Associated Graded Algebras To Filtered Algebras#
AUTHORS:
Travis Scrimshaw (2014-10-08): Initial version
- class sage.algebras.associated_graded.AssociatedGradedAlgebra(A, category=None)#
Bases:
CombinatorialFreeModule
The associated graded algebra/module \(\operatorname{gr} A\) of a filtered algebra/module with basis \(A\).
Let \(A\) be a filtered module over a commutative ring \(R\). Let \((F_i)_{i \in I}\) be the filtration of \(A\), with \(I\) being a totally ordered set. Define
\[G_i = F_i / \sum_{j < i} F_j\]for every \(i \in I\), and then
\[\operatorname{gr} A = \bigoplus_{i \in I} G_i.\]There are canonical projections \(p_i : F_i \to G_i\) for every \(i \in I\). Moreover \(\operatorname{gr} A\) is naturally a graded \(R\)-module with \(G_i\) being the \(i\)-th graded component. This graded \(R\)-module is known as the associated graded module (or, for short, just graded module) of \(A\).
Now, assume that \(A\) (endowed with the filtration \((F_i)_{i \in I}\)) is not just a filtered \(R\)-module, but also a filtered \(R\)-algebra. Let \(u \in G_i\) and \(v \in G_j\), and let \(u' \in F_i\) and \(v' \in F_j\) be lifts of \(u\) and \(v\), respectively (so that \(u = p_i(u')\) and \(v = p_j(v')\)). Then, we define a multiplication \(*\) on \(\operatorname{gr} A\) (not to be mistaken for the multiplication of the original algebra \(A\)) by
\[u * v = p_{i+j} (u' v').\]The associated graded algebra (or, for short, just graded algebra) of \(A\) is the graded algebra \(\operatorname{gr} A\) (endowed with this multiplication).
Now, assume that \(A\) is a filtered \(R\)-algebra with basis. Let \((b_x)_{x \in X}\) be the basis of \(A\), and consider the partition \(X = \bigsqcup_{i \in I} X_i\) of the set \(X\), which is part of the data of a filtered algebra with basis. We know (see
FilteredModulesWithBasis
) that \(A\) (being a filtered \(R\)-module with basis) is canonically (when the basis is considered to be part of the data) isomorphic to \(\operatorname{gr} A\) as an \(R\)-module. Therefore the \(k\)-th graded component \(G_k\) can be identified with the span of \((b_x)_{x \in X_k}\), or equivalently the \(k\)-th homogeneous component of \(A\). Suppose that \(u' v' = \sum_{k \leq i+j} m_k\) where \(m_k \in G_k\) (which has been identified with the \(k\)-th homogeneous component of \(A\)). Then \(u * v = m_{i+j}\). We also note that the choice of identification of \(G_k\) with the \(k\)-th homogeneous component of \(A\) depends on the given basis.The basis \((b_x)_{x \in X}\) of \(A\) gives rise to a basis of \(\operatorname{gr} A\). This latter basis is still indexed by the elements of \(X\), and consists of the images of the \(b_x\) under the \(R\)-module isomorphism from \(A\) to \(\operatorname{gr} A\). It makes \(\operatorname{gr} A\) into a graded \(R\)-algebra with basis.
In this class, the \(R\)-module isomorphism from \(A\) to \(\operatorname{gr} A\) is implemented as
to_graded_conversion()
and also as the default conversion from \(A\) to \(\operatorname{gr} A\). Its inverse map is implemented asfrom_graded_conversion()
. The projection \(p_i : F_i \to G_i\) is implemented asprojection()
(i)
.INPUT:
A
– a filtered module (or algebra) with basis
OUTPUT:
The associated graded module of \(A\), if \(A\) is just a filtered \(R\)-module. The associated graded algebra of \(A\), if \(A\) is a filtered \(R\)-algebra.
EXAMPLES:
Associated graded module of a filtered module:
sage: A = Modules(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: grA.category() Category of graded vector spaces with basis over Rational Field sage: x = A.basis()[Partition([3,2,1])] sage: grA(x) Bbar[[3, 2, 1]]
Associated graded algebra of a filtered algebra:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: grA.category() Category of graded algebras with basis over Rational Field sage: x,y,z = [grA.algebra_generators()[s] for s in ['x','y','z']] sage: x bar(U['x']) sage: y * x + z bar(U['x']*U['y']) + bar(U['z']) sage: A(y) * A(x) + A(z) U['x']*U['y']
We note that the conversion between
A
andgrA
is the canonicalQQ
-module isomorphism stemming from the fact that the underlyingQQ
-modules ofA
andgrA
are isomorphic:sage: grA(A.an_element()) bar(U['x']^2*U['y']^2*U['z']^3) + 2*bar(U['x']) + 3*bar(U['y']) + bar(1) sage: elt = A.an_element() + A.algebra_generators()['x'] + 2 sage: grelt = grA(elt); grelt bar(U['x']^2*U['y']^2*U['z']^3) + 3*bar(U['x']) + 3*bar(U['y']) + 3*bar(1) sage: A(grelt) == elt True
Todo
The algebra
A
must currently be an instance of (a subclass of)CombinatorialFreeModule
. This should work with any filtered algebra with a basis.Todo
Implement a version of associated graded algebra for filtered algebras without a distinguished basis.
REFERENCES:
- algebra_generators()#
Return the algebra generators of
self
.This assumes that the algebra generators of \(A\) provided by its
algebra_generators
method are homogeneous.EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: grA.algebra_generators() Finite family {'x': bar(U['x']), 'y': bar(U['y']), 'z': bar(U['z'])}
- degree_on_basis(x)#
Return the degree of the basis element indexed by
x
.EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: all(A.degree_on_basis(x) == grA.degree_on_basis(x) ....: for g in grA.algebra_generators() for x in g.support()) True
- gen(*args, **kwds)#
Return a generator of
self
.EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: grA.gen('x') bar(U['x'])
- one_basis()#
Return the basis index of the element \(1\) of \(\operatorname{gr} A\).
This assumes that the unity \(1\) of \(A\) belongs to \(F_0\).
EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: grA.one_basis() 1
- product_on_basis(x, y)#
Return the product on basis elements given by the indices
x
andy
.EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: grA = A.graded_algebra() sage: G = grA.algebra_generators() sage: x,y,z = G['x'], G['y'], G['z'] sage: x * y # indirect doctest bar(U['x']*U['y']) sage: y * x bar(U['x']*U['y']) sage: z * y * x bar(U['x']*U['y']*U['z'])