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?