Sets of morphisms between free modules¶

The class FreeModuleHomset implements sets of homomorphisms between two free modules of finite rank over the same commutative ring.

The subclass FreeModuleEndset implements the special case of sets of endomorphisms.

AUTHORS:

Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
Matthias Koeppe (2024): add FreeModuleEndset

REFERENCES:

Chaps. 13, 14 of R. Godement : Algebra [God1968]
Chap. 3 of S. Lang : Algebra [Lan2002]

class sage.tensor.modules.free_module_homset.FreeModuleEndset(fmodule, name, latex_name)[source]¶

Bases: FreeModuleHomset

Ring of endomorphisms of a free module of finite rank over a commutative ring.

Given a free modules \(M\) of rank \(n\) over a commutative ring \(R\), the class FreeModuleEndset implements the ring \(\mathrm{Hom}(M,M)\) of endomorphisms \(M\rightarrow M\).

This is a Sage parent class, whose element class is FiniteRankFreeModuleMorphism.

INPUT:

fmodule – free module \(M\) (domain and codomain of the endomorphisms), as an instance of FiniteRankFreeModule
name – (default: None) string; name given to the end-set; if none is provided, Hom(M,M) will be used
latex_name – (default: None) string; LaTeX symbol to denote the hom-set; if none is provided, \(\mathrm{Hom}(M,M)\) will be used

EXAMPLES:

The set of homomorphisms \(M\rightarrow M\), i.e. endomorphisms, is obtained by the function End():

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: End(M)
Set of Morphisms from Rank-3 free module M over the Integer Ring
 to Rank-3 free module M over the Integer Ring
 in Category of finite dimensional modules over Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> End(M)
Set of Morphisms from Rank-3 free module M over the Integer Ring
 to Rank-3 free module M over the Integer Ring
 in Category of finite dimensional modules over Integer Ring

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e')
End(M)

End(M) is actually identical to Hom(M,M):

Sage

sage: End(M) is Hom(M,M)
True

Python

>>> from sage.all import *
>>> End(M) is Hom(M,M)
True

Sage Live

End(M) is Hom(M,M)

The unit of the endomorphism ring is the identity map:

Sage

sage: End(M).one()
Identity endomorphism of Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> End(M).one()
Identity endomorphism of Rank-3 free module M over the Integer Ring

Sage Live

End(M).one()

whose matrix in any basis is of course the identity matrix:

Sage

sage: End(M).one().matrix(e)
[1 0 0]
[0 1 0]
[0 0 1]

Python

>>> from sage.all import *
>>> End(M).one().matrix(e)
[1 0 0]
[0 1 0]
[0 0 1]

Sage Live

End(M).one().matrix(e)

There is a canonical identification between endomorphisms of \(M\) and tensors of type \((1,1)\) on \(M\). Accordingly, coercion maps have been implemented between \(\mathrm{End}(M)\) and \(T^{(1,1)}(M)\) (the module of all type-\((1,1)\) tensors on \(M\), see TensorFreeModule):

Sage

sage: T11 = M.tensor_module(1,1) ; T11
Free module of type-(1,1) tensors on the Rank-3 free module M over
 the Integer Ring
sage: End(M).has_coerce_map_from(T11)
True
sage: T11.has_coerce_map_from(End(M))
True

Python

>>> from sage.all import *
>>> T11 = M.tensor_module(Integer(1),Integer(1)) ; T11
Free module of type-(1,1) tensors on the Rank-3 free module M over
 the Integer Ring
>>> End(M).has_coerce_map_from(T11)
True
>>> T11.has_coerce_map_from(End(M))
True

Sage Live

T11 = M.tensor_module(1,1) ; T11
End(M).has_coerce_map_from(T11)
T11.has_coerce_map_from(End(M))

See TensorFreeModule for examples of the above coercions.

There is a coercion \(\mathrm{GL}(M) \rightarrow \mathrm{End}(M)\), since every automorphism is an endomorphism:

Sage

sage: GL = M.general_linear_group() ; GL
General linear group of the Rank-3 free module M over the Integer Ring
sage: End(M).has_coerce_map_from(GL)
True

Python

>>> from sage.all import *
>>> GL = M.general_linear_group() ; GL
General linear group of the Rank-3 free module M over the Integer Ring
>>> End(M).has_coerce_map_from(GL)
True

Sage Live

GL = M.general_linear_group() ; GL
End(M).has_coerce_map_from(GL)

Of course, there is no coercion in the reverse direction, since only bijective endomorphisms are automorphisms:

Sage

sage: GL.has_coerce_map_from(End(M))
False

Python

>>> from sage.all import *
>>> GL.has_coerce_map_from(End(M))
False

Sage Live

GL.has_coerce_map_from(End(M))

The coercion \(\mathrm{GL}(M) \rightarrow \mathrm{End}(M)\) in action:

