Prefer symbols over strings as indeterminants

docs/src/AlgebraicGeometry/Schemes/AffineSchemes.md

@@ -29,7 +29,7 @@ See [`inclusion_morphism(::AbsAffineScheme, ::AbsAffineScheme)`](@ref) for a way
 ### Affine n-space
 
 ```@docs
-affine_space(kk::BRT, n::Int; variable_name="x") where {BRT<:Ring}
+affine_space(kk::BRT, n::Int; variable_name=:x) where {BRT<:Ring}
 affine_space(kk::BRT, var_names::AbstractVector{<:VarName}) where {BRT<:Ring}
 ```
 

docs/src/DeveloperDocumentation/styleguide.md

@@ -37,6 +37,11 @@ deviate from them in some cases; in that case just do so.
   matrices...
 - Input sanity checks should be enabled by default, they can then be disabled
   internally if they are known to be true, and manually by users.
+- All user-facing functions that expect some kind of indeterminant name etc.
+  (like `polynomial_ring(QQ, <indeterminant_name>)`) should accept a
+  `VarName = Union{Symbol, Char, String}`, and convert it to a symbol for internal
+  handling. Library and test code should (if possible) call such functions with
+  `Symbol` arguments, as this is the most efficient way.
 
 
 ## Naming conventions

experimental/BasisLieHighestWeight/src/WeylPolytope.jl

@@ -116,7 +116,7 @@ function compute_zero_coordinates(
   zero_coordinates = Int[]
   for c in n:-1:1
     length(non_zeros) == m && break
-    if !isempty(intersect(non_zeros, findall(!iszero, bir_sequence.weights_alpha[c])))
+    if !isdisjoint(non_zeros, findall(!iszero, bir_sequence.weights_alpha[c]))
       union!(non_zeros, findall(<(0), bir_sequence.weights_w[c]))
     else
       push!(zero_coordinates, c)

experimental/FTheoryTools/src/AbstractFTheoryModels/methods.jl

@@ -334,7 +334,7 @@ function put_over_concrete_base(m::AbstractFTheoryModel, concrete_data::Dict{Str
     all_appearing_exponents = hcat([collect(exponents(m))[1] for m in all_appearing_monomials]...)
     for k in 1:nrows(all_appearing_exponents)
       if any(!is_zero, all_appearing_exponents[k,:])
-        gen_name = string(gens(parent(polys[1]))[k])
+        gen_name = string(symbols(parent(polys[1]))[k])
         @req haskey(concrete_data, gen_name) "Required base section $gen_name not specified"
         @req parent(concrete_data[gen_name]) == cox_ring(concrete_data["base"]) "Specified sections must reside in Cox ring of given base"
         new_model_secs[gen_name] = concrete_data[gen_name]

experimental/FTheoryTools/src/HypersurfaceModels/constructors.jl

@@ -53,9 +53,9 @@ end
 
 function hypersurface_model(base::NormalToricVariety, fiber_ambient_space::NormalToricVariety, fiber_twist_divisor_classes::Vector{ToricDivisorClass}, p::String; completeness_check::Bool = true)
   # Consistency checks
-  gens_base_names = [string(g) for g in gens(cox_ring(base))]
-  gens_fiber_names = [string(g) for g in gens(cox_ring(fiber_ambient_space))]
-  if intersect(Set(gens_base_names), Set(gens_fiber_names)) != Set()
+  gens_base_names = symbols(cox_ring(base))
+  gens_fiber_names = symbols(cox_ring(fiber_ambient_space))
+  if !isdisjoint(gens_base_names, gens_fiber_names)
     @vprint :FTheoryModelPrinter 0 "Variable names duplicated between base and fiber coordinates.\n"
   end
   if completeness_check
@@ -183,7 +183,7 @@ function hypersurface_model(auxiliary_base_vars::Vector{String}, auxiliary_base_
 
