From 1ceb97748d650ce9ee6265042b7f70ace7765510 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Fri, 12 May 2023 17:02:24 +0200
Subject: [PATCH 01/14] Small fix in the localizations.

---
 src/Rings/mpolyquo-localizations.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Rings/mpolyquo-localizations.jl b/src/Rings/mpolyquo-localizations.jl
index bbc5f0e8164d..41843ecd64f2 100644
--- a/src/Rings/mpolyquo-localizations.jl
+++ b/src/Rings/mpolyquo-localizations.jl
@@ -1006,7 +1006,7 @@ function MPolyQuoLocalizedRingHom(
   return MPolyQuoLocalizedRingHom(L, S, hom(base_ring(L), S, a), check=check)
 end
 
-hom(L::MPolyQuoLocRing, S::Ring, res::Map; check::Bool=true) = MPolyQuoLocalizedRingHom(L, S, a, check=check)
+hom(L::MPolyQuoLocRing, S::Ring, res::Map; check::Bool=true) = MPolyQuoLocalizedRingHom(L, S, res, check=check)
 
 function hom(L::MPolyQuoLocRing, S::Ring, a::Vector{T}; check::Bool=true) where {T<:RingElem}
   R = base_ring(L)

From def46cc01d5fc257b0ee429320bb3413f1c15564 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Fri, 12 May 2023 17:02:51 +0200
Subject: [PATCH 02/14] Implement base change for affine schemes of the common
 types.

---
 .../Schemes/AffineSchemes/Objects/Methods.jl  | 54 +++++++++++++++++++
 .../Schemes/AffineSchemes.jl                  | 23 ++++++++
 2 files changed, 77 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
index d6396b550693..7f2ce9e89b3e 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
@@ -199,6 +199,60 @@ function components(X::AbsSpec)
   return result
 end
 
+########################################################################
+# Base change and reduction modulo p
+########################################################################
 
+function change_base(phi::Any, X::AbsSpec)
+  kk = base_ring(X)
+  kk_red = parent(phi(zero(kk)))
+  R = OO(X)
+  R_red, Phi = _change_base_ring(phi, R)
+  Y = Spec(R_red)
+  return Y, SpecMor(Y, X, Phi)
+end
 
+### Some helper functions
+function _change_base_ring(phi::Any, R::MPolyRing)
+  K = coefficient_ring(R)
+  kk = parent(phi(zero(K)))
+  P, _ = polynomial_ring(kk, symbols(R))
+  Phi = hom(R, P, phi, gens(P))
+  return P, Phi
+end
 
+function _change_base_ring(phi::Any, A::MPolyQuoRing)
+  R = base_ring(A)
+  I = modulus(A)
+  P, Phi = _change_base_ring(phi, R)
+  I_red = ideal(P, Phi.(gens(I)))
+  Q, pr = quo(P, I_red)
+  Phi_bar = hom(A, Q, phi, gens(Q), check=false)
+  return Q, Phi_bar
+end
+
+function _change_base_ring(phi::Any, 
+    W::MPolyLocRing{<:Any, <:Any, <:Any, <:Any, 
+                    <:MPolyPowersOfElement}
+  )
+  R = base_ring(W)
+  P, Phi = _change_base_ring(phi, R)
+  U = inverted_set(W)
+  U_red = MPolyPowersOfElement(P, Phi.(denominators(U)))
+  W_red, loc_map = localization(P, U_red)
+  return W_red, hom(W, W_red, compose(Phi, loc_map), check=false)
+end
+
+function _change_base_ring(phi::Any, 
+    L::MPolyQuoLocRing{<:Any, <:Any, <:Any, <:Any, 
+                       <:MPolyPowersOfElement}
+  )
+  R = base_ring(L)
+  W = localized_ring(L)
+  W_red, Phi_W = _change_base_ring(phi, W)
+  I = modulus(L)
+  I_red = ideal(W_red, Phi_W.(gens(I)))
+  L_red, pr = quo(W_red, I_red)
+  res = MapFromFunc(x->L_red(phi(x)),R, L_red) 
+  return L_red, hom(L, L_red, res, check=false)
+end
diff --git a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
index 2a816521c7e8..689f40d4c702 100644
--- a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
@@ -181,3 +181,26 @@ end
   X = Spec(R, I)
   @test !is_smooth(X)
 end
+
+@testset "reduction mod p" begin
+  IA2 = affine_space(ZZ, 2)
+  P = OO(IA2)
+  x, y = gens(P)
+
+  X = subscheme(IA2, x^2 + y^2)
+
+  U = hypersurface_complement(IA2, x)
+
+  V = hypersurface_complement(X, x)
+
+  kk, pr = quo(ZZ, 5)
+  IA2_red, phi1 = oscar.change_base(pr, IA2)
+  X_red, phi2 = oscar.change_base(pr, X)
+  U_red, phi3 = oscar.change_base(pr, U)
+  V_red, phi4 = oscar.change_base(pr, V)
+
+  m1 = compose(inclusion_morphism(V_red, IA2_red), phi1);
+  m2 = compose(phi4, inclusion_morphism(V, IA2));
+  @test m1 == m2
+end
+

From aee5d8c646d7704971a82b7849babea0bf3b6a3b Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Fri, 12 May 2023 17:09:34 +0200
Subject: [PATCH 03/14] Add docstring.

---
 .../Schemes/AffineSchemes/Objects/Methods.jl               | 7 +++++++
 1 file changed, 7 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
index 7f2ce9e89b3e..c79be3548c5f 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
@@ -203,6 +203,13 @@ end
 # Base change and reduction modulo p
 ########################################################################
 
+@doc raw"""
+    change_base(phi::Any, X::AbsSpec)
+
+For an affine scheme `X` over a `base_ring` ``𝕜`` and a morphism 
+``φ : 𝕜 → 𝕂`` this computes ``Y = X × Spec(𝕂)`` and returns a pair 
+`(Y, psi)` where `psi` is the canonical map ``Y → X``.
+"""
 function change_base(phi::Any, X::AbsSpec)
   kk = base_ring(X)
   kk_red = parent(phi(zero(kk)))

From e685dba0fb0add58a67355e2cebe4d90916e70f8 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 08:52:16 +0200
Subject: [PATCH 04/14] Base change for morphisms of affine schemes.

