$ \newcommand{\cn}{\colon} \newcommand{\<}{\leqslant} \newcommand{\>}{\geqslant} \newcommand{\ss}{\subset} \newcommand{\gr}{\mathrm{gr}} $ Let $A$ be associative commutative algebra with unity and filtration $0=F_{-1}\ss F_0\ss F_1\ss\ldots\ss A=\bigcup_i F_i$, $1\in F_0$, $\gr(A)=\bigoplus_{i\>0} F_i/F_{i-1}$ be noetherian associated graded algebra. I want to prove that $A$ is also noetherian (exercise 4 after chapter 1 from Kassel's Quantum Groups).
Let $A_i=F_i/F_{i-1}$. I see that $A_i$ and $B_j=\gr(A)/\left(\bigoplus_{i\>j}A_i\right)$ are noetherian but how it can be seen that $A$ is finitely generated over $A_i$ or $B_j$? Or maybe exists some epimorphism $\gr(A)\to A$ or $B_j\to A$ for some $j$? Can it help if $\gr(A)$ and $A$ are integral domains?