   # Conduct simple consistency checks
   @req d > 0 "The dimension of the base space must be positive"
-  @req intersect(set_base_vars, set_fiber_vars) == Set() "Variable names duplicated between base and fiber coordinates."
+  @req isdisjoint(set_base_vars, set_fiber_vars) "Variable names duplicated between base and fiber coordinates."
   @req union(set_base_vars, set_fiber_vars) == set_p_vars "Variables names for polynomial p do not match variable choice for base and fiber"
   @req ncols(auxiliary_base_grading) == length(auxiliary_base_vars) "Number of base variables does not match the number of provided base gradings"
 

experimental/FTheoryTools/src/HypersurfaceModels/methods.jl

@@ -120,7 +120,7 @@ function tune(h::HypersurfaceModel, input_sections::Dict{String, <:Any}; complet
   # 2. Compute the new hypersurface equation
   parametrized_hypersurface_equation = hypersurface_equation_parametrization(h)
   R = parent(parametrized_hypersurface_equation)
-  vars = [string(k) for k in gens(R)]
+  vars = [string(k) for k in symbols(R)]
   S = cox_ring(ambient_space(h))
   images = [k in secs_names ? eval_poly(string(explicit_secs[k]), S) : k == "Kbar" ? eval_poly("0", S) : eval_poly(k, S) for k in vars]
   map = hom(R, S, images; check=false)

experimental/FTheoryTools/src/LiteratureModels/constructors.jl

@@ -525,7 +525,7 @@ function _construct_literature_model_over_arbitrary_base(model_dict::Dict{String
   elseif model_dict["model_descriptors"]["type"] == "hypersurface"
 
     # Extract base variable names
-    auxiliary_base_vars = [string(g) for g in gens(auxiliary_base_ring)]
+    auxiliary_base_vars = [string(g) for g in symbols(auxiliary_base_ring)]
 
     # Extract fiber ambient space
     rays = [[a for a in b] for b in model_dict["model_data"]["fiber_ambient_space_rays"]]

experimental/FTheoryTools/src/TateModels/constructors.jl

@@ -61,7 +61,7 @@ function global_tate_model(base::NormalToricVariety,
   vs2 = collect(keys(defining_section_parametrization))
   @req all(in(["a1", "a2", "a3", "a4", "a6"]), vs2) "Only the Tate sections a1, a2, a3, a4, a6 must be parametrized"
 
-  gens_base_names = [string(g) for g in gens(cox_ring(base))]
+  gens_base_names = [string(g) for g in symbols(cox_ring(base))]
   if ("x" in gens_base_names) || ("y" in gens_base_names) || ("z" in gens_base_names)
     @vprint :FTheoryModelPrinter 0 "Variable names duplicated between base and fiber coordinates.\n"
   end
@@ -144,7 +144,7 @@ Global Tate model over a not fully specified base
 function global_tate_model(auxiliary_base_ring::MPolyRing, auxiliary_base_grading::Matrix{Int64}, d::Int, ais::Vector{T}) where {T<:MPolyRingElem}
 
   # Execute consistency checks
-  gens_base_names = [string(g) for g in gens(auxiliary_base_ring)]
+  gens_base_names = [string(g) for g in symbols(auxiliary_base_ring)]
   @req length(ais) == 5 "We expect exactly 5 Tate sections"
   @req all(k -> parent(k) == auxiliary_base_ring, ais) "All Tate sections must reside in the provided auxiliary base ring"
   @req d > 0 "The dimension of the base space must be positive"
@@ -170,7 +170,7 @@ function global_tate_model(auxiliary_base_ring::MPolyRing, auxiliary_base_gradin
 
   # Compute defining_section_parametrization
   defining_section_parametrization = Dict{String, MPolyRingElem}()
-  vars_S = [string(k) for k in gens(S)]
+  vars_S = [string(k) for k in symbols(S)]
   if !("a1" in vars_S) || (a1 != eval_poly("a1", parent(a1)))
     defining_section_parametrization["a1"] = a1
   end

experimental/FTheoryTools/src/TateModels/methods.jl

@@ -184,7 +184,7 @@ function tune(t::GlobalTateModel, input_sections::Dict{String, <:Any}; completen
   if !isempty(parametrization_keys) && !isempty(secs_names)
     R = parent(def_secs_param[parametrization_keys[1]])
     S = parent(explicit_secs[secs_names[1]])
-    vars = [string(k) for k in gens(R)]
+    vars = [string(k) for k in symbols(R)]
     images = [k in secs_names ? explicit_secs[k] : k == "Kbar" ? eval_poly("0", S) : eval_poly(k, S) for k in vars]
     map = hom(R, S, images)
     for section in tate_sections

experimental/FTheoryTools/src/WeierstrassModels/constructors.jl

@@ -55,8 +55,8 @@ function weierstrass_model(base::NormalToricVariety,
   vs2 = collect(keys(defining_section_parametrization))
   @req all(in(("f", "g")), vs2) "Only the Weierstrass sections f, g must be parametrized"
 
