Graded modules#

class sage.categories.graded_modules.GradedModules(base_category)[source]#

Bases: GradedModulesCategory

The category of graded modules.

We consider every graded module \(M = \bigoplus_i M_i\) as a filtered module under the (natural) filtration given by

\[F_i = \bigoplus_{j < i} M_j.\]

EXAMPLES:

Sage

sage: GradedModules(ZZ)
Category of graded modules over Integer Ring
sage: GradedModules(ZZ).super_categories()
[Category of filtered modules over Integer Ring]

Python

>>> from sage.all import *
>>> GradedModules(ZZ)
Category of graded modules over Integer Ring
>>> GradedModules(ZZ).super_categories()
[Category of filtered modules over Integer Ring]

The category of graded modules defines the graded structure which shall be preserved by morphisms:

Sage

sage: Modules(ZZ).Graded().additional_structure()
Category of graded modules over Integer Ring

Python

>>> from sage.all import *
>>> Modules(ZZ).Graded().additional_structure()
Category of graded modules over Integer Ring

class ElementMethods[source]#: Bases: object

class ParentMethods[source]#: Bases: object

class sage.categories.graded_modules.GradedModulesCategory(base_category)[source]#

Bases: RegressiveCovariantConstructionCategory, Category_over_base_ring

EXAMPLES:

Sage

sage: C = GradedAlgebras(QQ)
sage: C
Category of graded algebras over Rational Field
sage: C.base_category()
Category of algebras over Rational Field
sage: sorted(C.super_categories(), key=str)
[Category of filtered algebras over Rational Field,
 Category of graded vector spaces over Rational Field]

sage: AlgebrasWithBasis(QQ).Graded().base_ring()
Rational Field
sage: GradedHopfAlgebrasWithBasis(QQ).base_ring()
Rational Field

Python

>>> from sage.all import *
>>> C = GradedAlgebras(QQ)
>>> C
Category of graded algebras over Rational Field
>>> C.base_category()
Category of algebras over Rational Field
>>> sorted(C.super_categories(), key=str)
[Category of filtered algebras over Rational Field,
 Category of graded vector spaces over Rational Field]

>>> AlgebrasWithBasis(QQ).Graded().base_ring()
Rational Field
>>> GradedHopfAlgebrasWithBasis(QQ).base_ring()
Rational Field

classmethod default_super_categories(category, *args)[source]#

Return the default super categories of category.Graded().

Mathematical meaning: every graded object (module, algebra, etc.) is a filtered object with the (implicit) filtration defined by \(F_i = \bigoplus_{j \leq i} G_j\).

INPUT:

cls – the class GradedModulesCategory
category – a category

OUTPUT: a (join) category

In practice, this returns category.Filtered(), joined together with the result of the method RegressiveCovariantConstructionCategory.default_super_categories() (that is the join of category.Filtered() and cat for each cat in the super categories of category).

EXAMPLES:

Consider category=Algebras(), which has cat=Modules() as super category. Then, a grading of an algebra \(G\) is also a filtration of \(G\):

Sage

sage: Algebras(QQ).Graded().super_categories()
[Category of filtered algebras over Rational Field,
 Category of graded vector spaces over Rational Field]

Python

>>> from sage.all import *
>>> Algebras(QQ).Graded().super_categories()
[Category of filtered algebras over Rational Field,
 Category of graded vector spaces over Rational Field]

This resulted from the following call:

Sage

sage: sage.categories.graded_modules.GradedModulesCategory.default_super_categories(Algebras(QQ))
Join of Category of filtered algebras over Rational Field
 and Category of graded vector spaces over Rational Field

Python

>>> from sage.all import *
>>> sage.categories.graded_modules.GradedModulesCategory.default_super_categories(Algebras(QQ))
Join of Category of filtered algebras over Rational Field
 and Category of graded vector spaces over Rational Field