Graded modules with basis¶
- class sage.categories.graded_modules_with_basis.GradedModulesWithBasis(base_category)[source]¶
- Bases: - GradedModulesCategory- The category of graded modules with a distinguished basis. - EXAMPLES: - sage: C = GradedModulesWithBasis(ZZ); C Category of graded modules with basis over Integer Ring sage: sorted(C.super_categories(), key=str) [Category of filtered modules with basis over Integer Ring, Category of graded modules over Integer Ring] sage: C is ModulesWithBasis(ZZ).Graded() True - >>> from sage.all import * >>> C = GradedModulesWithBasis(ZZ); C Category of graded modules with basis over Integer Ring >>> sorted(C.super_categories(), key=str) [Category of filtered modules with basis over Integer Ring, Category of graded modules over Integer Ring] >>> C is ModulesWithBasis(ZZ).Graded() True - class ElementMethods[source]¶
- Bases: - object- degree_negation()[source]¶
- Return the image of - selfunder the degree negation automorphism of the graded module to which- selfbelongs.- The degree negation is the module automorphism which scales every homogeneous element of degree \(k\) by \((-1)^k\) (for all \(k\)). This assumes that the module to which - selfbelongs (that is, the module- self.parent()) is \(\ZZ\)-graded.- EXAMPLES: - sage: E.<a,b> = ExteriorAlgebra(QQ) # needs sage.modules sage: ((1 + a) * (1 + b)).degree_negation() # needs sage.modules a*b - a - b + 1 sage: E.zero().degree_negation() # needs sage.modules 0 sage: P = GradedModulesWithBasis(ZZ).example(); P # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Integer Ring sage: pbp = lambda x: P.basis()[Partition(list(x))] sage: p = pbp([3,1]) - 2 * pbp([2]) + 4 * pbp([1]) # needs sage.combinat sage.modules sage: p.degree_negation() # needs sage.combinat sage.modules -4*P[1] - 2*P[2] + P[3, 1] - >>> from sage.all import * >>> E = ExteriorAlgebra(QQ, names=('a', 'b',)); (a, b,) = E._first_ngens(2)# needs sage.modules >>> ((Integer(1) + a) * (Integer(1) + b)).degree_negation() # needs sage.modules a*b - a - b + 1 >>> E.zero().degree_negation() # needs sage.modules 0 >>> P = GradedModulesWithBasis(ZZ).example(); P # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Integer Ring >>> pbp = lambda x: P.basis()[Partition(list(x))] >>> p = pbp([Integer(3),Integer(1)]) - Integer(2) * pbp([Integer(2)]) + Integer(4) * pbp([Integer(1)]) # needs sage.combinat sage.modules >>> p.degree_negation() # needs sage.combinat sage.modules -4*P[1] - 2*P[2] + P[3, 1] 
 
 - class ParentMethods[source]¶
- Bases: - object- degree_negation(element)[source]¶
- Return the image of - elementunder the degree negation automorphism of the graded module- self.- The degree negation is the module automorphism which scales every homogeneous element of degree \(k\) by \((-1)^k\) (for all \(k\)). This assumes that the module - selfis \(\ZZ\)-graded.- INPUT: - element– element of the module- self
 - EXAMPLES: - sage: E.<a,b> = ExteriorAlgebra(QQ) # needs sage.modules sage: E.degree_negation((1 + a) * (1 + b)) # needs sage.modules a*b - a - b + 1 sage: E.degree_negation(E.zero()) # needs sage.modules 0 sage: P = GradedModulesWithBasis(ZZ).example(); P # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Integer Ring sage: pbp = lambda x: P.basis()[Partition(list(x))] sage: p = pbp([3,1]) - 2 * pbp([2]) + 4 * pbp([1]) # needs sage.combinat sage.modules sage: P.degree_negation(p) # needs sage.combinat sage.modules -4*P[1] - 2*P[2] + P[3, 1] - >>> from sage.all import * >>> E = ExteriorAlgebra(QQ, names=('a', 'b',)); (a, b,) = E._first_ngens(2)# needs sage.modules >>> E.degree_negation((Integer(1) + a) * (Integer(1) + b)) # needs sage.modules a*b - a - b + 1 >>> E.degree_negation(E.zero()) # needs sage.modules 0 >>> P = GradedModulesWithBasis(ZZ).example(); P # needs sage.combinat sage.modules An example of a graded module with basis: the free module on partitions over Integer Ring >>> pbp = lambda x: P.basis()[Partition(list(x))] >>> p = pbp([Integer(3),Integer(1)]) - Integer(2) * pbp([Integer(2)]) + Integer(4) * pbp([Integer(1)]) # needs sage.combinat sage.modules >>> P.degree_negation(p) # needs sage.combinat sage.modules -4*P[1] - 2*P[2] + P[3, 1] 
 - quotient_module(submodule, check=True, already_echelonized=False, category=None)[source]¶
- Construct the quotient module - self/- submodule.- INPUT: - submodule– a submodule with basis of- self, or something that can be turned into one via- self.submodule(submodule)
- check,- already_echelonized– passed down to- ModulesWithBasis.ParentMethods.submodule()
- category– (optional) the category of the quotient module
 - Warning - At this point, this only supports quotients by free submodules admitting a basis in unitriangular echelon form. In this case, the quotient is also a free module, with a basis consisting of the retract of a subset of the basis of - self.- EXAMPLES: - sage: # needs sage.combinat sage.modules sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: S = E.submodule([x + y, x*y - y*z, y]) sage: Q = E.quotient_module(S) sage: Q.category() Join of Category of quotients of graded modules with basis over Rational Field and Category of graded vector spaces with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3) >>> S = E.submodule([x + y, x*y - y*z, y]) >>> Q = E.quotient_module(S) >>> Q.category() Join of Category of quotients of graded modules with basis over Rational Field and Category of graded vector spaces with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field - See also - Modules.WithBasis.ParentMethods.submodule()
 
