Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any example; can one obtain a certain "description" for these "left phantom" objects? For example, what happens in the unbounded derived category of a ring of infinite cohomological dimension?


Note that the subcategory of objects satisfying this conditions is triangulated and closed with respect to coproducts. I am willing to localize by this category (if it is non-zero; I would actually prefer to impose certain extra "orthogonality" conditions on the subcategory I kill). Yet I do not know whether the hom-classes in the quotient are sets. The localization functor should respect coproducts; yet it does not seems to respect the compactness of objects. 



Left orthogonals to compact objects in triangulated categories: existence and "control"?
This happens in spectra. By [a theorem of Lin](http://www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0402738-5/), there are no maps from the Eilenberg-MacLane spectra $H\mathbb{F}\_p$ to finite spectra; the proof is an Adams spectral sequence computation using the fact that the Steenrod algebra is self-injective.


For a more algebraic example, let $A$ be ring containing an infinite regular sequence $(x\_0,x\_1,\dots)$ and let $M=A/(x\_0,x\_1,\dots)$. We can resolve $M$ by an infinite Koszul complex and compute that $\operatorname{Ext}^\*(M,A)=0$. It follows that in the derived category of $A$, there are no maps from $M$ to compact objects.


As for getting some kind of control on these objects, I don't really know much, but I know Luke Wolcott has thought a lot about pathology in derived categories of non-Noetherian rings. You might try taking a look at his work and seeing if you can find anything useful.


