Update localizations and schemes.

docs/src/CommutativeAlgebra/localizations.md

@@ -101,6 +101,24 @@ the general [Ring Interface](@ref) of Oscar! This has not been done to a full ex
 for the previous two examples, but for `MPolyLocalizedRing`; see below.
 
 
+### Homomorphisms for localized rings
+
+Homomorphisms from localized rings to arbitrary algebras are of type 
+```@docs
+    AbsLocalizedRingHom
+```
+Note that, in order to be well-defined, we must have 
+that ``\phi'(u) \in S`` must be a unit for every element ``u \in U``. 
+
+The getters associated to this type which need to be implemented are 
+```@docs
+    domain(f::AbsLocalizedRingHom) 
+    codomain(f::AbsLocalizedRingHom) 
+    restricted_map(f::AbsLocalizedRingHom) 
+```
+Any concrete instance `f` of `AbsLocalizedRingHom` can then be applied to elements 
+`a` of `domain(f)` by calling `f(a)`. 
+
 ### Ideals in localized rings
 
 One of the main reasons to implement localizations in the first place 
@@ -408,64 +426,3 @@ lifted_denominator(a::MPolyQuoLocalizedRingElem)
 fraction(a::MPolyQuoLocalizedRingElem)
 ```
 
-## Homomorphisms of localized affine algebras 
-Suppose we are given two localizations of polynomial algebras 
-by means of commutative diagrams 
-```math
-\begin{matrix}
-      R &  →    & P = R/I\\
-      ↓  & &       ↓ \\
-V = R[T⁻¹] & →  & P[T⁻¹]
-\end{matrix}
-```
-and 
-```math
-\begin{matrix}
-      S   & →   & Q = S/J\\
-      ↓ & &        ↓ \\
-W = S[U⁻¹] & →  & Q[U⁻¹].
-\end{matrix}
-```
-
-**Lemma:**
-For any homomorphism ``φ : P[T⁻¹] → Q[U⁻¹]`` the following holds. 
-```math
-\begin{matrix}
-            &φ& \\
-    P[T⁻¹]  & →  & Q[U⁻¹]\\
-      ↑      & &     ↑\\
-    R[T⁻¹] & \dashrightarrow & S[U⁻¹]\\
-      ↑    & ↗ ψ   &↑ ι\\
-      R     &→  &S[c⁻¹]\\
-            & η   & ↑ κ\\
-             & &     S 
-\end{matrix}
-```
-  1) The composition of maps ``R → Q[U⁻¹]`` completely determines ``φ`` by the images ``xᵢ ↦ [aᵢ]/[bᵢ]`` with ``aᵢ ∈ S``, ``bᵢ ∈ U``.
-
-  2) Let ``ψ : R → S[U⁻¹]`` be the map determined by some choice of the images ``xᵢ↦ aᵢ/b``ᵢ as above. Then ``ψ`` extends to a map ``R[T⁻¹] → S[U⁻¹]`` if and only if for all ``t ∈ T : ψ(t) ∈ U``. This is not necessarily the case as the lift of images ``φ(t) ∈ Q[U⁻¹]`` in ``S[U⁻¹]`` need only be elements of ``U + J``.
-
-  3) Choosing a common denominator ``c`` for all ``ψ(xᵢ)``, we obtain a ring homomorphism ``η : R → S[c⁻¹]`` such that ``ψ = ι ∘ η``.
-
-Upshot: In order to describe ``φ``, we may store some homomorphism 
-``ψ : R → S[U⁻¹]``
-lifting it and keep in mind the ambiguity of choices for such ``ψ``.
-The latter point 3) will be useful for reducing to a homomorphism 
-of finitely generated algebras.
