Space of morphisms between Ore modules

AUTHOR:

  • Xavier Caruso (2024-10)

class sage.modules.ore_module_homspace.OreModule_homspace(domain, codomain, category=None)[source]

Bases: UniqueRepresentation, HomsetWithBase

Class for hom spaces between Ore modules.

Element[source]

alias of OreModuleMorphism

identity()[source]

Return the identity morphism in this homspace.

EXAMPLES:

sage: K.<z> = GF(7^2)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^3 + z*X + 1)
sage: End(M).identity()
Ore module endomorphism of Ore module of rank 3 over Finite Field in z of size 7^2 twisted by z |--> z^7

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(2), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) + z*X + Integer(1))
>>> End(M).identity()
Ore module endomorphism of Ore module of rank 3 over Finite Field in z of size 7^2 twisted by z |--> z^7
matrix_space()[source]

Return the matrix space used to represent the morphisms in this homspace.

EXAMPLES:

sage: K.<z> = GF(7^2)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^3 + z*X + 1)
sage: End(M).matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Finite Field in z of size 7^2

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(2), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) + z*X + Integer(1))
>>> End(M).matrix_space()
Full MatrixSpace of 3 by 3 dense matrices over Finite Field in z of size 7^2


sage: N = S.quotient_module(X^2 + z)
sage: Hom(M, N).matrix_space()
Full MatrixSpace of 3 by 2 dense matrices over Finite Field in z of size 7^2

>>> from sage.all import *
>>> N = S.quotient_module(X**Integer(2) + z)
>>> Hom(M, N).matrix_space()
Full MatrixSpace of 3 by 2 dense matrices over Finite Field in z of size 7^2
zero()[source]

Return the zero morphism in this homspace.

EXAMPLES:

sage: K.<z> = GF(7^2)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^3 + z*X + 1)
sage: End(M).zero()
Ore module endomorphism of Ore module of rank 3 over Finite Field in z of size 7^2 twisted by z |--> z^7

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(2), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) + z*X + Integer(1))
>>> End(M).zero()
Ore module endomorphism of Ore module of rank 3 over Finite Field in z of size 7^2 twisted by z |--> z^7