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ModulesGraded.jl
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ModulesGraded.jl
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using Random
RNG = Random.MersenneTwister(42)
##################################################################
# Tests graded modules
##################################################################
@testset "Graded free modules constructors" begin
R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg = grade(R,[Z[1], Z[1], Z[1]])[1]
M1 = graded_free_module(Rg, 2)
@test degrees(M1) == [Z[0], Z[0]]
M2 = graded_free_module(Rg, [2,2,4,3,3,5])
@test degrees(M2) == [2*Z[1], 2*Z[1], 4*Z[1], 3*Z[1], 3*Z[1], 5*Z[1]]
M3 = grade(M2,[2,2,4,3,3,6])
@test degrees(M3) == [2*Z[1], 2*Z[1], 4*Z[1], 3*Z[1], 3*Z[1], 6*Z[1]]
@test degrees_of_generators(M3) == [2*Z[1], 2*Z[1], 4*Z[1], 3*Z[1], 3*Z[1], 6*Z[1]]
@test is_graded(M1)
@test is_graded(M2)
@test is_graded(M3)
end
@testset "Graded free modules hom constructor" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, 3)
G = graded_free_module(Rg, 2)
V = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
a = hom(F, G, V)
@test is_graded(a)
@test domain(a) == F
@test codomain(a) == G
@test a(F[1]) == V[1]
W = Rg[y 0; x y; 0 z]
a = hom(F, G, W)
@test is_graded(a)
@test domain(a) == F
@test codomain(a) == G
@test a(F[1]) == y*G[1]
end
@testset "Kernel and image free modules hom" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, 3)
G = graded_free_module(Rg, 2)
V = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
a = hom(F, G, V)
Mk = kernel(a)
Mi = image(a)
@test is_graded(Mk[1])
@test is_graded(Mi[1])
@test ngens(Mk[1]) == 1
@test ngens(Mi[1]) == 3
@test codomain(Mk[2]) == F
@test codomain(Mi[2]) == G
end
@testset "Presentations 1" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, 3)
G = graded_free_module(Rg, 2)
V = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
a = hom(F, G, V)
Mk = kernel(a)
Mk1 = Mk[1]
p = presentation(Mk1)
@test all(is_graded(p[i]) for i in -2:1)
@test degrees(p[0]) == [2 * Z[1]]
Mp, f = present_as_cokernel(Mk1, :all)
@test is_graded(Mp)
@test is_graded(f)
@test degrees_of_generators(Mp) == [2 * Z[1]]
end
@testset "Degrees of free module homomorphisms" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, 3)
G = graded_free_module(Rg, 2)
V = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
a = hom(F, G, V)
@test degree(a) == Z[1]
@test is_homogeneous(a) == false
F = graded_free_module(Rg, [1,1,1])
G = graded_free_module(Rg, [0,0])
V = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
a = hom(F, G, V)
@test is_graded(a)
@test degree(a) == Z[0]
@test is_homogeneous(a)
end
@testset "Subquotient constructors" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, [1,1,1])
G = graded_free_module(Rg, [0,0])
B = Rg[y 0; x y; 0 z]
b = hom(F, G, B)
F1 = graded_free_module(Rg, [2,2,2])
F2 = graded_free_module(Rg, [2])
G = graded_free_module(Rg, [1,1])
V1 = [y*G[1], (x+y)*G[1]+y*G[2], z*G[2]]
V2 = [z*G[2]+y*G[1]]
a1 = hom(F1, G, V1)
a2 = hom(F2, G, V2)
V = subquotient(a1,a2)
@test is_graded(V)
@test degrees_of_generators(V) == [2*Z[1], 2*Z[1], 2*Z[1]]
A1 = Rg[x y;
2*x^2 3*y^2]
A2 = Rg[x^3 x^2*y;
(2*x^2+x*y)*x (2*y^3+y*x^2)]
B = Rg[4*x*y^3 (2*x+y)^4]
F2 = graded_free_module(Rg, [0,0])
M1 = SubQuo(F2, A1, B)
@test degrees_of_generators(M1) == [Z[1], 2*Z[1]]
F = graded_free_module(Rg, 2)
O = [x*F[1]+y*F[2],y*F[2]]
M = SubquoModule(F, O)
@test degrees_of_generators(M) == [Z[1], Z[1]]
O1 = [x*F[1]+y*F[2],y*F[2]]
O2 = [x^2*F[1]+y^2*F[2],y^2*F[2]]
M = SubquoModule(F, O1, O2)
@test degrees_of_generators(M) == [Z[1], Z[1]]
@test degrees_of_generators(M.quo) == [2*Z[1], 2*Z[1]]
end
@testset "Graded modules from matrices constructors" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F2 = graded_free_module(Rg,[8,8])
A1 = Rg[x y;
2*x^2 3*y^2]
MM = graded_cokernel(F2, A1)
@test degrees_of_generators(MM) == [8*Z[1], 8*Z[1]]
MM = cokernel(F2, A1)
@test degrees_of_generators(MM) == [8*Z[1], 8*Z[1]]
MM = graded_cokernel(A1)
@test degrees_of_generators(MM) == [Z[0], Z[0]]
MM = graded_image(F2, A1)
@test degrees_of_generators(MM) == [9*Z[1], 10*Z[1]]
