Hecke modules¶
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class
sage.categories.hecke_modules.
HeckeModules
(R)¶ Bases:
sage.categories.category_types.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: 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]
# [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
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class
Homsets
(category, *args)¶ Bases:
sage.categories.homsets.HomsetsCategory
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class
ParentMethods
¶ Bases:
object
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extra_super_categories
()¶
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class
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class
ParentMethods
¶ Bases:
object
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super_categories
()¶ EXAMPLES:
sage: HeckeModules(QQ).super_categories() [Category of vector spaces with basis over Rational Field]
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class