---
 .../AffineSchemes/Morphisms/Methods.jl        | 47 +++++++++++++++++++
 .../Schemes/AffineSchemes/Objects/Methods.jl  |  2 +-
 .../Schemes/AffineSchemes.jl                  |  5 ++
 3 files changed, 53 insertions(+), 1 deletion(-)

diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
index 5bcd9f8803eb..805664a72d50 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
@@ -119,3 +119,50 @@ function Base.show(io::IO, f::AbsSpecMor)
   print(io, "$(pullback(f)(last(x)))")
 end
 
+########################################################################
+# (6) Base change
+########################################################################
+
+@doc raw"""
+    change_base(phi::Any, f::AbsSpecMor)
+        domain_map::AbsSpecMor=change_base(phi, domain(f))[2],
+        codomain_map::AbsSpecMor=change_base(phi, codomain(f))[2]
+      )
+
+For a morphism ``f : X → Y`` between two schemes over a `base_ring` ``𝕜`` 
+and a ring homomorphism ``φ : 𝕜 → 𝕂`` this returns a triple 
+`(b₁, F, b₂)` consisting of the maps in the commutative diagram 
+```
+             f
+    X        →    Y
+    ↑ b₁          ↑ b₂
+  X×ₖSpec(𝕂) → Y×ₖSpec(𝕂)
+             F
+```
+"""
+function change_base(phi::Any, f::AbsSpecMor; 
+    domain_map::AbsSpecMor=change_base(phi, domain(f))[2],
+    codomain_map::AbsSpecMor=change_base(phi, codomain(f))[2]
+  )
+  X = domain(f)
+  Y = codomain(f)
+  XX = domain(domain_map)
+  YY = domain(codomain_map)
+  pbf = pullback(f)
+  pb1 = pullback(domain_map)
+  pb2 = pullback(codomain_map)
+
+  R = OO(Y)
+  S = OO(X)
+  RR = OO(YY)
+  SS = OO(XX)
+
+  img_gens = [pb1(pbf(x)) for x in gens(R)]
+  # For the pullback of F no explicit coeff_map is necessary anymore 
+  # since both rings in domain and codomain have the same (extended/reduced)
+  # coefficient ring by now.
+  pbF = hom(RR, SS, img_gens, check=true) # TODO: Set to false after testing
+
+  return domain_map, SpecMor(XX, YY, pbF, check=true), codomain_map # TODO: Set to false after testing
+end
+
diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
index c79be3548c5f..a941387a97bb 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
@@ -260,6 +260,6 @@ function _change_base_ring(phi::Any,
   I = modulus(L)
   I_red = ideal(W_red, Phi_W.(gens(I)))
   L_red, pr = quo(W_red, I_red)
-  res = MapFromFunc(x->L_red(phi(x)),R, L_red) 
+  res = compose(restricted_map(Phi_W), pr)
   return L_red, hom(L, L_red, res, check=false)
 end
diff --git a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
index 689f40d4c702..dec7b501549b 100644
--- a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
@@ -202,5 +202,10 @@ end
   m1 = compose(inclusion_morphism(V_red, IA2_red), phi1);
   m2 = compose(phi4, inclusion_morphism(V, IA2));
   @test m1 == m2
+
+  # Testing morphisms 
+  inc_U = inclusion_morphism(V, X)
+  (a, b, c) = oscar.change_base(pr, inc_U)
+  @test compose(a, inc_U) == compose(b, c)
 end
 

From c11755333ce6292fcad1fe74195d28ff80cd203c Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 09:03:56 +0200
Subject: [PATCH 05/14] Base change for PrincipalOpenSubsets and SpecOpen.

---
 .../Schemes/PrincipalOpenSubset/Objects/Methods.jl   | 12 ++++++++++++
 .../Schemes/SpecOpen/Objects/Methods.jl              | 11 +++++++++++
 test/AlgebraicGeometry/Schemes/AffineSchemes.jl      |  9 +++++++++
 3 files changed, 32 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
index 08f0655a1e3b..e1dad537d7d4 100644
--- a/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
@@ -39,3 +39,15 @@ function generic_fraction(a::MPolyQuoLocRingElem, U::PrincipalOpenSubset)
   return lifted_numerator(a)//lifted_denominator(a)
 end
 
+########################################################################
+# Base change
+########################################################################
+
+function change_base(phi::Any, U::PrincipalOpenSubset)
+  Y, f = change_base(phi, ambient_scheme(U))
+  pbf = pullback(f)
+  h = pbf(complement_equation(U))
+  UU = PrincipalOpenSubset(Y, h)
+  return UU, restrict(f, UU, U, check=true) # TODO: Set to false after testing
+end
+
diff --git a/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
index b9736abb8cb9..d867181896d5 100644
--- a/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
@@ -144,3 +144,14 @@ function Base.show(io::IO, U::SpecOpen)
   print(io, "complement of zero locus of $(complement_equations(U)) in $(ambient_scheme(U))")
 end
 
+########################################################################
+# Base change
+########################################################################
+function change_base(phi::Any, U::SpecOpen)
+  Y, f = change_base(phi, ambient_scheme(U))
+  pbf = pullback(f)
+  h = pbf.(complement_equations(U))
+  UU = SpecOpen(Y, h)
+  return UU, restrict(f, UU, U, check=true) # TODO: Set to false after testing
+end
+
diff --git a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
index dec7b501549b..697a790b40d3 100644
--- a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
@@ -207,5 +207,14 @@ end
   inc_U = inclusion_morphism(V, X)
   (a, b, c) = oscar.change_base(pr, inc_U)
   @test compose(a, inc_U) == compose(b, c)
+
+  # Behaviour for special types
+  U = PrincipalOpenSubset(IA2, y)
+  UU, f = oscar.change_base(pr, U)
+  @test UU isa PrincipalOpenSubset
+
+  W = SpecOpen(IA2, [x, y])
+  WW, f = oscar.change_base(pr, W)
+  @test WW isa SpecOpen
 end
 

From e53a58a60e628f934757338cec9fec598c7d569b Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 09:49:31 +0200
Subject: [PATCH 06/14] Add base change for SimpleGlueings.

