Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.

Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(A_0,A_0) \neq 0 \}$?

Maybe one should ask this question under some mild further restrictions such as $A$ being noetherian and/or $A_{i+j}=A_i A_j$ for all $i,j \geq 0$.

This question was asked in the special case of quadratic algebras here: Quadratic algebras and Koszul algebras and a positive answer would have a nice applications.