Zeroth Derived Functors as Adjoints and Injective Resolutions of Coherent Functors

The classical definitions of zeroth derived functors require existence of injectives or projectives. In this paper, we give definitions of the zeroth derived functors that do not require the existence of injectives or projectives. The new definitions result in generalized definitions of projective and injective stabilization of functors. The category of coherent functors is shown to admit a zeroth right derived functor. An interesting result of this fact is a counterpart to the Yoneda lemma for coherent functors. Moreover, zeroth derived functors are seen more appropriately as approximations of functors by left exact or right exact functors. Under certain reasonable conditions, the category of coherent functors is shown to have enough injectives. This result was first shown by Ron Gentle. We give an alternate proof of this fact.