---
 .../AffineSchemes/Objects/Attributes.jl       |  6 ++-
 .../Schemes/Glueing/Methods.jl                | 45 +++++++++++++++++++
 .../PrincipalOpenSubset/Objects/Methods.jl    | 10 +++--
 .../ProjectiveSchemes/Objects/Attributes.jl   |  4 +-
 .../Schemes/SpecOpen/Objects/Methods.jl       | 10 +++--
 test/AlgebraicGeometry/Schemes/Glueing.jl     |  9 ++++
 6 files changed, 73 insertions(+), 11 deletions(-)

diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
index d4b4cc340369..074bab6eccf7 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
@@ -157,10 +157,14 @@ function ambient_space(X::AbsSpec{BRT, RT}) where {BRT, RT<:MPolyRing}
   return X
 end
 
-@attr function ambient_space(X::Spec{BRT,RT}) where {BRT, RT <: Union{MPolyQuoRing,MPolyLocRing,MPolyQuoLocRing}}
+@attr function ambient_space(X::Spec{BRT,RT}) where {BRT<:Field, RT <: Union{MPolyQuoRing,MPolyLocRing,MPolyQuoLocRing}}
   return affine_variety(Spec(ambient_coordinate_ring(X)), check=false)
 end
 
+@attr function ambient_space(X::Spec{BRT,RT}) where {BRT, RT <: Union{MPolyQuoRing,MPolyLocRing,MPolyQuoLocRing}}
+  return Spec(ambient_coordinate_ring(X))
+end
+
 @attr function ambient_space(X::AbsSpec{BRT,RT}) where {BRT, RT <: Union{MPolyQuoRing,MPolyLocRing,MPolyQuoLocRing}}
   return ambient_space(underlying_scheme(X))
 end
diff --git a/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl b/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl
index 35744fc029b0..2097c28cadfd 100644
--- a/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl
@@ -124,3 +124,48 @@ function ==(G::AbsGlueing, H::AbsGlueing)
   return true
 end
 
+########################################################################
+# Base change
+########################################################################
+struct BaseChangeGlueingData{T1, T2}
+  phi::T1
+  G::T2
+  patch_change1::AbsSpecMor
+  patch_change2::AbsSpecMor
+end
+
+function _compute_glueing_base_change(gd::BaseChangeGlueingData)
+  # extraction of the glueing data
+  G = gd.G
+  pc1 = gd.patch_change1
+  pc2 = gd.patch_change2
+  phi = gd.phi
+
+  # computation of the new glueing
+  (X, Y) = patches(G)
+  (U, V) = glueing_domains(G)
+  (f, g) = glueing_morphisms(G)
+  
+  XX = domain(pc1)
+  YY = domain(pc2)
+
+  UU, map_U = change_base(phi, U, ambient_map=pc1)
+  VV, map_V = change_base(phi, V, ambient_map=pc2)
+
+  # TODO: The following will not yet work for glueings along SpecOpens.
+  U isa PrincipalOpenSubset && V isa PrincipalOpenSubset || error("base change not implemented for glueings along SpecOpens")
+
+  _, ff, _ = change_base(phi, f, domain_map=map_U, codomain_map=map_V)
+  _, gg, _ = change_base(phi, g, domain_map=map_V, codomain_map=map_U)
+
+  return Glueing(XX, YY, ff, gg)
+end
+
+function change_base(phi::Any, G::AbsGlueing;
+    patch_change1::AbsSpecMor=change_base(phi, glueing_domains(G)[1])[2], # The base change morphism for 
+    patch_change2::AbsSpecMor=change_base(phi, glueing_domains(G)[2])[2]  # the two glueing patches
+  )
+  gd = BaseChangeGlueingData{typeof(phi), typeof(G)}(phi, G, patch_change1, patch_change2)
+  return LazyGlueing(domain(patch_change1), domain(patch_change2), _compute_glueing_base_change, gd)
+end
+
diff --git a/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
index e1dad537d7d4..ec91904e93ee 100644
--- a/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
@@ -43,11 +43,13 @@ end
 # Base change
 ########################################################################
 
-function change_base(phi::Any, U::PrincipalOpenSubset)
-  Y, f = change_base(phi, ambient_scheme(U))
-  pbf = pullback(f)
+function change_base(phi::Any, U::PrincipalOpenSubset;
+    ambient_map::AbsSpecMor=change_base(phi, ambient_scheme(U))[2] # the base change on the ambient scheme
+  )
+  Y = domain(ambient_map)
+  pbf = pullback(ambient_map)
   h = pbf(complement_equation(U))
   UU = PrincipalOpenSubset(Y, h)
-  return UU, restrict(f, UU, U, check=true) # TODO: Set to false after testing
+  return UU, restrict(ambient_map, UU, U, check=true) # TODO: Set to false after testing
 end
 
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl
index eaf2d544aed6..bb59ff74ee31 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl
@@ -291,7 +291,7 @@ end
 
 @attr function affine_cone(
     P::AbsProjectiveScheme{RT, <:MPolyQuoRing}
-  ) where {RT<:Field}
+  ) where {RT<:Union{Field, ZZRing}}
   S = homogeneous_coordinate_ring(P)
   PS = base_ring(S)
   PP = forget_grading(PS) # the ungraded polynomial ring
@@ -305,7 +305,7 @@ end
 
 @attr function affine_cone(
     P::AbsProjectiveScheme{RT, <:MPolyDecRing}
-  ) where {RT<:Field}
+  ) where {RT<:Union{Field, ZZRing}}
   S = homogeneous_coordinate_ring(P)
   PP = forget_grading(S) # the ungraded polynomial ring
   phi = hom(S, PP, gens(PP))
diff --git a/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
index d867181896d5..7ecffde1aaec 100644
--- a/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
@@ -147,11 +147,13 @@ end
 ########################################################################
 # Base change
 ########################################################################
-function change_base(phi::Any, U::SpecOpen)
-  Y, f = change_base(phi, ambient_scheme(U))
-  pbf = pullback(f)
+function change_base(phi::Any, U::SpecOpen;
+    ambient_map::AbsSpecMor=change_base(phi, ambient_scheme(U))[2] # the base change on the ambient scheme
+  )
+  Y = domain(ambient_map)
+  pbf = pullback(ambient_map)
   h = pbf.(complement_equations(U))
   UU = SpecOpen(Y, h)
-  return UU, restrict(f, UU, U, check=true) # TODO: Set to false after testing
+  return UU, restrict(ambient_map, UU, U, check=true) # TODO: Set to false after testing
 end
 
diff --git a/test/AlgebraicGeometry/Schemes/Glueing.jl b/test/AlgebraicGeometry/Schemes/Glueing.jl
index c3853cf26118..01d7b867ac5c 100644
--- a/test/AlgebraicGeometry/Schemes/Glueing.jl
+++ b/test/AlgebraicGeometry/Schemes/Glueing.jl
@@ -122,3 +122,12 @@ end
   @test glueing_domains(G1) == glueing_domains(DG)
   @test inverse(DG) == DG
 end
+
+@testset "base change" begin
+  IP1 = covered_scheme(projective_space(ZZ, 1))
+  C = default_covering(IP1)
+  G = first(values(glueings(C)))
+  GG = oscar.change_base(pr, G)
+  @test underlying_glueing(GG) isa SimpleGlueing
+end
+

From f781238ccb8f81d1c2d530bd07b774edd797a4d7 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 10:15:21 +0200
Subject: [PATCH 07/14] Add base change for CoveredSchemes.

