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Fixes for intersections and coverings #1571

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merged 4 commits into from
Sep 29, 2022
Merged

Fixes for intersections and coverings #1571

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ae1cb1f
fixes for intersections and coverings
Sep 27, 2022
09e48b2
make Oscar compile again
Sep 27, 2022
472ff5b
add Todo
Sep 27, 2022
bbc82bb
set back to false
Sep 28, 2022
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7 changes: 6 additions & 1 deletion experimental/Schemes/AffineSchemes.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -813,7 +813,9 @@ identity_map(X::AbsSpec{<:Any, <:MPolyQuoLocalizedRing}) = SpecMor(X, X, hom(OO(
identity_map(X::AbsSpec{<:Any, <:MPolyLocalizedRing}) = SpecMor(X, X, hom(OO(X), OO(X), gens(base_ring(OO(X))), check=false))
identity_map(X::AbsSpec{<:Any, <:MPolyRing}) = SpecMor(X, X, hom(OO(X), OO(X), gens(OO(X))))
identity_map(X::AbsSpec{<:Any, <:MPolyQuo}) = SpecMor(X, X, hom(OO(X), OO(X), gens(ambient_ring(X))))
inclusion_map(X::T, Y::T) where {T<:AbsSpec} = SpecMor(X, Y, gens(base_ring(OO(Y))))
inclusion_map(X::AbsSpec, Y::AbsSpec) = SpecMor(X, Y, gens(base_ring(OO(Y)))) # TODO: Remove
inclusion_morphism(X::AbsSpec, Y::AbsSpec; check::Bool=true) = SpecMor(X, Y, gens(base_ring(OO(Y))), check=check)
inclusion_morphism(X::SpecOpen, Y::AbsSpec; check::Bool=true) = inclusion_morphism(X, SpecOpen(Y), check=check)

function restrict(f::SpecMor, U::AbsSpec, V::AbsSpec; check::Bool=true)
if check
Expand All @@ -824,6 +826,9 @@ function restrict(f::SpecMor, U::AbsSpec, V::AbsSpec; check::Bool=true)
return SpecMor(U, V, OO(U).(pullback(f).(gens(domain(pullback(f))))), check=check)
end

restriction(f::SpecMor, U::AbsSpec, V::AbsSpec; check::Bool=true) = restrict(f,U,V,check=check)
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function compose(f::AbsSpecMor, g::AbsSpecMor; check::Bool=true)
codomain(f) == domain(g) || error("Morphisms can not be composed")
return SpecMor(domain(f), codomain(g), compose(pullback(g), pullback(f)), check=check)
Expand Down
40 changes: 18 additions & 22 deletions experimental/Schemes/CoveredScheme.jl
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Expand Up @@ -146,22 +146,22 @@ function intersect_in_covering(U::AbsSpec, V::AbsSpec, C::Covering)
(l, m, n) = indexin(V, C)
if i == l # U and V are affine opens of the same patch X in C
X = C[i]
if X == U == V
iso = identity_map(SpecOpen(X))
if X == U == V
iso = identity_map(X)
return iso, iso, iso, iso
end
f = one(base_ring(OO(X))) # For the case where U is already a basic patch
f = one(ambient_ring(X)) # For the case where U is already a basic patch
if j != 0 # In this case, U appears in some refinement
f = gens(affine_refinements(C)[C[i]][j][1])[k]
f = gens(affine_refinements(C)[X][j][1])[k]
end
g = one(base_ring(OO(X)))
g = one(ambient_ring(X))
if m != 0
g = gens(affine_refinements(C)[C[l]][m][1])[n]
g = gens(affine_refinements(C)[X][m][1])[n]
end
W = SpecOpen(X, [f*g])
W = PrincipalOpenSubset(X, [f*g])
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Let's try to find a test for this line, tomorrow. Also, we should run the whole method on covered schemes that use both types of glueings.

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We can write it when #1573 is merged.

isoW = identity_map(W)
incWtoU = inclusion_morphism(W, SpecOpen(U), check=false)
incWtoV = inclusion_morphism(W, SpecOpen(V), check=false)
incWtoU = inclusion_morphism(W, U, check=false)
incWtoV = inclusion_morphism(W, V, check=false)
return isoW, isoW, incWtoU, incWtoV
else
G = C[i, l]
Expand All @@ -172,8 +172,8 @@ function intersect_in_covering(U::AbsSpec, V::AbsSpec, C::Covering)
WV = intersect(preimU, V)
isoWUtoWV = restriction(f, WU, WV, check=false)
isoWVtoWU = restriction(g, WV, WU, check=false)
incWUtoU = inclusion_morphism(WU, SpecOpen(U), check=false)
incWVtoV = inclusion_morphism(WV, SpecOpen(V), check=false)
incWUtoU = inclusion_morphism(WU, U, check=false)
incWVtoV = inclusion_morphism(WV, V, check=false)
return isoWUtoWV, isoWVtoWU, incWUtoU, incWVtoV
end
end
Expand Down Expand Up @@ -285,29 +285,25 @@ end
for j in i+1:r+1
x = gens(base_ring(OO(U[i])))
y = gens(base_ring(OO(U[j])))
f = SpecOpenMor(U[i], x[j-1],
U[j], y[i],
Ui = PrincipalOpenSubset(U[i], OO(U[i])(x[j-1]))
Uj = PrincipalOpenSubset(U[j], OO(U[j])(y[i]))
f = SpecMor(Ui, Uj,
vcat([x[k]//x[j-1] for k in 1:i-1],
[1//x[j-1]],
[x[k-1]//x[j-1] for k in i+1:j-1],
[x[k]//x[j-1] for k in j:r],
x[r+1:end]),
check=false
check=true
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)
g = SpecOpenMor(U[j], y[i],
U[i], x[j-1],
g = SpecMor(Uj, Ui,
vcat([y[k]//y[i] for k in 1:i-1],
[y[k+1]//y[i] for k in i:j-2],
[1//y[i]],
[y[k]//y[i] for k in j:r],
y[r+1:end]),
check=false
check=true
)
# the following two lines are necessary to assure correct parenthood
# of elements
f = restriction(f, domain(f), domain(g), check=false)
g = restriction(g, domain(g), domain(f), check=false)
add_glueing!(result, Glueing(U[i], U[j], f, g, check=false))
add_glueing!(result, SimpleGlueing(U[i], U[j], f, g, check=false))
end
end
return result
Expand Down
11 changes: 11 additions & 0 deletions test/Schemes/ProjectiveSchemes.jl
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Expand Up @@ -63,3 +63,14 @@ end
a = OO(U)([x//y, z//x])
@test pullback(projection_to_base(Y))(a)*s0 == s1
end


@testset "projective schemes as covered schemes" begin
P3 = projective_space(QQ,3)
S = ambient_ring(P3)
F = subscheme(P3,ideal(S,S[1]^4+S[2]^4+S[3]^4+S[4]^4))
Fc = as_covered_scheme(F)
U = patches(Fc)[1]
V = patches(Fc)[2]
oscar.intersect_in_covering(U,V,Fc[1])
end