Operations on Modules

Kernel

kernelMethod
kernel(a::ModuleFPHom)

Return the kernel of a as an object of type SubQuo.

Additionally, if K denotes this object, return the inclusion map K $\rightarrow$ domain(a).

Examples

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> W = matrix(R, [y 0; x y; 0 z])
[y   0]
[x   y]
[0   z]

julia> a = hom(F, G, W);

julia> kernel(a)
(Submodule with 1 generator
1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
represented as subquotient with no relations., Map with following data
Domain:
=======
Submodule with 1 generator
1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
represented as subquotient with no relations.
Codomain:
=========
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field)
julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F1 = free_module(R, 1)
Free module of rank 1 over Multivariate Polynomial Ring in x, y, z over Rational Field

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

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

julia> M = SubQuo(F1, 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> N = M;

julia> V = [y^2*N[1], x*N[2]]
2-element Vector{SubQuoElem{fmpq_mpoly}}:
 x*y^2*e[1]
 x*y*e[1]

julia> b = hom(M, N, V);

julia> K, iota = kernel(b)
(Subquotient of Submodule with 3 generators
1 -> (-x + y^2)*e[1]
2 -> x*y*e[1]
3 -> -x*y*e[1]
by Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1], Map with following data
Domain:
=======
Subquotient of Submodule with 3 generators
1 -> (-x + y^2)*e[1]
2 -> x*y*e[1]
3 -> -x*y*e[1]
by Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]
Codomain:
=========
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])
source

Image

imageMethod
image(a::ModuleFPHom)

Return the image of a as an object of type SubQuo.

Additionally, if I denotes this object, return the inclusion map I $\rightarrow$ codomain(a).

Examples

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> W = matrix(R, [y 0; x y; 0 z])
[y   0]
[x   y]
[0   z]

julia> a = hom(F, G, W);

julia> I, iota = image(a)
(Submodule with 3 generators
1 -> y*e[1]
2 -> x*e[1] + y*e[2]
3 -> z*e[2]
represented as subquotient with no relations., Map with following data
Domain:
=======
Submodule with 3 generators
1 -> y*e[1]
2 -> x*e[1] + y*e[2]
3 -> z*e[2]
represented as subquotient with no relations.
Codomain:
=========
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field)
julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F1 = free_module(R, 1)
Free module of rank 1 over Multivariate Polynomial Ring in x, y, z over Rational Field

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

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

julia> M = SubQuo(F1, 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> N = M;

julia> V = [y^2*N[1], x*N[2]]
2-element Vector{SubQuoElem{fmpq_mpoly}}:
 x*y^2*e[1]
 x*y*e[1]

julia> b = hom(M, N, V);

julia> image(b)
(Subquotient of Submodule with 2 generators
1 -> x*y^2*e[1]
2 -> x*y*e[1]
by Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1], Map with following data
Domain:
=======
Subquotient of Submodule with 2 generators
1 -> x*y^2*e[1]
2 -> x*y*e[1]
by Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]
Codomain:
=========
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])
source

Cokernel

cokernelMethod
cokernel(a::ModuleFPHom)

Return the cokernel of a as an object of type SubQuo.

Examples

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> W = matrix(R, [y 0; x y; 0 z])
[y   0]
[x   y]
[0   z]

julia> a = hom(F, G, W);

julia> cokernel(a)
Subquotient of Submodule with 2 generators
1 -> e[1]
2 -> e[2]
by Submodule with 3 generators
1 -> y*e[1]
2 -> x*e[1] + y*e[2]
3 -> z*e[2]
julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F1 = free_module(R, 1)
Free module of rank 1 over Multivariate Polynomial Ring in x, y, z over Rational Field

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

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

julia> M = SubQuo(F1, 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> N = M;

julia> V = [y^2*N[1], x*N[2]]
2-element Vector{SubQuoElem{fmpq_mpoly}}:
 x*y^2*e[1]
 x*y*e[1]

julia> b = hom(M, N, V);

julia> cokernel(b)
Subquotient of Submodule with 2 generators
1 -> x*e[1]
2 -> y*e[1]
by Submodule with 5 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]
4 -> x*y^2*e[1]
5 -> x*y*e[1]
source

Direct Sums and Products

direct_sumMethod
direct_sum(M::ModuleFP{T}...; task::Symbol = :sum) where T

Given modules $M_1\dots M_n$, say, return the direct sum $\bigoplus_{i=1}^n M_i$.

