Homological Algebra

presentation(M::ModuleFP)

Return a free presentation of $M$.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> A = R[x; y];

julia> B = R[x^2; y^3; z^4];

julia> M = SubquoModule(A, B);

julia> P = presentation(M);

julia> rank(P[1])
5

julia> rank(P[0])
2

julia> R, _ = polynomial_ring(QQ, ["x", "y", "z"]);

julia> Z = abelian_group(0);

julia> Rg, (x, y, z) = grade(R, [Z[1],Z[1],Z[1]]);

julia> F = graded_free_module(Rg, [1,2,2]);

julia> p = presentation(F)
p_1 ----> p_0 ----> p_-1 ----> p_-2

julia> p[-2]
Graded free module Rg^0 of rank 0 over Rg

julia> p[-1]
Graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg

julia> p[0]
Graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg

julia> p[1]
Graded free module Rg^0 of rank 0 over Rg

julia> map(p,-1)
F -> 0
e[1] -> 0
e[2] -> 0
e[3] -> 0
Homogeneous module homomorphism

julia> map(p,0)
F -> F
e[1] -> e[1]
e[2] -> e[2]
e[3] -> e[3]
Homogeneous module homomorphism

julia> map(p,1)
0 -> F
Homogeneous module homomorphism

julia> F = graded_free_module(Rg, 1);

julia> A = Rg[x; y];

julia> B = Rg[x^2; y^3; z^4];

julia> M = SubquoModule(F, A, B);

julia> P = presentation(M)
P_1 ----> P_0 ----> P_-1 ----> P_-2

julia> P[-2]
Graded free module Rg^0 of rank 0 over Rg

julia> P[-1]
Graded subquotient of submodule of F generated by
1 -> x*e[1]
2 -> y*e[1]
by submodule of F generated by
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]

julia> P[0]
Graded free module Rg^2([-1]) of rank 2 over Rg

julia> P[1]
Graded free module Rg^2([-2]) + Rg^1([-3]) + Rg^2([-5]) of rank 5 over Rg

julia> map(P,-1)
M -> 0
x*e[1] -> 0
y*e[1] -> 0
Homogeneous module homomorphism

julia> map(P,0)
br^2 -> M
e[1] -> x*e[1]
e[2] -> y*e[1]
Homogeneous module homomorphism

julia> map(P,1)
br^5 -> br^2
e[1] -> x*e[1]
e[2] -> -y*e[1] + x*e[2]
e[3] -> y^2*e[2]
e[4] -> z^4*e[1]
e[5] -> z^4*e[2]
Homogeneous module homomorphism

present_as_cokernel(M::SubquoModule, task::Symbol = :none)

Return a subquotient C which is isomorphic to M, and whose generators are the standard unit vectors of its ambient free module.

Additionally,

• return an isomorphism M $\to$ C if task = :with_morphism,
• return and cache an isomorphism M $\to$ C if task = :cache_morphism,
• do none of the above if task = :none (default).

If task = :only_morphism, return only an isomorphism.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> A = R[x; y];

julia> B = R[x^2; y^3; z^4];

julia> M = SubquoModule(A, B)
Subquotient of Submodule with 2 generators
1 -> x*e[1]
2 -> y*e[1]
by Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]

julia> C = present_as_cokernel(M)
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 5 generators
1 -> x*e[1]
2 -> -y*e[1] + x*e[2]
3 -> y^2*e[2]
4 -> z^4*e[1]
5 -> z^4*e[2]

julia> R, _ = polynomial_ring(QQ, ["x", "y", "z"]);

julia> Z = abelian_group(0);

julia> Rg, (x, y, z) = grade(R, [Z[1],Z[1],Z[1]]);

julia> F = graded_free_module(Rg, 1);

julia> A = Rg[x; y];

julia> B = Rg[x^2; y^3; z^4];

julia> M = SubquoModule(F, A, B);

julia> present_as_cokernel(M, :with_morphism)
(Graded subquotient of submodule of br^2 generated by
1 -> e[1]
2 -> e[2]
by submodule of br^2 generated by
1 -> x*e[1]
2 -> -y*e[1] + x*e[2]
3 -> y^2*e[2]
4 -> z^4*e[1]
5 -> z^4*e[2], Graded subquotient of submodule of br^2 generated by
1 -> e[1]
2 -> e[2]
by submodule of br^2 generated by
1 -> x*e[1]
2 -> -y*e[1] + x*e[2]
3 -> y^2*e[2]
4 -> z^4*e[1]
5 -> z^4*e[2] -> M
e[1] -> x*e[1]
e[2] -> y*e[1]
Homogeneous module homomorphism)

free_resolution(M::SubquoModule{<:MPolyRingElem}; 
    ordering::ModuleOrdering = default_ordering(M),
    length::Int=0, algorithm::Symbol=:fres
  )

Return a free resolution of M.

