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?

thederived category of modules? – Reid Barton Nov 23 '09 at 3:45