On the global dimension of an algebra

V. E. Govorov

Matematičeskie zametki, Tome 14 (1973) no. 3, pp. 399-406.

Résumé

Let algebra $R=\Lambda/P$, where $\operatorname{w. gl. dim}R:=\{\min n|_{\forall R}\text{-modules }X,Y$, $\operatorname{Tor}_{n+1}^R(X,Y)=0\}$. In order that $\operatorname{w. gl. dim}R\le2n$ ($\operatorname{w. gl. dim}R\le2n+1$), it is necessary and sufficient that, for any two ideals of algebra $\Lambda$, a left ideal $A$ and a right ideal $B$, containing ideal $P$, the following equation holds: $$ AP^n\cap P^nB=AP^nB+P^{n+1} \quad (AP^nB\cap P^{n+1}=AP^{n+1}+P^{n+1}B). $$