Notes on Relative Extension Functors

Abstract

The objective of this paper is to study the relative homological properties of contravariantly and covariantly finite subcategories. Some sufficient conditions for ${\underset{¯}{\mathcal{E}xt}}_{\mathcal{Y}}^n\left(A,B\right)\simeq {\overline{\mathcal{E}xt}}_{\mathcal{X}}^n\left(A,B\right)$ are obtained. We also give the conditions under which the stable categories $\left(\mathcal{C}/\mathcal{W}\right)/\left(\mathcal{X}/\mathcal{W}\right)$ are one-side triangulated categories.