MM = image(F2, A1)
@test degrees_of_generators(MM) == [9*Z[1], 10*Z[1]]
MM = graded_image(A1)
@test degrees_of_generators(MM) == [Z[1], 2*Z[1]]
end
@testset "Graded morphisms from matrix constructor" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg,[1,2, 3])
A2 = Rg[x^3 x^2 x;
(2*x^2+x*y)*x^2 (2*y^2+x^2)*x x^2]
f = graded_map(F, A2)
@test degrees(domain(f)) == [4*Z[1], 5*Z[1]]
@test degrees(codomain(f)) == [Z[1], 2*Z[1], 3*Z[1]]
end
@testset "Graded morphisms of subquotients constructor" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, 1);
A = Rg[x; y]
B = Rg[x^2; y^3; z^4]
M = SubquoModule(F, A, B)
N = M;
V = [y^2*N[1], x^2*N[2]]
a = hom(M, N, V);
@test is_graded(a)
@test degree(a) == 2*Z[1]
end
@testset "Graded morphisms of subquotients and kernel" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, 1);
A = Rg[x; y]
B = Rg[x^2; y^3; z^4]
M = SubquoModule(F, A, B)
N = M;
V = [y^2*N[1], x^2*N[2]]
a = hom(M, N, V);
K, incl = kernel(a);
@test ngens(K) == 3
@test domain(incl) == K
end
@testset "Is isomorphic for graded free modules" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z)=grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg, [1,1,3,2])
G = graded_free_module(Rg, [1,1,2,3])
@test is_isomorphic(F, G)
F = graded_free_module(Rg, [1,1,5,2])
G = graded_free_module(Rg, [1,1,2,3])
@test !is_isomorphic(F, G)
end
@testset "Canonical isomorphism between graded subquotients" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F1 = graded_free_module(Rg,[2,3, 4])
A2 = Rg[x^3 x^2 x;
(2*x^2+x*y)*x^2 (2*y^2+x^2)*x x^2]
MM1 = image(F1, A2)
F2 = graded_free_module(Rg,[2,4, 3])
A2a = Rg[x^3 x x^2;
(2*x^2+x*y)*x^2 x^2 (2*y^2+x^2)*x]
MM2 = image(F2, A2a)
@test is_canonically_isomorphic(MM1.sub, MM2.sub)
_ , f = is_canonically_isomorphic_with_map(MM1, MM2)
@test f(MM1[1]) == MM2[1]
end
@testset "Being a subset and equality for graded subquotients" begin
R, _= polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]])
F = graded_free_module(Rg,2)
O1 = [x*F[1]+y*F[2],y*F[2]]
O1a = [x*F[1],y*F[2]]
O2 = [x^2*F[1]+y^2*F[2],y^2*F[2]]
M1 = SubquoModule(F, O1, O2)
M2 = SubquoModule(F, O1a, O2)
@test M1 == M2
@test is_subset(M1,M2)
@test is_subset(M2,M1)
@test M1.sub == M2.sub
@test is_subset(M1.sub,M2.sub)
@test is_subset(M2.sub,M1.sub)
end
@testset "Sum of graded subquotients" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
AM = Rg[x;]
BM = Rg[x^2; y^3; z^4]
M = SubquoModule(F, AM, BM)
AN = Rg[y;]
BN = Rg[x^2; y^3; z^4]
N = SubquoModule(F, AN, BN)
O, f1, f2 = sum(M, N)
@test ngens(O) == 2
@test codomain(f1) == O
@test codomain(f2) == O
@test domain(f1) == M
@test domain(f2) == N
end
@testset "Being well defined and equality for graded morphisms of subquotients" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; y^3; z^4]
M = SubquoModule(F, A, B)
N = M
V = [y^2*N[1], x^2*N[2]]
a = hom(M, N, V)
@test is_welldefined(a)
W = Rg[y^2 0; 0 x^2]
b = hom(M, N, W)
@test a == b
@test nrows(matrix(a)) == 2
@test ncols(matrix(a)) == 2
F = graded_free_module(Rg, 1)
W = [y*N[1], x*N[2]]
c = hom(M, N, W);
@test !is_welldefined(c)
end
@testset "Being zero for elements of graded subquotients" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; y^3; z^4]
M = SubquoModule(F, A, B)
@test !is_zero(M[1])
@test is_zero(x*M[1])
end
@testset "Hom module of graded free modules" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F1 = graded_free_module(Rg, [1,2,2])
F2 = graded_free_module(Rg, [3,5])
V, f = hom(F1, F2)
@test is_graded(V)
@test degrees(V) == [2*Z[1], 4*Z[1], Z[1], 3*Z[1], Z[1], 3*Z[1]]
F1 = graded_free_module(Rg, 3)
F2 = graded_free_module(Rg, 5)
V = hom(F1, F2)
g = V[2](V[1][1])
@test is_graded(g)
@test g(F1[1])==F2[1]
end
@testset "Presentations 2" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, [1,2,2]);
p = presentation(F)
@test rank(p[-2]) == 0
@test degrees(p[-1]) == [Z[1], 2*Z[1], 2*Z[1]]
@test degrees(p[0]) == [Z[1], 2*Z[1], 2*Z[1]]
@test rank(p[1]) == 0
@test is_zero(map(p,-1))
@test degree(map(p,0)) == Z[0]
@test is_zero(map(p,1))
F = graded_free_module(Rg, 1);
A = Rg[x; y];
B = Rg[x^2; y^3; z^4];
M = SubquoModule(F, A, B);
P = presentation(M);
@test rank(P[-2]) == 0
