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Noether normalization lemma

The Noether normalization lemma is a foundational in that states: for any k and any finitely generated commutative k-algebra A that is a , there exists a non-negative d (equal to the of A) and algebraically independent elements y_1, \dots, y_d \in A such that the k[y_1, \dots, y_d] is isomorphic to a and A is a finite module over it. Introduced by the German mathematician in 1926, the lemma originally assumed the base field was infinite, with the finite field case later established by Oscar Zariski in 1943. In its geometric formulation, the lemma implies that any affine algebraic variety over an is a finite surjective onto an of the same , providing a powerful tool to embed complex algebraic structures into simpler polynomial settings. The theorem's significance lies in its role as a bridge between algebra and geometry, enabling proofs of key results such as and the characterization of for finitely generated algebras. It facilitates the study of integral extensions and module-finiteness, which are central to understanding properties like and Cohen-Macaulay rings in higher-dimensional varieties. Extensions of the lemma appear in more general settings, such as for projective schemes or rings over arbitrary Noetherian domains, underscoring its enduring influence in modern and .

Introduction

Historical Context

The Noether normalization lemma emerged from Emmy Noether's foundational contributions to in the early . In , Noether introduced the lemma in her paper "Der Endlichkeitssatz der Invarianten endlicher Gruppen der Charakteristik p," published in the Nachrichten der Gesellschaft der Wissenschaften zu , Mathematisch-Physikalische Klasse. This work addressed the finiteness of invariant rings for actions in positive , where she employed the lemma—originally proved assuming an infinite base field—to show that such rings are finitely generated over the base field. The case was later established by Yasuo Akizuki in 1933. The lemma itself provided a key technique for embedding finitely generated algebras into integral extensions of polynomial rings, marking a significant advancement in ideal theory and . Noether's development of the lemma was deeply influenced by David Hilbert's earlier results on the structure of ideals in polynomial rings. Hilbert's basis theorem, established in 1890, proved that ideals in polynomial rings over fields admit finite bases, laying groundwork for Noether's abstraction of algebraic structures. These influences aligned with Noether's broader program, initiated in her 1921 paper "Idealtheorie in Ringbereichen," to axiomatize and generalize classical algebra through ring-theoretic concepts, moving away from computational approaches toward structural invariance. The lemma quickly became a of , enabling rigorous definitions of ring via transcendence degree and facilitating proofs in . Its impact extended to resolving of Hilbert's fourteenth problem on invariant finiteness, solidifying its role as a foundational tool for in the decades following its publication.

Overview and Intuition

The Noether normalization lemma provides a fundamental way to understand the structure of finitely generated algebras over a by showing that such an is essentially a finite extension of a in a smaller number of variables, where the number of variables corresponds to the . This intuitive idea reveals that even complex algebraic structures can be "reduced" to something resembling the familiar and well-behaved polynomial ring k[x_1, \dots, x_d], up to integrality, thereby capturing the of the without altering its . This perspective is particularly motivated by challenges in , where one seeks to simplify the study of rings arising from group actions on rings. By identifying a over which the full is , the lemma reduces the complexity of analyzing ideals, modules, and symmetries, making it easier to compute invariants and understand the ring's geometric or algebraic behavior. The approach assumes familiarity with concepts like finitely generated algebras, Noetherian rings, and integral extensions, which form the basic toolkit for such reductions. Geometrically, the lemma offers a non-technical to projecting a high-dimensional onto a lower-dimensional , such that the map is finite and surjective, preserving the variety's "finiteness" while allowing analysis in simpler coordinates. This highlights how algebraic objects can be visualized and studied through their relationship to , facilitating insights into dimension and morphisms without losing essential structure.

Formal Statement

Algebraic Version

The algebraic version of Noether's normalization lemma provides a fundamental structure theorem for finitely generated algebras over a . Let k be a and let A be a finitely generated k-algebra. Then there exist a non-negative d and elements x_1, \dots, x_d \in A that are algebraically independent over k such that A is a finite over the subring S = k[x_1, \dots, x_d]. The integer d equals the of A, which measures the supremum of the lengths of chains of prime ideals in A. This dimension d is uniquely determined by the algebra A, independent of the choice of the elements x_1, \dots, x_d, and coincides with the transcendence degree of the fraction field of A over k when A is an . The lemma applies in full generality to any finitely generated k-algebra A, without the requirement that A be an or reduced. In particular, since A is finite as an S-, there exists a of generators \{a_1, \dots, a_m\} \subseteq A such that every element of A can be expressed as an S- of these generators. As a consequence of the finite module structure, for every a \in A, there exists a monic polynomial f_a(T) \in S[T] such that f_a(a) = 0. This integral dependence ensures that A is over S when A is an , but the finite module condition holds more broadly. The elements x_1, \dots, x_d can often be chosen to be linear combinations of a given set of generators of A over k, facilitating explicit constructions.