If length != 0, the free resolution is only computed up to the length-th free module. algorithm can be set to :sres or :fres.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])

julia> A = R[x; y]
[x]
[y]

julia> B = R[x^2; x*y; y^2; z^4]
[x^2]
[x*y]
[y^2]
[z^4]

julia> M = SubquoModule(A, B)
Subquotient of Submodule with 2 generators
1 -> x*e[1]
2 -> y*e[1]
by Submodule with 4 generators
1 -> x^2*e[1]
2 -> x*y*e[1]
3 -> y^2*e[1]
4 -> z^4*e[1]

julia> fr = free_resolution(M, length=1)

rank   | 6  2
-------|------
degree | 1  0

julia> is_complete(fr)
false

julia> fr[4]
Free module of rank 0 over Multivariate polynomial ring in 3 variables over QQ

julia> fr

rank   | 0  2  6  6  2
-------|---------------
degree | 4  3  2  1  0

julia> is_complete(fr)
true

julia> fr = free_resolution(M, algorithm=:sres)

rank   | 0  2  6  6  2
-------|---------------
degree | 4  3  2  1  0

Note: Over rings other than polynomial rings, the method will default to a lazy, iterative kernel computation.

homology(C::ComplexOfMorphisms{<:ModuleFP})

Return the homology of C.

Examples

julia> R, (x,) = polynomial_ring(QQ, ["x"]);

julia> F = free_module(R, 1);

julia> A, _ = quo(F, [x^4*F[1]]);

julia> B, _ = quo(F, [x^3*F[1]]);

julia> a = hom(A, B, [x^2*B[1]]);

julia> b = hom(B, B, [x^2*B[1]]);

julia> C = ComplexOfMorphisms(ModuleFP, [a, b]);

julia> H = homology(C)
3-element Vector{SubquoModule{QQMPolyRingElem}}:
 Subquotient of Submodule with 1 generator
1 -> x*e[1]
by Submodule with 1 generator
1 -> x^4*e[1]
 Subquotient of Submodule with 1 generator
1 -> x*e[1]
by Submodule with 2 generators
1 -> x^3*e[1]
2 -> x^2*e[1]
 Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 2 generators
1 -> x^3*e[1]
2 -> x^2*e[1]

homology(C::ComplexOfMorphisms{<:ModuleFP}, i::Int)

Return the i-th homology module of C.

Examples

julia> R, (x,) = polynomial_ring(QQ, ["x"]);

julia> F = free_module(R, 1);

julia> A, _ = quo(F, [x^4*F[1]]);

julia> B, _ = quo(F, [x^3*F[1]]);

julia> a = hom(A, B, [x^2*B[1]]);

julia> b = hom(B, B, [x^2*B[1]]);

julia> C = ComplexOfMorphisms(ModuleFP, [a, b]);

julia> H = homology(C, 1)
Subquotient of Submodule with 1 generator
1 -> x*e[1]
by Submodule with 2 generators
1 -> x^3*e[1]
2 -> x^2*e[1]

hom(M::ModuleFP, N::ModuleFP)

Return the module Hom(M,N) as an object of type SubquoModule.

Additionally, if H is that object, return the map which sends an element of H to the corresponding homomorphism M $\to$ N.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = FreeMod(R, 2);

julia> V = [x*F[1], y^2*F[2]];

julia> M = quo(F, V)[1]
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 2 generators
1 -> x*e[1]
2 -> y^2*e[2]

julia> H = hom(M, M)[1]
hom of (M, M)

julia> gens(H)
2-element Vector{SubquoModuleElem{QQMPolyRingElem}}:
 (e[1] -> e[1])
 (e[2] -> e[2])

julia> relations(H)
4-element Vector{FreeModElem{QQMPolyRingElem}}:
 x*(e[1] -> e[1])
 y^2*(e[1] -> e[2])
 x*(e[2] -> e[1])
 y^2*(e[2] -> e[2])

element_to_homomorphism(f::ModuleFPElem)