@test is_graded(P[-2])
@test P[-1] == M
@test degrees(P[0]) == [Z[1], Z[1]]
@test degrees(P[1]) == [2*Z[1], 2*Z[1], 3*Z[1], 5*Z[1], 5*Z[1]]
@test is_homogeneous(map(P, -1))
@test is_homogeneous(map(P, 0))
@test is_homogeneous(map(P, 1))
@test domain(map(P, -1)) == M
@test domain(map(P, 0)) == P[0]
@test codomain(map(P, 0)) == M
@test codomain(map(P, 1)) == P[0]
@test domain(map(P, 1)) == P[1]
N, phi = present_as_cokernel(M, :all)
@test degrees_of_generators(N) == [Z[1], Z[1]]
@test is_homogeneous(phi)
@test domain(phi) == N
@test codomain(phi) == M
psi = phi.inverse_isomorphism
@test is_homogeneous(psi)
@test domain(psi) == M
@test codomain(psi) == N
@test is_bijective(phi)
@test is_bijective(psi)
end
@testset "Equality morphism for subquotients" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1);
A = Rg[x-y; x+y];
B = Rg[x^2; y^3; z^4];
M = SubquoModule(F, A, B);
A1 = Rg[x; y; x^2*y+y^3];
B1 = Rg[x^2; y^3; z^4];
M1 = SubquoModule(F, A1, B1);
f = is_equal_with_morphism(M, M1)
@test is_homogeneous(f)
@test domain(f) == M
@test codomain(f) == M1
@test is_bijective(f)
end
@testset "Free resolutions" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 3)
rs = free_resolution(F)
@test is_graded(rs)
@test rank(rs[0]) == 3
@test rank(rs[-1]) == 3
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
F = graded_free_module(Rg, 1)
M = SubquoModule(F, A, B)
fr = free_resolution_via_kernels(M)
@test is_graded(fr)
@test degrees(fr[0]) == [Z[1], Z[1]]
@test degrees(fr[1]) == vcat(fill(2*Z[1], 4), fill(5*Z[1], 2))
@test degrees(fr[2]) == vcat(fill(3*Z[1], 2), fill(6*Z[1], 4))
@test degrees(fr[3]) == fill(7*Z[1], 2)
end
@testset "Hom module of graded subquotient modules" begin
R, _ = polynomial_ring(QQ, ["x", "y"]);
Z = abelian_group(0)
Rg, (x,y) = grade(R,[Z[1], Z[1]])
F = graded_free_module(Rg, 2);
V = [x*F[1], y^2*F[2]];
M = quo(F, V)[1]
H, f = hom(M, M)
@test is_graded(H)
@test degrees_of_generators(H) == [Z[0], Z[0]]
@test degrees_of_generators(H.quo) == [Z[1], 2*Z[1], Z[1], 2*Z[1]]
@test is_homogeneous(f(H[1]))
a = element_to_homomorphism(x*H[1]+y*H[2])
@test matrix(a) == Rg[x 0; 0 y]
W = [x*M[1], y*M[2]];
a = hom(M, M, W);
@test degree(a) == Z[1]
@test iswelldefined(a)
m = homomorphism_to_element(H, a)
@test m == y*H[2]
end
@testset "Dual and double dual" begin
R, _ = polynomial_ring(QQ, ["x", "y", "z"]);
Z = abelian_group(0)
Rg, (x,y) = grade(R,[Z[1], Z[1], Z[1]])
F1 = graded_free_module(Rg, 1)
F2 = graded_free_module(Rg, 2)
F2v, ev = dual(F2, cod=F1)
@test ev(F2v[1])(F2[1]) == F1[1]
FF, psi = double_dual(F2)
@test degrees_of_generators(FF) == [Z[0], Z[0]]
@test is_injective(psi)
M, inc = sub(F2, [x*F2[1], y*F2[1]])
F1 = graded_free_module(Rg, 1)
Mv, ev = dual(M, cod=F1)
@test degrees_of_generators(Mv) == [Z[0]]
@test ev(Mv[1])(M[1]) == x*F1[1]
Mvv, psi = double_dual(M, cod=F1)
@test matrix(psi) == Rg[x; y]
F = graded_free_module(Rg, 2);
V = [x*F[1], y^2*F[2]];
M = quo(F, V)[1]
dM, f = dual(M)
@test is_graded(dM)
@test is_zero(dM)
@test domain(f)== dM
ddM, g = double_dual(M)
@test is_graded(ddM)
@test is_zero(ddM)
@test codomain(g)== ddM
F = graded_free_module(Rg, 1);
M = SubquoModule(F, [(x^2-y^2)*F[1], (x^2+y^2)*F[1]]);
dM, f = dual(M)
@test is_graded(dM)
@test degrees_of_generators(M) == [2*Z[1], 2*Z[1]]
@test domain(f)== dM
ddM, g = double_dual(M)
@test is_graded(ddM)
@test degrees_of_generators(ddM) == [Z[0]]
@test codomain(g)== ddM
@test is_injective(g)
end
@testset "Hom module element morphism conversion" begin
R0, _ = polynomial_ring(QQ, ["x", "y"])
Z = abelian_group(0)
Rg, (x, y) = grade(R0, [Z[1],Z[1]])
A1 = Rg[x^2+y^2 x*y; x^2+y^2 x*y]
B1 = Rg[x*y^3+x^4+y^2*x^2 x^4; y^4-3*x^4 x*y^3-x^4]
F = graded_free_module(Rg, 2)
M1 = SubquoModule(F, A1, B1)
A2 = Rg[x;]
B2 = Rg[y^3;]
F1 = graded_free_module(Rg, 1)
M2 = SubquoModule(F1, A2,B2)
SQ = hom(M1,M2)[1]
v = gens(SQ)[1]
@test v == homomorphism_to_element(SQ, element_to_homomorphism(v))
end
@testset "Multiplication morphism" begin
R0, _ = polynomial_ring(QQ, ["x", "y"])
Z = abelian_group(0)
Rg, (x, y) = grade(R0, [Z[1],Z[1]])
A1 = Rg[x^2+y^2 x*y; x^2+y^2 x*y]
B1 = Rg[x*y^3+x^4+y^2*x^2 x^4; y^4-3*x^4 x*y^3-x^4]
F = graded_free_module(Rg, 2)
