# Hecke modules¶

class sage.categories.hecke_modules.HeckeModules(R)

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


TODO: check that this is what we want:

sage: C.super_categories()
[Category of modules 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


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


TESTS:

sage: TestSuite(HeckeModules(ZZ)).run()

class HomCategory(category, name=None)

Initializes this HomCategory

INPUT:
• category – the category whose Homsets are the objects of this category.
• name – An optional name for this category.

EXAMPLES:

We need to skip one test, since the hierarchy of hom categories isn’t consistent yet:

sage: C = sage.categories.category.HomCategory(Rings()); C
Category of hom sets in Category of rings
sage: TestSuite(C).run(skip=['_test_category_graph'])

ParentMethods

alias of HomCategory.ParentMethods

base_ring()

EXAMPLES:

sage: HeckeModules(QQ).hom_category().base_ring()
Rational Field

extra_super_categories()

EXAMPLES:

sage: HeckeModules(ZZ).hom_category().extra_super_categories()
[]

class HeckeModules.ParentMethods
HeckeModules.super_categories()

EXAMPLES:

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


G-Sets

#### Next topic

Highest Weight Crystals