# Global dimension of a graded algebra

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.

According to Proposition 15, the category of graded left $$A$$-modules has certain nice properties whenever $$A_0$$ is a semi-primary ring. Then Theorem 13 can be reformulated as
$$\operatorname{gldim} (A)=\sup \{i \geq 0 | \operatorname{Tor}^A_i(A_0,A_0) \neq 0 \}.$$
$$\sup \{i \geq 0 | \operatorname{Tor}^A_i(A_0,A_0)\neq 0 \}\\=\sup \{i \geq 0 | \operatorname{Ext}_A^i(A_0,A_0)\neq 0 \}.$$