M1 = SubquoModule(F, A1, B1)
End_M = hom(M1,M1)[1]
R_as_module = graded_free_module(Rg,1)
phi = multiplication_induced_morphism(R_as_module, End_M)
@test is_homogeneous(phi)
@test matrix(phi) == Rg[1 0]
end
@testset "Graded tensor product of graded free modules" begin
R0, _ = polynomial_ring(QQ, ["x", "y"])
Z = abelian_group(0)
Rg, (x, y) = grade(R0, [Z[1],Z[1]])
F1 = graded_free_module(Rg, [1,2]);
F2 = graded_free_module(Rg, [3]);
F3 = graded_free_module(Rg, [5,6]);
tensor_product(F1, F2, F3)
M,t = tensor_product(F1, F2, F3 ,task=:map)
v = t((F1[1],F2[1],F3[1]))
@test degree(v) == 9*Z[1]
@test degrees_of_generators(M) == [9*Z[1], 10*Z[1], 10*Z[1], 11*Z[1]]
end
@testset "Graded tensor product of graded subquotients" begin
R0, _ = polynomial_ring(QQ, ["x", "y"])
Z = abelian_group(0)
Rg, (x, y) = grade(R0, [Z[1],Z[1]])
A1 = Rg[x^2+y^2 x*y; x^2+y^2 x*y]
B1 = Rg[x*y^3+x^4+y^2*x^2 x^4; y^4-3*x^4 x*y^3-x^4]
F = graded_free_module(Rg, 2)
M1 = SubquoModule(F, A1, B1)
A2 = Rg[x;]
B2 = Rg[y^3;]
F1 = graded_free_module(Rg, 1)
M2 = SubquoModule(F1, A2, B2)
M, pure_M = tensor_product(M1, M2, task=:map)
@test is_graded(M)
v = pure_M((M1[1],M2[1]))
@test degree(v) == 3*Z[1]
@test degrees_of_generators(M) == [3*Z[1], 3*Z[1]]
end
@testset "Betti tables 1" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
M = SubquoModule(F, A, B)
free_res1 = free_resolution(M, length=1)
free_res1a = free_resolution(M)
free_res2 = free_resolution(M, algorithm=:sres)
free_res3 = free_resolution_via_kernels(M)
@test first(chain_range(free_res1)) == 1
@test first(chain_range(free_res1a)) == 4
@test first(chain_range(free_res2)) == 4
@test first(chain_range(free_res3)) == 4
end
@testset "Betti tables 2" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
F = graded_free_module(Rg, 1)
M = SubquoModule(F, A, B)
fr = free_resolution(M)
T1 = Dict{Tuple{Int64, Any}, Int64}(
(0, Z[1]) => 2,
(3, 7*Z[1]) => 2,
(2, 3*Z[1]) => 2,
(1, 2*Z[1]) => 4,
(1, 5*Z[1]) => 2,
(2, 6*Z[1]) => 4)
T = betti_table(fr)
@test as_dictionary(T) == T1
T = betti(fr)
@test as_dictionary(T) == T1
reverse_direction!(T)
@test as_dictionary(T) == T1
fr2 = free_resolution_via_kernels(M)
@test as_dictionary(betti(fr2)) == T1
end
@testset "Resolution and Betti tables 2" begin
K = GF(31991)
R, _ = polynomial_ring(K, ["x", "y", "z", "u", "v"])
Z = abelian_group(0)
Rg, (x,y,z,u,v) = grade(R, [Z[1],Z[1],Z[1],Z[1],Z[1]])
I = ideal([10000*x^3-8565*x^2*y-7937*x*y^2+14060*x^2*z+1416*x*y*z-3771*x*z^2-15712*x^2*u-5990*x*y*u+3271*x*z*u-6141*x*u^2-14457*x^2*v-194*x*y*v-15529*y^2*v+12735*x*z*v+11618*y*z*v+9223*z^2*v+14387*x*u*v+7102*y*u*v+13097*z*u*v+3485*u^2*v+12211*x*v^2-4061*y*v^2+2878*z*v^2-6247*u*v^2-1256*v^3,-10000*x^2*y+8565*x*y^2+7937*y^3-14060*x*y*z-1416*y^2*z+3771*y*z^2+15712*x*y*u+5990*y^2*u-3271*y*z*u+6141*y*u^2-x^2*v+3340*x*y*v-797*y^2*v-7781*x*z*v+14337*y*z*v+7675*z^2*v+9192*x*u*v+1747*y*u*v-13195*z*u*v+9126*u^2*v-5013*x*v^2+15010*y*v^2-6899*z*v^2-6373*u*v^2-6580*v^3,-10000*x^2*z+8565*x*y*z+7937*y^2*z-14060*x*z^2-1416*y*z^2+3771*z^3+15712*x*z*u+5990*y*z*u-3271*z^2*u+6141*z*u^2+12149*x^2*v+669*x*y*v+9621*y^2*v+1543*x*z*v+8614*y*z*v-649*z^2*v-12278*x*u*v+14655*y*u*v+10594*z*u*v+10462*u^2*v-2455*x*v^2-10875*y*v^2-7761*z*v^2-10318*u*v^2-6636*v^3,-10000*x^2*u+8565*x*y*u+7937*y^2*u-14060*x*z*u-1416*y*z*u+3771*z^2*u+15712*x*u^2+5990*y*u^2-3271*z*u^2+6141*u^3-9289*x^2*v+13297*x*y*v+13024*y^2*v-12477*x*z*v+9258*y*z*v+10372*z^2*v-4128*x*u*v-3551*y*u*v+4570*z*u*v+15473*u^2*v+4446*x*v^2-7043*y*v^2-14100*z*v^2+5002*u*v^2+9799*v^3,-9289*x^3+13297*x^2*y+13024*x*y^2-12477*x^2*z+9258*x*y*z+10372*x*z^2+13406*x^2*u-3745*x*y*u-15529*y^2*u-14686*x*z*u+11618*y*z*u+9223*z^2*u-2131*x*u^2+7102*y*u^2+13097*z*u^2+3485*u^3+4446*x^2*v-7043*x*y*v-14100*x*z*v-14778*x*u*v-4061*y*u*v+2878*z*u*v-6247*u^2*v+9799*x*v^2-1256*u*v^2,9289*x^2*y-13297*x*y^2-13024*y^3+12477*x*y*z-9258*y^2*z-10372*y*z^2-x^2*u+7468*x*y*u+2754*y^2*u-7781*x*z*u+9767*y*z*u+7675*z^2*u+9192*x*u^2-13726*y*u^2-13195*z*u^2+9126*u^3-4446*x*y*v+7043*y^2*v+14100*y*z*v-5013*x*u*v+10008*y*u*v-6899*z*u*v-6373*u^2*v-9799*y*v^2-6580*u*v^2,9289*x^2*z-13297*x*y*z-13024*y^2*z+12477*x*z^2-9258*y*z^2-10372*z^3+12149*x^2*u+669*x*y*u+9621*y^2*u+5671*x*z*u