Associated Graded Algebras To Filtered Algebras

AUTHORS:

  • Travis Scrimshaw (2014-10-08): Initial version
class sage.algebras.associated_graded.AssociatedGradedAlgebra(A, category=None)

Bases: sage.combinat.free_module.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 as from_graded_conversion(). The projection \(p_i : F_i \to G_i\) is implemented as projection() (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 and grA is the canonical QQ-module isomorphism stemming from the fact that the underlying QQ-modules of A and grA 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 and y.

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'])