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Move dim from schemes to rings #4280

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merged 8 commits into from
Nov 13, 2024
Merged

Move dim from schemes to rings #4280

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a584ce2
Move `dim` from schemes to rings
paemurru Nov 6, 2024
747a65e
Add missing `return`
paemurru Nov 6, 2024
f5390f6
Separate dim for MPolyComplementOfKPointIdeal
paemurru Nov 13, 2024
c03b430
Apply suggestions from code review
paemurru Nov 13, 2024
5082bf7
Remove redundant functions
paemurru Nov 13, 2024
4b199f4
Fix bug when base ring is the integers
paemurru Nov 13, 2024
5f4c467
dim for zzModRing and ZZModRing
paemurru Nov 13, 2024
c9c18f3
Update src/Rings/mpolyquo-localizations.jl
simonbrandhorst Nov 13, 2024
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42 changes: 2 additions & 40 deletions src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -399,52 +399,14 @@ julia> dim(Y) # one dimension comes from ZZ and two from x1 and x2
3
```
"""
dim(X::AbsAffineScheme)

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyQuoLocRing})
error("Not implemented")
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyQuoLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyPowersOfElement}})
return dim(closure(X))
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyQuoLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:Union{MPolyComplementOfPrimeIdeal, MPolyComplementOfKPointIdeal}}})
# Spec (R / I)_P
R = OO(X)
P = prime_ideal(inverted_set(R))
I = saturated_ideal(modulus(R))
return dim(I) - dim(P)
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyLocRing})
error("Not implemented")
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyPowersOfElement}})
# zariski open subset of A^n
return dim(closure(X))
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:Union{MPolyComplementOfPrimeIdeal, MPolyComplementOfKPointIdeal}}})
P = prime_ideal(inverted_set(OO(X)))
return codim(P)
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyRing})
return dim(ideal(ambient_coordinate_ring(X), [zero(ambient_coordinate_ring(X))]))
end

@attr Any function dim(X::AbsAffineScheme{<:Ring, <:MPolyQuoRing})
return dim(modulus(OO(X)))
end
dim(X::AbsAffineScheme) = dim(OO(X))

@doc raw"""
codim(X::AbsAffineScheme)

Return the codimension of ``X`` in its ambient affine space.

Throws and error if ``X`` does not have an ambient affine space.
Throws an error if ``X`` does not have an ambient affine space.

# Examples
```jldoctest
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13 changes: 11 additions & 2 deletions src/Rings/mpoly-affine-algebras.jl
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Expand Up @@ -20,8 +20,17 @@ julia> dim(A)
```
"""
function dim(A::MPolyQuoRing)
I = A.I
return dim(I)
return dim(modulus(A))
end

function dim(A::zzModRing)
modulus(A) == 1 && error("Function `dim` gives wrong answers if the base ring is the zero ring.")
return 0
end

function dim(A::ZZModRing)
modulus(A) == 1 && error("Function `dim` gives wrong answers if the base ring is the zero ring.")
return 0
end

@doc raw"""
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31 changes: 30 additions & 1 deletion src/Rings/mpolyquo-localizations.jl
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Expand Up @@ -2565,7 +2565,36 @@ function small_generating_set(
end::Vector{elem_type(base_ring(I))}
end

dim(R::MPolyQuoLocRing{<:Field, <:FieldElem, <:MPolyRing, <:MPolyRingElem, <:MPolyComplementOfPrimeIdeal}) = dim(saturated_ideal(modulus(R))) - dim(prime_ideal(inverted_set(R)))
@attr Int function dim(R::MPolyLocRing)
error("Not implemented")
end

@attr Int function dim(R::MPolyQuoLocRing{<:Any, <:Any, <:MPolyRing, <:MPolyRingElem, <:Union{MPolyComplementOfPrimeIdeal, MPolyComplementOfKPointIdeal}})
P = prime_ideal(inverted_set(R))
I = saturated_ideal(modulus(R))
return dim(I) - dim(P)
end
Comment on lines +2572 to +2576
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Please handle them separately. For MPolyComplementOfKPointIdeal P is certified to be zero-dimensional so that we do not need to compute dim(P).

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P is certified to be zero-dimensional

It seems that this is not true.

