When may "summand of" be dropped from the definition of perfect dg module? Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of $\mathcal{S}\mathcal{F}(\mathcal{A})$ consisting of DG modules homotopy equivalent to a direct summand of a finitely generated semi-free DG module over $\mathcal{A}$. Here, $\mathcal{S}\mathcal{F}(\mathcal{A})$ is the DG category of semi-free DG modules over $\mathcal{A}$, and a module is considered ``free'' if it is a direct sum of shifts of representable modules.
Example 1.9 of Lunts-Orlov is the assertion that the bounded derived category of $\mathcal{A}$ is (the homotopy category of) $Perf(\mathcal{A})$.
Under what conditions on $\mathcal{A}$ are all objects of $Perf(\mathcal{A})$ actually homotopy equivalent (isomorphic in $H_0$) to finitely generated semi-free DG modules?
The motivation of this question is the case when $\mathcal{A}$ is a strands algebra from bordered Heegaard Floer homology (the structure as a small dg category comes from the distinguished idempotents of $\mathcal{A}$).The papers that decategorify such $\mathcal{A}$ typically identify the unbounded derived category of $\mathcal{A}$ with $H_0$ of the category of Type D structures homotopy equivalent to  operationally bounded Type D structures. The bounded derived category is identified with the full subcategory on Type D structures homotopy equivalent to finitely generated operationally bounded Type D structures.
An operationally bounded Type D structure is (more or less) the same as a semi-free dg module $X$ with a choice of decomposition of $X$ as $\mathcal{A} \otimes_{\mathcal{I}} N$ with respect to the $\mathcal{A}$-action, where $\mathcal{I}$ is the idempotent ring of $\mathcal{A}$ and $N$ is a left $\mathcal{I}$-module. Type D structures are isomorphic or homotopy equivalent iff the corresponding dg modules are isomorphic or homotopy equivalent. The same is true for finite generation.
So, based on the usual definition of perfect dg modules, I'd expect at first to have to use``Type D structures which split inject in $H_0$ into a finitely generated operationally bounded Type D structure'' as the bounded derived category. But this is less convenient for computing $K_0$, and I'd rather be able to replace an object of the bounded derived category with an isomorphic object which literally is a finitely generated bounded Type D structure.
In other words, my question is: are there any sufficient conditions one can place on dg algebras $\mathcal{A}$ for this to work, which are general enough to cover some interesting examples of $\mathcal{A}$ with nonzero differential?
 A: For a DGA $A$, all perfect modules are equivalent to semi-free modules if and only if $K(A)$ is generated by the class of the rank 1 free module. Furthermore, a particular perfect module is equivalent to a semi-free module if it is in the subgroup generated by the rank 1 free module. $K$-theory is exactly the gap between semi-free and perfect modules, so this does not seem to me useful for computing $K$-theory. You may as well ask: which DGAs have trivial reduced $K$-theory; maybe you should, but I think that is a different question.
It is easy to see that this $K$-theory condition is necessary. The definition of semi-free can be rephrased as that the module is generated from shifts of $A$ by a finite sequence of cones; so its class in $K(A)$ is in the subgroup generated by $[A]$. For example, for the product of two fields $A=E\times F$, an $A$-module is just a pair of $E$- and $F$-vector spaces. They could have different dimensions, so $K(A)=\mathbb Z^2$. But a semi-free module must have equal Euler characteristics. For example, the cone of $(1,0)\colon A\to A$ has homology zero on the $E$ side and nontrivial on the $F$ side, but its Euler characteristics are both zero. So $(E,0)$ is a perfect module not equivalent to a semi-free one.
Thomason proved that this necessary condition is actually sufficient. Given a triangulated category $C$, a triangulated subcategory $D$ (ie, a full subcategory closed under cones (and isomorphisms)) is called dense if every object of $C$ is a summand of an object of $D$. Thomason proved that dense subcategories of $C$ are determined by their $K$-groups, which necessarily inject into $K(C)$. The dense subcategory $D\subset C$ corresponding to a subgroup $H\subset K(C)$ consists of the objects whose class is in $H$. The semi-free modules are dense in the perfect modules by definition of the perfect modules, so a perfect module is equivalent to a semi-free module if and only if its class is in the subgroup generated by the ring $\langle [A]\rangle$. In particular, for every perfect module $M$, the module $M\oplus M[1]$ has trivial class and thus has a semi-free model.
