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  "body": "\n## Definition\n\nLet $X$ be a complex manifold and let $A$ be an [[analytic subset|VMDTGTMN9P]] of $X$.\nLet $\\operatorname{reg} A$ be the set of [[regular points|JPP8N6MZGP]] of $A$.\nAt a regular point $z$, the set $A$ is near $z$ a complex submanifold, and $\\dim_z A$ is its complex dimension at $z$.\nFor an arbitrary point $a \\in A$, the *dimension of $A$ at $a$* is\n$$\\dim_a A = \\limsup_{z \\to a,\\ z \\in \\operatorname{reg} A} \\dim_z A.$$\nThe *dimension of $A$* is $\\dim A = \\sup_{a \\in A} \\dim_a A$, with $\\dim \\emptyset = -\\infty$.\nThe set $A$ has *pure dimension* $k$ if $\\dim_a A = k$ for every $a \\in A$.\n\n## Remarks\n\nThe definition makes sense because $\\operatorname{reg} A$ is dense in $A$; at a regular point $a$ it gives back the manifold dimension.\nExample: let $A = \\set{z_1 z_3 = 0,\\ z_2 z_3 = 0}$ in $\\C^3$, the union of the plane $P = \\set{z_3 = 0}$ and the line $L = \\set{z_1 = z_2 = 0}$.\nThe regular points are the points of $P \\setminus \\set{0}$, of dimension $2$, and the points of $L \\setminus \\set{0}$, of dimension $1$.\nSo $\\dim_0 A = 2$, $\\dim_a A = 1$ for $a \\in L \\setminus \\set{0}$, and $\\dim A = 2$.\nNon-example of pure dimension: this set $A$ does not have pure dimension, since its local dimension takes the values $1$ and $2$.\nIt also shows that the dimension is not the number of variables minus the number of equations, which would be $1$ here.\nThere are equivalent forms of $\\dim_a A$ when $X = \\C^n$.\nIt is the Krull dimension of $\\mathcal{O}_{n,a}/\\mathcal{I}(A_a)$.\nIt is the largest of the integers $d$ given by the local parametrization theorem for the irreducible components of the germ $A_a$.\nIt is the smallest $k$ such that some complex affine subspace of codimension $k$ through $a$ meets $A$ in the isolated point $a$.\n",
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