Fix dim for affine schemes (oscar-system#2639)

Implemented dimension for localizations along complements of prime
ideals and powers of an element. Not implemented for other
localizations.

src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl

@@ -387,13 +387,37 @@ julia> dim(Y) # one dimension comes from ZZ and two from x1 and x2
 dim(X::AbsSpec)
 
 @attr function dim(X::AbsSpec{<:Ring, <:MPolyQuoLocRing})
-  return dim(saturated_ideal(modulus(OO(X))))
+  error("Not implemented")
+end
+
+@attr function dim(X::AbsSpec{<:Ring, <:MPolyQuoLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyPowersOfElement}})
+  println("QuoLoc PowersOfElement")
+  return dim(closure(X))
+end
+
+@attr function dim(X::AbsSpec{<:Ring, <:MPolyQuoLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyComplementOfPrimeIdeal}})
+  println("QuoLoc ComplementOfPrimeIdeal")
+  # 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 function dim(X::AbsSpec{<:Ring, <:MPolyLocRing})
-  # the following line is supposed to refer the problem to the
-  # algebra side of the problem
-  return dim(ideal(ambient_coordinate_ring(X), [zero(ambient_coordinate_ring(X))]))
+  error("Not implemented")
+end
+
+@attr function dim(X::AbsSpec{<:Ring, <:MPolyLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyPowersOfElement}})
+  println("Loc PowersOfElement")
+  # zariski open subset of A^n
+  return dim(closure(X))
+end
+
+@attr function dim(X::AbsSpec{<:Ring, <:MPolyLocRing{<:Any,<:Any,<:MPolyRing,<:MPolyRingElem, <:MPolyComplementOfPrimeIdeal}})
+  println("Loc ComplementOfPrimeIdeal")
+  P = prime_ideal(inverted_set(OO(X)))
+  return codim(P)
 end
 
 @attr function dim(X::AbsSpec{<:Ring, <:MPolyRing})
@@ -404,7 +428,6 @@ end
   return dim(modulus(OO(X)))
 end
 
-
 @doc raw"""
     codim(X::AbsSpec)
 

src/Rings/mpoly-localizations.jl

@@ -214,7 +214,7 @@ mutable struct MPolyComplementOfPrimeIdeal{
 
   function MPolyComplementOfPrimeIdeal(
       P::MPolyIdeal{RingElemType}; 
-      check::Bool=false
+      check::Bool=true
     ) where {RingElemType}
     R = base_ring(P)
     check && (is_prime(P) || error("the ideal $P is not prime"))
@@ -378,7 +378,7 @@ Complement
   in multivariate polynomial ring in 3 variables over QQ
 ```
 """
-complement_of_prime_ideal(P::MPolyIdeal; check::Bool=false) = MPolyComplementOfPrimeIdeal(P; check)
+complement_of_prime_ideal(P::MPolyIdeal; check::Bool=true) = MPolyComplementOfPrimeIdeal(P; check)
 
 @doc raw"""  
     powers_of_element(f::MPolyRingElem)
@@ -1178,7 +1178,7 @@ function Localization(
 end
 
 ### additional constructors
-MPolyLocRing(R::RingType, P::MPolyIdeal{RingElemType}) where {RingType, RingElemType} = MPolyLocRing(R, MPolyComplementOfPrimeIdeal(P))
+MPolyLocRing(R::RingType, P::MPolyIdeal{RingElemType}; check::Bool=true) where {RingType, RingElemType} = MPolyLocRing(R, MPolyComplementOfPrimeIdeal(P); check)
 
 Localization(R::MPolyRing, v::Vector{T}) where {T<:MPolyRingElem} = Localization(MPolyPowersOfElement(R, v))
 

test/AlgebraicGeometry/Schemes/AffineSchemes.jl

@@ -77,11 +77,22 @@
   @test !issubset(A3, X)
   @test issubset(A3,A3)
   @test issubset(intersect(A3,A3), A3)
+
+  # Tests for dimension when localizing with respect to either a prime
+  # ideal or powers of an element
   @test dim(A3) == 3
   @test dim(U) == 3
   @test dim(X) == 2
   @test codim(A3) == 0
   @test codim(X) == 1
+  disjoint_plane_and_line = subscheme(A3, [x*(x - 1), x*y])
+  line = hypersurface_complement(disjoint_plane_and_line, x)
+  plane = hypersurface_complement(disjoint_plane_and_line, y)
+  @test dim(line) == 1
+  @test dim(plane) == 2
+  A3_localized_along_line = Spec(Localization(R, complement_of_prime_ideal(ideal(R, [x, y])))[1])
+  @test dim(A3_localized_along_line) == 2
+  @test dim(standard_spec(A3_localized_along_line)) == 2
 end
 
 @testset "smoothness tests" begin