Induced map between associated graded algebra?

Question

I have just learned how one can take an filtered algebra and get an associated graded algebra, (here's the wikipedia article for reference). I have also seen in various places, (such as this question), the notation $gr(f)$, where $f:A\to B$ is some $k$-algebra homomorphism, (or maybe just a vector space homomorphism?). What exactly is this induced map? Also, is this a functor?

Answer 1

Let $A$ be a filtered $k$-algebra, so we have an increasing sequence $0\subset A_0\subset A_1\subset\cdots$ of $k$-submodules such that $A=\bigcup_nA_n$ and the multiplication satisfies $A_m\cdot A_n\subset A_{m+n}$. Then $\mathrm{gr}(A):=\bigoplus_n\bar A_n$, where $\bar A_n:=A_n/A_{n-1}$, is a graded $k$-algebra.

If $B=\bigcup_nB_n$ is another filtered algebra, then a graded algebra homomorphism $g\colon\mathrm{gr}(A)\to\mathrm{gr}(B)$ is an algebra homomorphism such that $g(\bar A_n)\subset \bar B_n$ for all $n$.

Call an algebra homomorphism $f\colon A\to B$ filtered if $f(A_n)\subset B_n$ for all $n$. Such a map induces an algebra homomorphism $\mathrm{gr}(f)\colon\mathrm{gr}(A)\to\mathrm{gr}(B)$. We therefore get a functor $\mathrm{gr}$ from the category of filtered algebras and filtered homomorphisms to the category of graded algebras and graded homomorphism.