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


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

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

class ElementMethods

Bases: object

class ParentMethods

Bases: object

EXAMPLES:

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]

Rational Field
Rational Field

classmethod default_super_categories(category, *args)

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: 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.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