-  gens_base_names = [string(g) for g in gens(cox_ring(base))]
-  if ("x" in gens_base_names) || ("y" in gens_base_names) || ("z" in gens_base_names)
+  gens_base_names = symbols(cox_ring(base))
+  if (:x in gens_base_names) || (:y in gens_base_names) || (:z in gens_base_names)
     @vprint :FTheoryModelPrinter 0 "Variable names duplicated between base and fiber coordinates.\n"
   end
 
@@ -125,7 +125,7 @@ Weierstrass model over a not fully specified base
 function weierstrass_model(auxiliary_base_ring::MPolyRing, auxiliary_base_grading::Matrix{Int64}, d::Int, weierstrass_f::MPolyRingElem, weierstrass_g::MPolyRingElem)
 
   # Execute consistency checks
-  gens_base_names = [string(g) for g in gens(auxiliary_base_ring)]
+  gens_base_names = [string(g) for g in symbols(auxiliary_base_ring)]
   @req ((parent(weierstrass_f) == auxiliary_base_ring) && (parent(weierstrass_g) == auxiliary_base_ring)) "All Weierstrass sections must reside in the provided auxiliary base ring"
   @req d > 0 "The dimension of the base space must be positive"
   if ("x" in gens_base_names) || ("y" in gens_base_names) || ("z" in gens_base_names)
@@ -150,7 +150,7 @@ function weierstrass_model(auxiliary_base_ring::MPolyRing, auxiliary_base_gradin
 
   # Compute defining_section_parametrization
   defining_section_parametrization = Dict{String, MPolyRingElem}()
-  vars_S = [string(k) for k in gens(S)]
+  vars_S = [string(k) for k in symbols(S)]
   if !("f" in vars_S) || (f != eval_poly("f", parent(f)))
     defining_section_parametrization["f"] = f
   end

experimental/FTheoryTools/src/WeierstrassModels/methods.jl

@@ -107,7 +107,7 @@ function tune(w::WeierstrassModel, input_sections::Dict{String, <:Any}; complete
   if !isempty(parametrization_keys) && !isempty(secs_names)
     R = parent(def_secs_param[parametrization_keys[1]])
     S = parent(explicit_secs[secs_names[1]])
-    vars = [string(k) for k in gens(R)]
+    vars = [string(k) for k in symbols(R)]
     images = [k in secs_names ? explicit_secs[k] : k == "Kbar" ? eval_poly("0", S) : eval_poly(k, S) for k in vars]
     map = hom(R, S, images)
     for section in weierstrass_sections

experimental/FTheoryTools/src/auxiliary.jl

@@ -9,13 +9,13 @@
   b_rays = matrix(ZZ, rays(base))
   b_cones = matrix(ZZ, ray_indices(maximal_cones(base)))
   b_grades = reduce(vcat, [elem.coeff for elem in cox_ring(base).d])
-  b_var_names = [string(k) for k in gens(cox_ring(base))]
+  b_var_names = symbols(cox_ring(base))
 
   # Extract information about the fiber ambient space
   f_rays = matrix(ZZ, rays(fiber_amb_space))
   f_cones = matrix(ZZ, ray_indices(maximal_cones(fiber_amb_space)))
   f_grades = reduce(vcat, [elem.coeff for elem in cox_ring(fiber_amb_space).d])
-  f_var_names = [string(k) for k in gens(cox_ring(fiber_amb_space))]
+  f_var_names = symbols(cox_ring(fiber_amb_space))
 
   # Extract coefficients of divisors D1, D2 and compute u_matrix
   fiber_twist_divisor_classes_coeffs = [divisor_class(D).coeff for D in fiber_twist_divisor_classes]
@@ -179,8 +179,8 @@
 
