In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Noether normalization lemma is a result of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, introduced by
Emmy Noether
Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ...
in 1926.
It states that for any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''k'', and any
finitely generated commutative
''k''-algebra ''A'', there exists a non-negative integer ''d'' and
algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically in ...
elements ''y''
1, ''y''
2, ..., ''y''
''d'' in ''A'' such that ''A'' is a
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts inclu ...
over the polynomial ring ''S'' = ''k''
1, ''y''2, ..., ''y''''d''">'y''1, ''y''2, ..., ''y''''d''
The integer ''d'' above is uniquely determined; it is the
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
of the ring ''A''. When ''A'' is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
, ''d'' is also the
transcendence degree
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
of the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ''A'' over ''k''.
The theorem has a geometric interpretation. Suppose ''A'' is integral. Let ''S'' be the
coordinate ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
of the ''d''-dimensional
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, and let ''A'' be the coordinate ring of some other ''d''-dimensional
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
''X''. Then the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
''S'' → ''A'' induces a surjective
finite morphism In algebraic geometry, a finite morphism between two affine varieties
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polyn ...
of
affine varieties
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
. The conclusion is that any
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
is a
branched covering In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In topology
In topology, a map is a ''branched covering'' if it is a covering map everywhere except for a nowhere dense set known as the branch set. ...
of affine space.
When ''k'' is infinite, such a branched covering map can be constructed by taking a general projection from an affine space containing ''X'' to a ''d''-dimensional subspace.
More generally, in the language of schemes, the theorem can equivalently be stated as follows: every affine ''k''-scheme (of finite type) ''X'' is
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
over an affine ''n''-dimensional space. The theorem can be refined to include a chain of ideals of ''R'' (equivalently, closed subsets of ''X'') that are finite over the affine coordinate subspaces of the appropriate dimensions.
The form of the Noether normalization lemma stated above can be used as an important step in proving Hilbert's
Nullstellensatz. This gives it further geometric importance, at least formally, as the Nullstellensatz underlies the development of much of classical
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. The theorem is also an important tool in establishing the notions of
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
for ''k''-algebras.
Proof
The following proof is due to Nagata and is taken from Mumford's red book. A proof in the geometric flavor is also given in the page 127 of the red book an
this mathoverflow thread
The ring ''A'' in the lemma is generated as a ''k''-algebra by elements, say,
. We shall induct on ''m''. If
, then the assertion is trivial. Assume now
. It is enough to show that there is a subring ''S'' of ''A'' that is generated by
elements, such that ''A'' is finite over ''S.'' Indeed, by the inductive hypothesis, we can find algebraically independent elements
of ''S'' such that ''S'' is finite over