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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $M$ be a nonzero [[$R$-module|0NZ8JA25WK]].\nThe *projective dimension* $\\operatorname{pd}_R M$ is the least integer $n \\ge 0$ such that $M$ has a [[projective resolution|HK2AA1YSQ2]]\n$$0 \\to P_n \\to P_{n-1} \\to \\cdots \\to P_0 \\to M \\to 0$$\nof length $n$.\nIf no such finite resolution exists, then $\\operatorname{pd}_R M = \\infty$.\n\n## Remarks\n\nOne has $\\operatorname{pd}_R M = 0$ if and only if $M$ is a [[projective module|BFFA03GE3V]].\nEquivalently, $\\operatorname{pd}_R M \\le n$ if and only if $\\operatorname{Ext}^{n+1}_R(M, N) = 0$ for every $R$-module $N$, where $\\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].\nExample: $\\operatorname{pd}_\\Z \\Z/n\\Z = 1$ for $n \\ge 2$; the resolution $0 \\to \\Z \\xrightarrow{\\cdot n} \\Z \\to \\Z/n\\Z \\to 0$ has length $1$, and $\\Z/n\\Z$ is not projective, because projective $\\Z$-modules are free and hence have no nonzero torsion.\nNon-example of finiteness: over $R = \\Z/4\\Z$, the module $\\Z/2\\Z$ has $\\operatorname{pd}_R \\Z/2\\Z = \\infty$.\nSome authors set $\\operatorname{pd}_R 0 = -\\infty$.\n",
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