Change non-scheme proj functions to something else

For biproducts, naming the output triple
    [obj], inj, pr
seems to be common. Among the changes was changing "proj" to "pr" in the
triples
    [obj], inj, proj

docs/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties.md

@@ -84,7 +84,7 @@ normal_toric_varieties_from_glsm(charges::ZZMatrix)
 
 ```@docs
 Base.:*(v::NormalToricVarietyType, w::NormalToricVarietyType)
-proj(E::ToricLineBundle...)
+projectivization(E::ToricLineBundle...)
 total_space(E::ToricLineBundle...)
 ```
 

docs/src/CommutativeAlgebra/affine_algebras.md

@@ -382,7 +382,7 @@ julia> D2, (a, b, c) = graded_polynomial_ring(QQ, ["a", "b", "c"]);
 
 julia> V2 = [p2(w-y), p2(x), p2(z)];
 
-julia> proj = hom(D2, C2, V2)
+julia> pr = hom(D2, C2, V2)
 Ring homomorphism
   from graded multivariate polynomial ring in 3 variables over QQ
   to quotient of multivariate polynomial ring by ideal (-x*z + y^2, -w*z + x*y, -w*y + x^2)
@@ -391,7 +391,7 @@ defined by
   b -> x
   c -> z
 
-julia> nodalCubic = kernel(proj)
+julia> nodalCubic = kernel(pr)
 Ideal generated by
   -a^2*c + b^3 - 2*b^2*c + b*c^2
 

experimental/LieAlgebras/src/LieAlgebraModuleHom.jl

@@ -45,7 +45,7 @@ end
 
 @doc raw"""
     matrix(h::LieAlgebraModuleHom) -> MatElem
-    
+
 Return the transformation matrix of `h` w.r.t. the bases of the domain and codomain.
 
 Note: The matrix operates on the coefficient vectors from the right.
@@ -201,7 +201,7 @@ end
 @doc raw"""
     hom(V1::LieAlgebraModule, V2::LieAlgebraModule, imgs::Vector{<:LieAlgebraModuleElem}; check::Bool=true) -> LieAlgebraModuleHom
 
-Construct the homomorphism from `V1` to `V2` by sending the `i`-th basis element of `V1` 
+Construct the homomorphism from `V1` to `V2` by sending the `i`-th basis element of `V1`
 to `imgs[i]` and extending linearly.
 All elements of `imgs` must lie in `V2`.
 Currently, `V1` and `V2` must be modules over the same Lie algebra.
@@ -260,7 +260,7 @@ julia> h = hom(V1, V2, matrix(QQ, 3, 1, [0, 0, 0]))
 Lie algebra module morphism
   from standard module of dimension 3 over gl_3
   to abstract Lie algebra module of dimension 1 over gl_3
-  
+
 julia> [(v, h(v)) for v in basis(V1)]
 3-element Vector{Tuple{LieAlgebraModuleElem{QQFieldElem}, LieAlgebraModuleElem{QQFieldElem}}}:
  (v_1, 0)
@@ -403,7 +403,7 @@ function canonical_projection(V::LieAlgebraModule, i::Int)
   @req fl "Module must be a direct sum"
   @req 1 <= i <= length(Vs) "Index out of bound"
   j = sum(dim(Vs[l]) for l in 1:(i - 1); init=0)
-  proj = hom(
+  projection = hom(
     V,
     Vs[i],
     [
@@ -413,15 +413,15 @@ function canonical_projection(V::LieAlgebraModule, i::Int)
     ];
     check=false,
   )
-  return proj
+  return projection
 end
 
 @doc raw"""
     hom_direct_sum(V::LieAlgebraModule{C}, W::LieAlgebraModule{C}, hs::Matrix{<:LieAlgebraModuleHom}) -> LieAlgebraModuleHom
     hom_direct_sum(V::LieAlgebraModule{C}, W::LieAlgebraModule{C}, hs::Vector{<:LieAlgebraModuleHom}) -> LieAlgebraModuleHom
 
