# Coverings

Coverings are the backbone data structure for CoveredSchemes in Oscar.

CoveringType
Covering

A covering of a scheme $X$ by affine charts $Uᵢ$ which are glued along isomorphisms $gᵢⱼ : Uᵢ⊃ Vᵢⱼ → Vⱼᵢ ⊂ Uⱼ$.

Note: The distinction between the different affine charts of the scheme is made from their hashes. Thus, an affine scheme must not appear more than once in any covering!

source

## Constructors

CoveringMethod
Covering(patches::Vector{<:AbsAffineScheme})

Return a Covering with pairwise disjoint affine charts $Uᵢ$ given by the entries of patches. This Covering will have no gluings except those gluings along the identity of every affine chart to itself.

Examples

julia> P1, (x,y) = QQ["x", "y"];

julia> P2, (u,v) = QQ["u", "v"];

julia> U1 = spec(P1);

julia> U2 = spec(P2);

julia> C = Covering([U1, U2]) # A Covering with two disjoint affine charts
Covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]
source
disjoint_unionMethod
disjoint_union(C1::Covering, C2::Covering)

Return the Covering corresponding to the disjoint union of C1 and C2.

The charts and gluings of the disjoint union are given by the disjoint union of the charts and gluings of the covers C1 and C2.

Examples

julia> P1, (x,y) = QQ["x", "y"];

julia> P2, (u,v) = QQ["u", "v"];

julia> U1 = spec(P1);

julia> U2 = spec(P2);

julia> C1 = Covering(U1) # Set up the trivial covering with only one patch
Covering
described by patches
1: affine 2-space
in the coordinate(s)
1: [x, y]

julia> C2 = Covering(U2)
Covering
described by patches
1: affine 2-space
in the coordinate(s)
1: [u, v]

julia> C = disjoint_union(C1, C2)
Covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]
source

## Attributes

affine_chartsMethod
affine_charts(C::Covering)

Return the list of affine charts that make up the Covering C.

source
gluingsMethod
gluings(C::Covering)

Return a dictionary of gluings of the affine_charts of C.

The keys are pairs (U, V) of affine_charts. One can also use C[U, V] to obtain the respective gluing.

Note: Gluings are lazy in the sense that they are in general only computed when asked for. This method only returns the internal cache, but does not try to compute new gluings.

source

## Methods

add_gluing!Method
add_gluing!(C::Covering, G::AbsGluing)

Add a gluing G to the covering C.

The patches of G must be among the affine_charts of C.

Examples

julia> P1, (x,y) = QQ["x", "y"];

julia> P2, (u,v) = QQ["u", "v"];

julia> U1 = spec(P1);

julia> U2 = spec(P2);

julia> C = Covering([U1, U2]) # A Covering with two disjoint affine charts
Covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]

julia> V1 = PrincipalOpenSubset(U1, x); # Preparations for gluing

julia> V2 = PrincipalOpenSubset(U2, u);

julia> f = morphism(V1, V2, [1//x, y//x]); # The gluing isomorphism

julia> g = morphism(V2, V1, [1//u, v//u]); # and its inverse

julia> G = Gluing(U1, U2, f, g); # Construct the gluing

julia> add_gluing!(C, G) # Make the gluing part of the Covering
Covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]

julia> C[U1, U2] == G # Check whether the gluing of U1 and U2 in C is G.
true
source

# Gluings

Gluings are used to identify open subsets $U \subset X$ and $V \subset Y$ of affine schemes along an isomorphism $f \colon U \leftrightarrow V \colon g$.

