Balance in Stable Categories

We study when the stable category ${mathcal A}/langle{mathcal T}rangleWe study when the stable category A/áT?{mathcal A}/langle{mathcal T}rangle of an abelian category A{mathcal A} modulo a full additive subcategory T{mathcal T} is balanced and, in case T{mathcal T} is functorially finite in A{mathcal A}, we study a weak version of balance for A/áT?{mathcal A}/langle{mathcal T}rangle. Precise necessary and sufficient conditions are given in case T{mathcal T} is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.