---
 .../Schemes/CoveredSchemes/Objects/Methods.jl | 10 ++++++++
 .../Schemes/Covering/Objects/Methods.jl       | 25 +++++++++++++++++++
 .../Schemes/CoveredScheme.jl                  |  5 ++++
 3 files changed, 40 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl
index 941e150f7fce..7e8b605e2274 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl
@@ -38,3 +38,13 @@ function Base.show(io::IO, X::AbsCoveredScheme)
   end
   print(io, "covered scheme with $(npatches(default_covering(X))) affine patches in its default covering")
 end
+
+########################################################################
+# Base change
+########################################################################
+function change_base(phi::Any, X::AbsCoveredScheme)
+  C = default_covering(X)
+  CC, f_CC = change_base(phi, C)
+  XX = CoveredScheme(CC)
+  return XX, CoveredSchemeMorphism(XX, X, f_CC)
+end
diff --git a/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl
index d4be19557ba3..06bd0d16e24a 100644
--- a/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl
@@ -324,4 +324,29 @@ function is_refinement(D::Covering, C::Covering)
   return true, CoveringMorphism(D, C, map_dict, check=false)
 end
 
+########################################################################
+# Base change
+########################################################################
+function change_base(phi::Any, C::Covering)
+  U = patches(C)
+  patch_change = [change_base(phi, V) for V in U]
+
+  glueing_dict = IdDict{Tuple{AbsSpec, AbsSpec}, AbsGlueing}()
+  for i in 1:length(U)
+    (A, map_A) = patch_change[i]
+    for j in 1:length(U)
+      (B, map_B) = patch_change[j]
+      G = C[U[i], U[j]] # the glueing
+      GG = change_base(phi, G, patch_change1=map_A, patch_change2=map_B)
+      glueing_dict[A, B] = GG
+    end
+  end
+  CC = Covering([V for (V, _) in patch_change], glueing_dict)
 
+  mor_dict = IdDict{AbsSpec, AbsSpecMor}()
+  for (V, phi) in patch_change
+    mor_dict[V] = phi
+  end
+
+  return CC, CoveringMorphism(CC, C, mor_dict, check=true) # TODO: Set to false after testing
+end
diff --git a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
index 30f5d6439bed..172bc3525b4a 100644
--- a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
+++ b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
@@ -217,3 +217,8 @@ end
       @test underlying_glueing(glueings(dom_cov)[k]) isa SimpleGlueing
   end
 end
+
+@testset "base change" begin
+  IP1 = covered_scheme(projective_space(ZZ, 1))
+  IP1_red, red_map = oscar.change_base(pr, IP1)
+end

From 44a14d97eb58dcc91aa605eb6ed3acc638789718 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 11:06:13 +0200
Subject: [PATCH 08/14] Add base change for morphisms of CoveredSchemes.

---
 .../CoveredSchemes/Morphisms/Methods.jl       | 30 +++++++++++++++++++
 .../Schemes/Covering/Morphisms/Methods.jl     | 24 +++++++++++++++
 .../Schemes/CoveredScheme.jl                  | 12 ++++++++
 test/AlgebraicGeometry/Schemes/Glueing.jl     |  1 +
 4 files changed, 67 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
index 51f252a9ba65..c61a91d40384 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
@@ -65,3 +65,33 @@ function maps_with_given_codomain(f::AbsCoveredSchemeMorphism, V::AbsSpec)
   return result
 end
 
+########################################################################
+# Comparison
+########################################################################
+function ==(f::AbsCoveredSchemeMorphism, g::AbsCoveredSchemeMorphism)
+  domain(f) === domain(g) || return false
+  codomain(f) === codomain(g) || return false
+  f_cov = covering_morphism(f)
+  g_cov = covering_morphism(g)
+  domain(f_cov) === domain(g_cov) || error("comparison across refinements not implemented")
+  codomain(f_cov) === codomain(g_cov) || error("comparison across refinements not implemented")
+  return all(U->(f_cov[U] == g_cov[U]), patches(domain(f_cov)))
+end
+
+########################################################################
+# Base change
+########################################################################
+function change_base(phi::Any, f::AbsCoveredSchemeMorphism;
+    domain_map::AbsCoveredSchemeMorphism=change_base(phi, domain(f))[2],
+    codomain_map::AbsCoveredSchemeMorphism=change_base(phi, codomain(f))[2]
+  )
+  f_cov = covering_morphism(f)
+  dom_cov = covering_morphism(domain_map)
+  cod_cov = covering_morphism(codomain_map)
+  _, ff_cov_map, _ = change_base(phi, f_cov, domain_map=dom_cov, codomain_map=cod_cov)
+  X = domain(f)
+  Y = codomain(f)
+  XX = domain(domain_map)
+  YY = domain(codomain_map)
+  return domain_map, CoveredSchemeMorphism(XX, YY, ff_cov_map, check=true), codomain_map #TODO: Set to false after testing.
+end
diff --git a/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
index db552d9a92ef..84a8a545a12c 100644
--- a/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
@@ -49,3 +49,27 @@ function simplify(C::Covering)
   return Cnew, i_cov_mor, j_cov_mor
 end
 
+########################################################################
+# Base change
+########################################################################
+function change_base(phi::Any, f::CoveringMorphism;
+    domain_map::CoveringMorphism=change_base(phi, domain(f))[2],
+    codomain_map::CoveringMorphism=change_base(phi, codomain(f))[2]
+  )
+  D = domain(f)
+  C = codomain(f)
+  DD = domain(domain_map)
+  CC = domain(codomain_map)
+  mor_dict = IdDict{AbsSpec, AbsSpecMor}()
+  for UU in patches(DD)
+    U = codomain(domain_map[UU])
+    V = codomain(f[U])
+    g_V = first(maps_with_given_codomain(codomain_map, V)) # The result must be unique as it arises 
+                                                           # from a base change.
+    _, ff, _ = change_base(phi, f[U], domain_map=domain_map[UU], codomain_map=g_V)
+    mor_dict[UU] = ff
+  end
+
+  return domain_map, CoveringMorphism(DD, CC, mor_dict, check=true), codomain_map # TODO: Set to false after testing.
+end
+
diff --git a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
index 172bc3525b4a..9277ad2cea51 100644
--- a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
+++ b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
@@ -219,6 +219,18 @@ end
 end
 
