Hecke modules#

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

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)[source]#

Bases: HomsetsCategory

class ParentMethods[source]#: Bases: object

extra_super_categories()[source]#

class ParentMethods[source]#: Bases: object

super_categories()[source]#

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]