If f is an element of a module created via hom(M,N), for some modules M and N, return the homomorphism M $\to$ N corresponding to f.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = FreeMod(R, 2);

julia> V = [x*F[1], y^2*F[2]];

julia> M = quo(F, V)[1]
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 2 generators
1 -> x*e[1]
2 -> y^2*e[2]

julia> H = hom(M, M)[1];

julia> gens(H)
2-element Vector{SubquoModuleElem{QQMPolyRingElem}}:
 (e[1] -> e[1])
 (e[2] -> e[2])

julia> relations(H)
4-element Vector{FreeModElem{QQMPolyRingElem}}:
 x*(e[1] -> e[1])
 y^2*(e[1] -> e[2])
 x*(e[2] -> e[1])
 y^2*(e[2] -> e[2])

julia> a = element_to_homomorphism(H[1]+y*H[2])
Map with following data
Domain:
=======
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 2 generators
1 -> x*e[1]
2 -> y^2*e[2]
Codomain:
=========
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 2 generators
1 -> x*e[1]
2 -> y^2*e[2]

julia> matrix(a)
[1   0]
[0   y]

homomorphism_to_element(H::ModuleFP, a::ModuleFPHom)

If the module H is created via hom(M,N), for some modules M and N, and a: M $\to$ N is a homomorphism, then return the element of H corresponding to a.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = FreeMod(R, 2);

julia> V = [x*F[1], y^2*F[2]];

julia> M = quo(F, V)[1]
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 2 generators
1 -> x*e[1]
2 -> y^2*e[2]

julia> H = hom(M, M)[1];

julia> gens(H)
2-element Vector{SubquoModuleElem{QQMPolyRingElem}}:
 (e[1] -> e[1])
 (e[2] -> e[2])

julia> relations(H)
4-element Vector{FreeModElem{QQMPolyRingElem}}:
 x*(e[1] -> e[1])
 y^2*(e[1] -> e[2])
 x*(e[2] -> e[1])
 y^2*(e[2] -> e[2])

julia> W =  [M[1], y*M[2]];

julia> a = hom(M, M, W);

julia> iswelldefined(a)
true

julia> matrix(a)
[1   0]
[0   y]

julia> m = homomorphism_to_element(H, a)
(e[1] -> e[1]) + y*(e[2] -> e[2])

ext(M::ModuleFP, N::ModuleFP, i::Int)

Return $\text{Ext}^i(M,N)$.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = FreeMod(R, 1);

julia> V = [x*F[1], y*F[1]];

julia> M = quo(F, V)[1]
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 2 generators
1 -> x*e[1]
2 -> y*e[1]

julia> ext(M, M, 0)
Subquotient of Submodule with 1 generator
1 -> (e[1] -> e[1])
by Submodule with 2 generators
1 -> x*(e[1] -> e[1])
2 -> y*(e[1] -> e[1])

julia> ext(M, M, 1)
Subquotient of Submodule with 2 generators
1 -> (e[2] -> e[1])
2 -> (e[1] -> e[1])
by Submodule with 4 generators
1 -> x*(e[1] -> e[1])
2 -> y*(e[1] -> e[1])
3 -> x*(e[2] -> e[1])
4 -> y*(e[2] -> e[1])

julia> ext(M, M, 2)
Subquotient of Submodule with 1 generator
1 -> (e[1] -> e[1])
by Submodule with 3 generators
1 -> x*(e[1] -> e[1])
2 -> y*(e[1] -> e[1])
3 -> -y*(e[1] -> e[1])

julia> ext(M, M, 3)
Submodule with 0 generators
represented as subquotient with no relations.

tensor_product(M::ModuleFP...; task::Symbol = :none)

Given a collection of modules, say, $M_1, \dots, M_n$ over a ring $R$, return $M_1\otimes_R \cdots \otimes_R M_n$.

If task = :map, additionally return the map which sends a tuple $(m_1,\dots, m_n)$ of elements $m_i\in M_i$ to the pure tensor $m_1\otimes\dots\otimes m_n$.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> F = free_module(R, 1);

julia> A = R[x; y];

julia> B = R[x^2; y^3; z^4];

julia> M = SubquoModule(F, A, B);

julia> gens(M)
2-element Vector{SubquoModuleElem{QQMPolyRingElem}}:
 x*e[1]
 y*e[1]

julia> T, t = tensor_product(M, M; task = :map);

julia> gens(T)
4-element Vector{SubquoModuleElem{QQMPolyRingElem}}:
 x^2*e[1] \otimes e[1]
 x*y*e[1] \otimes e[1]
 x*y*e[1] \otimes e[1]
 y^2*e[1] \otimes e[1]

julia> domain(t)
parent of tuples of type Tuple{SubquoModuleElem{QQMPolyRingElem}, SubquoModuleElem{QQMPolyRingElem}}

julia> t((M[1], M[2]))
x*y*e[1] \otimes e[1]

tor(M::ModuleFP, N::ModuleFP, i::Int)

