−→
−→
f (x)∈V
x∈U
By definition, we know limf (x)∈V OY (V ) = OY,f (x) and limx∈U OX (U ) = OX,x . We obtain
−→
−→
#
a ring homomorphism fx : OY,f (x) → OX,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# : OY,f (x) → OX,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, OSpec A ) for some commutative ring A. A scheme is a locally ringed space (X, OX )
such that there exists an open covering {Ui }i∈I of X such that (Ui , OX |Ui ) is an affine
scheme for all i ∈ I, or equivalently, a scheme is a locally ringed space (X, OX ) with the
property that every point has an open neighborhood U such that (U, OX |U ) is an affine
scheme. A morphism of schemes is a morphism of locally ringed spaces.
If (X, OX ) is a scheme and U is an open subset of X such that (U, OX |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 and m respectively. A ring homomorphism ϕ : A → B is a
A
B
local homomorphism if ϕ−1 (mB ) = mA .