On Auslander–Reiten translates in functorially finite subcategories and applications
We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander–Reiten sequences. We also prove an Auslander–Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category has finite type. The class of these algebras contains all blocks of Schur algebras $S(2,r)$.