# Theory of representation rt: quadratic algebras and Koszul algebras

Leave $$A$$ be a quadratic algebra and $$B$$ the external algebra of $$A$$.
In case $$A$$ is a Koszul algebra, we should have the global dimension of $$A$$ plus one equals the length of Loewy's $$B$$ (Is there any 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 represents the length of Loewy and $$A_0$$ The degree zero is part of the graduated algebra. $$A$$.
I'm not sure in general about the first equality here (at least it should be valid for $$A$$ finite dimensional), but the second equality must be correct since $$B$$ It is generated in degrees 0 and 1.

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

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

Question 2: It is a quadratic algebra Koszul iff $$gldim (A) + 1 = LL (B)$$ Holds?