On the global dimension of an algebra

Abstract

Let algebra R = Λ/P, where Λ is a free algebra over a field w. gl. dim R: = {min n ¦∀ R-modules X, Y, Tor ^R _n+1 (X, Y)=0}. In order that w. gl. dim R≤2n (w. gl. dim R≤2n+1), it is necessary and sufficient that, for any two ideals of algebra Λ, a left ideal A and a right ideal B, containing ideal P, the following equation holds:

$$AP^n \cap P^n B = AP^n B + P^{n + 1} (AP^n B \cap P^{n + 1} = AP^{n + 1} + P^{n + 1} B).$$