weak global dimension

Let $R$ be a ring. The (right) weak global dimension of $R$ is defined as

$$\mathrm{w}.\mathrm{gl}.\dim R=\sup \{{\mathrm{wd}}_{R}M|M\text{ is a right module}\}.$$

Unlike global dimension of $R$ the definition of the weak global dimension is left/right symmetric. This follows from the fact that for every left module $M$ and right module $N$ there is an isomorphism

$${\mathrm{Tor}}_n^R(M,N)\simeq {\mathrm{Tor}}_n^R(N,M).$$

Thus we simply say that $R$ has the weak global dimension. Note that this does not hold for Ext functors, so (generally) the definition of global dimension is not left/right symmetric.

The following proposition is a simple consequence of the fact that every projective module is flat:

Proposition. For any ring $R$ we have

$$\mathrm{w}.\mathrm{gl}.\dim R⩽\min \{\mathrm{l}.\mathrm{gl}.\dim R,\mathrm{r}.\mathrm{gl}.\dim R\},$$

where $\mathrm{l}.\mathrm{gl}.\dim$ and $\mathrm{r}.\mathrm{gl}.\dim$ denote the left global dimension and right global dimension respectively.