Balance functors and relative tilting modules

Let $$\mathfrak {C}$$ C and $$\mathfrak {D}$$ D be two classes of left R-modules, $$l_{\mathfrak {D}}\mathfrak {C}$$ l D C = the class of all left R-modules admitting exact left $$\mathfrak {C}$$ C -resolutions which are $$\mathrm{Hom}(-,\mathfrak {D})$$ Hom ( - , D ) -exact, $$r_{\mathfrak {C}}\mathfrak {D}$$ r C D = the class of all left R-modules admitting exact right $$\mathfrak {D}$$ D -resolutions which are $$\mathrm{Hom}(\mathfrak {C},-)$$ Hom ( C , - ) -exact. We first study some properties of $$l_{\mathfrak {D}}\mathfrak {C}$$ l D C and $$r_{\mathfrak {C}}\mathfrak {D}$$ r C D . Then, using the Hom balance functor determined by the above two special classes of modules, we introduce and investigate F-(Wakamatsu) tilting and F-(Wakamatsu) cotilting modules which are possibly infinitely generated over arbitrary rings for an additive subfunctor F of $$\mathrm{Ext}^{1}(-,-)$$ Ext 1 ( - , - ) . Some classical results are extended.