Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$. In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a reference for this?).

Namely we should have $gldim(A)= \inf \{ i \geq 0 | Ext_A^i(A_0,A_0)=0 \} = LL(B)$, where LL stands for Loewy length and $A_0$ is the degree zero part of the graded algebra $A$. Im not sure in general about the first equality here (it should at least hold for $A$ finite dimensional), but the second equality should be correct since $B$ is generated in degree 0 and 1.

Thus $gldim(A)+1=LL(B)$.

Question 1: Is $gldim(A)= \inf \{ i \geq 0 | Ext_A^i(A_0,A_0)=0 \}$ true in general? Is there a reference?

Question 2: Is a quadratic algebra Koszul iff $gldim(A)+1=LL(B)$ holds?