Dimension and Noether normalization

The Krull dimension of a ring is the supremum of lengths of strict chains of prime ideals, where a chain with d inclusions has length d. Under the reversed ideal–geometry correspondence, it equals the dimension measured by chains of irreducible closed subsets. For a finitely generated k-domain A, it is also the transcendence degree of its fraction field over k. This comparison requires finite generation; the chain definition remains available much more generally, but naive parameter counting need not capture arbitrary rings.

For a reduced algebraic set with several components, dimension is the maximum of their dimensions. A smaller component should not be erased merely because it does not affect that maximum. Nor is dimension automatically ambient dimension minus the number of displayed generators. For instance z−xy imposes one independent relation on three coordinates and allows x,y as free parameters, whereas adding twice the same equation imposes no new condition. Heights and regular sequences explain when a codimension count is justified.

Noether normalization says that a nonzero finitely generated k-algebra A contains algebraically independent elements z₁ through z_d such that A is a finite module over their polynomial ring. The integer d is dim A. This algebraic statement works over any field and does not require A to be reduced; our geometric applications use a reduced coordinate ring over an algebraically closed field. Over infinite fields one can often arrange suitable linear combinations of ambient coordinates, but the theorem should not be replaced by an unsupported claim that every coordinate projection works.

Finite module generation implies integrality: every element satisfies a monic equation over the polynomial subring. Geometrically the chosen functions define a finite surjective morphism to affine d-space. Surjectivity follows from lying over and the point correspondence over the algebraically closed field. Individual fibres are finite, with scheme multiplicities allowed, but they need not contain a constant number of distinct points. Branching can merge points without contradicting module finiteness. Finding the monic equations is a concrete way to verify the stronger finite-map assertion.

The hyperbola xy=1 has coordinate ring k[x,x inverse]. Projection to x is injective on coordinate rings and dominant, but its point image misses zero. It cannot be finite and surjective onto the line. Algebraically, x inverse is not integral over k[x]: a monic equation for it, multiplied by a large enough power of x and evaluated at zero, would give one equal to zero. Thus x is an independent parameter that does not make the whole coordinate ring finite.

Another parameter can repair the situation. Taking the sum of x and y forces x to satisfy a monic quadratic and recovers y from that sum and x. The resulting finite map has two points in a general fibre and fewer distinct points at special values in characteristic zero. The map is not a normalization of the hyperbola; Noether normalization chooses a finite map to affine space, while integral-closure normalization later changes the source ring inside its own function field. Their shared name should not conceal their different inputs and outputs.