### Relative homological algebra for the proper class omega_f, by Grigory Garkusha

[This paper has been updated again by the author.]

The proper class of f-monomorphisms is introduced. The corresponding
relatively injective (respectively FP-injective and flat) modules are
pure-injective fp-injective (respectively fp-injective and fp-flat)
modules. The relative (co)homological functors $\Ext_f$ and $\Tor^f$ as well
as the corresponding dimensions for rings and modules are studied. Moreover,
it is shown that the relative derived category $D^+_f(R)$ and the derived
category of the locally coherent Grothendieck category (mod-R,Ab)/lim S^R as
well as K-theory K'(R)=K(R-mod) for the category of finitely presented left
modules and K-theory of the abelian category of coherent objects of
(mod-R,Ab)/lim S^R are naturally equivalent.

First version: Apr 22, 2001; second version: Jul 23, 2001; third version:
Nov 13, 2002.

Grigory Garkusha <ggarkusha@mail.ru>