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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $M$ be a left $R$-[[module|0NZ8JA25WK]].\nA *flat resolution* of length $n$ of $M$ is an [[exact sequence|JB7ZBFBA9C]]\n$$0 \\to F_n \\to F_{n-1} \\to \\cdots \\to F_0 \\to M \\to 0$$\nin which every $F_i$ is a [[flat|73RYYEVDKA]] left $R$-module.\nThe *weak dimension* (or *flat dimension*) $\\operatorname{wd}(M)$ is the least $n \\ge 0$ such that $M$ has a flat resolution of length $n$.\nIf there is no such $n$, then $\\operatorname{wd}(M) = \\infty$.\nThe *weak global dimension* of $R$ is the supremum of $\\operatorname{wd}(M)$ over all left $R$-modules $M$.\n\n## Remarks\n\nEquivalently, $\\operatorname{wd}(M) \\le n$ if and only if $\\operatorname{Tor}^R_{n+1}(N, M) = 0$ for every right $R$-module $N$, with [[Tor|DDQRERAQCV]] the derived functors of the tensor product.\nSo 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 $M$.\nIt is the same for left and right modules.\nExample: over $R = \\Z$, the module $\\Z/2$ has weak dimension $1$.\nThe sequence $0 \\to \\Z \\xrightarrow{2} \\Z \\to \\Z/2 \\to 0$ is a flat resolution of length $1$.\nThe 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.\nNon-example of finite dimension: over $R = \\Z/4$, the module $\\Z/2$ has infinite weak dimension.\nIts 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 $n \\ge 1$.\nThe weak dimension of a module is at most its [[projective dimension|MR9QCH1PNA]], so the weak global dimension is at most the left and the right [[global dimension|GWA0T1WKCN]].\nFor a left Noetherian ring it equals the left global dimension.\nA ring has weak global dimension $0$ if and only if it is von Neumann regular.\n",
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