4.1.1.20 HeckeModules Objects

class HeckeModules
The category of Hecke modules.

A Hecke module is a module $ M$ over the 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 $ \mathbf{T}$ -module homomorphisms; but they are homomorphisms over the anemic Hecke algebra.

We create the category of Hecke modules over $ \mathbf{Q}$ .

sage: C = HeckeModules(RationalField()); C
Category of Hecke modules 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()))
Category of Hecke modules over Univariate Polynomial Ring in x over Integer
Ring
HeckeModules( R)

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