Filtered Modules¶
A filtered module over a ring \(R\) with a totally ordered indexing set \(I\) (typically \(I = \NN\)) is an \(R\)-module \(M\) equipped with a family \((F_i)_{i \in I}\) of \(R\)-submodules satisfying \(F_i \subseteq F_j\) for all \(i,j \in I\) having \(i \leq j\), and \(M = \bigcup_{i \in I} F_i\). This family is called a filtration of the given module \(M\).
Todo
Implement a notion for decreasing filtrations: where \(F_j \subseteq F_i\) when \(i \leq j\).
Todo
Implement filtrations for all concrete categories.
Todo
Implement \(\operatorname{gr}\) as a functor.
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sage.categories.filtered_modules.
FilteredModules
¶ The category of filtered modules over a given ring \(R\).
A filtered module over a ring \(R\) with a totally ordered indexing set \(I\) (typically \(I = \NN\)) is an \(R\)-module \(M\) equipped with a family \((F_i)_{i \in I}\) of \(R\)-submodules satisfying \(F_i \subseteq F_j\) for all \(i,j \in I\) having \(i \leq j\), and \(M = \bigcup_{i \in I} F_i\). This family is called a filtration of the given module \(M\).
EXAMPLES:
sage: Modules(ZZ).Filtered() Category of filtered modules over Integer Ring sage: Modules(ZZ).Filtered().super_categories() [Category of modules over Integer Ring]
REFERENCES:
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class
sage.categories.filtered_modules.
FilteredModulesCategory
(base_category)¶ Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
,sage.categories.category_types.Category_over_base_ring
EXAMPLES:
sage: C = Algebras(QQ).Filtered() sage: C Category of filtered 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 filtered modules over Rational Field] sage: AlgebrasWithBasis(QQ).Filtered().base_ring() Rational Field sage: HopfAlgebrasWithBasis(QQ).Filtered().base_ring() Rational Field