-Given modules `V` and `W` which are direct sums with `r` respective `s` summands,  
-say $M = M_1 \oplus \cdots \oplus M_r$, $N = N_1 \oplus \cdots \oplus N_s$, and given a $r \times s$ matrix 
+Given modules `V` and `W` which are direct sums with `r` respective `s` summands,
+say $M = M_1 \oplus \cdots \oplus M_r$, $N = N_1 \oplus \cdots \oplus N_s$, and given a $r \times s$ matrix
 `hs` of homomorphisms $h_{ij} : V_i \to W_j$, return the homomorphism
 $V \to W$ with $ij$-components $h_{ij}$.
 
@@ -471,7 +471,7 @@ end
 
 Given modules `V` and `W` which are tensor products with the same number of factors,
 say $V = V_1 \otimes \cdots \otimes V_r$, $W = W_1 \otimes \cdots \otimes W_r$,
-and given a vector `hs` of homomorphisms $a_i : V_i \to W_i$, return 
+and given a vector `hs` of homomorphisms $a_i : V_i \to W_i$, return
 $a_1 \otimes \cdots \otimes a_r$.
 
 This works for $r$th tensor powers as well.

experimental/LieAlgebras/test/LieAlgebraModuleHom-test.jl

@@ -119,7 +119,7 @@
       @test canonical_injection(V, i) * canonical_projection(V, i) == identity_map(Vs[i])
     end
     @test sum(
-      proj * inj for (proj, inj) in zip(canonical_projections(V), canonical_injections(V))
+      pr * inj for (pr, inj) in zip(canonical_projections(V), canonical_injections(V))
     ) == identity_map(V)
   end
 

experimental/QuadFormAndIsom/src/embeddings.jl

@@ -1005,8 +1005,8 @@ function _classes_isomorphic_subgroups(q::TorQuadModule,
     push!(blocks, compose(__f, j))
     push!(ni, ngens(domain(__f)))
   end
-  D, inj, proj = biproduct(domain.(blocks))
-  phi = hom(D, q, TorQuadModuleElem[sum([blocks[i](proj[i](a)) for i in 1:length(pds)]) for a in gens(D)])
+  D, inj, pr = biproduct(domain.(blocks))
+  phi = hom(D, q, TorQuadModuleElem[sum([blocks[i](pr[i](a)) for i in 1:length(pds)]) for a in gens(D)])
   @hassert :ZZLatWithIsom 1 is_isometry(phi)
 
   list_can = Vector{Tuple{TorQuadModuleMap, AutomorphismGroup{TorQuadModule}}}[]

experimental/QuadFormAndIsom/src/hermitian_miranda_morrison.jl

@@ -228,19 +228,20 @@ function _get_product_quotient(E::Hecke.RelSimpleNumField, Fac::Vector{Tuple{Abs
     push!(Ps, P)
   end
 
-  G, proj, inj = biproduct(groups...)
+  # Is this correct? The order is usually "G, inj, pr", not "G, pr, inj"
+  G, pr, inj = biproduct(groups...)
 
   function dlog(x::Vector{<:Hecke.RelSimpleNumFieldElem})
     if length(x) == 1
-      return sum(inj[i](dlogs[i](x[1])) for i in 1:length(Fac)) 
+      return sum(inj[i](dlogs[i](x[1])) for i in 1:length(Fac))
     else
       @hassert :ZZLatWithIsom 1 length(x) == length(Fac)
       return sum(inj[i](dlogs[i](x[i])) for i in 1:length(Fac))
     end
   end
 
   function exp(x::FinGenAbGroupElem)
-    v = elem_type(E)[exps[i](proj[i](x)) for i in 1:length(Fac)]
+    v = elem_type(E)[exps[i](pr[i](x)) for i in 1:length(Fac)]
     @hassert :ZZLatWithIsom 1 dlog(v) == x
     return v
   end
@@ -263,10 +264,10 @@ end
 # (D^{-1}L^#/L)_p is not unimodular.
 #
 # According to [BH23], the quotient D^{-1}L^#/L is unimodular at p
-# if and only if 
+# if and only if
 #  - either L is unimodular at p, and D and p are coprime
 #  - or L is P^{-a}-modular where P is largest prime ideal
-#    over p fixed the canonical involution, and a is the valuation of D at P. 
+#    over p fixed the canonical involution, and a is the valuation of D at P.
 
