MISSING TITLE

class sage.categories.ore_modules.OreModules(ore)

Bases: Category_over_base_ring

Category of Ore modules.

ore_ring(var='x')

Return the underlying Ore polynomial ring.

INPUT:

• var (default; x) – the variable name

EXAMPLES:

Sage

sage: from sage.categories.ore_modules import OreModules
sage: K.<a> = GF(5^3)
sage: Frob = K.frobenius_endomorphism()
sage: cat = OreModules(K, Frob)
sage: cat.ore_ring()
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

sage: cat.ore_ring('y')
Ore Polynomial Ring in y over Finite Field in a of size 5^3 twisted by a |--> a^5

Python

>>> from sage.all import *
>>> from sage.categories.ore_modules import OreModules
>>> K = GF(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> Frob = K.frobenius_endomorphism()
>>> cat = OreModules(K, Frob)
>>> cat.ore_ring()
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

>>> cat.ore_ring('y')
Ore Polynomial Ring in y over Finite Field in a of size 5^3 twisted by a |--> a^5

super_categories()

Return the immediate super categories of this category.

EXAMPLES:

Sage

sage: from sage.categories.ore_modules import OreModules
sage: K.<a> = GF(5^3)
sage: Frob = K.frobenius_endomorphism()
sage: cat = OreModules(K, Frob)
sage: cat.super_categories()
[Category of vector spaces over Finite Field in a of size 5^3]

Python

>>> from sage.all import *
>>> from sage.categories.ore_modules import OreModules
>>> K = GF(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> Frob = K.frobenius_endomorphism()
>>> cat = OreModules(K, Frob)
>>> cat.super_categories()
[Category of vector spaces over Finite Field in a of size 5^3]

twisting_derivation()

Return the underlying twisting derivation.

EXAMPLES:

Sage

sage: from sage.categories.ore_modules import OreModules
sage: R.<t> = QQ[]
sage: d = R.derivation()
sage: cat = OreModules(R, d)
sage: cat.twisting_derivation()
d/dt

Python

>>> from sage.all import *
>>> from sage.categories.ore_modules import OreModules
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> d = R.derivation()
>>> cat = OreModules(R, d)
>>> cat.twisting_derivation()
d/dt

If the twising derivation is zero, nothing is returned:

Sage

sage: K.<a> = GF(5^3)
sage: Frob = K.frobenius_endomorphism()
sage: cat = OreModules(K, Frob)
sage: cat.twisting_derivation()

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> Frob = K.frobenius_endomorphism()
>>> cat = OreModules(K, Frob)
>>> cat.twisting_derivation()

twisting_morphism()

Return the underlying twisting morphism.

EXAMPLES:

Sage

sage: from sage.categories.ore_modules import OreModules
sage: K.<a> = GF(5^3)
sage: Frob = K.frobenius_endomorphism()
sage: cat = OreModules(K, Frob)
sage: cat.twisting_morphism()
Frobenius endomorphism a |--> a^5 on Finite Field in a of size 5^3

Python

>>> from sage.all import *
>>> from sage.categories.ore_modules import OreModules
>>> K = GF(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> Frob = K.frobenius_endomorphism()
>>> cat = OreModules(K, Frob)
>>> cat.twisting_morphism()
Frobenius endomorphism a |--> a^5 on Finite Field in a of size 5^3

If the twising morphism is the identity, nothing is returned:

Sage

sage: R.<t> = QQ[]
sage: d = R.derivation()
sage: cat = OreModules(R, d)
sage: cat.twisting_morphism()

Python

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> d = R.derivation()
>>> cat = OreModules(R, d)
>>> cat.twisting_morphism()