# Hecke modules#

class sage.categories.hecke_modules.HeckeModules(R)[source]#

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

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


TODO: check that this is what we want:

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

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

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

>>> 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)[source]#
class ParentMethods[source]#

Bases: object

extra_super_categories()[source]#
class ParentMethods[source]#

Bases: object

super_categories()[source]#

EXAMPLES:

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

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