-
-
-A homomorphism of localized affine algebras is stored in 
-```@docs
-MPolyQuoLocalizedRingHom{
-  BaseRingType, 
-  BaseRingElemType, 
-  RingType, 
-  RingElemType, 
-  DomainMultSetType, 
-  CodomainMultSetType
-}
-```
-An additional getter method is 
-```@docs
-images(f::MPolyQuoLocalizedRingHom)
-```

docs/src/Experimental/elliptic_curves.md

@@ -104,7 +104,7 @@ A = ResidueRing(ZZ, ZZ(n))
 S, (x,y,z) = PolynomialRing(A, ["x", "y", "z"])
 T, _ = grade(S)
 E = Oscar.ProjEllipticCurve(T(y^2*z - x^3 - 10*x*z^2 + 2*z^3))
-PP = projective_space(A, 2)
+PP = proj_space(A, 2)
 Q = Oscar.Geometry.ProjSpcElem(PP[1], [A(1), A(3), A(1)])
 P = Oscar.Point_EllCurve(E, Q)
 P2 = Oscar.IntMult_Point_EllCurveZnZ(ZZ(2), P)

docs/src/Experimental/plane_curves.md

@@ -80,7 +80,7 @@ the projective plane as follows, where `K` is the base ring:
 #### Example
 ```@repl oscar
 K = QQ
-PP = projective_space(K, 2)
+PP = proj_space(K, 2)
 ```
 
 Then, one can define a projective point as follows:

experimental/Experimental.jl

@@ -5,3 +5,9 @@ include("PlaneCurve.jl")
 include("InvariantTheory.jl")
 include("GITFans.jl")
 include("GModule.jl")
+
+include("Schemes/AffineSchemes.jl")
+include("Schemes/SpecOpen.jl")
+include("Schemes/Glueing.jl")
+include("Schemes/ProjectiveSchemes.jl")
+

experimental/PlaneCurve/DivisorCurve.jl

@@ -81,7 +81,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2 + y*z + x^2))
 Projective plane curve defined by x^2 + y^2 + y*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -302,7 +302,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2 + y*z + x^2))
 Projective plane curve defined by x^2 + y^2 + y*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -424,7 +424,7 @@ end
 
 function divisor(C::ProjPlaneCurve{S}, F::Oscar.MPolyElem_dec{S}) where S <: FieldElem
     R = parent(C.eq)
-    PP = projective_space(R.R.base_ring, 2)
+    PP = proj_space(R.R.base_ring, 2)
     return divisor(PP[1], C, F)
 end
 
@@ -449,7 +449,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2 + y*z + x^2))
 Projective plane curve defined by x^2 + y^2 + y*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -471,7 +471,7 @@ end
 
 function divisor(C::ProjPlaneCurve{S}, phi::AbstractAlgebra.Generic.Frac{T})  where {S <: FieldElem, T <: Oscar.MPolyElem_dec{S}}
     R = parent(C.eq)
-    PP = projective_space(R.R.base_ring, 2)
+    PP = proj_space(R.R.base_ring, 2)
     return divisor(PP[1], C, phi)
 end
 
@@ -596,7 +596,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2*z - x*(x-z)*(x+3*z)))
 Projective plane curve defined by -x^3 - 2*x^2*z + 3*x*z^2 + y^2*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -695,7 +695,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2*z - x*(x-z)*(x+3*z)))
 Projective plane curve defined by -x^3 - 2*x^2*z + 3*x*z^2 + y^2*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -739,7 +739,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2*z - x*(x-z)*(x+3*z)))
 Projective plane curve defined by -x^3 - 2*x^2*z + 3*x*z^2 + y^2*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -781,7 +781,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(T(y^2*z - x*(x-z)*(x+3*z)))
 Projective plane curve defined by -x^3 - 2*x^2*z + 3*x*z^2 + y^2*z
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 

experimental/PlaneCurve/ProjCurve.jl

@@ -91,7 +91,7 @@ julia> C = Oscar.ProjCurve(I)
 Projective curve defined by the ideal(x^2, y^2*z, z^2)
 
 
-julia> PP = projective_space(QQ, 3)
+julia> PP = proj_space(QQ, 3)
 (Projective space of dim 3 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2], x[3]])
 