Return $\text{Tor}_i(M,N)$.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> A = R[x; y];

julia> B = R[x^2; y^3; z^4];

julia> M = SubquoModule(A, B);

julia> F = free_module(R, 1);

julia> Q, _ = quo(F, [x*F[1]]);

julia> T0 = tor(Q, M, 0)
Subquotient of Submodule with 2 generators
1 -> x*e[1] \otimes e[1]
2 -> y*e[1] \otimes e[1]
by Submodule with 4 generators
1 -> x^2*e[1] \otimes e[1]
2 -> y^3*e[1] \otimes e[1]
3 -> z^4*e[1] \otimes e[1]
4 -> x*y*e[1] \otimes e[1]

julia> T1 = tor(Q, M, 1)
Subquotient of Submodule with 1 generator
1 -> -x*e[1] \otimes e[1]
by Submodule with 3 generators
1 -> x^2*e[1] \otimes e[1]
2 -> y^3*e[1] \otimes e[1]
3 -> z^4*e[1] \otimes e[1]

julia> T2 =  tor(Q, M, 2)
Submodule with 0 generators
represented as subquotient with no relations.

 fitting_ideal(M::ModuleFP{T}, i::Int) where T <: MPolyRingElem

Return the i-th Fitting ideal of M.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = free_module(R, 2);

julia> o = zero(R)
0

julia> U = matrix([x^3-y^2 o; o x^3-y^2; -x^2 y; -y x])
[x^3 - y^2           0]
[        0   x^3 - y^2]
[     -x^2           y]
[       -y           x]

julia> M = quo(F,U)[1]
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 4 generators
1 -> (x^3 - y^2)*e[1]
2 -> (x^3 - y^2)*e[2]
3 -> -x^2*e[1] + y*e[2]
4 -> -y*e[1] + x*e[2]

julia> fitting_ideal(M, -1)
ideal(0)

julia> fitting_ideal(M, 0)
ideal(x^3 - y^2)

julia> fitting_ideal(M, 1)
ideal(y, x)

julia> fitting_ideal(M, 2)
ideal(1)

 is_flat(M::ModuleFP{T}) where T <: MPolyRingElem

Return true if M is flat, false otherwise.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = free_module(R, 2);

julia> o = zero(R)
0

julia> U = matrix([x^3-y^2 o; o x^3-y^2; -x^2 y; -y x])
[x^3 - y^2           0]
[        0   x^3 - y^2]
[     -x^2           y]
[       -y           x]

julia> M = quo(F,U)[1]
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 4 generators
1 -> (x^3 - y^2)*e[1]
2 -> (x^3 - y^2)*e[2]
3 -> -x^2*e[1] + y*e[2]
4 -> -y*e[1] + x*e[2]

julia> is_flat(M)
false

 non_flat_locus(M::ModuleFP{T}) where T <: MPolyRingElem

Return an ideal of base_ring(M) which defines the non-flat-locus of M in the sense that the localization of M at a prime ideal of base_ring(M) is non-flat iff the prime ideal contains the returned ideal.

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> F = free_module(R, 2);

julia> o = zero(R)
0

julia> U = matrix([x^3-y^2 o; o x^3-y^2; -x^2 y; -y x])
[x^3 - y^2           0]
[        0   x^3 - y^2]
[     -x^2           y]
[       -y           x]

julia> M = quo(F,U)[1]
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 4 generators
1 -> (x^3 - y^2)*e[1]
2 -> (x^3 - y^2)*e[2]
3 -> -x^2*e[1] + y*e[2]
4 -> -y*e[1] + x*e[2]

julia> non_flat_locus(M)
ideal(x^3 - y^2)

 is_regular_sequence(V::Vector{T}, M::ModuleFP{T}) where T <: MPolyRingElem

Return true if the elements of V form, in the given order, a regular sequence on M. Return false, otherwise.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> F = free_module(R, 1);

julia> V =  [x*z-z, x*y-y, x]
3-element Vector{QQMPolyRingElem}:
 x*z - z
 x*y - y
 x

julia> is_regular_sequence(V, F)
false

julia> W = [x*z-z, x, x*y-y]
3-element Vector{QQMPolyRingElem}:
 x*z - z
 x
 x*y - y

julia> is_regular_sequence(W, F)
true

 koszul_matrix(V::Vector{T}, p::Int) where T <: MPolyRingElem

If $f_1, \dots, f_r$ are the entries of V in the given order, return the matrix representing the p-th map of the Koszul complex $K(f_1, \dots, f_r)$.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> V = gens(R)
3-element Vector{QQMPolyRingElem}:
 x
 y
 z

