Derived category with total cohomology finite dimensional: is there a better name for it? One of the annoying things about derived categories is that they come with a host of different finiteness conditions, which are all subtlely different, and for each situation you want a particular one.  
For a project I'm doing right now, I think I want to use the derived category of modules over a particular finite-dimensional algebra composed of bounded above complexes (so I have projective resolutions) with finite-dimensional cohomology modules, which are only non-zero in finitely many degrees. 
Is there a name for this category I should know about? 
 A: Hopefully there is not a name for this category nor will you invent one, either. The usual «put the conditions on the homology of complexes as a subindex to the $D$ and say "derived category of complexes with finite dimensional cohomology"» approach does wonders in introducing no new notation and no new names. We have too many of those already.
If in your paper you will deal exclusively with that category, make a local convention and say something to the point that «in this paper the phrase "derived category" will mean "derived category of complexes with finite dimensional cohomology", and the notation $D(A)$ will stand for that category» somewhere at the end of the introduction (do use the explicit long name in the abstract and in the introduction itself, though)
A: I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation though - if your algebra is $A$ then I think
$$D^{-,b}_{\mathrm{fd}}(A)$$ 
is pretty much standard - the fd for finite dimensional maybe not so much but conditions on the cohomology traditionally go there. If want the components finite dimensional as well then you could drop the fd and use $A$-$\mathrm{mod}$ instead of $A$. You probably already know all of this though...
Post Coffee Edit: Actually the above is slightly silly - it is perfectly resonable to consider the homotopy category of bounded above complexes (usually of projectives) with bounded finite dimensional cohomology but in the derived case you may as well just look at $D_{\mathrm{fd}}(A)$. Unless you have a particularly good reason for wanting bounded above complexes you may as well take all isomorphs in $D(A)$ - this is correct philosophically. You can then drop the bounded (since finite dimensional total cohomology implies boundedness) and just call it the derived category of $A$-modules with finite dimensional total cohomology.
A: Following an analogy of Drinfeld's, this is like complaining that there are lots of different function spaces (all the Sobolev spaces, L^p spaces etc) with different growth conditions.. these kind of restrictions are the algebraic analog of functional analysis..
In algebraic geometry what you describe is basically the bounded coherent category 
(bounded cohomologies, each of which is a finitely presented module)..?
