Make Cox rings of linear quotients work in greater generality (oscar-…

…system#2669)

experimental/LinearQuotients/README.md

@@ -19,8 +19,6 @@ with the spectrum of the invariant ring. Following the (more or less implicit)
 convention in the invariant theory code, this means that we actually construct the
 linear quotient $V^\ast/G$, that is, we work with the dual representation.
 We should decide, whether this is as desired.
-* The function `cox_ring(::LinearQuotient)` should work for arbitrary groups, not just
-for groups not containing reflections
 * The function `cox_ring_of_qq_factorial_terminalization(::LinearQuotient)` should work
 for arbitrary subgroups of $\operatorname{SL}_n$, not just for groups generated by
 junior elements

experimental/LinearQuotients/docs/src/cox_rings.md

@@ -9,8 +9,6 @@ CurrentModule = Oscar
 By a theorem of Arzhantsev and Gaifullin [AG10](@cite), the Cox ring of a linear
 quotient $V/G$ is graded isomorphic to the invariant ring $K[V]^{[G,G]}$, where
 $[G,G]$ is the derived subgroup of $G$.
-This functionality is so far only available if the group does not contain any
-reflections.
 ```@docs
 cox_ring(L::LinearQuotient)
 ```

experimental/LinearQuotients/docs/src/linear_quotients.md

@@ -26,7 +26,7 @@ linear_quotient(G::MatrixGroup)
 
