Krull dimension
this is regularity, which is an algebraic formulation of the fact that affine -space is a variety without singularities. In fact the following generalization holds: Let formula_57 be a Noetherian ring. Then formula_57 has finite global dimension if and only if formula_57 is regular and the Krull dimension of formula_57 is finite; in that case the global dimension of formula_57 is equal to the Krull dimension. This result may be proven using Serre's theorem on regular local rings.