 - submodule(gens, check=True, already_echelonized=False, unitriangular=False, support_order=None, category=None, *args, **opts)[source]¶
- Return the submodule spanned by a finite set of elements. - INPUT: - gens– list or family of elements of- self
- check– boolean (default:- True); whether to verify that the elements of- gensare in- self
- already_echelonized– boolean (default:- False); whether
- the elements of - gensare already in (not necessarily reduced) echelon form
 
- unitriangular– boolean (default:- False); whether the lift morphism is unitriangular
- support_order– (optional) either something that can be converted into a tuple or a key function
- category– (optional) the category of the submodule
 - If - already_echelonizedis- False, then the generators are put in reduced echelon form using- echelonize(), and reindexed by \(0,1,...\).- Warning - At this point, this method only works for finite dimensional submodules and if matrices can be echelonized over the base ring. - If in addition - unitriangularis- True, then the generators are made such that the coefficients of the pivots are 1, so that lifting map is unitriangular.- The basis of the submodule uses the same index set as the generators, and the lifting map sends \(y_i\) to \(gens[i]\). - See also - ModulesWithBasis.FiniteDimensional.ParentMethods.quotient_module()
 - EXAMPLES: - A graded submodule of a graded module generated by homogeneous elements is naturally graded: - sage: # needs sage.combinat sage.modules sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: S = E.submodule([x + y, x*y - y*z]) sage: S.category() Join of Category of graded vector spaces with basis over Rational Field and Category of subobjects of filtered modules with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field sage: S.basis()[0].degree() 1 sage: S.basis()[1].degree() 2 - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3) >>> S = E.submodule([x + y, x*y - y*z]) >>> S.category() Join of Category of graded vector spaces with basis over Rational Field and Category of subobjects of filtered modules with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field >>> S.basis()[Integer(0)].degree() 1 >>> S.basis()[Integer(1)].degree() 2 - We check on the echelonized basis: - sage: Sp = E.submodule([1, x + y + 5, x*y - y*z + x + y - 2]) # needs sage.combinat sage.modules sage: Sp.category() # needs sage.combinat sage.modules Join of Category of graded vector spaces with basis over Rational Field and Category of subobjects of filtered modules with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field - >>> from sage.all import * >>> Sp = E.submodule([Integer(1), x + y + Integer(5), x*y - y*z + x + y - Integer(2)]) # needs sage.combinat sage.modules >>> Sp.category() # needs sage.combinat sage.modules Join of Category of graded vector spaces with basis over Rational Field and Category of subobjects of filtered modules with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field - If it is generated by inhomogeneous elements, then it is filtered by default: - sage: F = E.submodule([x + y*z, x*z + y*x]) # needs sage.combinat sage.modules sage: F.category() # needs sage.combinat sage.modules Join of Category of subobjects of filtered modules with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field and Category of filtered vector spaces with basis over Rational Field - >>> from sage.all import * >>> F = E.submodule([x + y*z, x*z + y*x]) # needs sage.combinat sage.modules >>> F.category() # needs sage.combinat sage.modules Join of Category of subobjects of filtered modules with basis over Rational Field and Category of finite dimensional filtered modules with basis over