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:
- 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 - FreeModuleEndsetimplements 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: 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 - >>> 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 - End(M)is actually identical to- Hom(M,M):- sage: End(M) is Hom(M,M) True - >>> from sage.all import * >>> End(M) is Hom(M,M) True - The unit of the endomorphism ring is the identity map: - sage: End(M).one() Identity endomorphism of Rank-3 free module M over the Integer Ring - >>> from sage.all import * >>> End(M).one() Identity endomorphism of Rank-3 free module M over the Integer Ring - whose matrix in any basis is of course the identity matrix: - sage: End(M).one().matrix(e) [1 0 0] [0 1 0] [0 0 1] - >>> from sage.all import * >>> End(M).one().matrix(e) [1 0 0] [0 1 0] [0 0 1] - 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: 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 - >>> 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 - See - TensorFreeModulefor examples of the above coercions.- There is a coercion \(\mathrm{GL}(M) \rightarrow \mathrm{End}(M)\), since every automorphism is an endomorphism: - 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 - >>> 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 - Of course, there is no coercion in the reverse direction, since only bijective endomorphisms are automorphisms: - sage: GL.has_coerce_map_from(End(M)) False - >>> from sage.all import * >>> GL.has_coerce_map_from(End(M)) False - The coercion \(\mathrm{GL}(M) \rightarrow \mathrm{End}(M)\) in action: - 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] - >>> 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] - Element[source]¶
- alias of - FiniteRankFreeModuleEndomorphism
 - one()[source]¶
- Return the identity element of - selfconsidered 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: 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 - >>> 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 - 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: 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 - >>> 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 - Conversely, one can recover - M.identity_map()from- H.one()by means of a conversion \(\mathrm{End}(M)\rightarrow \mathrm{GL}(M)\):- sage: GL = M.general_linear_group() sage: M.identity_map() == GL(H.one()) True - >>> from sage.all import * >>> GL = M.general_linear_group() >>> M.identity_map() == GL(H.one()) True 
 
- 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 - FreeModuleHomsetimplements 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: 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 - >>> 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 - Hom-sets are cached: - sage: H is Hom(M,N) True - >>> from sage.all import * >>> H is Hom(M,N) True - The LaTeX formatting is: - sage: latex(H) \mathrm{Hom}\left(M,N\right) - >>> from sage.all import * >>> latex(H) \mathrm{Hom}\left(M,N\right) - 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: 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 - >>> 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 - An example of construction from a matrix w.r.t. bases that are not the default ones: - 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 - >>> 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 - The zero element: - 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] - >>> 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] - The test suite for H is passed: - sage: TestSuite(H).run() - >>> from sage.all import * >>> TestSuite(H).run() - Element[source]¶
- alias of - FiniteRankFreeModuleMorphism