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On the global dimension of an algebra

V. E. Govorov

Moscow Institute of Electronic Engineering

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


UDC: 519.4

Received: 10.04.1972

 English version: Mathematical Notes, 1973, 14:3, 789–792

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