 @testset "base change" begin
+  kk, pr = quo(ZZ, 5)
   IP1 = covered_scheme(projective_space(ZZ, 1))
   IP1_red, red_map = oscar.change_base(pr, IP1)
+  
+  IP2 = projective_space(ZZ, 2)
+  S = homogeneous_coordinate_ring(IP2)
+  (x, y, z) = gens(S)
+  I = ideal(S, x^2 + y^2 + z^2)
+  IP2_cov = covered_scheme(IP2)
+  II = IdealSheaf(IP2, I)
+
+  inc_X = oscar.CoveredClosedEmbedding(IP2_cov, II)
+  (a, b, c) = oscar.change_base(pr, inc_X)
+  @test compose(a, inc_X) == compose(b, c)
 end
diff --git a/test/AlgebraicGeometry/Schemes/Glueing.jl b/test/AlgebraicGeometry/Schemes/Glueing.jl
index 01d7b867ac5c..119f2a6ec9d3 100644
--- a/test/AlgebraicGeometry/Schemes/Glueing.jl
+++ b/test/AlgebraicGeometry/Schemes/Glueing.jl
@@ -124,6 +124,7 @@ end
 end
 
 @testset "base change" begin
+  kk, pr = quo(ZZ, 5)
   IP1 = covered_scheme(projective_space(ZZ, 1))
   C = default_covering(IP1)
   G = first(values(glueings(C)))

From c8b76e52fc697d1151c8c01205fc935b35e07fa1 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 11:39:39 +0200
Subject: [PATCH 09/14] Add documentation for generic methods.

---
 docs/src/AlgebraicGeometry/Schemes/GeneralSchemes.md | 9 +++++++++
 src/AlgebraicGeometry/Schemes/main.jl                | 1 +
 2 files changed, 10 insertions(+)

diff --git a/docs/src/AlgebraicGeometry/Schemes/GeneralSchemes.md b/docs/src/AlgebraicGeometry/Schemes/GeneralSchemes.md
index 099c476789d2..21fc54fc074f 100644
--- a/docs/src/AlgebraicGeometry/Schemes/GeneralSchemes.md
+++ b/docs/src/AlgebraicGeometry/Schemes/GeneralSchemes.md
@@ -13,3 +13,12 @@ Morphisms of schemes shall be derived from the abstract type
 ```@docs
     SchemeMor{DomainType, CodomainType, MorphismType, BaseMorType}
 ```
+
+## Change of base
+```@docs
+    base_change(phi::Any, X::Scheme)
+    base_change(phi::Any, f::SchemeMor;
+        domain_map::AbsSchemeMor, codomain_map::AbsSchemeMor
+      )
+```
+   
diff --git a/src/AlgebraicGeometry/Schemes/main.jl b/src/AlgebraicGeometry/Schemes/main.jl
index 510eba47bb78..e3d7de4f358e 100644
--- a/src/AlgebraicGeometry/Schemes/main.jl
+++ b/src/AlgebraicGeometry/Schemes/main.jl
@@ -8,6 +8,7 @@ import Base: intersect
 # what follows                                                         #
 ########################################################################
 include("Types.jl")
+include("Methods.jl")
 
 include("AffineSchemes/Objects/Types.jl")
 include("AffineSchemes/Morphisms/Types.jl")

From 3f4e5246560029b7cecca95ced09645902a5f5b9 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 11:45:09 +0200
Subject: [PATCH 10/14] Rename change_base -> base_change.

---
 .../Schemes/AffineSchemes/Morphisms/Methods.jl     | 12 ++++++------
 .../Schemes/AffineSchemes/Objects/Methods.jl       |  4 ++--
 .../Schemes/CoveredSchemes/Morphisms/Methods.jl    |  8 ++++----
 .../Schemes/CoveredSchemes/Objects/Methods.jl      |  4 ++--
 .../Schemes/Covering/Morphisms/Methods.jl          |  8 ++++----
 .../Schemes/Covering/Objects/Methods.jl            |  6 +++---
 src/AlgebraicGeometry/Schemes/Glueing/Methods.jl   | 14 +++++++-------
 .../Schemes/PrincipalOpenSubset/Objects/Methods.jl |  4 ++--
 .../Schemes/SpecOpen/Objects/Methods.jl            |  4 ++--
 test/AlgebraicGeometry/Schemes/AffineSchemes.jl    | 14 +++++++-------
 test/AlgebraicGeometry/Schemes/CoveredScheme.jl    |  4 ++--
 test/AlgebraicGeometry/Schemes/Glueing.jl          |  2 +-
 12 files changed, 42 insertions(+), 42 deletions(-)

diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
index 805664a72d50..467f99c1a737 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
@@ -124,9 +124,9 @@ end
 ########################################################################
 
 @doc raw"""
-    change_base(phi::Any, f::AbsSpecMor)
-        domain_map::AbsSpecMor=change_base(phi, domain(f))[2],
-        codomain_map::AbsSpecMor=change_base(phi, codomain(f))[2]
+    base_change(phi::Any, f::AbsSpecMor)
+        domain_map::AbsSpecMor=base_change(phi, domain(f))[2],
+        codomain_map::AbsSpecMor=base_change(phi, codomain(f))[2]
       )
 