 function _elementary_divisors(L::HermLat, D::Hecke.RelNumFieldOrderIdeal)
   Ps = collect(keys(factor(D)))

experimental/QuadFormAndIsom/src/lattices_with_isometry.jl

@@ -177,7 +177,7 @@ minimal_polynomial(Lf::ZZLatWithIsom) = minimal_polynomial(isometry(Lf))
 @doc raw"""
     genus(Lf::ZZLatWithIsom) -> ZZGenus
 
-Given a lattice with isometry $(L, f)$, return the genus of the underlying 
+Given a lattice with isometry $(L, f)$, return the genus of the underlying
 lattice $L$.
 
 # Examples
@@ -1301,7 +1301,7 @@ Integer lattice of rank 5 and degree 5
   [0    0    1    0    0]
   [0    0    0    1    0]
 
-julia> Lh, proj = direct_product(Lf, Lg)
+julia> Lh, pr = direct_product(Lf, Lg)
 (Integer lattice with isometry of finite order 10, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])
 
 julia> Lh
@@ -1321,9 +1321,9 @@ Integer lattice of rank 10 and degree 10
 ```
 """
 function direct_product(x::Vector{ZZLatWithIsom})
-  Vf, proj = direct_product(ambient_space.(x))
+  Vf, pr = direct_product(ambient_space.(x))
   Bs = block_diagonal_matrix(basis_matrix.(x))
-  return lattice(Vf, Bs; check=false), proj
+  return lattice(Vf, Bs; check=false), pr
 end
 
 direct_product(x::Vararg{ZZLatWithIsom}) = direct_product(collect(x))
@@ -1385,7 +1385,7 @@ Integer lattice of rank 5 and degree 5
   [0    0    1    0    0]
   [0    0    0    1    0]
 
-julia> Lh, inj, proj = biproduct(Lf, Lg)
+julia> Lh, inj, pr = biproduct(Lf, Lg)
 (Integer lattice with isometry of finite order 10, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space], AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])
 
 julia> Lh
@@ -1403,14 +1403,14 @@ Integer lattice of rank 10 and degree 10
   [ 0    0    0    0    0   0    0    1    0    0]
   [ 0    0    0    0    0   0    0    0    1    0]
 
-julia> matrix(compose(inj[1], proj[1]))
+julia> matrix(compose(inj[1], pr[1]))
 [1   0   0   0   0]
 [0   1   0   0   0]
 [0   0   1   0   0]
 [0   0   0   1   0]
 [0   0   0   0   1]
 
-julia> matrix(compose(inj[1], proj[2]))
+julia> matrix(compose(inj[1], pr[2]))
 [0   0   0   0   0]
 [0   0   0   0   0]
 [0   0   0   0   0]
@@ -1419,9 +1419,9 @@ julia> matrix(compose(inj[1], proj[2]))
 ```
 """
 function biproduct(x::Vector{ZZLatWithIsom})
-  Vf, inj, proj = biproduct(ambient_space.(x))
+  Vf, inj, pr = biproduct(ambient_space.(x))
   Bs = block_diagonal_matrix(basis_matrix.(x))
-  return lattice(Vf, Bs; check=false), inj, proj
+  return lattice(Vf, Bs; check=false), inj, pr
 end
 
 biproduct(x::Vararg{ZZLatWithIsom}) = biproduct(collect(x))
@@ -1529,7 +1529,7 @@ Integer lattice of rank 4 and degree 5
 julia> H = hermitian_structure(M)
 Hermitian lattice of rank 1 and degree 1
   over relative maximal order of Relative number field of degree 2 over maximal real subfield of cyclotomic field of order 5
-  with pseudo-basis 
+  with pseudo-basis
   (1, 1//1 * <1, 1>)
   (z_5, 1//1 * <1, 1>)
 