Additionally, return

  • a vector containing the canonical injections $M_i\rightarrow\bigoplus_{i=1}^n M_i$ if task = :sum (default),
  • a vector containing the canonical projections $\bigoplus_{i=1}^n M_i\rightarrow M_i$ if task = :prod,
  • two vectors containing the canonical injections and projections, respectively, if task = :both,
  • none of the above maps if task = :none.
source
direct_productMethod
direct_product(M::ModuleFP{T}...; task::Symbol = :prod) where T

Given modules $M_1\dots M_n$, say, return the direct product $\prod_{i=1}^n M_i$.

Additionally, return

  • a vector containing the canonical projections $\prod_{i=1}^n M_i\rightarrow M_i$ if task = :prod (default),
  • a vector containing the canonical injections $M_i\rightarrow\prod_{i=1}^n M_i$ if task = :sum,
  • two vectors containing the canonical projections and injections, respectively, if task = :both,
  • none of the above maps if task = :none.
source

Presentations

presentationMethod
presentation(M::ModuleFP)

Return a free presentation of $M$.

Examples

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

julia> A = R[x; y];

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

julia> M = SubQuo(A, B);

julia> P = presentation(M);

julia> rank(P[1])
5

julia> rank(P[0])
2
source

Syzygies and Free Resolutions

free_resolutionMethod
free_resolution(M::SubQuo; 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) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[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 = SubQuo(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 x, y, z over Rational Field

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
source

Homology

homologyMethod
homology(C::ChainComplex{<:ModuleFP})

Return the homology of C.

Examples

julia> R, (x,) = PolynomialRing(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 = ChainComplex(ModuleFP, [a, b]);

julia> H = homology(C)
3-element Vector{SubQuo{fmpq_mpoly}}:
 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]
source
homologyMethod
homology(C::ChainComplex{<:ModuleFP}, i::Int)

Return the i-th homology module of C.

Examples

julia> R, (x,) = PolynomialRing(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 = ChainComplex(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]
source

Hom and Ext

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

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

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) = PolynomialRing(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 (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], 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> gens(H)
2-element Vector{SubQuoElem{fmpq_mpoly}}:
 (e[1] -> e[1])
 (e[2] -> e[2])

julia> relations(H)
4-element Vector{FreeModElem{fmpq_mpoly}}:
 x*(e[1] -> e[1])
 y^2*(e[1] -> e[2])
 x*(e[2] -> e[1])
 y^2*(e[2] -> e[2])
source
element_to_homomorphismMethod
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) = PolynomialRing(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{SubQuoElem{fmpq_mpoly}}:
 (e[1] -> e[1])
 (e[2] -> e[2])

julia> relations(H)
4-element Vector{FreeModElem{fmpq_mpoly}}:
 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]
source
homomorphism_to_elementMethod
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) = PolynomialRing(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{SubQuoElem{fmpq_mpoly}}:
 (e[1] -> e[1])
 (e[2] -> e[2])

julia> relations(H)
4-element Vector{FreeModElem{fmpq_mpoly}}:
 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])
source
extMethod
ext(M::ModuleFP, N::ModuleFP, i::Int)

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

Examples

julia> R, (x, y) = PolynomialRing(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.
source

Tensorproduct and Tor

tensor_productMethod
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) = PolynomialRing(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 = SubQuo(F, A, B);

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

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

julia> gens(T)
4-element Vector{SubQuoElem{fmpq_mpoly}}:
 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{SubQuoElem{fmpq_mpoly}, SubQuoElem{fmpq_mpoly}}

julia> t((M[1], M[2]))
x*y*e[1] \otimes e[1]
source
torMethod
tor(M::ModuleFP, N::ModuleFP, i::Int)

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

Examples

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

julia> A = R[x; y];

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

julia> M = SubQuo(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.
source