 ## Class group
 
-The divisor class group of a linear quotient $V/G$ is controlled by the reflections
+The divisor class group of a linear quotient $V/G$ is controlled by the pseudo-reflections
 contained in the group $G$, see [Ben93](@cite).
 ```@docs
 class_group(L::LinearQuotient)

experimental/LinearQuotients/src/cox_rings.jl

@@ -195,27 +195,31 @@ end
 #
 ################################################################################
 
-# Let G < GL(V) be a finite group and GtoAbG the map G -> Ab(G) = G/[G, G].
-# The invariant ring K[V]^{[G,G]} has a well-defined action by Ab(G) and is hence
-# graded by the characters of Ab(G). The group of irreducible characters of the
-# abelian group Ab(G) is isomorphic to Ab(G), so we assume that K[V]^{[G, G]} is
-# graded by Ab(G). To choose a canonical isomorphism, we require a root of unity
-# of order at least exponent(Ab(G)). This is the input zeta: it is the root of
-# unity and its order (this is in the context of fixed_root_of_unity(::LinearQuotient)).
-# Return the degree of the polynomial f as an element of codomain(GtoAbG),
+# Let G < GL(V) be a finite group and GtoA a surjective map G -> A to an abelian
+# group A. Let H be the kernel of GtoA. The invariant ring K[V]^H has a
+# well-defined action by A and is hence graded by the characters of A. The group
+# of irreducible characters of the abelian group A is isomorphic to A, so we
+# assume that K[V]^H is graded by A. To choose a canonical isomorphism, we
+# require a root of unity of order at least exponent(A). This is the input zeta:
+# it is the root of unity and its order (this is in the context of
+# fixed_root_of_unity(::LinearQuotient)).
+# This is intended to be used in the situation, where f is an element of the Cox
+# ring of the linear quotient V/G and GtoA is the map from G to the class group
+# of V/G.
+# Return the degree of the polynomial f as an element of codomain(GtoA),
 # assuming that f is homogeneous.
-function ab_g_degree(GtoAbG::Map, f::MPolyRingElem, zeta::Tuple{<:FieldElem, Int})
-  AbG = codomain(GtoAbG)
+function ab_g_degree(GtoA::Map, f::MPolyRingElem, zeta::Tuple{<:FieldElem, Int})
+  A = codomain(GtoA)
   K = coefficient_ring(parent(f))
   @assert K === parent(zeta[1])
 
   powers_of_zeta = _powers_of_root_of_unity(zeta...)
   l = zeta[2]
 
-  c = zeros(ZZRingElem, ngens(AbG))
-  eldivs = elementary_divisors(AbG)
-  for i in 1:ngens(AbG)
-    fi = right_action(f, GtoAbG\AbG[i])
+  c = zeros(ZZRingElem, ngens(A))
+  eldivs = elementary_divisors(A)
+  for i in 1:ngens(A)
+    fi = right_action(f, GtoA\A[i])
     q, r = divrem(fi, f)
     @assert is_zero(r) "Polynomial is not homogeneous"
     z = first(AbstractAlgebra.coefficients(q))
@@ -224,7 +228,7 @@ function ab_g_degree(GtoAbG::Map, f::MPolyRingElem, zeta::Tuple{<:FieldElem, Int
     @assert is_integral(k)
     c[i] = numerator(k)
   end
-  return AbG(c)
+  return A(c)
 end
 
 @doc raw"""
@@ -233,17 +237,21 @@ end
 Return the Cox ring of the linear quotient `L` in a presentation as a graded affine
 algebra (`MPolyQuoRing`) and an injective map from this ring into a polynomial ring.
 
-By a theorem of Arzhantsev--Gaifullin [AG10](@cite) the Cox ring is graded isomorphic
-to the invariant ring of the derived subgroup of `group(L)`.
+Let `G = group(L)` and let `H` be the subgroup generated by the pseudo-reflections
+contained in `G`. By a theorem of Arzhantsev--Gaifullin [AG10](@cite), the Cox
+ring is graded isomorphic to the invariant ring of the group `H[G,G]`, where
+`[G,G]` is the derived subgroup of `G`.
 We use ideas from [DK17](@cite) to find homogeneous generators of the invariant ring.
 To get a map from `group(G)` to the grading group of the returned ring, use
 [`class_group`](@ref).
 """
 function cox_ring(L::LinearQuotient; algo_gens::Symbol = :default, algo_rels::Symbol = :groebner_basis)
   G = group(L)
-  is_small_group(G) || error("Only implemented for groups not containing reflections")
-
+  Grefl, GrefltoG = subgroup_of_pseudo_reflections(G)
   H, HtoG = derived_subgroup(G)
+  # Compute H*Grefl
+  H = matrix_group(base_ring(G), degree(G),
+                   vcat(map(HtoG, gens(H)), map(GrefltoG, gens(Grefl))))
   A, GtoA = class_group(L)
 
   RH = invariant_ring(H)
@@ -300,7 +308,7 @@ function cox_ring(L::LinearQuotient; algo_gens::Symbol = :default, algo_rels::Sy
   T = grade(T, degrees)[1]
 
   # The relations are in general not homogeneous
-  relsT = reduce(vcat, [ collect(values(homogeneous_components(T(f)))) for f in relsT ])
+  relsT = reduce(vcat, [ collect(values(homogeneous_components(T(f)))) for f in relsT ], init = elem_type(T)[])
   Q, TtoQ = quo(T, ideal(T, relsT))
   QtoR = hom(Q, Rgraded, [ Rgraded(f) for f in hom_invars ])
 

experimental/LinearQuotients/src/linear_quotients.jl

@@ -37,9 +37,19 @@ function fixed_root_of_unity(L::LinearQuotient)
     K isa AnticNumberField || throw(Hecke.NotImplemented())
     fl, l = Hecke.is_cyclotomic_type(K)
     fl || throw(Hecke.NotImplemented())
-    fl, q = divides(l, e)
+    if is_odd(l)
+      ll = 2l
+    else
+      ll = l
+    end
+    fl, q = divides(ll, e)
     fl || error("$(base_ring(L)) does not contain a $(e)-th root of unity")
-    L.root_of_unity = (gen(K)^q, e)
+    if is_odd(l)
+      zeta = (-gen(K))^q
+    else
+      zeta = gen(K)^q
+    end
+    L.root_of_unity = (zeta, e)
   end
   return L.root_of_unity
 end
@@ -57,15 +67,15 @@ Return the class group of the linear quotient `L` and a map from `group(L)` to
