Epimorphisms and descent for locally ringed spaces determined by sections #
A morphism f : X ⟶ Y of locally ringed spaces which is surjective on points and whose maps on
sections Γ(Y, U) ⟶ Γ(X, f⁻¹ U) are all monomorphisms is an epimorphism. A morphism of locally
ringed spaces is determined by its underlying morphism of presheafed spaces, so this follows from
the presheafed-space criterion
AlgebraicGeometry.PresheafedSpace.epi_of_base_surjective_of_c_app_mono. It applies when
𝒪_Y ⟶ f_* 𝒪_X is an isomorphism.
Let now c : X ⟶ Y be a morphism of locally ringed spaces which is a topological quotient map and
for which 𝒪_Y → c_* 𝒪_X is an isomorphism, that is, every Γ(Y, U) ⟶ Γ(X, c⁻¹ U) is an
isomorphism. Then c is a categorical quotient for maps which are constant on its fibres: a
morphism h : X ⟶ Z to an arbitrary locally ringed space Z whose underlying map is constant on
the fibres of c factors uniquely through c.
The underlying continuous map of the descended morphism g : Y ⟶ Z is the map induced by the
quotient map c. On sections over an open W ⊆ Z, it is the composite
Γ(Z, W) ⟶ Γ(X, h⁻¹ W) ⟶ Γ(X, c⁻¹ g⁻¹ W) ≅ Γ(Y, g⁻¹ W), using c⁻¹ g⁻¹ W ⊆ h⁻¹ W and the
inverse of c on sections. The composite c ≫ g is then h as a morphism of presheafed spaces.
Every point of Y is c x for some x, and the stalk map of h at x is the stalk map of g
at c x followed by that of c at x, so the stalk maps of g are local because those of h
are. Uniqueness holds because c is surjective and injective on sections, hence an epimorphism.
The construction is compatible with any commutative square, in particular with base change: if
c' : X' ⟶ Y' has the same two properties and u ≫ c = c' ≫ v, then the descent of u ≫ h
along c' is v followed by the descent of h along c. The scheme versions of these results,
which apply to contractions of curves, are in TauCeti.AlgebraicGeometry.Descent.Contraction.
Main definitions #
AlgebraicGeometry.LocallyRingedSpace.Hom.descOfIsQuotientMap: the morphismY ⟶ Zof locally ringed spaces induced by a morphismX ⟶ Zwhich is constant on the fibres ofc.
Main results #
AlgebraicGeometry.LocallyRingedSpace.epi_of_base_surjective_of_c_app_mono: a morphism of locally ringed spaces which is surjective on points and a monomorphism on sections over every open is an epimorphism.AlgebraicGeometry.LocallyRingedSpace.Hom.comp_descOfIsQuotientMap: the descended morphism factorsh.AlgebraicGeometry.LocallyRingedSpace.Hom.eq_descOfIsQuotientMap: it is the only such factorization.AlgebraicGeometry.LocallyRingedSpace.Hom.existsUnique_comp_eq_of_isQuotientMap: the universal property.AlgebraicGeometry.LocallyRingedSpace.Hom.descOfIsQuotientMap_comp_eq: descent preserves morphisms over a baseS.AlgebraicGeometry.LocallyRingedSpace.Hom.descOfIsQuotientMap_of_commSq: compatibility with commutative squares, and so with base change.
References #
- The Stacks Project, Tag 0E7B (contractions of nodal curves) and Tag 0E8A (the stabilization of a family of nodal curves, where morphisms are descended along such contractions).
A morphism of locally ringed spaces which is surjective on points and a monomorphism on sections is an epimorphism.
Descent along a contraction. Let c : X ⟶ Y be a topological quotient map for which
𝒪_Y → c_* 𝒪_X is an isomorphism. A morphism h : X ⟶ Z of locally ringed spaces whose
underlying map is constant on the fibres of c induces the morphism Y ⟶ Z through which it
factors (AlgebraicGeometry.LocallyRingedSpace.Hom.comp_descOfIsQuotientMap).
Instances For
The morphism descended along c factors the original morphism.
The morphism descended along c factors the original morphism.
On points, the morphism descended along c sends c x to h x.
The morphism descended along c is the only morphism through which h factors.
Descending a morphism already expressed as a composite through c recovers that morphism.
The universal property of a contraction. If c : X ⟶ Y is a topological quotient map for
which 𝒪_Y → c_* 𝒪_X is an isomorphism, then every morphism of locally ringed spaces
h : X ⟶ Z whose underlying map is constant on the fibres of c factors uniquely through c.
Descent along c preserves morphisms over a base: if h lies over S through Z ⟶ S and
c lies over S through Y ⟶ S, then so does the descended morphism.
Descent along c preserves morphisms over a base: if h lies over S through Z ⟶ S and
c lies over S through Y ⟶ S, then so does the descended morphism.
Given a commutative square u ≫ c = c' ≫ v, if h is constant on the fibres of c, then
u ≫ h is constant on the fibres of c'.
Descent commutes with base change. Given a commutative square u ≫ c = c' ≫ v in which
c' is also a topological quotient map that is an isomorphism on sections, the descent of
u ≫ h along c' is v followed by the descent of h along c.