Filtrations, Algebra Objects, and the Associated Graded

Filtrations and the Associated Graded Functor

Filtered algebras admit a categorical description: they are monoid objects in a suitable symmetric monoidal category of filtered vector spaces. The associated graded construction is a strong symmetric monoidal functor, which explains why it naturally transports algebra structures.

Throughout, $k$$k$ is a field. We use nonnegative, increasing, exhaustive filtrations.

Filtered vector spaces and sketches

A filtered vector space is a vector space $V$$V$ equipped with

$$0=F_{-1}V\subseteq F_{0}V\subseteq F_{1}V\subseteq \cdots ,V=\bigcup _{n\ge 0}{F}_{n}V.$$

Its morphisms are linear maps satisfying $f(F_{n}V)\subseteq F_{n}W$$f(F_nV)\subseteq F_nW$. Denote this category by ${\mathrm{Fil}}_{\ge 0}(k)$$\operatorname{Fil}_{\geq0}(k)$.

This structure can be described by a sketch interpreted in ${\mathbf{Vect}}_k$$\mathbf{Vect}_k$. Start with a chain

$$S_{-1}\stackrel{i_{-1}}{\to }{S}_0\stackrel{i_0}{\to }{S}_1⟶\cdots$$

and a cocone $j_n:S_n\to S_{\mathrm{\infty }}$$j_n:S_n\to S_\infty$. Require $S_{-1}$$S_{-1}$ to be terminal, hence zero, and require the cocone to be a colimit cocone.

To express that each $i_n$$i_n$ is monic, designate the following square as a pullback:

$$\begin{array}{ccc}{S}_n& \stackrel{\mathrm{id}}{\to }& S_n\\ \mathrm{id}\downarrow & & \downarrow i_n\\ S_n& \stackrel{i_n}{\to }& S_{n+1}.\end{array}$$

This square is a pullback exactly when $i_n$$i_n$ is a monomorphism. In a vector-space model, we can therefore identify the stages with subspaces of their colimit. Natural transformations between models become precisely the filtration-preserving maps. Thus

$${\mathrm{Mod}}_{{\mathbf{Vect}}_k}(\mathcal{S})\simeq {\mathrm{Fil}}_{\ge 0}(k).$$

This sketch uses finite limits and a countable colimit. The fixed lower bound is part of our convention.

The convolution tensor product

The monoidal structure is additional data on this category. Equip $V{\otimes }_{k}W$$V\otimes_kW$ with the convolution filtration

$$F_n(V\otimes W)=\sum _{p+q=n}{F}_{p}V\otimes F_{q}W.$$

The unit is $k$$k$ with $F_{n}k=k$$F_nk=k$ for $n\ge 0$$n\geq0$ and zero in negative degrees.

A map $V\otimes W\to U$$V\otimes W\to U$ is filtered exactly when the corresponding bilinear map $b$$b$ satisfies

$$b(F_{p}V,F_{q}W)\subseteq F_{p+q}U.$$

The usual associator and flip preserve these filtrations. Hence ${\mathrm{Fil}}_{\ge 0}(k)$$\operatorname{Fil}_{\geq0}(k)$ is symmetric monoidal.

Filtered algebras are monoid objects

A monoid object consists of filtered maps

$$\mu :A\otimes A⟶A,\eta :k⟶A$$

satisfying associativity and the unit laws. Their underlying maps make $A$$A$ a unital associative $k$$k$-algebra.

By the definition of the tensor filtration, $\mu$$\mu$ and $\eta$$\eta$ are filtered precisely when

$$F_{p}A\cdot F_{q}A\subseteq F_{p+q}A,1_A\in F_{0}A.$$

These are exactly the axioms of a filtered algebra. Consequently,

$$\mathrm{Mon}({\mathrm{Fil}}_{\ge 0}(k))\simeq {\mathrm{FiltAlg}}_{\ge 0}(k).$$

Associated graded is strong symmetric monoidal

Define

$$\mathrm{gr}V=\underset{n\ge 0}{⨁}{F}_{n}V/F_{n-1}V.$$

Filtered maps induce maps on these quotients, giving a functor

$$\mathrm{gr}:{\mathrm{Fil}}_{\ge 0}(k)⟶{\mathrm{Gr}}_{\ge 0}(k).$$

The target has tensor product

$$(X\otimes Y{)}_n=\underset{p+q=n}{⨁}{X}_p\otimes Y_q,$$

with the ordinary flip as symmetry, without Koszul signs.

There is a canonical natural map

$${\mathrm{\Phi }}_{V,W}:\mathrm{gr}V\otimes \mathrm{gr}W⟶\mathrm{gr}(V\otimes W),$$

given by

$$(v+F_{p-1}V)\otimes (w+F_{q-1}W)⟼v\otimes w+F_{p+q-1}(V\otimes W).$$

Changing either representative by a lower-degree element changes the tensor by a lower-degree element, so this map is well-defined.

To prove that it is an isomorphism, choose complements giving

$$F_{n}V=\underset{p\le n}{⨁}{V}_p,F_{n}W=\underset{q\le n}{⨁}{W}_q.$$

Then

$$F_n(V\otimes W)=\underset{p+q\le n}{⨁}{V}_p\otimes W_q,$$

and therefore

$${\mathrm{gr}}_n(V\otimes W)\cong \underset{p+q=n}{⨁}{V}_p\otimes W_q.$$

Under these identifications, ${\mathrm{\Phi }}_{V,W}$$\Phi_{V,W}$ is the identity on each summand. The choices establish invertibility; the map itself is canonical.

Together with the unit isomorphism $\mathrm{gr}(k)\cong k$$\operatorname{gr}(k)\cong k$, these maps make $\mathrm{gr}$$\operatorname{gr}$ strong symmetric monoidal. Coherence follows by checking pure tensors: all associativity composites give the same tensor class, and the ordinary flips commute with $\mathrm{\Phi }$$\Phi$.

For a filtered algebra $A$$A$, its associated graded multiplication is consequently the composite

$$\mathrm{gr}A\otimes \mathrm{gr}A\stackrel{{\mathrm{\Phi }}_{A,A}}{\to }\mathrm{gr}(A\otimes A)\stackrel{\mathrm{gr}(\mu )}{\to }\mathrm{gr}A.$$

In representatives, this is the familiar formula

$$(a+F_{p-1}A)(b+F_{q-1}A)=ab+F_{p+q-1}A.$$

Its associativity and unit laws follow from the monoidal functoriality. The associated graded algebra is the image of an algebra object under $\mathrm{gr}$$\operatorname{gr}$.