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  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$.\nThen the [[Gelfand–Kirillov dimension|MDVSY2NMD2]] of $A$ is not in the open interval $(1, 2)$.\nTogether with the elementary fact that it is not in $(0, 1)$, this gives\n$$\\operatorname{GKdim} A \\in \\set{0} \\cup \\set{1} \\cup [2, \\infty].$$\n\n## Remarks\n\nThe gap between $0$ and $1$ is easy: if a finitely generated algebra is infinite-dimensional, then $V^n \\subsetneq V^{n+1}$ for all $n$, so $\\dim V^n \\ge n + 1$.\nThe gap between $1$ and $2$ is harder.\nOne reduces to algebras given by monomial relations, whose growth is governed by counting words in the generators.\nThen a combinatorial argument on words applies.\nIf the number of nonzero words of some length $n$ is at most $n$, then the number of nonzero words of each length stays bounded.\nSo the growth is at most linear.\nSince the dimension of a general algebra is the supremum over its finitely generated subalgebras, the gaps pass to all algebras.\n\nExample: $k[x]$ has Gelfand–Kirillov dimension $1$, and the [[polynomial ring|0SDC934R52]] $k[x, y]$ has dimension $2$.\nSo both ends of the gap $(1, 2)$ are attained.\nNo further gap exists: every real number $r \\ge 2$ is the Gelfand–Kirillov dimension of some finitely generated algebra.\nThis contrasts with the [[Krull dimension|MVGBS3J1P7]] of commutative rings, which takes only integer values or $\\infty$.\n\n## Explanations\n\n### L1 human\n\nThe Gelfand–Kirillov dimension measures how fast an algebra grows, and one might expect any value to be possible.\nBergman's theorem says that values strictly between $1$ and $2$ never occur.\nIf the growth of an algebra is measured by an exponent $d$, so that it grows roughly like $n^d$, then $d$ is $0$, $1$, or at least $2$.\n\n### L2 human\n\nThe theorem is a statement about the [[growth|EHY4PYSMGZ]] of algebras, and its heart is combinatorics on words.\nIt shows that Gelfand–Kirillov dimension behaves like an integer-valued dimension at the bottom of the scale, while above $2$ every real value can occur.\nIt is a basic input in the study of algebras of small Gelfand–Kirillov dimension, such as algebras of dimension $1$ and $2$, which are often compared with curves and surfaces.\n",
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