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
Module homomorphism
  from K
  to F

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
Module homomorphism
  from K
  to M

julia> Rg, (x, y, z) = graded_polynomial_ring(QQ, [: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 graded submodule of F with 2 generators
  1: x*e[1]
  2: y*e[1]
by graded submodule of F 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^2*N[2]];

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

julia> kernel(a)
(Graded subquotient of graded submodule of F with 2 generators
  1: y*e[1]
  2: -x*y*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1], Hom: graded subquotient of graded submodule of F with 2 generators
  1: y*e[1]
  2: -x*y*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1] -> M)

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
Module homomorphism
  from I
  to G

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
Module homomorphism
  from I
  to M

julia> Rg, (x, y, z) = graded_polynomial_ring(QQ, [: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 graded submodule of F with 2 generators
  1: x*e[1]
  2: y*e[1]
by graded submodule of F 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^2*N[2]];

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

julia> image(a)
(Graded subquotient of graded submodule of F with 2 generators
  1: x*y^2*e[1]
  2: x^2*y*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1], Hom: graded subquotient of graded submodule of F with 2 generators
  1: x*y^2*e[1]
  2: x^2*y*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1] -> M)

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> Rg, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

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)
Graded module homomorphism of degree [1]
  from F
  to G
defined by
  e[1] -> y*e[1]
  e[2] -> x*e[1] + y*e[2]
  e[3] -> z*e[2]

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

source

cokernel(F::FreeMod{R}, A::MatElem{R}) where R

Return the cokernel of A as an object of type SubquoModule with ambient free module F.

Examples

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

julia> F = free_module(R, 2)
Free module of rank 2 over R

julia> A = R[x y; 2*x^2 3*y^2]
[    x       y]
[2*x^2   3*y^2]
 
julia> M = cokernel(F, A)
Subquotient of submodule with 2 generators
  1: e[1]
  2: e[2]
by submodule with 2 generators
  1: x*e[1] + y*e[2]
  2: 2*x^2*e[1] + 3*y^2*e[2]

julia> ambient_free_module(M) === F
true

julia> Rg, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> F = graded_free_module(Rg, [8,8])
Graded free module Rg^2([-8]) of rank 2 over Rg

julia> A = Rg[x y; 2*x^2 3*y^2]
[    x       y]
[2*x^2   3*y^2]
 
julia> M = cokernel(F, A)
Graded subquotient of graded submodule of F with 2 generators
  1: e[1]
  2: e[2]
by graded submodule of F with 2 generators
  1: x*e[1] + y*e[2]
  2: 2*x^2*e[1] + 3*y^2*e[2]

julia> ambient_free_module(M) === F
true

julia> degrees_of_generators(M)
2-element Vector{FinGenAbGroupElem}:
 [8]
 [8]

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(I::ModuleFP, g::FinGenAbGroupElem, 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 graded submodule of F with 10 generators
  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 graded submodule of F with 3 generators
  1: x*e[1]
  2: y^4*e[1]
  3: z^5*e[1]

source

Twists

In the graded case, we have:

twist — Method

twist(M::ModuleFP{T}, g::FinGenAbGroupElem) where {T<:MPolyDecRingElem}

Return the twisted module M(g).

twist(M::ModuleFP{T}, W::Vector{<:IntegerUnion}) where {T<:MPolyDecRingElem}

Given a module M over a $\mathbb Z^m$-graded polynomial ring and a vector W of $m$ integers, convert W into an element g of the grading group of the ring and proceed as above.

twist(M::ModuleFP{T}, d::IntegerUnion) where {T<:MPolyDecRingElem}

Given a module M over a $\mathbb Z$-graded polynomial ring and an integer d, convert d into an element g of the grading group of the ring and proceed as above.

Examples

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

julia> I = ideal(R, [zero(R)])
Ideal generated by
  0

julia> M = quotient_ring_as_module(I)
Graded submodule of R^1 with 1 generator
  1: e[1]
represented as subquotient with no relations

julia> degree(gen(M, 1))
[0]

julia> N = twist(M, 2)
Graded submodule of R^1 with 1 generator
  1: e[1]
represented as subquotient with no relations

julia> degree(gen(N, 1))
[-2]

source