experimental/PlaneCurve/ProjEllipticCurve.jl

@@ -1,6 +1,6 @@
 export ProjEllipticCurve, discriminant, issmooth, j_invariant,
        Point_EllCurve, curve, weierstrass_form, toweierstrass,
-       iselliptic, list_rand, proj_space
+       iselliptic, list_rand
 
 ################################################################################
 # Helping functions
@@ -155,7 +155,7 @@ julia> T, _ = grade(S)
 julia> F = T(-x^3 - 3*x^2*y - 3*x*y^2 - x*z^2 - y^3 + y^2*z - y*z^2 - 4*z^3)
 -x^3 - 3*x^2*y - 3*x*y^2 - x*z^2 - y^3 + y^2*z - y*z^2 - 4*z^3
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -186,7 +186,7 @@ mutable struct ProjEllipticCurve{S} <: ProjectivePlaneCurve{S}
     v = shortformtest(eq.f)
     T = parent(eq)
     K = T.R.base_ring
-    PP = projective_space(K, 2)
+    PP = proj_space(K, 2)
     V = gens(T)
     new{S}(eq, 3, Dict{ProjEllipticCurve{S}, Int}(), Oscar.Geometry.ProjSpcElem(PP[1], [K(0), K(1), K(0)]), [hom(T, T, V), hom(T, T, V)], Hecke.EllipticCurve(v[2], v[1]))
   end
@@ -218,7 +218,7 @@ mutable struct ProjEllipticCurve{S} <: ProjectivePlaneCurve{S}
     K = T.R.base_ring
     n = modulus(K)
     gcd(Hecke.data(d), n) == ZZ(1) || error("The discriminant is not invertible")
-    PP = projective_space(K, 2)
+    PP = proj_space(K, 2)
     V = gens(T)
     E = new{S}()
     E.eq = eq
@@ -274,7 +274,7 @@ julia> T, _ = grade(S)
 julia> F = T(-x^3 - 3*x^2*y - 3*x*y^2 - x*z^2 - y^3 + y^2*z - y*z^2 - 4*z^3)
 -x^3 - 3*x^2*y - 3*x*y^2 - x*z^2 - y^3 + y^2*z - y*z^2 - 4*z^3
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -462,7 +462,7 @@ end
 
 Return the projective space to which the point `P` belongs.
 """
-function proj_space(P::Point_EllCurve{S}) where S <: FieldElem
+function Oscar.Geometry.proj_space(P::Point_EllCurve{S}) where S <: FieldElem
    return P.Pt.parent
 end
 
@@ -564,7 +564,7 @@ julia> T, _ = grade(S)
   y -> [1]
   z -> [1], MPolyElem_dec{fmpq, fmpq_mpoly}[x, y, z])
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -638,7 +638,7 @@ end
 
 Return the projective space to which the base point of the elliptic curve `E` belongs.
 """
-function proj_space(E::ProjEllipticCurve{S}) where S <: FieldElem
+function Oscar.Geometry.proj_space(E::ProjEllipticCurve{S}) where S <: FieldElem
    return base_point(E).parent
 end
 

experimental/PlaneCurve/ProjPlaneCurve.jl

@@ -32,7 +32,7 @@ julia> T, _ = grade(S)
 julia> C = Oscar.ProjPlaneCurve(x^2*(x+y)*(y^3-x^2*z))
 Projective plane curve defined by -x^5*z - x^4*y*z + x^3*y^3 + x^2*y^4
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -74,7 +74,7 @@ julia> T, _ = grade(S)
   y -> [1]
   z -> [1], MPolyElem_dec{fmpq, fmpq_mpoly}[x, y, z])
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -155,7 +155,7 @@ julia> T, _ = grade(S)
   y -> [1]
   z -> [1], MPolyElem_dec{fmpq, fmpq_mpoly}[x, y, z])
 
-julia> PP = projective_space(QQ, 2)
+julia> PP = proj_space(QQ, 2)
 (Projective space of dim 2 over Rational Field
 , MPolyElem_dec{fmpq, fmpq_mpoly}[x[0], x[1], x[2]])
 
@@ -223,7 +223,7 @@ end
 
 function curve_intersect(C::ProjectivePlaneCurve{S}, D::ProjectivePlaneCurve{S}) where S <: FieldElem
    R = parent(C.eq)
-   PP = projective_space(R.R.base_ring, 2)
+   PP = proj_space(R.R.base_ring, 2)
    curve_intersect(PP[1], C, D)
 end
 
@@ -298,7 +298,7 @@ end
 
 function curve_singular_locus(C::ProjectivePlaneCurve)
    R = parent(C.eq)
-   PP = projective_space(R.R.base_ring, 2)
+   PP = proj_space(R.R.base_ring, 2)
    curve_singular_locus(PP[1], C)
 end