Morphisms between finitely generated modules over a PID#

AUTHOR:

William Stein, 2009

sage.modules.fg_pid.fgp_morphism.FGP_Homset(X, Y)[source]#

EXAMPLES:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W
sage: Q.Hom(Q)           # indirect doctest
Set of Morphisms from Finitely generated module V/W over Integer Ring with invariants (4, 12) to Finitely generated module V/W over Integer Ring with invariants (4, 12) in Category of modules over Integer Ring
sage: True # Q.Hom(Q) is Q.Hom(Q)
True
sage: type(Q.Hom(Q))
<class 'sage.modules.fg_pid.fgp_morphism.FGP_Homset_class_with_category'>

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)]); Q = V/W
>>> Q.Hom(Q)           # indirect doctest
Set of Morphisms from Finitely generated module V/W over Integer Ring with invariants (4, 12) to Finitely generated module V/W over Integer Ring with invariants (4, 12) in Category of modules over Integer Ring
>>> True # Q.Hom(Q) is Q.Hom(Q)
True
>>> type(Q.Hom(Q))
<class 'sage.modules.fg_pid.fgp_morphism.FGP_Homset_class_with_category'>

class sage.modules.fg_pid.fgp_morphism.FGP_Homset_class(X, Y, category=None)[source]#

Bases: Homset

Homsets of FGP_Module

Element[source]#: alias of FGP_Morphism

class sage.modules.fg_pid.fgp_morphism.FGP_Morphism(parent, phi, check=True)[source]#

Bases: Morphism

A morphism between finitely generated modules over a PID.

EXAMPLES:

An endomorphism:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi = Q.hom([Q.0+3*Q.1, -Q.1]); phi
Morphism from module over Integer Ring with invariants (4, 12) to module with invariants (4, 12) that sends the generators to [(1, 3), (0, 11)]
sage: phi(Q.0) == Q.0 + 3*Q.1
True
sage: phi(Q.1) == -Q.1
True

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> phi = Q.hom([Q.gen(0)+Integer(3)*Q.gen(1), -Q.gen(1)]); phi
Morphism from module over Integer Ring with invariants (4, 12) to module with invariants (4, 12) that sends the generators to [(1, 3), (0, 11)]
>>> phi(Q.gen(0)) == Q.gen(0) + Integer(3)*Q.gen(1)
True
>>> phi(Q.gen(1)) == -Q.gen(1)
True

A morphism between different modules V1/W1 —> V2/W2 in different ambient spaces:

Sage

sage: V1 = ZZ^2; W1 = V1.span([[1,2],[3,4]]); A1 = V1/W1
sage: V2 = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W2 = V2.span([2*V2.0+4*V2.1, 9*V2.0+12*V2.1, 4*V2.2]); A2=V2/W2
sage: A1
Finitely generated module V/W over Integer Ring with invariants (2)
sage: A2
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi = A1.hom([2*A2.0])
sage: phi(A1.0)
(2, 0)
sage: 2*A2.0
(2, 0)
sage: phi(2*A1.0)
(0, 0)

Python

>>> from sage.all import *
>>> V1 = ZZ**Integer(2); W1 = V1.span([[Integer(1),Integer(2)],[Integer(3),Integer(4)]]); A1 = V1/W1
>>> V2 = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W2 = V2.span([Integer(2)*V2.gen(0)+Integer(4)*V2.gen(1), Integer(9)*V2.gen(0)+Integer(12)*V2.gen(1), Integer(4)*V2.gen(2)]); A2=V2/W2
>>> A1
Finitely generated module V/W over Integer Ring with invariants (2)
>>> A2
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> phi = A1.hom([Integer(2)*A2.gen(0)])
>>> phi(A1.gen(0))
(2, 0)
>>> Integer(2)*A2.gen(0)
(2, 0)
>>> phi(Integer(2)*A1.gen(0))
(0, 0)

im_gens()[source]#

Return tuple of the images of the generators of the domain under this morphism.

EXAMPLES:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W
sage: phi = Q.hom([Q.0,Q.0 + 2*Q.1])
sage: phi.im_gens()
((1, 0), (1, 2))
sage: phi.im_gens() is phi.im_gens()
True

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)]); Q = V/W
>>> phi = Q.hom([Q.gen(0),Q.gen(0) + Integer(2)*Q.gen(1)])
>>> phi.im_gens()
((1, 0), (1, 2))
>>> phi.im_gens() is phi.im_gens()
True

image()[source]#

Compute the image of this homomorphism.

EXAMPLES:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: Q.hom([Q.0+3*Q.1, -Q.1]).image()
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: Q.hom([3*Q.1, Q.1]).image()
Finitely generated module V/W over Integer Ring with invariants (12)

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> Q.hom([Q.gen(0)+Integer(3)*Q.gen(1), -Q.gen(1)]).image()
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> Q.hom([Integer(3)*Q.gen(1), Q.gen(1)]).image()
Finitely generated module V/W over Integer Ring with invariants (12)

inverse_image(A)[source]#

Given a submodule A of the codomain of this morphism, return the inverse image of A under this morphism.