Sage

sage: a = GL.an_element() ; a
Automorphism of the Rank-3 free module M over the Integer Ring
sage: a.matrix(e)
[ 1  0  0]
[ 0 -1  0]
[ 0  0  1]
sage: ea = End(M)(a) ; ea
Generic endomorphism of Rank-3 free module M over the Integer Ring
sage: ea.matrix(e)
[ 1  0  0]
[ 0 -1  0]
[ 0  0  1]

Python

>>> from sage.all import *
>>> a = GL.an_element() ; a
Automorphism of the Rank-3 free module M over the Integer Ring
>>> a.matrix(e)
[ 1  0  0]
[ 0 -1  0]
[ 0  0  1]
>>> ea = End(M)(a) ; ea
Generic endomorphism of Rank-3 free module M over the Integer Ring
>>> ea.matrix(e)
[ 1  0  0]
[ 0 -1  0]
[ 0  0  1]

Sage Live

a = GL.an_element() ; a
a.matrix(e)
ea = End(M)(a) ; ea
ea.matrix(e)

Element[source]¶: alias of FiniteRankFreeModuleEndomorphism

one()[source]¶

Return the identity element of self considered as a monoid.

OUTPUT:

the identity element of \(\mathrm{End}(M) = \mathrm{Hom}(M,M)\), as an instance of FiniteRankFreeModuleMorphism

EXAMPLES:

Identity element of the set of endomorphisms of a free module over \(\ZZ\):

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: H = End(M)
sage: H.one()
Identity endomorphism of Rank-3 free module M over the Integer Ring
sage: H.one().matrix(e)
[1 0 0]
[0 1 0]
[0 0 1]
sage: H.one().is_identity()
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> H = End(M)
>>> H.one()
Identity endomorphism of Rank-3 free module M over the Integer Ring
>>> H.one().matrix(e)
[1 0 0]
[0 1 0]
[0 0 1]
>>> H.one().is_identity()
True

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
e = M.basis('e')
H = End(M)
H.one()
H.one().matrix(e)
H.one().is_identity()

NB: mathematically, H.one() coincides with the identity map of the free module \(M\). However the latter is considered here as an element of \(\mathrm{GL}(M)\), the general linear group of \(M\). Accordingly, one has to use the coercion map \(\mathrm{GL}(M) \rightarrow \mathrm{End}(M)\) to recover H.one() from M.identity_map():

Sage

sage: M.identity_map()
Identity map of the Rank-3 free module M over the Integer Ring
sage: M.identity_map().parent()
General linear group of the Rank-3 free module M over the Integer Ring
sage: H.one().parent()
Set of Morphisms from Rank-3 free module M over the Integer Ring
 to Rank-3 free module M over the Integer Ring
 in Category of finite dimensional modules over Integer Ring
sage: H.one() == H(M.identity_map())
True

Python

>>> from sage.all import *
>>> M.identity_map()
Identity map of the Rank-3 free module M over the Integer Ring
>>> M.identity_map().parent()
General linear group of the Rank-3 free module M over the Integer Ring
>>> H.one().parent()
Set of Morphisms from Rank-3 free module M over the Integer Ring
 to Rank-3 free module M over the Integer Ring
 in Category of finite dimensional modules over Integer Ring
>>> H.one() == H(M.identity_map())
True

Sage Live

M.identity_map()
M.identity_map().parent()
H.one().parent()
H.one() == H(M.identity_map())

Conversely, one can recover M.identity_map() from H.one() by means of a conversion \(\mathrm{End}(M)\rightarrow \mathrm{GL}(M)\):

Sage

sage: GL = M.general_linear_group()
sage: M.identity_map() == GL(H.one())
True

Python

>>> from sage.all import *
>>> GL = M.general_linear_group()
>>> M.identity_map() == GL(H.one())
True

Sage Live

GL = M.general_linear_group()
M.identity_map() == GL(H.one())

class sage.tensor.modules.free_module_homset.FreeModuleHomset(fmodule1, fmodule2, name, latex_name)[source]¶

Bases: Homset

Set of homomorphisms between free modules of finite rank over a commutative ring.

Given two free modules \(M\) and \(N\) of respective ranks \(m\) and \(n\) over a commutative ring \(R\), the class FreeModuleHomset implements the set \(\mathrm{Hom}(M,N)\) of homomorphisms \(M\rightarrow N\). The set \(\mathrm{Hom}(M,N)\) is actually a free module of rank \(mn\) over \(R\), but this aspect is not taken into account here.

This is a Sage parent class, whose element class is FiniteRankFreeModuleMorphism.

The case \(M=N\) (endomorphisms) is delegated to the subclass FreeModuleEndset.

