Operations on Modules

Kernel

kernel — Method

kernel(a::ModuleFPHom)

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

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

Examples

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

julia> F = free_module(R, 3);

julia> G = free_module(R, 2);

julia> W = R[y 0; x y; 0 z]
[y   0]
[x   y]
[0   z]

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

julia> K, incl = kernel(a);

julia> K
Submodule with 1 generator
1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
represented as subquotient with no relations.

julia> incl
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 3 variables over QQ

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

julia> F = free_module(R, 1);

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

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

julia> M = SubquoModule(F, 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{SubquoModuleElem{QQMPolyRingElem}}:
 x*y^2*e[1]
 x*y*e[1]

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

julia> K, incl = kernel(a);

julia> K
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]

julia> incl
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]

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

julia> Z = abelian_group(0)
GrpAb: Z

julia> Rg, (x, y, z) = grade(R, [Z[1],Z[1],Z[1]])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])

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)
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> N = M;

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

julia> a = hom(M, N, V)
M -> M
x*e[1] -> x*y^2*e[1]
y*e[1] -> x^2*y*e[1]
Graded module homomorphism of degree [2]

julia> kernel(a)
(Graded subquotient of submodule of F generated by
1 -> -y*e[1]
2 -> (x^2 - y^2)*e[1]
3 -> -x*y*e[1]
by submodule of F generated by
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1], Graded subquotient of submodule of F generated by
1 -> -y*e[1]
2 -> (x^2 - y^2)*e[1]
3 -> -x*y*e[1]
by submodule of F generated by
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1] -> M
-y*e[1] -> -y*e[1]
(x^2 - y^2)*e[1] -> (x^2 - y^2)*e[1]
-x*y*e[1] -> -x*y*e[1]
Homogeneous module homomorphism)

source

Image

image — Method

image(a::ModuleFPHom)

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

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

Examples

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

julia> F = free_module(R, 3);

julia> G = free_module(R, 2);

julia> W = R[y 0; x y; 0 z]
[y   0]
[x   y]
[0   z]

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

julia> I, incl = image(a);

julia> I
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.

julia> incl
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 3 variables over QQ

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

julia> F = free_module(R, 1);

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

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

julia> M = SubquoModule(F, 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{SubquoModuleElem{QQMPolyRingElem}}:
 x*y^2*e[1]
 x*y*e[1]

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

julia> I, incl = image(a);

julia> I
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]

julia> incl
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]

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

julia> Z = abelian_group(0)
GrpAb: Z

julia> Rg, (x, y, z) = grade(R, [Z[1],Z[1],Z[1]])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])

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)
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> N = M;

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

julia> a = hom(M, N, V)
M -> M
x*e[1] -> x*y^2*e[1]
y*e[1] -> x^2*y*e[1]
Graded module homomorphism of degree [2]

julia> image(a)
(Graded subquotient of submodule of F generated by
1 -> x*y^2*e[1]
2 -> x^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], Graded subquotient of submodule of F generated by
1 -> x*y^2*e[1]
2 -> x^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] -> M
x*y^2*e[1] -> x*y^2*e[1]
x^2*y*e[1] -> x^2*y*e[1]
Homogeneous module homomorphism)

source

Cokernel

cokernel — Method

cokernel(a::ModuleFPHom)

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

Examples

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

julia> F = free_module(R, 3);

julia> G = free_module(R, 2);

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

julia> F = free_module(R, 1);

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

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

julia> M = SubquoModule(F, 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{SubquoModuleElem{QQMPolyRingElem}}:
 x*y^2*e[1]
 x*y*e[1]

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

julia> cokernel(a)
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]

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, 3);

julia> G = graded_free_module(Rg, 2);

julia> W = Rg[y 0; x y; 0 z]
[y   0]
[x   y]
[0   z]

julia> a = hom(F, G, W)
F -> G
e[1] -> y*e[1]
e[2] -> x*e[1] + y*e[2]
e[3] -> z*e[2]
Graded module homomorphism of degree [1]

julia> M = cokernel(a)
Graded subquotient of submodule of G generated by
1 -> e[1]
2 -> e[2]
by submodule of G generated by
1 -> y*e[1]
2 -> x*e[1] + y*e[2]
3 -> z*e[2]

source

Direct Sums and Products

direct_sum — Method

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 $M_i\to\bigoplus_{i=1}^n M_i$ if task = :sum (default),
a vector containing the canonical projections $\bigoplus_{i=1}^n M_i\to M_i$ 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_product — Method

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

source

Truncation

truncate — Function

truncate(M::ModuleFP, g::GrpAbFinGenElem, task::Symbol = :with_morphism)

Given a finitely presented graded module M over a $\mathbb Z$-graded multivariate polynomial ring with positive weights, return the truncation of M at degree g.

Put more precisely, return the truncation as an object of type SubquoModule.

Additionally, if N denotes this object,

return the inclusion map N $\to$ M if task = :with_morphism (default),
return and cache the inclusion map N $\to$ M if task = :cache_morphism,
do none of the above if task = :none.

If task = :only_morphism, return only the inclusion map.

truncate(M::ModuleFP, d::Int, task::Symbol = :with_morphism)

Given a module M as above, and given an integer d, convert d into an element g of the grading group of base_ring(I) and proceed as above.

Examples

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

julia> F = graded_free_module(R, 1)
Graded free module R^1([0]) of rank 1 over R

julia> V = [x*F[1]; y^4*F[1]; z^5*F[1]];

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

julia> M[1]
e[1]

julia> MT = truncate(M, 3);

julia> MT[1]
Graded subquotient of submodule of F generated by
1 -> z^3*e[1]
2 -> y*z^2*e[1]
3 -> y^2*z*e[1]
4 -> y^3*e[1]
5 -> x*z^2*e[1]
6 -> x*y*z*e[1]
7 -> x*y^2*e[1]
8 -> x^2*z*e[1]
9 -> x^2*y*e[1]
10 -> x^3*e[1]
by submodule of F generated by
1 -> x*e[1]
2 -> y^4*e[1]
3 -> z^5*e[1]

source

Operations on Modules

Subquotients Related to Homomorphisms

Kernel

Image

Cokernel

Direct Sums and Products

Truncation