 For a morphism ``f : X → Y`` between two schemes over a `base_ring` ``𝕜`` 
@@ -140,9 +140,9 @@ and a ring homomorphism ``φ : 𝕜 → 𝕂`` this returns a triple
              F
 ```
 """
-function change_base(phi::Any, f::AbsSpecMor; 
-    domain_map::AbsSpecMor=change_base(phi, domain(f))[2],
-    codomain_map::AbsSpecMor=change_base(phi, codomain(f))[2]
+function base_change(phi::Any, f::AbsSpecMor; 
+    domain_map::AbsSpecMor=base_change(phi, domain(f))[2],
+    codomain_map::AbsSpecMor=base_change(phi, codomain(f))[2]
   )
   X = domain(f)
   Y = codomain(f)
diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
index a941387a97bb..45cbcd56bace 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl
@@ -204,13 +204,13 @@ end
 ########################################################################
 
 @doc raw"""
-    change_base(phi::Any, X::AbsSpec)
+    base_change(phi::Any, X::AbsSpec)
 
 For an affine scheme `X` over a `base_ring` ``𝕜`` and a morphism 
 ``φ : 𝕜 → 𝕂`` this computes ``Y = X × Spec(𝕂)`` and returns a pair 
 `(Y, psi)` where `psi` is the canonical map ``Y → X``.
 """
-function change_base(phi::Any, X::AbsSpec)
+function base_change(phi::Any, X::AbsSpec)
   kk = base_ring(X)
   kk_red = parent(phi(zero(kk)))
   R = OO(X)
diff --git a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
index c61a91d40384..3ce0c281bbaf 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
@@ -81,14 +81,14 @@ end
 ########################################################################
 # Base change
 ########################################################################
-function change_base(phi::Any, f::AbsCoveredSchemeMorphism;
-    domain_map::AbsCoveredSchemeMorphism=change_base(phi, domain(f))[2],
-    codomain_map::AbsCoveredSchemeMorphism=change_base(phi, codomain(f))[2]
+function base_change(phi::Any, f::AbsCoveredSchemeMorphism;
+    domain_map::AbsCoveredSchemeMorphism=base_change(phi, domain(f))[2],
+    codomain_map::AbsCoveredSchemeMorphism=base_change(phi, codomain(f))[2]
   )
   f_cov = covering_morphism(f)
   dom_cov = covering_morphism(domain_map)
   cod_cov = covering_morphism(codomain_map)
-  _, ff_cov_map, _ = change_base(phi, f_cov, domain_map=dom_cov, codomain_map=cod_cov)
+  _, ff_cov_map, _ = base_change(phi, f_cov, domain_map=dom_cov, codomain_map=cod_cov)
   X = domain(f)
   Y = codomain(f)
   XX = domain(domain_map)
diff --git a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl
index 7e8b605e2274..aa58ea906296 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl
@@ -42,9 +42,9 @@ end
 ########################################################################
 # Base change
 ########################################################################
-function change_base(phi::Any, X::AbsCoveredScheme)
+function base_change(phi::Any, X::AbsCoveredScheme)
   C = default_covering(X)
-  CC, f_CC = change_base(phi, C)
+  CC, f_CC = base_change(phi, C)
   XX = CoveredScheme(CC)
   return XX, CoveredSchemeMorphism(XX, X, f_CC)
 end
diff --git a/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
index 84a8a545a12c..ce9e7458930a 100644
--- a/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Covering/Morphisms/Methods.jl
@@ -52,9 +52,9 @@ end
 ########################################################################
 # Base change
 ########################################################################
-function change_base(phi::Any, f::CoveringMorphism;
-    domain_map::CoveringMorphism=change_base(phi, domain(f))[2],
-    codomain_map::CoveringMorphism=change_base(phi, codomain(f))[2]
+function base_change(phi::Any, f::CoveringMorphism;
+    domain_map::CoveringMorphism=base_change(phi, domain(f))[2],
+    codomain_map::CoveringMorphism=base_change(phi, codomain(f))[2]
   )
   D = domain(f)
   C = codomain(f)
@@ -66,7 +66,7 @@ function change_base(phi::Any, f::CoveringMorphism;
     V = codomain(f[U])
     g_V = first(maps_with_given_codomain(codomain_map, V)) # The result must be unique as it arises 
                                                            # from a base change.
-    _, ff, _ = change_base(phi, f[U], domain_map=domain_map[UU], codomain_map=g_V)
+    _, ff, _ = base_change(phi, f[U], domain_map=domain_map[UU], codomain_map=g_V)
     mor_dict[UU] = ff
   end
 
diff --git a/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl
index 06bd0d16e24a..e455603372ee 100644
--- a/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl
@@ -327,9 +327,9 @@ end
 ########################################################################
 # Base change
 ########################################################################
-function change_base(phi::Any, C::Covering)
+function base_change(phi::Any, C::Covering)
   U = patches(C)
-  patch_change = [change_base(phi, V) for V in U]
+  patch_change = [base_change(phi, V) for V in U]
 
   glueing_dict = IdDict{Tuple{AbsSpec, AbsSpec}, AbsGlueing}()
   for i in 1:length(U)
@@ -337,7 +337,7 @@ function change_base(phi::Any, C::Covering)
     for j in 1:length(U)
       (B, map_B) = patch_change[j]
       G = C[U[i], U[j]] # the glueing
-      GG = change_base(phi, G, patch_change1=map_A, patch_change2=map_B)
+      GG = base_change(phi, G, patch_change1=map_A, patch_change2=map_B)
       glueing_dict[A, B] = GG
     end
   end
diff --git a/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl b/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl
index 2097c28cadfd..62953afcf93c 100644
--- a/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Glueing/Methods.jl
@@ -149,21 +149,21 @@ function _compute_glueing_base_change(gd::BaseChangeGlueingData)
   XX = domain(pc1)
   YY = domain(pc2)
 
-  UU, map_U = change_base(phi, U, ambient_map=pc1)
-  VV, map_V = change_base(phi, V, ambient_map=pc2)
+  UU, map_U = base_change(phi, U, ambient_map=pc1)
+  VV, map_V = base_change(phi, V, ambient_map=pc2)
 
   # TODO: The following will not yet work for glueings along SpecOpens.
   U isa PrincipalOpenSubset && V isa PrincipalOpenSubset || error("base change not implemented for glueings along SpecOpens")
 