Geometric Version

The geometric version of the Noether normalization lemma reformulates the algebraic result in the language of algebraic geometry, providing a bridge between ring theory and the study of varieties. For an affine algebraic variety X over a field k with coordinate ring A = k[X], where \dim X = d, there exists a finite surjective morphism \pi: X \to \mathbb{A}^d_k. This morphism is dominant, meaning its image is dense in \mathbb{A}^d_k, and it arises from embedding a polynomial subring k[x_1, \dots, x_d] \subset A such that A is a finite module over this subring. Geometrically, this implies that X is a finite cover of affine d-space, capturing the idea that complex varieties can be "unfolded" onto simpler Euclidean-like spaces while preserving essential dimensional properties. The \pi is , meaning it satisfies the universal property of integral extensions in the category of schemes, and its fibers over points in the image are finite sets (possibly empty over points outside the image). This finiteness ensures that the generic fiber has dimension zero, aligning the dimension of X with that of \mathbb{A}^d_k. The coordinate ring of the target \mathbb{A}^d_k is precisely the k[x_1, \dots, x_d], which injects into A, reflecting the transcendence degree of the function field of X over k. Such projections are not unique and can often be chosen generically, avoiding singular loci to maintain smoothness in fibers where possible. In the more general scheme-theoretic setting, the extends to any X of finite type over k with \dim X = d: there exists a X \to \mathbb{A}^d_k that is surjective on underlying topological spaces and has finite fibers. This version applies beyond reduced varieties, accommodating non-reduced structures while preserving the core finite extension property. The role in dimension theory is pivotal, as the ensures that chains of irreducible subvarieties in X correspond to those in \mathbb{A}^d_k, thereby equating the Krull of the coordinate ring with the geometric defined via subvariety chains.

Proof

Preliminary Concepts

A ring R is called Noetherian if every ideal of R is finitely generated, or equivalently, if R satisfies the ascending chain condition on ideals. This property ensures that ideals cannot "grow indefinitely" in a controlled manner, which is fundamental for studying finitely generated structures in commutative algebra. A key consequence is Hilbert's basis theorem, which states that if k is a field, then the polynomial ring k[x_1, \dots, x_n] is Noetherian for any n \geq 0. More generally, any finitely generated algebra over a Noetherian ring is itself Noetherian, implying that finitely generated algebras over fields are Noetherian. In the context of field extensions, algebraic independence plays a central role. A subset S of a K/k is algebraically independent over k if no nonzero in k[t_s \mid s \in S] vanishes when evaluated at elements of S. The transcendence degree of K over k, denoted \operatorname{trdeg}_k K, is the of a maximal algebraically independent subset of K over k, also called a transcendence basis. This degree measures the "transcendental" dimension of the extension and remains invariant under algebraic closures or separable extensions. An x in an extension B of a A is over A if there exists a f(t) = t^n + a_{n-1} t^{n-1} + \dots + a_0 with coefficients a_i \in A such that f(x) = 0. This monic condition ensures that integrality is well-defined without scaling issues and captures elements that behave like of polynomials over A. A extension B/A is if every of B is over A. For finite field extensions E/F, the primitive element theorem asserts that there exists \alpha \in E such that E = F(\alpha), provided the extension is separable (which holds automatically if \operatorname{char} F = 0 or if F is finite). Such an \alpha is called a primitive element, simplifying the description of the extension as a simple extension. To handle module finiteness in Noetherian settings, Nakayama's lemma provides a criterion: if M is a finitely generated module over a local ring (R, \mathfrak{m}) and \overline{M} = M / \mathfrak{m}M = 0, then M = 0. More generally, if a set of elements generates M modulo \mathfrak{m}M, it generates M. This lemma is crucial for lifting generators from special fibers to the whole module. Complementarily, the Artin-Rees lemma states that for a Noetherian ring R, ideal I \subset R, and finite modules N \subset M, there exists c > 0 such that I^n M \cap N = I^{n-c} (I^c M \cap N) for all n \geq c. This controls intersections in powers of ideals, aiding proofs of finiteness in completions or filtrations. These concepts form the foundational toolkit for establishing the Noether normalization lemma, which asserts that for a finitely generated A over a k, there exist algebraically independent elements x_1, \dots, x_d \in A such that A is over k[x_1, \dots, x_d].

