Applications of stratifying systems to the finitistic dimension

Given an Ext-injective stratifying system of ΛΛ-modules (θ,Y̲,≼) satisfying that the projective dimension of YY is finite, we prove that the finitistic dimension of the algebra ΛΛ is equal to the finitistic dimension of the category I(θ)={X∈modΛ:ExtΛ1(-,X)|F(θ)=0}. Moreover, using the theory of stratifying systems we obtain bounds for the finitistic dimension of ΛΛ. In particular, we get the optimal bound 2n-22n-2 for the finitistic dimension of a standardly stratified algebra with nn simples.