+12165*y*z*u-5219*z^2*u-12278*x*u^2+14655*y*u^2-4879*z*u^2+10462*u^3-4446*x*z*v+7043*y*z*v+14100*z^2*v-2455*x*u*v-10875*y*u*v-12763*z*u*v-10318*u^2*v-9799*z*v^2-6636*u*v^2,12149*x^3+669*x^2*y+9621*x*y^2-12914*x^2*z+8420*x*y*z-15529*y^2*z+12086*x*z^2+11618*y*z^2+9223*z^3-12278*x^2*u+14655*x*y*u-7010*x*z*u+7102*y*z*u+13097*z^2*u+10462*x*u^2+3485*z*u^2-2455*x^2*v-10875*x*y*v+4450*x*z*v-4061*y*z*v+2878*z^2*v-10318*x*u*v-6247*z*u*v-6636*x*v^2-1256*z*v^2,-12149*x^2*y-669*x*y^2-9621*y^3-x^2*z+1797*x*y*z-9411*y^2*z-7781*x*z^2+14986*y*z^2+7675*z^3+12278*x*y*u-14655*y^2*u+9192*x*z*u-8847*y*z*u-13195*z^2*u-10462*y*u^2+9126*z*u^2+2455*x*y*v+10875*y^2*v-5013*x*z*v-9220*y*z*v-6899*z^2*v+10318*y*u*v-6373*z*u*v+6636*y*v^2-6580*z*v^2,x^3+11117*x^2*y+991*x*y^2+15529*y^3+7781*x^2*z+4919*x*y*z-11618*y^2*z-7675*x*z^2-9223*y*z^2-9192*x^2*u+15857*x*y*u-7102*y^2*u+13195*x*z*u-13097*y*z*u-9126*x*u^2-3485*y*u^2+5013*x^2*v+4770*x*y*v+4061*y^2*v+6899*x*z*v-2878*y*z*v+6373*x*u*v+6247*y*u*v+6580*x*v^2+1256*y*v^2])
V = gens(I)
F = graded_free_module(Rg,1)
V1 = [p * F[1] for p in V]
M = quo(F, V1)[1]
free_res0 = free_resolution(M)
@test is_graded(free_res0)
@test_broken all(iszero, homology(free_res0.C))
free_res0a = free_resolution_via_kernels(M)
@test is_graded(free_res0a)
@test all(iszero, homology(free_res0a.C))
# a minimal generating set of I as produced by the old `minimal_generating_set` function
V = [x^2*u + 25341*x^2*v + 18282*x*y*u + 28685*x*y*v + 29881*x*z*u + 6569*x*z*v + 6499*x*u^2 + 31326*x*u*v + 24012*x*v^2 + 12818*y^2*u + 5322*y^2*v + 28229*y*z*u + 30832*y*z*v + 15643*y*u^2 + 21879*y*u*v + 3926*y*v^2 + 24252*z^2*u + 29725*z^2*v + 25334*z*u^2 + 23257*z*u*v + 15677*z*v^2 + 2082*u^3 + 14404*u^2*v + 10307*u*v^2 + 27796*v^3, x^2*z + 30515*x^2*v + 18282*x*y*z + 16824*x*y*v + 29881*x*z^2 + 6499*x*z*u + 26466*x*z*v + 16451*x*u*v + 7982*x*v^2 + 12818*y^2*z + 22172*y^2*v + 28229*y*z^2 + 15643*y*z*u + 6890*y*z*v + 5453*y*u*v + 25194*y*v^2 + 24252*z^3 + 25334*z^2*u + 22269*z^2*v + 2082*z*u^2 + 6186*z*u*v + 30485*z*v^2 + 29693*u^2*v + 8645*u*v^2 + 30085*v^3, x^2*z + 18213*x^2*u + 3291*x*y*z + 9316*x*y*u + 12069*x*z^2 + 11593*x*z*u + 1353*x*z*v + 20876*x*u^2 + 10094*x*u*v + 9101*y^2*z + 22652*y^2*u + 7221*y*z^2 + 5429*y*z*u + 3059*y*z*v + 31352*y*u^2 + 19564*y*u*v + 1559*z^3 + 11151*z^2*u + 26923*z^2*v + 27410*z*u^2 + 24499*z*u*v + 23597*z*v^2 + 22504*u^3 + 22395*u^2*v + 23649*u*v^2, x^2*y + 2844*x^2*v + 18282*x*y^2 + 29881*x*y*z + 6499*x*y*u + 2367*x*y*v + 23383*x*z*v + 26590*x*u*v + 20977*x*v^2 + 12818*y^3 + 28229*y^2*z + 15643*y^2*u + 27298*y^2*v + 24252*y*z^2 + 25334*y*z*u + 14097*y*z*v + 2082*y*u^2 + 22128*y*u*v + 19545*y*v^2 + 22153*z^2*v + 1137*z*u*v + 10273*z*v^2 + 22348*u^2*v + 17906*u*v^2 + 30776*v^3, x^2*y + 799*x^2*u + 3291*x*y^2 + 12069*x*y*z + 15385*x*y*u + 1353*x*y*v + 10765*x*z*u + 13522*x*u^2 + 6512*x*u*v + 9101*y^3 + 7221*y^2*z + 6933*y^2*u + 3059*y^2*v + 1559*y*z^2 + 1971*y*z*u + 26923*y*z*v + 26152*y*u^2 + 1358*y*u*v + 23597*y*v^2 + 9947*z^2*u + 17766*z*u^2 + 9849*z*u*v + 2274*u^3 + 5458*u^2*v + 10896*u*v^2, x^2*y + 3123*x^2*z + 9872*x*y^2 + 18385*x*y*z + 13015*x*y*u + 10875*x*y*v + 18894*x*z^2 + 21302*x*z*u + 12000*x*z*v + 6834*y^3 + 22815*y^2*z + 20435*y^2*u + 11817*y^2*v + 1555*y*z^2 + 20948*y*z*u + 2160*y*z*v + 10015*y*u^2 + 23814*y*u*v + 5940*y*v^2 + 24225*z^3 + 3577*z^2*u + 15634*z^2*v + 3483*z*u^2 + 4477*z*u*v + 11118*z*v^2, x^3 + 18282*x^2*y + 29881*x^2*z + 6499*x^2*u + 24718*x^2*v + 12818*x*y^2 + 28229*x*y*z + 15643*x*y*u + 24102*x*y*v + 24252*x*z^2 + 25334*x*z*u + 4528*x*z*v + 2082*x*u^2 + 139*x*u*v + 17849*x*v^2 + 15095*y^2*v + 26880*y*z*v + 11767*y*u*v + 31258*y*v^2 + 29583*z^2*v + 10344*z*u*v + 27327*z*v^2 + 26121*u^2*v + 20528*u*v^2 + 10928*v^3]
F = graded_free_module(Rg,1)
V1 = [p * F[1] for p in V]
M = quo(F, V1)[1]
# Free resolution via Singular gives wrong result
free_res1 = free_resolution(M)
@test is_graded(free_res1)
@test_broken all(iszero, homology(free_res1.C))
free_res2 = free_resolution(M, algorithm=:sres)
@test is_graded(free_res2)
@test_broken all(iszero, homology(free_res2.C))
free_res3 = free_resolution_via_kernels(M)
@test is_graded(free_res3)
B = betti(free_res3)
