the Krull dimension
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We say a curve is one-dimensional, a surface two-dimensional, all-space three-dimensional. The Krull dimension makes this counting purely algebraic, by measuring how many times you can nest irreducible shapes inside one another, each strictly smaller than the last. A point sits inside a curve sits inside a surface sits inside a solid — the longest such chain is the dimension.
Geometrically, the dimension of an irreducible variety X is the largest d for which there is a chain of irreducible closed subvarieties X_0 strictly inside X_1 strictly inside ... strictly inside X_d = X, each properly containing the previous. Translating through the coordinate ring, the irreducible closed subsets correspond to prime ideals, so this is the same as the longest chain of prime ideals p_0 strictly inside p_1 strictly inside ... strictly inside p_d in k[X] — and that length is precisely the Krull dimension of the ring. Two further descriptions coincide for varieties over a field: the dimension equals the transcendence degree of the function field k(X) over k, and (at a smooth point) the dimension of the Zariski tangent space. So 'count nested subvarieties,' 'count nested primes,' and 'count independent coordinates the function field needs' all give the same number.
Dimension is the most basic invariant of a variety and the backbone of everything from intersection theory (codimension counts equations) to the smoothness criterion (tangent space has the right dimension). Two honest caveats. First, dimension is local on components: a reducible variety can have components of different dimensions, and 'the dimension' is the maximum, so always ask whether a variety is pure-dimensional. Second, the agreement of Krull dimension, transcendence degree, and tangent-space dimension is a theorem for finitely generated algebras over a field; for general Noetherian rings these can come apart, and Krull dimension can even be infinite.
Affine 3-space A^3 has dimension 3, witnessed by the chain {point} inside {line} inside {plane} inside A^3, and matched by the prime chain (x, y, z) inside (x, y) inside (x) inside (0) in k[x, y, z]. The transcendence degree of k(x, y, z) over k is also 3 — three independent coordinates.
The same dimension shows up as a chain of subvarieties, a chain of primes, and a transcendence degree.
The equality of Krull dimension, transcendence degree of k(X), and tangent-space dimension holds for finitely generated algebras over a field, and the last only at smooth points; at a singular point the tangent space is strictly bigger.