Graded modules¶
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class
sage.categories.graded_modules.
GradedModules
(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategory
The category of graded modules.
EXAMPLES:
sage: GradedModules(ZZ) Category of graded modules over Integer Ring sage: GradedModules(ZZ).super_categories() [Category of 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
TESTS:
sage: TestSuite(GradedModules(ZZ)).run()
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class
Connected
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
TESTS:
sage: C = Modules(ZZ).FiniteDimensional(); C Category of finite dimensional modules over Integer Ring sage: type(C) <class 'sage.categories.modules.Modules.FiniteDimensional_with_category'> sage: type(C).__base__.__base__ <class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'> sage: TestSuite(C).run()
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class
GradedModules.
ElementMethods
¶
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class
GradedModules.
ParentMethods
¶
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class
GradedModules.
SubcategoryMethods
¶ -
Connected
()¶ Return the full subcategory of the connected objects of
self
.EXAMPLES:
sage: Modules(ZZ).Graded().Connected() Category of graded connected modules over Integer Ring sage: Coalgebras(QQ).Graded().Connected() Join of Category of graded connected modules over Rational Field and Category of coalgebras over Rational Field sage: GradedAlgebrasWithBasis(QQ).Connected() Category of graded connected algebras with basis over Rational Field
TESTS:
sage: TestSuite(Modules(ZZ).Graded().Connected()).run() sage: Coalgebras(QQ).Graded().Connected.__module__ 'sage.categories.graded_modules'
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-
GradedModules.
extra_super_categories
()¶ Adds
VectorSpaces
to the super categories ofself
if the base ring is a field.EXAMPLES:
sage: Modules(QQ).Graded().extra_super_categories() [Category of vector spaces over Rational Field] sage: Modules(ZZ).Graded().extra_super_categories() []
This makes sure that
Modules(QQ).Graded()
returns an instance ofGradedModules
and not a join category of an instance of this class and ofVectorSpaces(QQ)
:sage: type(Modules(QQ).Graded()) <class 'sage.categories.graded_modules.GradedModules_with_category'>
Todo
Get rid of this workaround once there is a more systematic approach for the alias
Modules(QQ)
->VectorSpaces(QQ)
. Probably the later should be a category with axiom, and covariant constructions should play well with axioms.
-
class
-
class
sage.categories.graded_modules.
GradedModulesCategory
(base_category)¶ Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
,sage.categories.category_types.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 algebras over Rational Field, Category of graded modules over Rational Field] sage: AlgebrasWithBasis(QQ).Graded().base_ring() Rational Field sage: GradedHopfAlgebrasWithBasis(QQ).base_ring() Rational Field
TESTS:
sage: GradedModules(ZZ) Category of graded modules over Integer Ring sage: Modules(ZZ).Graded() Category of graded modules over Integer Ring sage: GradedModules(ZZ) is Modules(ZZ).Graded() True