B1 = Dict{Tuple{Int64, Any}, Int64}(
(0, Z[0]) => 1,
(1, 3 * Z[1]) => 7,
(2, 4 * Z[1]) => 10,
(3, 5 * Z[1]) => 5,
(4, 6 * Z[1]) => 1)
@test as_dictionary(B) == B1
@test all(iszero, homology(free_res3.C))
end
@testset "Tensor module resolution" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
M = SubquoModule(F, A, B)
free_res = free_resolution_via_kernels(M)
F1 = graded_free_module(Rg, 1)
N = SubquoModule(F1, Rg[x*y+2*x^2; x+y], Rg[z^4;])
tensor_resolution = tensor_product(N,free_res)
@test chain_range(tensor_resolution) == chain_range(free_res)
for i in Hecke.map_range(tensor_resolution)
f = map(free_res,i)
M_i = domain(f)
tensored_f = map(tensor_resolution,i)
to_pure_tensors_i = get_attribute(domain(tensored_f),:tensor_pure_function)
to_pure_tensors_i_plus_1 = get_attribute(codomain(tensored_f), :tensor_pure_function)
for (n,mi) in zip(gens(N),gens(M_i))
@test tensored_f(to_pure_tensors_i((n,mi))) == to_pure_tensors_i_plus_1(n,f(mi))
end
end
end
@testset "Tensor resolution module" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
M = SubquoModule(F, A, B)
free_res = free_resolution_via_kernels(M)
F1 = graded_free_module(Rg, 1)
N = SubquoModule(F1, Rg[x+2*x^2*z; x+y-z], Rg[z^4;])
tensor_resolution = tensor_product(free_res,N)
@test chain_range(tensor_resolution) == chain_range(free_res)
for i in Hecke.map_range(tensor_resolution)
f = map(free_res,i)
M_i = domain(f)
tensored_f = map(tensor_resolution,i)
to_pure_tensors_i = get_attribute(domain(tensored_f),:tensor_pure_function)
to_pure_tensors_i_plus_1 = get_attribute(codomain(tensored_f), :tensor_pure_function)
for (mi,n) in zip(gens(M_i),gens(N))
@test tensored_f(to_pure_tensors_i((mi,n))) == to_pure_tensors_i_plus_1(f(mi),n)
end
end
end
@testset "Hom module resolution" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
M = SubquoModule(F, A, B)
free_res = free_resolution_via_kernels(M)
F1 = graded_free_module(Rg, 1)
N = SubquoModule(F1, Rg[x*y+2*x^2; x+y], Rg[z^4;])
hom_resolution = hom(N,free_res)
@test chain_range(hom_resolution) == chain_range(free_res)
for i in Hecke.map_range(hom_resolution)
f = map(free_res,i)
hom_f = map(hom_resolution,i)
hom_N_M_i = domain(hom_f)
for v in gens(hom_N_M_i)
@test element_to_homomorphism(hom_f(v)) == element_to_homomorphism(v)*f
end
end
N = SubquoModule(F1, Rg[x*y+2*x^2; x+y], Rg[z^4; x^2-y*z])
hom_resolution = hom(free_res,N)
@test last(chain_range(hom_resolution)) == first(chain_range(free_res))
@test first(chain_range(hom_resolution)) == last(chain_range(free_res))
for i in Hecke.map_range(hom_resolution)
f = map(free_res,i+1) #f[i]: M[i] -> M[i-1]
hom_f = map(hom_resolution,i) #f[i]: M[i] -> M[i+1]
hom_M_i_N = domain(hom_f)
for v in gens(hom_M_i_N)
@test element_to_homomorphism(hom_f(v)) == f*element_to_homomorphism(v)
end
end
hom_hom_resolution = hom(hom_resolution,N)
@test chain_range(hom_hom_resolution) == chain_range(free_res)
end
@testset "Hom resolution module" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
M = SubquoModule(F, A, B)
free_res = free_resolution_via_kernels(M)
F1 = graded_free_module(Rg, 1)
N = SubquoModule(F1, Rg[x*y+2*x^2; x+y], Rg[z^4;])
hom_resolution = hom(free_res,N)
@test last(chain_range(hom_resolution)) == first(chain_range(free_res))
@test first(chain_range(hom_resolution)) == last(chain_range(free_res))
for i in Hecke.map_range(hom_resolution)
f = map(free_res,i+1) #f[i]: M[i] -> M[i-1]
hom_f = map(hom_resolution,i) #f[i]: M[i] -> M[i+1]
hom_M_i_N = domain(hom_f)
for v in gens(hom_M_i_N)
@test element_to_homomorphism(hom_f(v)) == f*element_to_homomorphism(v)
end
end
hom_hom_resolution = hom(hom_resolution,N)
@test chain_range(hom_hom_resolution) == chain_range(free_res)
hom_resolution = hom_without_reversing_direction(free_res,N)
@test last(chain_range(hom_resolution)) == -first(chain_range(free_res))
@test first(chain_range(hom_resolution)) == -last(chain_range(free_res))
for i in Hecke.map_range(hom_resolution)
f = map(free_res,-i+1)
hom_f = map(hom_resolution,i)
hom_M_i_N = domain(hom_f)
for v in gens(hom_M_i_N)
@test element_to_homomorphism(hom_f(v)) == f*element_to_homomorphism(v)
end
end
hom_hom_resolution = hom_without_reversing_direction(hom_resolution,N)