     # Create new ring with auxiliary variable to construct the monodromy polynomial
     R = parent(f)
-    S, (_psi, ) = polynomial_ring(QQ, ["_psi"; [string(v) for v in gens(R)]], cached = false)
-    ring_map = hom(R, S, gens(S)[2:end])
+    S, (_psi,), _old_gens = polynomial_ring(QQ, [:_psi], symbols(R); cached = false)
+    ring_map = hom(R, S, _old_gens)
     poly_f = ring_map(f)
     poly_g = ring_map(g)
     locus = ring_map(gens(id)[1])
@@ -206,17 +206,17 @@
 
       # Get the grading matrix and the coordinates of the arbitrary base
       grading = weights(base_space(w))
-      base_coords = gens(coordinate_ring(base_space(w)))
-      @req (length(base_coords) == length(grading[1, :])) "The number of columns in the weight matrix does not match the number of base cooordinates"
+      base_coords_symbols = symbols(coordinate_ring(base_space(w)))
+      @req (length(base_coords_symbols) == length(grading[1, :])) "The number of columns in the weight matrix does not match the number of base cooordinates"
 
       # Choose explicit sections for all parameters of the model,
       # and then put the model over the concrete base using these data
-      concrete_data = merge(Dict(string(base_coords[i]) => generic_section(KBar^grading[1, i] * prod(hyperplane_bundle^grading[j, i] for j in 2:length(grading[:, 1]))) for i in eachindex(base_coords)), Dict("base" => concrete_base))
+      concrete_data = merge(Dict(string(base_coords_symbols[i]) => generic_section(KBar^grading[1, i] * prod(hyperplane_bundle^grading[j, i] for j in 2:length(grading[:, 1]))) for i in eachindex(base_coords_symbols)), Dict("base" => concrete_base))
       w = put_over_concrete_base(w, concrete_data)
 
       # We also need to determine the gauge locus over the new base
       # by using the explicit forms of all of the sections chosen above
-      list_of_sections = [concrete_data[string(base_coords[i])] for i in eachindex(base_coords)]
+      list_of_sections = [concrete_data[string(base_coords_symbols[i])] for i in eachindex(base_coords_symbols)]
       id = ideal([evaluate(p, list_of_sections) for p in gens(id)])
     end
 
@@ -315,7 +315,7 @@
   lin = ideal(map(hom(base_ring(lin), R, collect(1:ngens(R))), gens(lin)))
 
   # Create new base ring for the blown up ideal and a map between the rings
-  S, S_gens = polynomial_ring(QQ, [string("e_", index); [string("b_", index, "_", i) for i in 1:center_size]; [string(v) for v in gens(R)]], cached = false)
+  S, S_gens = polynomial_ring(QQ, [Symbol("e_", index); [Symbol("b_", index, "_", i) for i in 1:center_size]; symbols(R)], cached = false)
   (_e, new_coords...) = S_gens[1:center_size + 1]
   ring_map = hom(R, S, S_gens[center_size + 2:end])
 
@@ -471,7 +471,7 @@
   _e = gen(S, index_of_new_ray(bd))
   images = MPolyRingElem[]
   g_list = gens(S)
-  g_center = [string(k) for k in gens(ideal_in_cox_ring(center_unnormalized(bd)))]
+  g_center = [string(k) for k in symbols(ideal_in_cox_ring(center_unnormalized(bd)))]
   for v in g_list
     v == _e && continue
     if string(v) in g_center
@@ -490,7 +490,7 @@
 function _strict_transform(bd::ToricBlowupMorphism, tate_poly::MPolyRingElem)
   S = cox_ring(domain(bd))
   _e = gen(S, index_of_new_ray(bd))
-  g_list = string.(gens(S))
+  g_list = string.(symbols(S))
   g_center = [string(k) for k in gens(ideal_in_cox_ring(center_unnormalized(bd)))]
   position_of_center_variables = [findfirst(==(g), g_list) for g in g_center]
   pos_of_e = findfirst(==(string(_e)), g_list)

