Hecke modules - Category Framework

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# Hecke modules

class sage.categories.hecke_modules.HeckeModules(R)

  Bases: `Category_module`

  The category of Hecke modules.

  A Hecke module is a module \(M\) over the emph{anemic} Hecke algebra, i.e., the Hecke algebra generated by Hecke operators \(T_n\) with \(n\) coprime to the level of \(M\). (Every Hecke module defines a level function, which is a positive integer.) The reason we require that \(M\) only be a module over the anemic Hecke algebra is that many natural maps, e.g., degeneracy maps, Atkin-Lehner operators, etc., are \(\Bold{T}\)-module homomorphisms; but they are homomorphisms over the anemic Hecke algebra.

  EXAMPLES:

  We create the category of Hecke modules over \(\QQ\):

  Sage

      sage: C = HeckeModules(RationalField()); C
      Category of Hecke modules over Rational Field

  Python

      >>> from sage.all import *
      >>> C = HeckeModules(RationalField()); C
      Category of Hecke modules over Rational Field

  TODO: check that this is what we want:

  Sage

      sage: C.super_categories()
      [Category of vector spaces with basis over Rational Field]

  [Python]

      >>> from sage.all import *
      >>> C.super_categories()
      [Category of vector spaces with basis over Rational Field]

  # [Category of vector spaces over Rational Field]

  Note that the base ring can be an arbitrary commutative ring:

  Sage

      sage: HeckeModules(IntegerRing())
      Category of Hecke modules over Integer Ring
      sage: HeckeModules(FiniteField(5))
      Category of Hecke modules over Finite Field of size 5

  Python

      >>> from sage.all import *
      >>> HeckeModules(IntegerRing())
      Category of Hecke modules over Integer Ring
      >>> HeckeModules(FiniteField(Integer(5)))
      Category of Hecke modules over Finite Field of size 5

  The base ring doesn’t have to be a principal ideal domain:

  Sage

      sage: HeckeModules(PolynomialRing(IntegerRing(), 'x'))
      Category of Hecke modules over Univariate Polynomial Ring in x over Integer Ring

  [Python]

      >>> from sage.all import *
      >>> HeckeModules(PolynomialRing(IntegerRing(), 'x'))
      Category of Hecke modules over Univariate Polynomial Ring in x over Integer Ring

  class Homsets(category, *args)

    Bases: `HomsetsCategory`

    class ParentMethods

      Bases: `object`

    extra_super_categories()
  class ParentMethods

    Bases: `object`

  super_categories()

    EXAMPLES:

    Sage

        sage: HeckeModules(QQ).super_categories()
        [Category of vector spaces with basis over Rational Field]

    Python

        >>> from sage.all import *
        >>> HeckeModules(QQ).super_categories()
        [Category of vector spaces with basis over Rational Field]