Graded modules - Category Framework

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# Graded modules

class sage.categories.graded_modules.GradedModules(base_category)

  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

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

  Python

      >>> 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

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

  [Python]

      >>> from sage.all import *
      >>> Modules(ZZ).Graded().additional_structure()
      Category of graded modules over Integer Ring

  class ElementMethods

    Bases: `object`

  class ParentMethods

    Bases: `object`

class sage.categories.graded_modules.GradedModulesCategory(base_category)

  Bases: `RegressiveCovariantConstructionCategory`, `Category_over_base_ring`

  EXAMPLES:

  Sage

      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

  Python

      >>> 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)

    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

        sage: Algebras(QQ).Graded().super_categories()
        [Category of filtered algebras over Rational Field,
         Category of graded vector spaces over Rational Field]

    Python

        >>> 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: 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

    [Python]

        >>> 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