Noether normalization lemma
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In mathematics the Noether normalization lemma is a technical result of commutative algebra, attributed to Emmy Noether. In a simple version, it states that for any field K, and any commutative K-algebra A that is finitely generated over K and an integral domain, there are algebraically independent elements
- y1, y2, ..., yd,
in A, such that A is an integral extension of
- K[y1, y2, ..., yd] = B.
In geometric terms, B is the coordinate ring of affine space of dimension d, and A is the coordinate ring of some algebraic variety which must be of the same dimension (i.e. d is the transcendence degree of the field of fractions of A, and so is determined by A). The inclusion map of B into A gives as affine schemes a finite morphism
- φ: Spec(A) → Spec(B).
The conclusion is that any variety is a branched covering of affine space.
