# Associated Graded Algebras To Filtered Algebras¶

AUTHORS:

• Travis Scrimshaw (2014-10-08): Initial version

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.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.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.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: 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.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.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()