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: GradedModules(ZZ) Category of graded modules over Integer Ring sage: GradedModules(ZZ).super_categories() [Category of filtered modules over Integer Ring] - >>> 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: Modules(ZZ).Graded().additional_structure() Category of graded modules over Integer Ring - >>> from sage.all import * >>> Modules(ZZ).Graded().additional_structure() Category of graded modules over Integer Ring 
- class sage.categories.graded_modules.GradedModulesCategory(base_category)[source]¶
- Bases: - RegressiveCovariantConstructionCategory,- Category_over_base_ring- 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] sage: AlgebrasWithBasis(QQ).Graded().base_ring() Rational Field sage: GradedHopfAlgebrasWithBasis(QQ).base_ring() Rational Field - >>> 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- catfor each- catin 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] - >>> 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.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 - >>> 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