julia> R, (x, y) = ZZ[:x, :y]
(Multivariate polynomial ring in 2 variables over ZZ, ZZMPolyRingElem[x, y])

julia> S = complement_of_point_ideal(R, [0, 0])
Complement
  of maximal ideal corresponding to rational point with coordinates (0, 0)
  in multivariate polynomial ring in 2 variables over ZZ

julia> I = ideal(R, 0)
Ideal generated by
  0

julia> L, _ = localization(R, S)
(Localization of multivariate polynomial ring in 2 variables over ZZ at complement of maximal ideal of point (0, 0), Hom: R -> localized ring)

julia> W, _ = quo(L, L(I))
(Localization of quotient of multivariate polynomial ring at complement of maximal ideal, hom: Localized ring -> Localized quotient of multivariate polynomial ring)

julia> P = prime_ideal(inverted_set(W))
Ideal generated by
  x
  y

julia> dim(P)
1

So I will leave it as it is for now.

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My guess is that complement_of_point_ideal silently assumes that we are working over a field? Perhaps the type and constructor should be restricted accordingly what do you think @afkafkafk13 ?

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@afkafkafk13 afkafkafk13 Nov 13, 2024 •
edited
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julia> S = complement_of_point_ideal(R, [0, 0])
Complement
of maximal ideal corresponding to rational point with coordinates (0, 0)
in multivariate polynomial ring in 2 variables over ZZ

This is indeed a problem with complement_of_point_ideal:
This ideal is NOT maximal as we do not get a field as the quotient of the ring by it. This needs to be handled in complement_of_point_ideal. My suggestion: check dimension (not number of entries!!) of the ideal before doing anything else in complement_of_point_ideal, throw an error if it is not maximal (i.e. dim(ideal) is not zero -- this should already be working in Oscar) and point the user to complement_of_prime_ideal.

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It is a bit fuzzy what a "point" is here. Will we get anything computationally effective for the "point" given by the maximal ideal (2,x,y) in Z[x,y]? I impression was that ComplementOfKPointIdeal just assumes a base field.

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I impression was that ComplementOfKPointIdeal just assumes a base field.

I think it does assume but does not check for this (although it should).

I would leave fixing complement_of_point_ideal to a separate issue.

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@afkafkafk13 afkafkafk13 Nov 13, 2024 •
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ok. fine with me. But we urgently need to fix complement_of_point_ideal. Maybe we can just check

@simonbrandhorst no, we won't get anything computationally effective currently. You are right. We should just tell the user to use complement_of_prime_ideal instead, whenever we are not over a field.

Edit: This is a strategic decision and breaks some tests. I opened PR We need to discuss this. Over a ring this is not a point ideal and hence should not be used via this localization, anyway. Moreover, the localization at <x_1,\dots,x_n> in ZZ[x_1,\dots,x_n] is not what is produced here, as we are localizing w.r.t. a monomial ordering and continue working over ZZ and do not pass to QQ, even though ZZ \cap <x_1,\dots,x_n> = 0.


@attr Int function dim(R::MPolyQuoLocRing{<:Any, <:Any, <:MPolyRing, <:MPolyRingElem, <:MPolyPowersOfElement})
return dim(saturated_ideal(modulus(R)))
end

@attr Int function dim(R::MPolyLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyPowersOfElement})
# zariski open subset of A^n
return dim(base_ring(R))
end

@attr Int function dim(R::MPolyLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyComplementOfPrimeIdeal})
P = prime_ideal(inverted_set(R))
return codim(P)
end


@attr Int function dim(R::MPolyLocRing{<:Field,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyComplementOfKPointIdeal})
# localization of a polynomial ring over a field at a maximal ideal does not change the dimension
# because all maximal ideals have the same dimension in this case.
return dim(base_ring(R))
end

########################################################################
# Localizations of graded rings #
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