I was wondering whether there is a characterization of perfect DG modules over a DG algebra as there is one for modules over a ring. Namely, an object in $D(R)$, where $R$ is a ring, is perfect if and only if it is isomorphic to a bounded complex of finitely generated projective modules. Is there any similar characterization when we take $R$ a dga and $D(R)$ the derived category of DG modules over it?



Perfect DG modules
Well by definition a DG module $X$ is perfect iff it is a direct summand of a module which is a finite iterated extension of free modules (it's then a nontrivial result that this property is the same as being compact). If $Y$ is a dg module and $Y'$ is an extension of $Y$ by $R[n],$ this extension is classified by an element $\xi\in Ext^{1-n}(R, Y) = H^{1-n}(Y),$ so (choosing a cycle representing $\xi$) we see $Y'$ can be represented by a cone of the corresponding explicit arrow $R[n-1]\to Y.$ This lets you view any dg module which is a finite iterated extension of free modules as the total module of a bicomplex whose row complex is free and finite-dimensional. Now a perfect complex is a direct summand (equivalently, retract) of such a complex. 