julia> koszul_matrix(V, 3)
[z   -y   x]

julia> koszul_matrix(V, 2)
[-y    x   0]
[-z    0   x]
[ 0   -z   y]

julia> koszul_matrix(V, 1)
[x]
[y]
[z]

 koszul_complex(V::Vector{T}) where T <: MPolyRingElem

If $f_1, \dots, f_r$ are the entries of V in the given order, return the Koszul complex $K(f_1, \dots, f_r)$.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> V = gens(R)
3-element Vector{QQMPolyRingElem}:
 x
 y
 z

julia> K = koszul_complex(V);

julia> matrix(map(K, 2))
[-y    x   0]
[-z    0   x]
[ 0   -z   y]

julia> Kd = hom(K, free_module(R, 1));

julia> matrix(map(Kd, 1))
[-y   -z    0]
[ x    0   -z]
[ 0    x    y]

 koszul_homology(V::Vector{T}, M::ModuleFP{T}, p::Int) where T <: MPolyRingElem

If $f_1, \dots, f_r$ are the entries of V in the given order, return the p-th homology module of the complex $K(f_1, \dots, f_r)\otimes_R M$, where $K(f_1, \dots, f_r)$ is the Koszul complex defined by $f_1, \dots, f_r$.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> F = free_module(R, 1);

julia> V =  [x*y, x*z, y*z]
3-element Vector{QQMPolyRingElem}:
 x*y
 x*z
 y*z

julia> koszul_homology(V, F, 0)
Submodule with 3 generators
1 -> y*z*e[1]
2 -> x*z*e[1]
3 -> x*y*e[1]
represented as subquotient with no relations.

julia> koszul_homology(V, F, 1)
Submodule with 3 generators
1 -> -z*e[1] + z*e[2]
2 -> y*e[2]
3 -> x*e[1]
represented as subquotient with no relations.

julia> koszul_homology(V, F, 2)
Submodule with 1 generator
1 -> 0
represented as subquotient with no relations.

julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"]);

julia> TC = ideal(R, [x*z-y^2, w*z-x*y, w*y-x^2]);

julia> F = free_module(R, 1);

julia> koszul_homology(gens(TC), F, 0)
Submodule with 3 generators
1 -> (-x*z + y^2)*e[1]
2 -> (-w*z + x*y)*e[1]
3 -> (-w*y + x^2)*e[1]
represented as subquotient with no relations.

julia> koszul_homology(gens(TC), F, 1)
Submodule with 3 generators
1 -> y*e[1] - z*e[2]
2 -> x*e[1] - y*e[2]
3 -> w*e[1] - x*e[2]
represented as subquotient with no relations.

julia> koszul_homology(gens(TC), F, 2)
Submodule with 1 generator
1 -> 0
represented as subquotient with no relations.

 depth(I::MPolyIdeal{T}, M::ModuleFP{T}) where T <: MPolyRingElem

Return the depth of I on M.

Examples

julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"]);

julia> TC = ideal(R, [x*z-y^2, w*z-x*y, w*y-x^2]);

julia> dim(TC)
2

julia> F = free_module(R, 1);

julia> U = collect(gen(TC, i)*F[1] for i in 1:ngens(TC));

julia> M, _ = quo(F, U);

julia> I = ideal(R, gens(R))
ideal(w, x, y, z)

julia> depth(I, M)
2

julia> S, x, y = polynomial_ring(QQ, "x" => 1:3, "y" => 1:5);

julia> W = [y[1]-x[1]^2,  y[2]-x[2]^2,   y[3]-x[3]^2, y[4]-x[2]*(x[1]-x[3]),  y[5]-(x[1]-x[2])*x[3]];

julia> J = eliminate(ideal(S, W), x);

julia> R, y = polynomial_ring(QQ, "y" => 1:5);

julia> W = append!(repeat([zero(R)], 3), gens(R))
8-element Vector{QQMPolyRingElem}:
 0
 0
 0
 y[1]
 y[2]
 y[3]
 y[4]
 y[5]

julia> P = hom(S, R, W);

julia> VP4 = P(J);

julia> dim(VP4)
3

julia> F = free_module(R, 1);

julia> U = collect(gen(VP4, i)*F[1] for i in 1:ngens(VP4));

julia> M, _ = quo(F, U);

julia> I = ideal(R, gens(R))
ideal(y[1], y[2], y[3], y[4], y[5])

julia> depth(I, M)
1