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.

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:

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