# weak global dimension

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

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

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

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

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

Title weak global dimension WeakGlobalDimension 2013-03-22 19:18:42 2013-03-22 19:18:42 joking (16130) joking (16130) 4 joking (16130) Definition msc 16E05