# mathlibdocumentation

algebraic_geometry.locally_ringed_space

# The category of locally ringed spaces #

We define (bundled) locally ringed spaces (as SheafedSpace CommRing along with the fact that the stalks are local rings), and morphisms between these (morphisms in SheafedSpace with is_local_ring_hom on the stalk maps).

## Future work #

• Define the restriction along an open embedding
@[nolint]
structure algebraic_geometry.LocallyRingedSpace  :
Type (u_1+1)
• to_SheafedSpace :
• local_ring :

A LocallyRingedSpace is a topological space equipped with a sheaf of commutative rings such that all the stalks are local rings.

A morphism of locally ringed spaces is a morphism of ringed spaces such that the morphims induced on stalks are local ring homomorphisms.

The underlying topological space of a locally ringed space.

Equations

The structure sheaf of a locally ringed space.

Equations

A morphism of locally ringed spaces is a morphism of ringed spaces such that the morphims induced on stalks are local ring homomorphisms.

Equations
@[instance]
Equations
@[ext]
theorem algebraic_geometry.LocallyRingedSpace.hom_ext (f g : X.hom Y) (w : f.val = g.val) :
f = g

The stalk of a locally ringed space, just as a CommRing.

Equations

A morphism of locally ringed spaces f : X ⟶ Y induces a local ring homomorphism from Y.stalk (f x) to X.stalk x for any x : X.

Equations
@[instance]
@[simp]

The identity morphism on a locally ringed space.

Equations
@[instance]
Equations
@[simp]
theorem algebraic_geometry.LocallyRingedSpace.comp_coe (f : X.hom Y) (g : Y.hom Z) :
def algebraic_geometry.LocallyRingedSpace.comp (f : X.hom Y) (g : Y.hom Z) :
X.hom Z

Composition of morphisms of locally ringed spaces.

Equations
@[instance]

The category of locally ringed spaces.

Equations

The forgetful functor from LocallyRingedSpace to SheafedSpace CommRing.

Equations

The global sections, notated Gamma.

Equations
@[simp]