I have a field extension $F/F'$ and an algebra $L'$ over $F'$. Let $L$ be the induced $F$-algebra $F\otimes_{F'}L'$ and $C_*$ a chain complex over $L$.

*Is there a good way to decide whether $C_*$ is quasiisomorphic to a chain complex of the form  $L\otimes_{L'}D_*$ for a chain complex $D_*$ over $L'$?*

For example, if $L'=F'[x]$ and thus $L=F[x]$, and we look at a chain complex of the  form $0\rightarrow F[x]\stackrel{x-f}{\rightarrow}F[x]\rightarrow 0$, we get that $H_0(C_*)$ is the $F[x]$-module $F_f$. This module is $F$ where $x$ acts as multiplication by $f$. If $C_*$ is quasiisomorpic to an induced up chain complex, then $H_0(C_*)$ would also be obtained from $H_0(D_*)$ by induction, and this is the case exactly iff $f\in F'\subset F$.

*This gives a list of obstructions, but are these the only ones?*