Dimension of an algebraic variety

Summary

The dimension of an algebraic variety is the length of the longest chain of nested irreducible closed subsets in it. For an affine variety it equals the Krull dimension of the coordinate ring. For any variety it equals the transcendence degree of the function field over the base field. These different descriptions agree, which makes dimension one of the most robust invariants in algebraic geometry: points have dimension $0$, curves $1$, surfaces $2$.