Balance functors and relative tilting modules

Abstract

Let $\mathfrak {C}$ and $\mathfrak {D}$ be two classes of left R-modules, $l_{\mathfrak {D}}\mathfrak {C}$ = the class of all left R-modules admitting exact left $\mathfrak {C}$-resolutions which are $\mathrm{Hom}(-,\mathfrak {D})$-exact, $r_{\mathfrak {C}}\mathfrak {D}$ = the class of all left R-modules admitting exact right $\mathfrak {D}$-resolutions which are $\mathrm{Hom}(\mathfrak {C},-)$-exact. We first study some properties of $l_{\mathfrak {D}}\mathfrak {C}$ and $r_{\mathfrak {C}}\mathfrak {D}$. 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 $\mathrm{Ext}^{1}(-,-)$. Some classical results are extended.