Suppose we have a homomorphism $f : A^{\bullet} \longrightarrow B^{\bullet}$ of differential graded algebras over a field $k$, and consider the filtration


\begin{align\*}
A^{\bullet} \supseteq F^0A^{\bullet} \supseteq F^1A^{\bullet} \supseteq F^2A^{\bullet} \supseteq \cdots \hspace{0.5cm} (\*)
\end{align\*}


of $A$. I'm interested in "pushing forward" or "extending" $(\*)$ to a filtration on $B^{\bullet}$.


Let $F^n(A^{\bullet})^e$ be the two-sided ideal of $B^{\bullet}$ generated by the image of the ideal $F^n(A^{\bullet})$ of $A^{\bullet}$ by $f$. Then we get the sequence of ideals
\begin{align\*}
B^{\bullet} \supseteq F^0(A^{\bullet})^e \supseteq F^1(A^{\bullet})^e \supseteq F^2(A^{\bullet})^e \supseteq \cdots \hspace{0.5cm} (\*\*)
\end{align\*}
of $B$. I'd like to know the following about $(\*\*)$ :


1. If the filtration $(\*)$ is exhaustive (that is, $A^{\bullet} = F^0A^{\bullet}$), is $(\*\*)$ also exhaustive ?
2. If the filtration $(\*)$ is separated/Hausdorff (that is, $\cap\_n F^nA = \{0\}$) , is $(\*\*)$ also separated/Hausdorff ?
3. If the filtration $(\*)$ is complete (that is, the map $A^{\bullet} \longrightarrow \lim\_{n \rightarrow \infty} A^{\bullet} / F^nA^{\bullet}$ is an isomorphism), is $(\*\*)$ also complete ?


Are there relatively easy checks that one can do to answer any of these questions ? Alternatively, if there are any references that answer any of these questions, I'd like to know about them.


Thanks !



Properties of filtrations preserved by a DG-algebra homomorphism
1. Yes, if $A = F^0A$ then $1 \in F^0A$ so $f(1) = 1 \in F^0 B$ and so the ideal $F^0 B$ contains all of $B$. If you do not require unital dgas/maps, then the answer is no, just take the zero map and any exhaustive filtration at the source.
2. No, consider $A = k[x]$, $F^n A = (x^n)$, $B = k$, and $f : A \to B$, $f(x) = 1$. Then $\bigcap\_n F^n A = 0$ but $\bigcap\_n F^n B = k$.


Note sure about 3 but my instinct tells me no.


