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  "body": "\n## Summary\n\nThe weak dimension, or flat dimension, of a [[module|0NZ8JA25WK]] is the shortest length of a resolution of it by [[flat modules|73RYYEVDKA]].\nThe weak global dimension of a [[ring|0B1261B5ED]] is the supremum of the weak dimensions of its modules.\nIt is measured by the vanishing of the [[Tor functors|DDQRERAQCV]] and is the same for left and right modules.\nIt is at most the [[global dimension|GWA0T1WKCN]] and often easier to compute; rings of weak global dimension $0$ are exactly the von Neumann regular rings.\n\n## Explanations\n\n### L1 human\n\nFlat modules are the modules that behave well under tensor products, and every free module is flat.\nA general module can be approximated by flat modules in a chain, called a flat resolution.\nThe weak dimension counts how many steps such a chain needs at least.\nOver the integers, a module like $\\Z/2$ needs one step: $0 \\to \\Z \\to \\Z \\to \\Z/2 \\to 0$, so its weak dimension is $1$.\nA ring of small weak dimension has a simple homological theory.\n",
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