experimental/FTheoryTools/test/hypersurface_models.jl

@@ -16,7 +16,7 @@ h = hypersurface_model(B3, ambient_space_of_fiber, [D1, D2, D3], p; completeness
   @test dim(base_space(h)) == dim(B3)
   @test fiber_ambient_space(h) == ambient_space_of_fiber
   @test is_smooth(fiber_ambient_space(h)) == false
-  @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h)))] == ["x", "y", "z"]
+  @test symbols(cox_ring(fiber_ambient_space(h))) == [:x, :y, :z]
   @test toric_variety(calabi_yau_hypersurface(h)) == ambient_space(h)
   @test is_base_space_fully_specified(h) == true
 end
@@ -102,7 +102,7 @@ end
   @test dim(base_space(h3)) == 2
   @test is_smooth(fiber_ambient_space(h3)) == false
   @test is_simplicial(fiber_ambient_space(h3)) == true
-  @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h3)))] == ["u", "w", "v"]
+  @test symbols(cox_ring(fiber_ambient_space(h3))) == [:u, :w, :v]
   @test is_base_space_fully_specified(h3) == true
   @test is_partially_resolved(h3) == false
 end
@@ -131,7 +131,7 @@ h4 = hypersurface_model(auxiliary_base_vars, auxiliary_base_grading, d, ambient_
   @test fiber_ambient_space(h4) == ambient_space_of_fiber_2
   @test is_smooth(fiber_ambient_space(h4)) == false
   @test is_simplicial(fiber_ambient_space(h4)) == true
-  @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h4)))] == ["x", "y", "z"]
+  @test symbols(cox_ring(fiber_ambient_space(h4))) == [:x, :y, :z]
   @test is_base_space_fully_specified(h4) == false
   @test is_partially_resolved(h4) == false
 end
@@ -148,7 +148,7 @@ h5 = literature_model(arxiv_id = "1208.2695", equation = "B.5")
   @test dim(base_space(h5)) == 2
   @test is_smooth(fiber_ambient_space(h5)) == false
   @test is_simplicial(fiber_ambient_space(h5)) == true
-  @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h5)))] == ["u", "w", "v"]
+  @test symbols(cox_ring(fiber_ambient_space(h5))) == [:u, :w, :v]
   @test is_base_space_fully_specified(h5) == false
   @test is_partially_resolved(h5) == false
 end

experimental/FTheoryTools/test/literature_models.jl

@@ -363,7 +363,7 @@ h = literature_model(arxiv_id = "1507.05954", equation = "3.4")
   @test parent(hypersurface_equation(h)) == coordinate_ring(ambient_space(h))
   @test dim(base_space(h)) == 2
   @test is_smooth(fiber_ambient_space(h)) == true
-  @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h)))] == ["u", "v", "w"]
+  @test symbols(cox_ring(fiber_ambient_space(h))) == [:u, :v, :w]
   @test is_base_space_fully_specified(h) == false
   @test is_partially_resolved(h) == false
   @test string.(zero_section(h)) == ["0", "-b1", "a1"]

experimental/GroebnerWalk/examples/ku10.jl

@@ -1,7 +1,7 @@
 #ku
 using Oscar
 
-R, (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = polynomial_ring(QQ, ["x$index" for index in 1:10])
+R, (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = polynomial_ring(QQ, "x#" => 1:10)
 
 I = ideal([
   5*x1*x2+ 5*x1+ 3*x2+ 55,

experimental/GroebnerWalk/examples/zero-dimensional/cyclic7.jl

@@ -1,7 +1,7 @@
 #cyclic7
 
 using Oscar
-R, (z0, z1, z2, z3, z4, z5, z6) = polynomial_ring(QQ, ["z$index" for index in  0:6 ])
+R, (z0, z1, z2, z3, z4, z5, z6) = polynomial_ring(QQ, "z#" => 0:6)
 
 I = ideal([
   z0 + z1 + z2 + z3 + z4 + z5 + z6,

experimental/IntersectionTheory/src/Bott.jl

@@ -125,7 +125,7 @@
 function _get_ring(F::TnBundle)
   if get_attribute(F, :R) === nothing
     r = min(F.parent.dim, F.rank)
-    R, _ = graded_polynomial_ring(QQ, _parse_symbol("c", 1:r), collect(1:r))
+    R, _ = graded_polynomial_ring(QQ, :c => 1:r, collect(1:r))
     set_attribute!(R, :abstract_variety_dim => F.parent.dim)
     set_attribute!(F, :R => R)
   end
@@ -168,7 +168,7 @@
 # utility function that parses the weight specification
 function _parse_weight(n::Int, w)
   w == :int && return ZZ.(collect(1:n))
-  w == :poly && return polynomial_ring(QQ, ["u$i" for i in 1:n])[2]
+  w == :poly && return polynomial_ring(QQ, "u#" => 1:n)[2]
   if (w isa AbstractUnitRange) w = collect(w) end
   w isa Vector && length(w) == n && return w
   error("incorrect specification for weights")