-  _, ff, _ = change_base(phi, f, domain_map=map_U, codomain_map=map_V)
-  _, gg, _ = change_base(phi, g, domain_map=map_V, codomain_map=map_U)
+  _, ff, _ = base_change(phi, f, domain_map=map_U, codomain_map=map_V)
+  _, gg, _ = base_change(phi, g, domain_map=map_V, codomain_map=map_U)
 
   return Glueing(XX, YY, ff, gg)
 end
 
-function change_base(phi::Any, G::AbsGlueing;
-    patch_change1::AbsSpecMor=change_base(phi, glueing_domains(G)[1])[2], # The base change morphism for 
-    patch_change2::AbsSpecMor=change_base(phi, glueing_domains(G)[2])[2]  # the two glueing patches
+function base_change(phi::Any, G::AbsGlueing;
+    patch_change1::AbsSpecMor=base_change(phi, glueing_domains(G)[1])[2], # The base change morphism for 
+    patch_change2::AbsSpecMor=base_change(phi, glueing_domains(G)[2])[2]  # the two glueing patches
   )
   gd = BaseChangeGlueingData{typeof(phi), typeof(G)}(phi, G, patch_change1, patch_change2)
   return LazyGlueing(domain(patch_change1), domain(patch_change2), _compute_glueing_base_change, gd)
diff --git a/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
index ec91904e93ee..dfe27b5dad36 100644
--- a/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Methods.jl
@@ -43,8 +43,8 @@ end
 # Base change
 ########################################################################
 
-function change_base(phi::Any, U::PrincipalOpenSubset;
-    ambient_map::AbsSpecMor=change_base(phi, ambient_scheme(U))[2] # the base change on the ambient scheme
+function base_change(phi::Any, U::PrincipalOpenSubset;
+    ambient_map::AbsSpecMor=base_change(phi, ambient_scheme(U))[2] # the base change on the ambient scheme
   )
   Y = domain(ambient_map)
   pbf = pullback(ambient_map)
diff --git a/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
index 7ecffde1aaec..0845420cea16 100644
--- a/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/SpecOpen/Objects/Methods.jl
@@ -147,8 +147,8 @@ end
 ########################################################################
 # Base change
 ########################################################################
-function change_base(phi::Any, U::SpecOpen;
-    ambient_map::AbsSpecMor=change_base(phi, ambient_scheme(U))[2] # the base change on the ambient scheme
+function base_change(phi::Any, U::SpecOpen;
+    ambient_map::AbsSpecMor=base_change(phi, ambient_scheme(U))[2] # the base change on the ambient scheme
   )
   Y = domain(ambient_map)
   pbf = pullback(ambient_map)
diff --git a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
index 697a790b40d3..cc4d961e7f73 100644
--- a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
@@ -194,10 +194,10 @@ end
   V = hypersurface_complement(X, x)
 
   kk, pr = quo(ZZ, 5)
-  IA2_red, phi1 = oscar.change_base(pr, IA2)
-  X_red, phi2 = oscar.change_base(pr, X)
-  U_red, phi3 = oscar.change_base(pr, U)
-  V_red, phi4 = oscar.change_base(pr, V)
+  IA2_red, phi1 = oscar.base_change(pr, IA2)
+  X_red, phi2 = oscar.base_change(pr, X)
+  U_red, phi3 = oscar.base_change(pr, U)
+  V_red, phi4 = oscar.base_change(pr, V)
 
   m1 = compose(inclusion_morphism(V_red, IA2_red), phi1);
   m2 = compose(phi4, inclusion_morphism(V, IA2));
@@ -205,16 +205,16 @@ end
 
   # Testing morphisms 
   inc_U = inclusion_morphism(V, X)
-  (a, b, c) = oscar.change_base(pr, inc_U)
+  (a, b, c) = oscar.base_change(pr, inc_U)
   @test compose(a, inc_U) == compose(b, c)
 
   # Behaviour for special types
   U = PrincipalOpenSubset(IA2, y)
-  UU, f = oscar.change_base(pr, U)
+  UU, f = oscar.base_change(pr, U)
   @test UU isa PrincipalOpenSubset
 
   W = SpecOpen(IA2, [x, y])
-  WW, f = oscar.change_base(pr, W)
+  WW, f = oscar.base_change(pr, W)
   @test WW isa SpecOpen
 end
 
diff --git a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
index 9277ad2cea51..dfd7bf2f36b8 100644
--- a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
+++ b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
@@ -221,7 +221,7 @@ end
 @testset "base change" begin
   kk, pr = quo(ZZ, 5)
   IP1 = covered_scheme(projective_space(ZZ, 1))
-  IP1_red, red_map = oscar.change_base(pr, IP1)
+  IP1_red, red_map = oscar.base_change(pr, IP1)
   
   IP2 = projective_space(ZZ, 2)
   S = homogeneous_coordinate_ring(IP2)
@@ -231,6 +231,6 @@ end
   II = IdealSheaf(IP2, I)
 
   inc_X = oscar.CoveredClosedEmbedding(IP2_cov, II)
-  (a, b, c) = oscar.change_base(pr, inc_X)
+  (a, b, c) = oscar.base_change(pr, inc_X)
   @test compose(a, inc_X) == compose(b, c)
 end
diff --git a/test/AlgebraicGeometry/Schemes/Glueing.jl b/test/AlgebraicGeometry/Schemes/Glueing.jl
index 119f2a6ec9d3..2b6eab060c76 100644
--- a/test/AlgebraicGeometry/Schemes/Glueing.jl
+++ b/test/AlgebraicGeometry/Schemes/Glueing.jl
@@ -128,7 +128,7 @@ end
   IP1 = covered_scheme(projective_space(ZZ, 1))
   C = default_covering(IP1)
   G = first(values(glueings(C)))
-  GG = oscar.change_base(pr, G)
+  GG = oscar.base_change(pr, G)
   @test underlying_glueing(GG) isa SimpleGlueing
 end
 

From a989799dc81f80c2c0218a9df6cf2fd1f7d9f166 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 11:46:46 +0200
Subject: [PATCH 11/14] Fix error.

---
 src/AlgebraicGeometry/Schemes/Methods.jl | 39 ++++++++++++++++++++++++
 1 file changed, 39 insertions(+)
 create mode 100644 src/AlgebraicGeometry/Schemes/Methods.jl

diff --git a/src/AlgebraicGeometry/Schemes/Methods.jl b/src/AlgebraicGeometry/Schemes/Methods.jl
new file mode 100644
index 000000000000..3245d252c45f
--- /dev/null
+++ b/src/AlgebraicGeometry/Schemes/Methods.jl
@@ -0,0 +1,39 @@
+@doc raw"""
+    base_change(phi::Any, X::Scheme)
+
+For a `Scheme` ``X`` over a `base_ring` ``𝕜`` and a map ``φ : 𝕜 → R`` 
+we compute ``X' = X ×ₖ Spec(R)`` and return a pair `(X', f)` where 
+``f : X' → X`` is the canonical morphism.
+
+!!! note
+We do not restrict `phi` to be a `Hecke.Map` so that one can also use 
+coercion, anonymous functions, etc. 
+"""
+function base_change(phi::Any, X::Scheme)
+  error("base_change not implemented for input of type $(typeof(X))")
+end
+
+@doc raw"""
+    base_change(phi::Any, f::SchemeMor;
