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  "body": "\n## Definition\n\nLet $X$ be a smooth [[algebraic variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] $n$ over a field.\nLet $V, W \\subset X$ be irreducible closed subvarieties that meet *properly*: every irreducible component of $V \\cap W$ has dimension $\\dim V + \\dim W - n$.\nLet $Z$ be an irreducible component of $V \\cap W$, and let $A = \\mathcal{O}_{X, Z}$ be the local ring of $X$ at the generic point of $Z$.\nLet $I_V, I_W \\subset A$ be the ideals of $V$ and $W$.\nThe *intersection multiplicity* of $V$ and $W$ along $Z$ is\n$$i(Z; V \\cdot W) = \\sum_{j \\ge 0} (-1)^j \\, \\ell_A\\big(\\operatorname{Tor}_j^A(A/I_V, A/I_W)\\big),$$\nwhere $\\operatorname{Tor}$ is the [[Tor functor|QAGW4V0S62]] and $\\ell_A$ is the [[length|V6JBKT168T]] of an $A$-module.\n\n## Remarks\n\nThe sum is finite because $A$ is a [[regular local ring|63XXTF6B5M]], and each term has finite length because $Z$ is a component of $V \\cap W$.\nThis is Serre's Tor formula, and $i(Z; V \\cdot W) \\ge 1$.\nWhen $A/I_V$ and $A/I_W$ are Cohen–Macaulay, the higher Tor terms vanish and $i(Z; V \\cdot W) = \\ell_A\\big(A/(I_V + I_W)\\big)$.\nFor two plane curves $f = 0$ and $g = 0$ without common component, meeting at a point $P$, this gives $i_P(f, g) = \\dim_k \\mathcal{O}_{\\mathbb{A}^2, P}/(f, g)$.\nExample: for $f = y - x^2$ and $g = y$ at $P = (0, 0)$, $\\mathcal{O}_{\\mathbb{A}^2, P}/(y - x^2, y) \\cong k[x]_{(x)}/(x^2)$, of dimension $2$.\nNon-example: if $V = W$ is a line in $X = \\mathbb{A}^2$, then $V \\cap W$ has dimension $1$, not $1 + 1 - 2 = 0$, so the intersection is not proper and the multiplicity is not defined.\n",
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