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: 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]¶
- Bases: - HomsetsCategory