EXAMPLES:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi = Q.hom([0, Q.1])
sage: phi.inverse_image(Q.submodule([]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.kernel()
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.inverse_image(phi.codomain())
Finitely generated module V/W over Integer Ring with invariants (4, 12)

sage: phi.inverse_image(Q.submodule([Q.0]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.inverse_image(Q.submodule([Q.1]))
Finitely generated module V/W over Integer Ring with invariants (4, 12)

sage: phi.inverse_image(ZZ^3)
Traceback (most recent call last):
...
TypeError: A must be a finitely generated quotient module
sage: phi.inverse_image(ZZ^3 / W.scale(2))
Traceback (most recent call last):
...
ValueError: A must be a submodule of the codomain

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)]); Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> phi = Q.hom([Integer(0), Q.gen(1)])
>>> phi.inverse_image(Q.submodule([]))
Finitely generated module V/W over Integer Ring with invariants (4)
>>> phi.kernel()
Finitely generated module V/W over Integer Ring with invariants (4)
>>> phi.inverse_image(phi.codomain())
Finitely generated module V/W over Integer Ring with invariants (4, 12)

>>> phi.inverse_image(Q.submodule([Q.gen(0)]))
Finitely generated module V/W over Integer Ring with invariants (4)
>>> phi.inverse_image(Q.submodule([Q.gen(1)]))
Finitely generated module V/W over Integer Ring with invariants (4, 12)

>>> phi.inverse_image(ZZ**Integer(3))
Traceback (most recent call last):
...
TypeError: A must be a finitely generated quotient module
>>> phi.inverse_image(ZZ**Integer(3) / W.scale(Integer(2)))
Traceback (most recent call last):
...
ValueError: A must be a submodule of the codomain

kernel()[source]#

Compute the kernel of this homomorphism.

EXAMPLES:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: Q.hom([0, Q.1]).kernel()
Finitely generated module V/W over Integer Ring with invariants (4)
sage: A = Q.hom([Q.0, 0]).kernel(); A
Finitely generated module V/W over Integer Ring with invariants (12)
sage: Q.1 in A
True
sage: phi = Q.hom([Q.0-3*Q.1, Q.0+Q.1])
sage: A = phi.kernel(); A
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi(A)
Finitely generated module V/W over Integer Ring with invariants ()

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> Q.hom([Integer(0), Q.gen(1)]).kernel()
Finitely generated module V/W over Integer Ring with invariants (4)
>>> A = Q.hom([Q.gen(0), Integer(0)]).kernel(); A
Finitely generated module V/W over Integer Ring with invariants (12)
>>> Q.gen(1) in A
True
>>> phi = Q.hom([Q.gen(0)-Integer(3)*Q.gen(1), Q.gen(0)+Q.gen(1)])
>>> A = phi.kernel(); A
Finitely generated module V/W over Integer Ring with invariants (4)
>>> phi(A)
Finitely generated module V/W over Integer Ring with invariants ()

lift(x)[source]#

Given an element x in the codomain of self, if possible find an element y in the domain such that self(y) == x. Raise a ValueError if no such y exists.

INPUT:

x – element of the codomain of self.

EXAMPLES:

Sage

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q=V/W; phi = Q.hom([2*Q.0, Q.1])
sage: phi.lift(Q.1)
(0, 1)
sage: phi.lift(Q.0)
Traceback (most recent call last):
...
ValueError: no lift of element to domain
sage: phi.lift(2*Q.0)
(1, 0)
sage: phi.lift(2*Q.0+Q.1)
(1, 1)
sage: V = span([[5, -1/2]],ZZ); W = span([[20,-2]],ZZ); Q = V/W; phi=Q.hom([2*Q.0])
sage: x = phi.image().0; phi(phi.lift(x)) == x
True

Python

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q=V/W; phi = Q.hom([Integer(2)*Q.gen(0), Q.gen(1)])
>>> phi.lift(Q.gen(1))
(0, 1)
>>> phi.lift(Q.gen(0))
Traceback (most recent call last):
...
ValueError: no lift of element to domain
>>> phi.lift(Integer(2)*Q.gen(0))
(1, 0)
>>> phi.lift(Integer(2)*Q.gen(0)+Q.gen(1))
(1, 1)
>>> V = span([[Integer(5), -Integer(1)/Integer(2)]],ZZ); W = span([[Integer(20),-Integer(2)]],ZZ); Q = V/W; phi=Q.hom([Integer(2)*Q.gen(0)])
>>> x = phi.image().gen(0); phi(phi.lift(x)) == x
True