Weak dimension

Remarks

Equivalently, $\operatorname{wd}(M) \le n$ if and only if $\operatorname{Tor}^R_{n+1}(N, M) = 0$ for every right $R$-module $N$, with Tor the derived functors of the tensor product. So the weak global dimension of $R$ is the supremum of the $n$ with $\operatorname{Tor}^R_n(N, M) \ne 0$ for some right module $N$ and left module $M$. It is the same for left and right modules. Example: over $R = \Z$, the module $\Z/2$ has weak dimension $1$. The sequence $0 \to \Z \xrightarrow{2} \Z \to \Z/2 \to 0$ is a flat resolution of length $1$. The module $\Z/2$ is not flat, since tensoring the injective map $\Z \xrightarrow{2} \Z$ with $\Z/2$ gives the zero map on $\Z/2$, which is not injective. Non-example of finite dimension: over $R = \Z/4$, the module $\Z/2$ has infinite weak dimension. Its resolution $\cdots \to \Z/4 \xrightarrow{2} \Z/4 \xrightarrow{2} \Z/4 \to \Z/2 \to 0$ gives $\operatorname{Tor}^R_n(\Z/2, \Z/2) \cong \Z/2 \ne 0$ for every $n \ge 1$. The weak dimension of a module is at most its projective dimension, so the weak global dimension is at most the left and the right global dimension. For a left Noetherian ring it equals the left global dimension. A ring has weak global dimension $0$ if and only if it is von Neumann regular.