+        domain_map::SchemeMor, codomain_map::SchemeMor
+      )
+
+For a morphism ``f : X → Y`` with both ``X`` and ``Y`` defined over a 
+`base_ring` ``𝕜`` and a map ``φ : 𝕜 → R`` return a triple `(a, F, b)` 
+where ``a : X' → X`` is the morphism from `base_change(phi, X)`, 
+``b : Y' → Y`` the one for ``Y``, and ``F : X' → Y'`` the induced 
+morphism on those fiber products.
+
+!!! note
+We do not restrict `phi` to be a `Hecke.Map` so that one can also use 
+coercion, anonymous functions, etc. 
+
+!!! note 
+The morphisms ``a`` and ``b`` can be passed as the optional arguments 
+`domain_map` and `codomain_map`, respectively. 
+"""
+function base_change(phi::Any, f::SchemeMor;
+    domain_map::SchemeMor, codomain_map::SchemeMor
+  )
+  error("base_change not implemented for input of type $(typeof(f))")
+end

From 19c918058c490f526624315b3fe8b5e442604a9c Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Mon, 15 May 2023 16:42:29 +0200
Subject: [PATCH 12/14] Fix docstrings.

---
 src/AlgebraicGeometry/Schemes/Methods.jl | 9 +++------
 1 file changed, 3 insertions(+), 6 deletions(-)

diff --git a/src/AlgebraicGeometry/Schemes/Methods.jl b/src/AlgebraicGeometry/Schemes/Methods.jl
index 3245d252c45f..a4205bd60da1 100644
--- a/src/AlgebraicGeometry/Schemes/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/Methods.jl
@@ -6,8 +6,7 @@ we compute ``X' = X ×ₖ Spec(R)`` and return a pair `(X', f)` where
 ``f : X' → X`` is the canonical morphism.
 
 !!! note
-We do not restrict `phi` to be a `Hecke.Map` so that one can also use 
-coercion, anonymous functions, etc. 
+    We do not restrict `phi` to be a `Hecke.Map` so that one can also use coercion, anonymous functions, etc. 
 """
 function base_change(phi::Any, X::Scheme)
   error("base_change not implemented for input of type $(typeof(X))")
@@ -25,12 +24,10 @@ where ``a : X' → X`` is the morphism from `base_change(phi, X)`,
 morphism on those fiber products.
 
 !!! note
-We do not restrict `phi` to be a `Hecke.Map` so that one can also use 
-coercion, anonymous functions, etc. 
+    We do not restrict `phi` to be a `Hecke.Map` so that one can also use coercion, anonymous functions, etc. 
 
 !!! note 
-The morphisms ``a`` and ``b`` can be passed as the optional arguments 
-`domain_map` and `codomain_map`, respectively. 
+    The morphisms ``a`` and ``b`` can be passed as the optional arguments `domain_map` and `codomain_map`, respectively. 
 """
 function base_change(phi::Any, f::SchemeMor;
     domain_map::SchemeMor, codomain_map::SchemeMor

From e6ff891b64f6058e31c55a3dee12f30261ca0154 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Wed, 17 May 2023 11:06:28 +0200
Subject: [PATCH 13/14] Remove oscar. stuff.

---
 test/AlgebraicGeometry/Schemes/AffineSchemes.jl | 14 +++++++-------
 test/AlgebraicGeometry/Schemes/CoveredScheme.jl |  4 ++--
 test/AlgebraicGeometry/Schemes/Glueing.jl       |  2 +-
 3 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
index cc4d961e7f73..ecf5bc594f10 100644
--- a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
@@ -194,10 +194,10 @@ end
   V = hypersurface_complement(X, x)
 
   kk, pr = quo(ZZ, 5)
-  IA2_red, phi1 = oscar.base_change(pr, IA2)
-  X_red, phi2 = oscar.base_change(pr, X)
-  U_red, phi3 = oscar.base_change(pr, U)
-  V_red, phi4 = oscar.base_change(pr, V)
+  IA2_red, phi1 = base_change(pr, IA2)
+  X_red, phi2 = base_change(pr, X)
+  U_red, phi3 = base_change(pr, U)
+  V_red, phi4 = base_change(pr, V)
 
   m1 = compose(inclusion_morphism(V_red, IA2_red), phi1);
   m2 = compose(phi4, inclusion_morphism(V, IA2));
@@ -205,16 +205,16 @@ end
 
   # Testing morphisms 
   inc_U = inclusion_morphism(V, X)
-  (a, b, c) = oscar.base_change(pr, inc_U)
+  (a, b, c) = base_change(pr, inc_U)
   @test compose(a, inc_U) == compose(b, c)
 
   # Behaviour for special types
   U = PrincipalOpenSubset(IA2, y)
-  UU, f = oscar.base_change(pr, U)
+  UU, f = base_change(pr, U)
   @test UU isa PrincipalOpenSubset
 
   W = SpecOpen(IA2, [x, y])
-  WW, f = oscar.base_change(pr, W)
+  WW, f = base_change(pr, W)
   @test WW isa SpecOpen
 end
 
diff --git a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
index dfd7bf2f36b8..4de1758fa16d 100644
--- a/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
+++ b/test/AlgebraicGeometry/Schemes/CoveredScheme.jl
@@ -221,7 +221,7 @@ end
 @testset "base change" begin
   kk, pr = quo(ZZ, 5)
   IP1 = covered_scheme(projective_space(ZZ, 1))
-  IP1_red, red_map = oscar.base_change(pr, IP1)
+  IP1_red, red_map = base_change(pr, IP1)
   
   IP2 = projective_space(ZZ, 2)
   S = homogeneous_coordinate_ring(IP2)
@@ -231,6 +231,6 @@ end
   II = IdealSheaf(IP2, I)
 
   inc_X = oscar.CoveredClosedEmbedding(IP2_cov, II)
-  (a, b, c) = oscar.base_change(pr, inc_X)
+  (a, b, c) = base_change(pr, inc_X)
   @test compose(a, inc_X) == compose(b, c)
 end
diff --git a/test/AlgebraicGeometry/Schemes/Glueing.jl b/test/AlgebraicGeometry/Schemes/Glueing.jl
index 2b6eab060c76..8ce72dd104f9 100644
--- a/test/AlgebraicGeometry/Schemes/Glueing.jl
+++ b/test/AlgebraicGeometry/Schemes/Glueing.jl
@@ -128,7 +128,7 @@ end
   IP1 = covered_scheme(projective_space(ZZ, 1))
   C = default_covering(IP1)
   G = first(values(glueings(C)))
-  GG = oscar.base_change(pr, G)
+  GG = base_change(pr, G)
   @test underlying_glueing(GG) isa SimpleGlueing
 end
 

From 26e292aa76ece05c28770e2ead150acaa219e1a5 Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Wed, 17 May 2023 11:08:24 +0200
Subject: [PATCH 14/14] Add comment about optional arguments.

---
 .../Schemes/AffineSchemes/Morphisms/Methods.jl                 | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
index 467f99c1a737..692ec677cdc6 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl
@@ -138,6 +138,9 @@ and a ring homomorphism ``φ : 𝕜 → 𝕂`` this returns a triple
     ↑ b₁          ↑ b₂
   X×ₖSpec(𝕂) → Y×ₖSpec(𝕂)
              F
+
+The optional arguments `domain_map` and `codomain_map` can be used 
+to specify the morphisms `b₁` and `b₂`, respectively. 
 ```
 """
 function base_change(phi::Any, f::AbsSpecMor;