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Cleanup and tests for covered schemes. #1632

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Oct 18, 2022
Merged

Cleanup and tests for covered schemes. #1632

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dfd8159
Cleanup and tests for covered schemes.
HechtiDerLachs Oct 17, 2022
6338ea0
as_covered_scheme -> covered_scheme.
HechtiDerLachs Oct 17, 2022
daf77e1
Add documentation for simplify.
HechtiDerLachs Oct 18, 2022
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19 changes: 18 additions & 1 deletion experimental/Schemes/AffineSchemes.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -58,6 +58,13 @@ base_ring_type(X::AbsSpec) = base_ring_type(typeof(X))
base_ring_elem_type(::Type{SpecType}) where {BRT, RT, SpecType<:AbsSpec{BRT, RT}} = elem_type(BRT)
base_ring_elem_type(X::AbsSpec) = base_ring_elem_type(typeof(X))

poly_type(::Type{SpecType}) where {BRT, RT<:MPolyRing, SpecType<:AbsSpec{BRT, RT}} = elem_type(RT)
poly_type(::Type{SpecType}) where {BRT, T, RT<:MPolyQuo{T}, SpecType<:AbsSpec{BRT, RT}} = T
poly_type(::Type{SpecType}) where {BRT, T, RT<:MPolyLocalizedRing{<:Any, <:Any, <:Any, T}, SpecType<:AbsSpec{BRT, RT}} = T
poly_type(::Type{SpecType}) where {BRT, T, RT<:MPolyQuoLocalizedRing{<:Any, <:Any, <:Any, T}, SpecType<:AbsSpec{BRT, RT}} = T
poly_type(X::AbsSpec) = poly_type(typeof(X))


### generically derived getters
@Markdown.doc """
base_ring(X::AbsSpec)
Expand Down Expand Up @@ -1035,7 +1042,17 @@ end

strict_modulus(X::Spec) = saturated_ideal(modulus(OO(X)))

function simplify(X::Spec)
@Markdown.doc """
simplify(X::AbsSpec{<:Field})

Given an affine scheme ``X`` with coordinate ring ``R = 𝕜[x₁,…,xₙ]/I``
(or a localization thereof), use `Singular`'s `elimpart` to try
to eliminate variables ``xᵢ`` to arrive at a simpler presentation
``R ≅ R' = 𝕜[y₁,…,yₘ]/J`` for some ideal ``J``; return
the triple ``(Y, f, g)`` where ``Y = Spec(R')`` and ``f : Y ↔ X : g``
are the identifying isomorphisms.
"""
function simplify(X::AbsSpec{<:Field})
L, f, g = simplify(OO(X))
Y = Spec(L)
YtoX = SpecMor(Y, X, f)
Expand Down
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