## Types

The abstract type of any such gluing is

AbsGluingType
AbsGluing

A gluing of two affine schemes $X$ and $Y$ (the patches) along open subsets $U$ in $X$ and $V$ in $Y$ (the gluing_domains) along mutual isomorphisms $f : U ↔ V : g$ (the gluing_morphisms).

source

The available concrete types are

GluingType
Gluing

Concrete instance of an AbsGluing for gluings of affine schemes $X ↩ U ≅ V ↪ Y$ along open subsets $U$ and $V$ of type AffineSchemeOpenSubscheme.

source
SimpleGluingType
SimpleGluing

Concrete instance of an AbsGluing for gluings of affine schemes $X ↩ U ≅ V ↪ Y$ along open subsets $U$ and $V$ of type PrincipalOpenSubset.

source

## Constructors

GluingMethod
Gluing(X::AbsAffineScheme, Y::AbsAffineScheme, f::SchemeMor, g::SchemeMor)

Glue two affine schemes $X$ and $Y$ along mutual isomorphisms $f$ and $g$ of open subsets $U$ of $X$ and $V$ of $Y$.

Examples

julia> P1, (x,y) = QQ["x", "y"]; P2, (u,v) = QQ["u", "v"];

julia> U1 = spec(P1); U2 = spec(P2);

julia> V1 = PrincipalOpenSubset(U1, x); # Preparations for gluing

julia> V2 = PrincipalOpenSubset(U2, u);

julia> f = morphism(V1, V2, [1//x, y//x]); # The gluing isomorphism

julia> g = morphism(V2, V1, [1//u, v//u]); # and its inverse

julia> G = Gluing(U1, U2, f, g) # Construct the gluing
Gluing
of affine 2-space
and affine 2-space
along the open subsets
[x, y]   AA^2 \ scheme(x)
[u, v]   AA^2 \ scheme(u)
given by the pullback function
u -> 1/x
v -> y/x

julia> G isa SimpleGluing # Since the gluing domains were PrincipalOpenSubsets, this defaults to a SimpleGluing
true

julia> # Alternative using AffineSchemeOpenSubschemes as gluing domains:

julia> W1 = AffineSchemeOpenSubscheme(U1, [x]); W2 = AffineSchemeOpenSubscheme(U2, [u]);

julia> h1 = AffineSchemeOpenSubschemeMor(W1, W2, [1//x, y//x]);

julia> h2 = AffineSchemeOpenSubschemeMor(W2, W1, [1//u, v//u]);

julia> H = Gluing(U1, U2, h1, h2)
Gluing
of affine 2-space
and affine 2-space
along the open subsets
[x, y]   complement to V(x) in affine scheme with coordinates [x, y]
[u, v]   complement to V(u) in affine scheme with coordinates [u, v]
defined by the map
affine scheme morphism
from [x, y]  AA^2 \ scheme(x)
to   [u, v]  affine 2-space
given by the pullback function
u -> 1/x
v -> y/x

julia> H isa Gluing
true
source

## Attributes

patchesMethod
patches(G::AbsGluing)

Return a pair of affine schemes (X, Y) which are glued by G.

source
gluing_domainsMethod
gluing_domains(G::AbsGluing)

Return a pair of open subsets $U$ and $V$ of the respective patches of G which are glued by G along the gluing_morphisms of G.

source
gluing_morphismsMethod
gluing_morphisms(G::AbsGluing)

Return a pair of mutually inverse isomorphisms (f, g) of open subsets $U$ and $V$ of the respective patches of G which are used for the gluing identification.

source
inverseMethod
inverse(G::AbsGluing)

Return the gluing H with patches, gluing_domains, and gluing_morphisms in opposite order compared to G.

source

## Methods

composeMethod
compose(G::AbsGluing, H::AbsGluing)

Given gluings X ↩ U ≅ V ↪ Y and Y ↩ V' ≅ W ↪ Z, return the gluing X ↩ V ∩ V' ↪ Z.

WARNING: In general such a gluing will not provide a separated scheme. Use maximal_extension to extend the gluing.

source
maximal_extensionMethod
maximal_extension(G::Gluing)

Given a gluing X ↩ U ≅ V ↪ Y, try to find the maximal extension to an open subset U' ⊃ U in X and V' ⊃ V in Y so that the resulting scheme is separated.

source
restrictMethod
restrict(G::AbsGluing, f::AbsAffineSchemeMor, g::AbsAffineSchemeMor; check::Bool=true)

Given a gluing $X ↩ U ≅ V ↪ Y$ and isomorphisms $f : X → X'$ and $g: Y → Y'$, return the induced gluing of $X'$ and $Y'$.

source