𝜏-tilting theory in abelian categories

Abstract

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