experimental/IntersectionTheory/src/Main.jl

@@ -1295,7 +1295,7 @@ function schur_functor(F::AbstractBundle, λ::Partition)
   λ = conjugate(λ)
   X = F.parent
   w = _wedge(sum(λ), chern_character(F))
-  S, ei = polynomial_ring(QQ, ["e$i" for i in 1:length(w)])
+  S, ei = polynomial_ring(QQ, "e#" => 1:length(w))
   e = i -> i < 0 ? S() : ei[i+1]
   M = [e(λ[i]-i+j) for i in 1:length(λ), j in 1:length(λ)]
   sch = det(matrix(S, M)) # Jacobi-Trudi

experimental/LinearQuotients/src/cox_rings.jl

@@ -457,7 +457,7 @@ function minimal_parts(I::MPolyIdeal, w::Vector{ZZRingElem})
   @assert nvars(R) == length(w)
 
   w1 = push!(copy(w), ZZRingElem(-1))
-  S, t = graded_polynomial_ring(K, ["t$i" for i in 1:(nvars(R) + 1)], w1)
+  S, t = graded_polynomial_ring(K, "t#" => 1:(nvars(R) + 1), w1)
 
   Ihom = homogenize_at_last_variable(I, S)
 
@@ -487,7 +487,7 @@ function g_homogeneous_ideal(I::MPolyIdeal, weights::Vector{ZZRingElem}, order::
   R = base_ring(I)
 
   w = push!(copy(weights), ZZRingElem(1))
-  S, t = graded_polynomial_ring(coefficient_ring(R), ["t$i" for i in 1:(nvars(R) + 1)], w)
+  S, t = graded_polynomial_ring(coefficient_ring(R), "t#" => 1:(nvars(R) + 1), w)
 
   Ihom = homogenize_at_last_variable(I, S)
 

experimental/LinearQuotients/src/linear_quotients.jl

@@ -161,7 +161,7 @@ function weights_of_action(
   end
 
   to_eig = right_action(R, inv(V))
-  S, t = graded_polynomial_ring(K, ["t$i" for i in 1:ngens(R)], weights)
+  S, t = graded_polynomial_ring(K, "t#" => 1:ngens(R), weights)
   # The images of the generators of R are in general not homogeneous in S, so
   # we have to turn of the check, if we want to build this map...
   RtoS = hom(R, S, [to_eig(x)(t...) for x in gens(R)]; check=false)

experimental/LinearQuotients/src/misc.jl

@@ -125,7 +125,7 @@ function homogenize_at_last_variable(I::MPolyIdeal, S::MPolyDecRing)
   Rp = polynomial_ring(coefficient_ring(R), :t => 1:ngens(R))[1]
   RtoRp = hom(R, Rp, [gen(Rp, findfirst(isequal(i), p)) for i in 1:ngens(Rp)])
 
-  Sp, _ = graded_polynomial_ring(coefficient_ring(S), ["t$i" for i in 1:ngens(S)], w_perm)
+  Sp, _ = graded_polynomial_ring(coefficient_ring(S), "t#" => 1:ngens(S), w_perm)
   SptoS = hom(Sp, S, [gens(S)[p[i]] for i in 1:ngens(S)])
 
   # Homogenize the generators

experimental/Schemes/src/CoveredProjectiveSchemes.jl

@@ -362,7 +362,7 @@ end
 #  PW = projective_space(W, var_names)
 #  PWC = affine_cone(PW)
 #  prW = projection_to_base(PW)
-#  WA1, pW, pA = product(W, affine_space(base_ring(base_ring(OO(W))), 1, variable_name="t"))
+#  WA1, pW, pA = product(W, affine_space(base_ring(base_ring(OO(W))), 1, variable_name="t#"))
 #  t = pullback(pA)(OO(codomain(pA))(base_ring(OO(codomain(pA)))[1]))
 #  imgs = vcat((x->t*x).(pullback(pW).(gens(I))), pullback(pW).(gens(base_ring(OO(W)))))
 #  inner_phi = hom(base_ring(OO(PWC)), OO(WA1), imgs)