The associated graded algebra of a f.g filtered algebra is f.g? Let $k$ be a field and let $A$ be a (non)commutative filtered $k$-algebra. If, as $k$-algebra, $gr(A)$ is finitely generated, then $A$ is finitely generated too. 
My question: is it true that if $A$ is finitely generated, then $gr(A)$ is finitely generated too? 
I think the answer depends on the filtration. For example, if we consider $A$ with the standard filtration of a finitely generated algebra, then the answer to my question is positive. But I can't think of any counter-example.      
 A: As you suspect, the answer is no. Before getting on to an example, the philosophy is the following: the Rees construction allows one to think of the associated graded algebra as being the special fiber in a flat family of algebras---in some sense, the "worst behaved" member of the family. And if the bad seed is relatively speaking good, all the kids are good. But the converse need not hold in general.
Let $A=k[x]$ and define a filtration by setting $F_d(A)$ equal to the set of polynomials of degree at most $2^{d-1}$ (and with $F_0(A)$ the constants). This gives an increasing filtration on $A$ such that the dimension of the $d$th graded piece of the associated graded algebra is $2^{d-2}$. 
Now a commutative finitely generated graded $k$-algebra is a quotient of a polynomial ring in $n$ variables, and the dimension of the $d$th graded piece of a polynomial ring is polynomial in $d$ (given by a certain binomial coefficient). So the associated graded we have constructed is not finitely generated.
