On good filtration dimensions for standardly stratified algebras

$\nabla$-good filtration dimensions of modules and of algebras are introduced by Parker for quasi-hereditary algebras. These concepts are now generalized to the setting of standardly stratified algebras. Let $A$ be a standardly stratified algebra. The $\bar{\nabla}$-good filtration dimension of $A$ is proved to be the projective dimension of the characteristic module of $A$. Several characterizations of $\bar{\nabla}$-good filtration dimensions and $\bar{\Delta}$-good filtration dimensions are given for properly stratified algebras. Finally we give an application of these results to the global dimensions of quasi-hereditary algebras with exact Borel subalgebra.