@@ -1592,11 +1592,11 @@ Finite quadratic module
 Abelian group: Z/6
 Bilinear value module: Q/Z
 Quadratic value module: Q/2Z
-Gram matrix quadratic form: 
+Gram matrix quadratic form:
 [5//6]
 
 julia> qf
-Isometry of Finite quadratic module: Z/6 -> Q/2Z defined by 
+Isometry of Finite quadratic module: Z/6 -> Q/2Z defined by
 [1]
 
 julia> f = matrix(QQ, 5, 5, [ 1  0  0  0  0;
@@ -1608,7 +1608,7 @@ julia> f = matrix(QQ, 5, 5, [ 1  0  0  0  0;
 julia> Lf = integer_lattice_with_isometry(L, f);
 
 julia> discriminant_group(Lf)[2]
-Isometry of Finite quadratic module: Z/6 -> Q/2Z defined by 
+Isometry of Finite quadratic module: Z/6 -> Q/2Z defined by
 [5]
 ```
 """

experimental/QuadFormAndIsom/src/spaces_with_isometry.jl

@@ -592,7 +592,7 @@ Quadratic space of dimension 2
   [1    1]
   [0   -1]
 
-julia> Vf3, proj = direct_product(Vf1, Vf2)
+julia> Vf3, pr = direct_product(Vf1, Vf2)
 (Quadratic space with isometry of finite order 2, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])
 
 julia> Vf3
@@ -615,9 +615,9 @@ with gram matrix
 ```
 """
 function direct_product(x::Vector{T}) where T <: QuadSpaceWithIsom
-  V, proj = direct_product(space.(x))
+  V, pr = direct_product(space.(x))
   f = block_diagonal_matrix(isometry.(x))
-  return quadratic_space_with_isometry(V, f; check = false), proj
+  return quadratic_space_with_isometry(V, f; check = false), pr
 end
 
 direct_product(x::Vararg{QuadSpaceWithIsom}) = direct_product(collect(x))
@@ -676,7 +676,7 @@ Quadratic space of dimension 2
   [1    1]
   [0   -1]
 
-julia> Vf3, inj, proj = biproduct(Vf1, Vf2)
+julia> Vf3, inj, pr = biproduct(Vf1, Vf2)
 (Quadratic space with isometry of finite order 2, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space], AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])
 
 julia> Vf3
@@ -697,19 +697,19 @@ with gram matrix
 [0   0    2   -1]
 [0   0   -1    2]
 
-julia> matrix(compose(inj[1], proj[1]))
+julia> matrix(compose(inj[1], pr[1]))
 [1   0]
 [0   1]
 
-julia> matrix(compose(inj[1], proj[2]))
+julia> matrix(compose(inj[1], pr[2]))
 [0   0]
 [0   0]
 ```
 """
 function biproduct(x::Vector{T}) where T <: QuadSpaceWithIsom
-  V, inj, proj = biproduct(space.(x))
+  V, inj, pr = biproduct(space.(x))
   f = block_diagonal_matrix(isometry.(x))
-  return quadratic_space_with_isometry(V, f; check = false), inj, proj
+  return quadratic_space_with_isometry(V, f; check = false), inj, pr
 end
 
 biproduct(x::Vararg{QuadSpaceWithIsom}) = biproduct(collect(x))

experimental/Schemes/ProjectiveModules.jl

@@ -33,13 +33,13 @@ function is_projective(M::SubquoModule; check::Bool=true)
   R = base_ring(M)
   r = ngens(M)
   Rr = FreeMod(R, r)
-  proj = hom(Rr, M, gens(M))
+  pr = hom(Rr, M, gens(M))
 
   if check
-    iszero(annihilator(M)) || return false, proj, nothing
+    iszero(annihilator(M)) || return false, pr, nothing
   end
 
-  K, inc = kernel(proj)
+  K, inc = kernel(pr)
   s = ngens(K)
   Rs = FreeMod(R, s)
   a = compose(hom(Rs, K, gens(K)), inc)