INPUT:

fmodule1 – free module \(M\) (domain of the homomorphisms), as an instance of FiniteRankFreeModule
fmodule2 – free module \(N\) (codomain of the homomorphisms), as an instance of FiniteRankFreeModule
name – (default: None) string; name given to the hom-set; if none is provided, Hom(M,N) will be used
latex_name – (default: None) string; LaTeX symbol to denote the hom-set; if none is provided, \(\mathrm{Hom}(M,N)\) will be used

EXAMPLES:

Set of homomorphisms between two free modules over \(\ZZ\):

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: N = FiniteRankFreeModule(ZZ, 2, name='N')
sage: H = Hom(M,N) ; H
Set of Morphisms from Rank-3 free module M over the Integer Ring
 to Rank-2 free module N over the Integer Ring
 in Category of finite dimensional modules over Integer Ring
sage: type(H)
<class 'sage.tensor.modules.free_module_homset.FreeModuleHomset_with_category_with_equality_by_id'>
sage: H.category()
Category of homsets of modules over Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> N = FiniteRankFreeModule(ZZ, Integer(2), name='N')
>>> H = Hom(M,N) ; H
Set of Morphisms from Rank-3 free module M over the Integer Ring
 to Rank-2 free module N over the Integer Ring
 in Category of finite dimensional modules over Integer Ring
>>> type(H)
<class 'sage.tensor.modules.free_module_homset.FreeModuleHomset_with_category_with_equality_by_id'>
>>> H.category()
Category of homsets of modules over Integer Ring

Sage Live

M = FiniteRankFreeModule(ZZ, 3, name='M')
N = FiniteRankFreeModule(ZZ, 2, name='N')
H = Hom(M,N) ; H
type(H)
H.category()

Hom-sets are cached:

Sage

sage: H is Hom(M,N)
True

Python

>>> from sage.all import *
>>> H is Hom(M,N)
True

Sage Live

H is Hom(M,N)

The LaTeX formatting is:

Sage

sage: latex(H)
\mathrm{Hom}\left(M,N\right)

Python

>>> from sage.all import *
>>> latex(H)
\mathrm{Hom}\left(M,N\right)

Sage Live

latex(H)

As usual, the construction of an element is performed by the __call__ method; the argument can be the matrix representing the morphism in the default bases of the two modules:

Sage

sage: e = M.basis('e')
sage: f = N.basis('f')
sage: phi = H([[-1,2,0], [5,1,2]]) ; phi
Generic morphism:
  From: Rank-3 free module M over the Integer Ring
  To:   Rank-2 free module N over the Integer Ring
sage: phi.parent() is H
True

Python

>>> from sage.all import *
>>> e = M.basis('e')
>>> f = N.basis('f')
>>> phi = H([[-Integer(1),Integer(2),Integer(0)], [Integer(5),Integer(1),Integer(2)]]) ; phi
Generic morphism:
  From: Rank-3 free module M over the Integer Ring
  To:   Rank-2 free module N over the Integer Ring
>>> phi.parent() is H
True

Sage Live

e = M.basis('e')
f = N.basis('f')
phi = H([[-1,2,0], [5,1,2]]) ; phi
phi.parent() is H

An example of construction from a matrix w.r.t. bases that are not the default ones:

Sage

sage: ep = M.basis('ep', latex_symbol=r"e'")
sage: fp = N.basis('fp', latex_symbol=r"f'")
sage: phi2 = H([[3,2,1], [1,2,3]], bases=(ep,fp)) ; phi2
Generic morphism:
  From: Rank-3 free module M over the Integer Ring
  To:   Rank-2 free module N over the Integer Ring

Python

>>> from sage.all import *
>>> ep = M.basis('ep', latex_symbol=r"e'")
>>> fp = N.basis('fp', latex_symbol=r"f'")
>>> phi2 = H([[Integer(3),Integer(2),Integer(1)], [Integer(1),Integer(2),Integer(3)]], bases=(ep,fp)) ; phi2
Generic morphism:
  From: Rank-3 free module M over the Integer Ring
  To:   Rank-2 free module N over the Integer Ring

Sage Live

ep = M.basis('ep', latex_symbol=r"e'")
fp = N.basis('fp', latex_symbol=r"f'")
phi2 = H([[3,2,1], [1,2,3]], bases=(ep,fp)) ; phi2

The zero element:

Sage

sage: z = H.zero() ; z
Generic morphism:
  From: Rank-3 free module M over the Integer Ring
  To:   Rank-2 free module N over the Integer Ring
sage: z.matrix(e,f)
[0 0 0]
[0 0 0]

Python

>>> from sage.all import *
>>> z = H.zero() ; z
Generic morphism:
  From: Rank-3 free module M over the Integer Ring
  To:   Rank-2 free module N over the Integer Ring
>>> z.matrix(e,f)
[0 0 0]
[0 0 0]

Sage Live

z = H.zero() ; z
z.matrix(e,f)

The test suite for H is passed:

Sage

sage: TestSuite(H).run()

Python

>>> from sage.all import *
>>> TestSuite(H).run()

Sage Live

TestSuite(H).run()

Element[source]¶: alias of FiniteRankFreeModuleMorphism