@test chain_range(hom_hom_resolution) == chain_range(free_res)
end
@testset "Tor and Ext" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
A = Rg[x; y]
B = Rg[x^2; x*y; y^2; z^4]
F = graded_free_module(Rg, 1)
M = SubquoModule(F, A, B)
Q, _ = quo(F, [x*F[1]])
G = free_module(Rg, 2)
M_coker = present_as_cokernel(M)
T0 = tor(Q, M, 0)
T1 = tor(Q, M, 1)
T2 = tor(Q, M, 2)
@test is_canonically_isomorphic(T0, M)
@test ngens(present_as_cokernel(T1)) == ngens(M_coker)
# Todo twist
@test iszero(T2)
T0 = tor(M, Q, 0)
T1 = tor(M, Q, 1)
T2 = tor(M, Q, 2)
@test is_canonically_isomorphic(present_as_cokernel(T0), M_coker)
# todo simplify
@test iszero(T2)
E0 = ext(Q, M, 0)
E1 = ext(Q, M, 1)
E2 = ext(Q, M, 2)
@test is_canonically_isomorphic(present_as_cokernel(E0), M_coker)
@test ngens(present_as_cokernel(E1)) == ngens(M_coker)
@test iszero(E2)
E0 = ext(M, Q, 0)
E1 = ext(M, Q, 1)
E2 = ext(M, Q, 2)
E3 = ext(M, Q, 3)
E4 = ext(M, Q, 4)
@test iszero(E0)
@test iszero(E1)
# Todo simplify
@test ngens(E3) == ngens(M_coker)
# Todo twist
@test iszero(E4)
end
@testset "Groebner bases graded" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
J = SubquoModule(F, [x*F[1], (x^2)*F[1], (x+y)*F[1]])
@test leading_module(J) == SubquoModule(F, [x*F[1], y*F[1]])
J = SubquoModule(F, [(x*y^2+x*y^2)*F[1], (x^2*y+x^2*z-y^3)*F[1]])
@test leading_module(J) == SubquoModule(F, [x*y^2*F[1], x^2*y*F[1], y^4*F[1], x^3*z^2*F[1]])
F = graded_free_module(Rg, 2)
lp = lex(gens(base_ring(F)))*lex(gens(F))
M = SubquoModule(F, [(x^2*y^2*F[1]+y*z^3*F[2]), x*z*F[1]+z^2*F[2]])
@test leading_module(M,lp) == SubquoModule(F, [x*z*F[1], x^2*y^2*F[1], x*y^2*z^2*F[2]])
R, _ = polynomial_ring(QQ, ["x_"*string(i) for i=1:4])
Z = abelian_group(0)
Rg, x = grade(R, [Z[1],Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
lp = lex(gens(base_ring(F)))*lex(gens(F))
J = SubquoModule(F, [(x[1]+x[2])*F[1], (x[1]+x[2]+2*x[3]+2*x[4])*F[1],(x[1]+x[2]+x[3]+x[4])*F[1]])
@test reduced_groebner_basis(J, lp).O == Oscar.ModuleGens([(x[3]+x[4])*F[1], (x[1]+x[2])*F[1]], F).O
@test haskey(J.groebner_basis, lp)
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F = graded_free_module(Rg, 1)
lp = lex(gens(base_ring(F)))*lex(gens(F))
I = SubquoModule(F, [(x-y)*F[1], (y^2-z^2)*F[1]])
f = (x*y^2+y*z^2)*F[1]
@test Oscar.reduce(f, I) == 2*y*z^2*F[1]
F = graded_free_module(Rg, 2)
A = Rg[(x+z)^2 y*z+x^2; (y*x+2*z^2) z^2]
B = Rg[2*z*(x*z+y^2) x*z^2]
M = SubquoModule(F, A, B)
gb = groebner_basis(M)
P = sum(Oscar.SubModuleOfFreeModule(F, gb), Oscar.SubModuleOfFreeModule(F, gb.quo_GB))
Q = Oscar.SubModuleOfFreeModule(F, groebner_basis(M.sum))
@test P == Q
v = x*((x^2 + 2*x*z + z^2)*F[1] + (x^2 + y*z)*F[2]) + (2*x*z^2 + 2*y^2*z)*F[1] + x*z^2*F[2]
@test represents_element(v, M)
end
@testset "Tensor product of morphisms graded" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F2 = graded_free_module(Rg,2)
F3 = graded_free_module(Rg,3)
F4 = graded_free_module(Rg,4)
A1 = Rg[6*x*y^2*z^2 + 4*y*z^4 + 12*y^2*z^3 6*x^2*y^2*z + 11*x^2*z^3; x y; y*z 3*x*z + 3*x*y + 5*y*z]
B1 = Rg[12*z^2 x*y]
A2 = Rg[8*x^2*y*z 6*x^2*z^2 4*y^4;
9*x*z^5 3*x^2*y^2*z^2 11*y*z^5;
8*x*y*z 15*y^3 9*x*z^2]
B2 = Rg[0 14*x*y 6*x^2]
M1 = SubquoModule(F2,A1,B1)
M2 = SubquoModule(F3,A2,B2)
M, pure_M = tensor_product(M1,M2, task=:map)
@test is_graded(M)
phi = hom_tensor(M, M, [identity_map(M1), identity_map(M2)])
@test is_homogeneous(phi)
v = M[1]
@test phi(v) == v
A3 = Rg[7*x^2*y^2 11*x^2*y*z;
6*x^2*y 4*x^2*z]
M3 = SubquoModule(Oscar.SubModuleOfFreeModule(F2, A3))
N,pure_N = tensor_product(M3,F4, task=:map)
@test is_graded(N)
M3_to_M1 = SubQuoHom(M3, M1, [M1[1], x^3*M1[2]])
@test is_welldefined(M3_to_M1)
F4_to_M2 = FreeModuleHom(F4,M2, [x^2*M2[1], M2[2], y^3*M2[3], z^3*M2[3]])
phi = hom_tensor(N,M,[M3_to_M1,F4_to_M2])
u1 = M3[1]
u2 = F4[1]
@test phi(pure_N((u1,u2))) == pure_M((M3_to_M1(u1),F4_to_M2(u2)))
end
@testset "direct product graded" begin
R0, _ = polynomial_ring(QQ, ["x", "y", "z"])
Z = abelian_group(0)
Rg, (x, y, z) = grade(R0, [Z[1],Z[1],Z[1]])
F2 = graded_free_module(Rg, 2)
F3 = graded_free_module(Rg, 3)
A1 = Rg[6*x*y^2*z^2 + 4*y*z^4 + 12*y^2*z^3 6*x^2*y^2*z + 11*x^2*z^3; x y; y*z 3*x*z + 3*x*y + 5*y*z]