Rational Field and Category of filtered vector spaces with basis over Rational Field - If - categoryis specified, then it does not give any extra structure to the submodule (we can think of this as applying the forgetful functor):- sage: # needs sage.combinat sage.modules sage: SM = E.submodule([x + y, x*y - y*z], ....: category=ModulesWithBasis(QQ)) sage: SM.category() Join of Category of finite dimensional vector spaces with basis over Rational Field and Category of subobjects of sets sage: FM = E.submodule([x + 1, x*y - x*y*z], ....: category=ModulesWithBasis(QQ)) sage: FM.category() Join of Category of finite dimensional vector spaces with basis over Rational Field and Category of subobjects of sets - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> SM = E.submodule([x + y, x*y - y*z], ... category=ModulesWithBasis(QQ)) >>> SM.category() Join of Category of finite dimensional vector spaces with basis over Rational Field and Category of subobjects of sets >>> FM = E.submodule([x + Integer(1), x*y - x*y*z], ... category=ModulesWithBasis(QQ)) >>> FM.category() Join of Category of finite dimensional vector spaces with basis over Rational Field and Category of subobjects of sets - If we have specified that this is a graded submodule of a graded module, then the echelonized elements must be homogeneous: - sage: Cat = ModulesWithBasis(QQ).Graded().Subobjects() sage: E.submodule([x + y, x*y - 1], category=Cat) # needs sage.combinat sage.modules Traceback (most recent call last): ... ValueError: all of the generators must be homogeneous sage: E.submodule([x + y, x*y - x - y], category=Cat) # needs sage.combinat sage.modules Free module generated by {0, 1} over Rational Field - >>> from sage.all import * >>> Cat = ModulesWithBasis(QQ).Graded().Subobjects() >>> E.submodule([x + y, x*y - Integer(1)], category=Cat) # needs sage.combinat sage.modules Traceback (most recent call last): ... ValueError: all of the generators must be homogeneous >>> E.submodule([x + y, x*y - x - y], category=Cat) # needs sage.combinat sage.modules Free module generated by {0, 1} over Rational Field 
 
 - class Quotients(category, *args)[source]¶
- Bases: - QuotientsCategory- class ElementMethods[source]¶
- Bases: - object- degree()[source]¶
- Return the degree of - self.- EXAMPLES: - sage: # needs sage.combinat sage.modules sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: S = E.submodule([x + y, x*y - y*z, y]) sage: Q = E.quotient_module(S) sage: B = Q.basis() sage: [B[i].lift() for i in Q.indices()] [1, z, x*z, y*z, x*y*z] sage: [B[i].degree() for i in Q.indices()] [0, 1, 2, 2, 3] - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3) >>> S = E.submodule([x + y, x*y - y*z, y]) >>> Q = E.quotient_module(S) >>> B = Q.basis() >>> [B[i].lift() for i in Q.indices()] [1, z, x*z, y*z, x*y*z] >>> [B[i].degree() for i in Q.indices()] [0, 1, 2, 2, 3] 
 
 - class ParentMethods[source]¶
- Bases: - object- degree_on_basis(m)[source]¶
- Return the degree of the basis element indexed by - min- self.- EXAMPLES: - sage: # needs sage.combinat sage.modules sage: E.<x,y,z> = ExteriorAlgebra(QQ) sage: S = E.submodule([x + y, x*y - y*z, y]) sage: Q = E.quotient_module(S) sage: B = Q.basis() sage: [B[i].lift() for i in Q.indices()] [1, z, x*z, y*z, x*y*z] sage: [Q.degree_on_basis(i) for i in Q.indices()] [0, 1, 2, 2, 3] - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> E = ExteriorAlgebra(QQ, names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3) >>> S = E.submodule([x + y, x*y - y*z, y]) >>> Q = E.quotient_module(S) >>> B = Q.basis() >>> [B[i].lift() for i in Q.indices()] [1, z, x*z, y*z, x*y*z] >>> [Q.degree_on_basis(i) for i in Q.indices()] [0, 1, 2, 2, 3]