Coverage for local/lib/python2.7/site-packages/sage/categories/hecke_modules.py : 48%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
r""" Hecke modules """ from __future__ import absolute_import #***************************************************************************** # Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu> # William Stein <wstein@math.ucsd.edu> # 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from sage.categories.category_types import Category_module from sage.categories.homsets import HomsetsCategory from sage.categories.modules_with_basis import ModulesWithBasis
class HeckeModules(Category_module): 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 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
TESTS::
sage: TestSuite(HeckeModules(ZZ)).run() """ def __init__(self, R): """ TESTS::
sage: TestSuite(HeckeModules(ZZ)).run()
sage: HeckeModules(Partitions(3)).run() Traceback (most recent call last): ... TypeError: R (=Partitions of the integer 3) must be a commutative ring """
def super_categories(self): """ EXAMPLES::
sage: HeckeModules(QQ).super_categories() [Category of vector spaces with basis over Rational Field] """
def _repr_object_names(self): """ Return the names of the objects of this category.
.. SEEALSO:: :meth:`Category._repr_object_names`
EXAMPLES::
sage: HeckeModules(QQ)._repr_object_names() 'Hecke modules over Rational Field' sage: HeckeModules(QQ) Category of Hecke modules over Rational Field """
class ParentMethods:
def _Hom_(self, Y, category): r""" Returns the homset from ``self`` to ``Y`` in the category ``category``
INPUT:
- ``Y`` -- an Hecke module - ``category`` -- a subcategory of :class:`HeckeModules`() or None
The sole purpose of this method is to construct the homset as a :class:`~sage.modular.hecke.homspace.HeckeModuleHomspace`. If ``category`` is specified and is not a subcategory of :class:`HeckeModules`, a ``TypeError`` is raised instead
This method is not meant to be called directly. Please use :func:`sage.categories.homset.Hom` instead.
EXAMPLES::
sage: M = ModularForms(Gamma0(7), 4) sage: H = M._Hom_(M, category = HeckeModules(QQ)); H Set of Morphisms from Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) of weight 4 over Rational Field to Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) of weight 4 over Rational Field in Category of Hecke modules over Rational Field sage: H.__class__ <class 'sage.modular.hecke.homspace.HeckeModuleHomspace_with_category'> sage: TestSuite(H).run(skip=["_test_elements", "_test_an_element", "_test_elements_eq", ....: "_test_elements_eq_reflexive", "_test_elements_eq_transitive", ....: "_test_elements_eq_symmetric", "_test_elements_neq", "_test_some_elements", ....: "_test_zero", "_test_additive_associativity", ....: "_test_one", "_test_associativity", "_test_prod"])
Fixing :meth:`_test_zero` (``__call__`` should accept a function as input) and :meth:`_test_elements*` (modular form morphisms elements should inherit from categories) is :trac:`12879`.
TESTS::
sage: H = M._Hom_(M, category = HeckeModules(GF(5))); H Traceback (most recent call last): ... TypeError: Category of Hecke modules over Finite Field of size 5 is not a subcategory of Category of Hecke modules over Rational Field
""" # TODO: double check that it's the correct HeckeModules category below:
class Homsets(HomsetsCategory): """ TESTS::
sage: TestSuite(HeckeModules(ZZ).Homsets()).run() """
def base_ring(self): """ EXAMPLES::
sage: HeckeModules(QQ).Homsets().base_ring() Rational Field """
def extra_super_categories(self): """ TESTS:
Check that Hom sets of Hecke modules are in the correct category (see :trac:`17359`)::
sage: HeckeModules(ZZ).Homsets().super_categories() [Category of modules over Integer Ring, Category of homsets] sage: HeckeModules(QQ).Homsets().super_categories() [Category of vector spaces over Rational Field, Category of homsets] """
class ParentMethods: pass |