 this group.
 
 If `G = group(L)`, then the class group is `Ab(G/H)`, where `H` is the subgroup
-of `G` generated by the reflections.
+of `G` generated by the pseudo-reflections.
 """
 function class_group(L::LinearQuotient)
   if isdefined(L, :class_group)
     return L.class_group
   end
 
   G = group(L)
-  H = subgroup_of_reflections(G)
+  H, _ = subgroup_of_pseudo_reflections(G)
   if !is_trivial(H)
     K, GtoK = quo(G, H)
   else

experimental/LinearQuotients/src/misc.jl

@@ -1,29 +1,40 @@
-# Try to 'update' the base_ring of G
-function map_entries(K::Field, G::MatrixGroup)
-  g = dense_matrix_type(K)[]
-  for h in gens(G)
-    push!(g, map_entries(K, h.elm))
-  end
-  return matrix_group(g)
-end
+@doc raw"""
+    is_pseudo_reflection(g::MatrixGroupElem)
 
-function is_reflection(g::MatrixGroupElem)
+Return `true` if `g` is a pseudo-reflection, `false` otherwise. By a
+pseudo-reflection, we mean a matrix with fixed space of codimension 1.
+"""
+function is_pseudo_reflection(g::MatrixGroupElem)
   return rank(g.elm - one(parent(g)).elm) == 1
 end
 
-function subgroup_of_reflections(G::MatrixGroup)
+@doc raw"""
+    subgroup_of_pseudo_reflections(G::MatrixGroup)
+
+Return the subgroup `H` of `G` generated by the pseudo-reflections in `G` and an
+embedding `H \to G`.
+
+See also [`is_pseudo_reflection`](@ref).
+"""
+function subgroup_of_pseudo_reflections(G::MatrixGroup)
   g = elem_type(G)[]
   for c in conjugacy_classes(G)
-    if is_reflection(representative(c))
+    if is_pseudo_reflection(representative(c))
       append!(g, collect(c))
     end
   end
-  return matrix_group(base_ring(G), degree(G), g)
+  return sub(G, g, check = false)
 end
 
-# Check if G contains reflections
+@doc raw"""
+    is_small_group(G::MatrixGroup)
+
+Return `true` if `G` does not contain any pseudo-reflections and `false` otherwise.
+
+See also [`is_pseudo_reflection`](@ref).
+"""
 function is_small_group(G::MatrixGroup)
-  return !any(c -> is_reflection(representative(c)), conjugacy_classes(G))
+  return !any(c -> is_pseudo_reflection(representative(c)), conjugacy_classes(G))
 end
 
 # Somehow one needs to look up powers of a given root of unity quite often...
@@ -151,8 +162,3 @@ function homogenize_at_last_variable(I::MPolyIdeal, S::MPolyDecRing)
   end
   return SptoS(ideal(Sp, res))
 end
-
-function ideal(S::MPolyDecRing, I::MPolyIdeal)
-  @assert base_ring(I) === forget_grading(S)
-  return ideal(S, [ S(f) for f in gens(I) ])
-end

experimental/LinearQuotients/test/cox_rings-test.jl

@@ -50,6 +50,22 @@
   for i = 1:ngens(R)
     @test Oscar.ab_g_degree(GtoA, RtoS(gen(R, i)), Oscar.fixed_root_of_unity(L)) == degree(gen(R, i))
   end
+
+  # An example containing reflections
+  K, a = cyclotomic_field(6, "a")
+  g1 = matrix(K, 3, 3, [ -1 0 0; 0 1 0; 0 0 1 ])
+  g2 = matrix(K, 3, 3, [ 0 0 1; 1 0 0; 0 1 0 ])
+  G = matrix_group(g1, g2)
+  L = linear_quotient(G)
+  R, RtoS = cox_ring(L)
+  @test ngens(R) == 3
+  @test grading_group(R).snf == ZZRingElem[ 3 ]
+  A, GtoA = class_group(L)
+  @test A === grading_group(R)
+  for i = 1:ngens(R)
+    @test Oscar.ab_g_degree(GtoA, RtoS(gen(R, i)), Oscar.fixed_root_of_unity(L)) == degree(gen(R, i))
+  end
+  @test is_zero(modulus(R))
 end
 
 @testset "Cox rings of QQ-factorial terminalizations" begin

experimental/LinearQuotients/test/linear_quotients-test.jl

@@ -62,4 +62,16 @@
   @test val(x[2]) == 1
   @test val(x[1] + x[2]) == 4
   @test val(x[1] - x[2]) == 1
+
+  # An example containing pseudo-reflections
+  K, a = cyclotomic_field(6, "a")
+  g1 = matrix(K, 3, 3, [ -1 0 0; 0 1 0; 0 0 1 ])
+  g2 = matrix(K, 3, 3, [ 0 0 1; 1 0 0; 0 1 0 ])
+  G = matrix_group(g1, g2)
+  L = linear_quotient(G)
+  A, GtoA = class_group(L)
+  @test is_snf(A)
+  @test A.snf == ZZRingElem[ 3 ]
+  @test is_zero(GtoA(G(g1)))
+  @test GtoA(G(g2)) == A([ 1 ])
 end

src/Groups/matrices/MatGrp.jl

@@ -104,6 +104,13 @@ function Base.deepcopy_internal(x::MatrixGroupElem, dict::IdDict)
   error("$x has neither :X nor :elm")
 end
 
+function change_base_ring(R::Ring, G::MatrixGroup)
+  g = dense_matrix_type(R)[]
+  for h in gens(G)
+    push!(g, map_entries(R, h.elm))
+  end
+  return matrix_group(g)
+end
 
 ########################################################################
 #

src/Rings/mpoly-ideals.jl

@@ -54,6 +54,12 @@ function ideal(g::Vector{T}) where {T <: MPolyRingElem}
   return ideal(parent(g[1]), g)
 end
 
+# Coerce an ungraded ideal in a graded ring
+function ideal(S::MPolyDecRing, I::MPolyIdeal)
+  @req base_ring(I) === forget_grading(S) "Rings do not coincide"
+  return ideal(S, [ S(f) for f in gens(I) ])
+end
+
 function is_graded(I::MPolyIdeal)
   return is_graded(Hecke.ring(I))
 end

test/Groups/matrixgroups.jl

@@ -311,6 +311,11 @@ end
    @testset for x in gens(G)
        @test iseven(rank(matrix(x)-1))
    end
+
+   G = matrix_group(matrix(QQ, 2, 2, [ -1 0 ; 0 -1 ]))
+   K, _ = cyclotomic_field(4)
+   H = change_base_ring(K, G)
+   @test H == matrix_group(matrix(K, 2, 2, [ -1 0 ; 0 -1 ]))
 end
 
 

test/Rings/mpoly-graded.jl

@@ -323,6 +323,10 @@ end
   I = ideal(S, [ f ])
   @test forget_decoration(I) == ideal(R, [ x + y ])
   @test forget_grading(I) == ideal(R, [ x + y ])
+  @test ideal(S, forget_decoration(I)) == I
+
+  T, _ = graded_polynomial_ring(QQ, [ "t" ], [ 1 ])
+  @test_throws ArgumentError ideal(T, forget_decoration(I))
 end
 
 @testset "Verify homogenization bugfix" begin