This is my first MO question, so please go easy on me if you think this is too vague.

Is there anything to say about the collection of chain complexes with flat homology? Is there a name for them, or a different characterization of them? Maybe there's a way to build them out of some other simpler objects?

Specifically, I'm working in the derived category of a (non-Noetherian) commutative ring. The collection of objects with flat homology seems closed under retracts and coproducts, but not under the formation of triangles. So they don't make any triangulated subcategory, but maybe there's something else to be said about them?

My motivation: there's a spectral sequence (according to Weibel) with $E^2$ term $E^2_{p,q} = \bigoplus_{q'+q''=q} \mbox{Tor}_p^R\left( H_{q'}(A),H_{q''}(B)\right)$ that always converges to $H_{p+q}(A\otimes_L^R B)$.

I've constructed an object, say $B$, whose homology in each degree is the same - it's the vector-space dual $R^*$ of the ring $R$ (which is a truncated polynomial algebra on infinitely many generators). I'm interested in the Bousfield class of this object, i.e. the objects $A$ such that $A\otimes_L^R B=0$.

According to the spectral sequence, when A has flat homology, things collapse, and all I need to do is look at the Bousfield class of $R^*$, which is known. So that's why I care about such objects.