B1 = Rg[12*z^2 x*y]
A2 = Rg[8*x^2*y*z 6*x^2*z^2 4*y^4;
9*x*z^5 3*x^2*y^2*z^2 11*y*z^5;
8*x*y*z 15*y^3 9*x*z^2]
B2 = Rg[0 14*x*y 6*x^2]
M1 = SubquoModule(F2, A1, B1)
M2 = SubquoModule(F3, A2, B2)
sum_M, emb = direct_sum(M1, M2)
@test is_graded(sum_M)
@test is_homogeneous(emb[1])
@test is_homogeneous(emb[2])
@test domain(emb[1]) === M1
@test domain(emb[2]) === M2
@test codomain(emb[1]) === sum_M
@test codomain(emb[2]) === sum_M
sum_M, proj = direct_sum(M1,M2, task=:prod)
@test is_graded(sum_M)
@test is_homogeneous(proj[1])
@test is_homogeneous(proj[2])
@test codomain(proj[1]) === M1
@test codomain(proj[2]) === M2
@test domain(proj[1]) === sum_M
@test domain(proj[2]) === sum_M
prod_M, emb, proj = direct_sum(M1,M2,task=:both)
@test is_graded(sum_M)
@test is_homogeneous(emb[1])
@test is_homogeneous(emb[2])
@test is_homogeneous(proj[1])
@test is_homogeneous(proj[2])
@test length(proj) == length(emb) == 2
@test ngens(prod_M) == ngens(M1) + ngens(M2)
for g in gens(prod_M)
@test g == sum([emb[i](proj[i](g)) for i=1:length(proj)])
end
for g in gens(M1)
@test g == proj[1](emb[1](g))
end
for g in gens(M2)
@test g == proj[2](emb[2](g))
end
A1 = Rg[4*x*y^2*z^2 + 6*x*z^4 9*x^2*z^3;
5*x^2*y*z + 12*x*y^3 8*x^2*y^2 + z^4;
11*x^2*z^2 8*y^4 + 4*y*z^3]
B1 = Rg[10*x^2*y*z 15*y^2*z^2 + 3*y*z^3]
N1 = SubquoModule(F2,A1,B1)
A2 = Rg[ 14*x^4 4*x*y^2*z 8*x^2*y^2;
2*x*y^2*z 4*x^2*z^2 15*y^4]
B2 = Rg[12*x*y*z^2 13*x^2*z^2 11*x*z^3]
N2 = SubquoModule(F3,A2,B2)
prod_N = direct_product(N1,N2,task=:none)
@test is_graded(prod_N)
@test ngens(prod_M) == ngens(M1) + ngens(M2)
for g in gens(prod_N)
@test g == sum([Hecke.canonical_injection(prod_N,i)(Hecke.canonical_projection(prod_N,i)(g)) for i=1:2])
end
for g in gens(N1)
@test g == Hecke.canonical_projection(prod_N,1)(Hecke.canonical_injection(prod_N,1)(g))
end
for g in gens(N2)
@test g == Hecke.canonical_projection(prod_N,2)(Hecke.canonical_injection(prod_N,2)(g))
end
M1_to_N1 = SubQuoHom(M1,N1,zero_matrix(Rg,3,3))
@test is_homogeneous(M1_to_N1)
H12 = hom(M1,N2)[1]
@test is_graded(H12)
H21 = hom(M2,N1)[1]
@test is_graded(H21)
M1_to_N2 = iszero(H12) ? SubQuoHom(M1,N2,zero_matrix(R,3,2)) : element_to_homomorphism(H12[1])
M2_to_N1 = iszero(H21) ? SubQuoHom(M2,N1,zero_matrix(R,2,3)) : element_to_homomorphism(x^3*H21[1])
M2_to_N2 = SubQuoHom(M2, N2, [0*N2[1],0*N2[1],0*N2[1]])
@assert degree(M1_to_N1) == Z[0]
@assert degree(M1_to_N2) == 6*Z[1]
@assert degree(M2_to_N1) == 6*Z[1]
@assert degree(M2_to_N2) == Z[0]
@assert is_welldefined(M1_to_N1)
@assert is_welldefined(M1_to_N2)
@assert is_welldefined(M2_to_N1)
@assert is_welldefined(M2_to_N2)
phi = hom_product(prod_M,prod_N,[M1_to_N1 M1_to_N2; M2_to_N1 M2_to_N2])
@test degree(phi) == 6*Z[1]
for g in gens(M1)
@test M1_to_N1(g) == Hecke.canonical_projection(prod_N,1)(phi(emb[1](g)))
@test M1_to_N2(g) == Hecke.canonical_projection(prod_N,2)(phi(emb[1](g)))
end
for g in gens(M2)
@test M2_to_N1(g) == Hecke.canonical_projection(prod_N,1)(phi(emb[2](g)))
@test M2_to_N2(g) == Hecke.canonical_projection(prod_N,2)(phi(emb[2](g)))
end
prod_FN,prod,emb = direct_product(F2,N2,task=:both)
@test is_graded(prod_FN)
@test ngens(prod_FN) == ngens(F2) + ngens(N2)
for g in gens(prod_FN)
@test g == sum([emb[i](prod[i](g)) for i=1:2])
end
for g in gens(F2)
@test g == prod[1](emb[1](g))
end
for g in gens(N2)
@test g == prod[2](emb[2](g))
end
end
@testset "Coordinates" begin
Z3, a = FiniteField(3,1,"a")
R, (x,y) = polynomial_ring(Z3, ["x", "y"])
Z = abelian_group(0)
Rg, (x, y) = grade(R, [Z[1],Z[1]])
coeffs = [Z3(i) for i=0:1]
A = Rg[x*y x^2+y^2; y^2 x*y; x^2+y^2 y^2]
B = Rg[2*x^2 (x+y)^2; x^2+y^2 x^2+2*x*y]
F = graded_free_module(Rg, 2)
M = SubquoModule(F, A, B)
monomials = [x,y]
coeff_generator = ([c1,c2] for c1 in coeffs for c2 in coeffs)
for coefficients in ([c1,c2,c3] for c1 in coeff_generator for c2 in coeff_generator for c3 in coeff_generator)
v = sparse_row(Rg, [(i,sum(coefficients[i][j]*monomials[j] for j=1:2)) for i=1:3])
v_as_FreeModElem = sum([v[i]*repres(M[i]) for i=1:ngens(M)])
elem1 = SubquoModuleElem(v_as_FreeModElem,M)
elem2 = SubquoModuleElem(v,M)
@test elem1 == elem2
end
end
######################################################################