$\tau$-tilting theory in abelian categories

Abstract

Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $\tau$-rigid subcategory is contained in a support $\tau$-tilting subcategory. We also show that support $\tau$-tilting subcategories are in bijection with certain finitely generated torsion classes. Some applications of our main results are also given.