Weak dimension

Summary

The weak dimension, or flat dimension, of a module is the shortest length of a resolution of it by flat modules. The weak global dimension of a ring is the supremum of the weak dimensions of its modules. It is measured by the vanishing of the Tor functors and is the same for left and right modules. It is at most the global dimension and often easier to compute; rings of weak global dimension $0$ are exactly the von Neumann regular rings.