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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $v_1, \\dots, v_k \\in \\Z^n$.\nLet $C = \\set{\\lambda_1 v_1 + \\cdots + \\lambda_k v_k : \\lambda_1, \\dots, \\lambda_k \\ge 0}$ be the rational polyhedral [[convex cone|6VE4KA12VD]] they generate in $\\R^n$.\nThen the [[monoid|0N6R902FG1]] $C \\cap \\Z^n$, with the addition of $\\Z^n$, is finitely generated.\n\n## Remarks\n\nLet $P = \\set{t_1 v_1 + \\cdots + t_k v_k : 0 \\le t_i \u003c 1}$.\nIt is a bounded set, so $P \\cap \\Z^n$ is finite.\nEvery $x \\in C \\cap \\Z^n$ can be written $x = \\sum_i t_i v_i$ with $t_i \\ge 0$, and then $x = \\sum_i \\floor{t_i} v_i + p$ with $p = \\sum_i (t_i - \\floor{t_i}) v_i \\in P \\cap \\Z^n$.\nSo the finite set $\\set{v_1, \\dots, v_k} \\cup (P \\cap \\Z^n)$ generates $C \\cap \\Z^n$.\n\nExample: $v_1 = (1, 0)$ and $v_2 = (1, 2)$ in $\\Z^2$.\nHere $P \\cap \\Z^2 = \\set{(0,0), (1,1)}$, so $C \\cap \\Z^2$ is generated by $(1,0)$, $(1,1)$, $(1,2)$.\nThe point $(1,1)$ is needed: it lies in $C$ but is not a combination of $(1,0)$ and $(1,2)$ with nonnegative integer coefficients.\n\nRationality is needed.\nFor the cone $C = \\set{(x, y) : x \\ge 0, \\ 0 \\le y \\le \\sqrt{2}\\, x}$ the monoid $C \\cap \\Z^2$ is not finitely generated.\nFinitely many generators in $C \\cap \\Z^2$ only produce points of slope at most $q$, the largest slope of a generator, and $q \u003c \\sqrt{2}$ since $q$ is rational.\nBut $C$ contains lattice points $(b, a)$ with $q \u003c a/b \u003c \\sqrt{2}$.\n\n## Explanations\n\n### L1 human\n\nTake a cone in space whose edges point along whole-number directions.\nThe lattice points inside the cone can be added together, and the lemma says that finitely many of them generate all the others by addition.\nThe generators are the edge vectors together with the few lattice points in a small box spanned by them.\n\n### L2 human\n\nThe lemma is the combinatorial input of the theory of affine toric varieties.\nFor a rational cone $\\sigma$, the semigroup algebra of $\\sigma^\\vee \\cap M$ is then finitely generated, and its spectrum is the [[toric variety|YWB981MEF2]] of $\\sigma$.\nThe minimal generating set of $C \\cap \\Z^n$ for a pointed cone is its Hilbert basis.\nIn invariant theory the same statement is used to show that rings of invariants of tori are finitely generated.\n",
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