Detailed Construction

The detailed construction of Noether's normalization lemma begins with a finitely generated algebra A over an infinite k, presented as A = k[y_1, \dots, y_n]/I where I is a proper ideal. The proof proceeds by induction on n, the number of generators. If I = (0), then A \cong k[y_1, \dots, y_n] and the elements x_i = y_i (for i = 1, \dots, n) are algebraically independent with A finite over k[x_1, \dots, x_n] as the identity map. Assume n > 0 and I \neq (0). Select a nonzero f \in I. Since k is infinite, there exists a linear change of variables x_i = \sum_{j=1}^n a_{ij} y_j (with coefficients a_{ij} \in k forming an ) such that, in the new coordinates, f becomes monic in x_n: f(x_1, \dots, x_n) = x_n^e + \sum_{j=0}^{e-1} g_j(x_1, \dots, x_{n-1}) x_n^j, where e = \deg f and g_j \in k[x_1, \dots, x_{n-1}]. This choice avoids the in the parameter space \mathbb{A}^{n^2}_k defined by the vanishing of the resultant (or leading homogeneous form evaluation) associated to f, ensuring the leading coefficient is a nonzero constant. Under this automorphism of k[y_1, \dots, y_n], the ideal I now contains the in x_n. Thus, A is generated as a over the B = k[x_1, \dots, x_{n-1}] / (I \cap k[x_1, \dots, x_{n-1}]) by the basis \{1, x_n, \dots, x_n^{e-1}\}, as higher powers of x_n reduce via the monic relation I. Hence, A is finite over B. By the inductive applied to B, which is finitely generated by n-1 elements, there exist algebraically independent z_1, \dots, z_d \in B such that B is finite over k[z_1, \dots, z_d], where d = \dim B = \dim A (since finite extensions preserve Krull dimension). By transitivity of finite module extensions, A is finite over k[z_1, \dots, z_d]. The original generators y_i satisfy monic linear equations over k[x_1, \dots, x_n] due to the invertible linear change: each y_i = \sum b_{ij} x_j for some b_{ij} \in k, so y_i - \sum b_{ij} x_j = 0. Thus, the y_i (and hence all of A) are integral over k[z_1, \dots, z_d], confirming the finiteness as A is finitely generated and integral over the polynomial subring. For finite fields k, replace the linear change with a powering substitution: set x_i = y_i - y_n^{e_i} for i = 1, \dots, n-1, choosing strictly decreasing large exponents e_1 \gg e_2 \gg \dots \gg e_{n-1} to ensure unique leading monomials in a suitable grading, making f monic in y_n over k[x_1, \dots, x_{n-1}] via distinct multi-index valuations. The remaining steps follow analogously, with integrality from monic relations y_n^e + \lower terms = 0 and y_i = x_i + y_n^{e_i} (satisfying X - x_i - y_n^{e_i} = 0). If A is not a domain, the construction applies directly as no integrality assumption is needed.

Refinements and Generalizations

Refinement for Finite Extensions

A refinement of Noether's normalization lemma exists for finitely generated algebras over a k where the extension of fraction fields is finite and separable. In this case, if A is an finitely generated over k with \operatorname{tr.deg}_k A = d and the fraction field K(A) is a finite of k(x_1, \dots, x_d) for some algebraically independent x_1, \dots, x_d \in A, then one can select these elements such that A is a over the polynomial k[x_1, \dots, x_d] of equal to the [K(A) : k(x_1, \dots, x_d)]. This strengthens the standard version by ensuring not only finiteness but also freeness, meaning A has a basis \{b_1, \dots, b_r\} over k[x_1, \dots, x_d] with no torsion elements, so every element of A can be uniquely expressed as \sum_{i=1}^r f_i b_i for polynomials f_i \in k[x_1, \dots, x_d]. When k is infinite, the choice of x_1, \dots, x_d can be made via generic linear combinations of a transcendence basis. Specifically, starting from algebraically independent elements y_1, \dots, y_d \in A, one forms x_i = \sum_{j=1}^m \lambda_{ij} y_j for generic \lambda_{ij} \in [k](/page/K); this ensures the natural map k[x_1, \dots, x_d] \to A is injective and that A remains finite over the image, preserving the integral extension while avoiding relations that would introduce dependencies in the fibers. The freeness follows from the generic freeness lemma applied to the finite module A over the : since k is , there exists an open dense subset of \mathbb{A}^d_k where the fiber modules are free of constant rank equal to the generic rank, which matches the field extension degree under separability. This proof sketch differs from the basic construction by incorporating generic projections to guarantee constant across fibers, leveraging separability to ensure the minimal discriminants are non-zero generically and prevent zero-divisors or varying dimensions in the special fibers. In zero, all algebraic extensions are separable, so the refinement always applies. However, in positive , it fails without separability: inseparable extensions can lead to non-constant ranks or torsion in the structure over the subring, as the fibers may have nilpotents or components that disrupt freeness.

Generalizations to Noetherian Rings

The Noether normalization lemma extends to the case where the base ring R is Noetherian and A is a finitely generated R-algebra. In this setting, there exist elements f_1, \dots, f_d \in A such that the ring map R[f_1, \dots, f_d] \to A is quasi-finite, where d is the relative dimension of A over R, defined as the maximum of the fibers of \mathrm{Spec}(A) \to \mathrm{Spec}(R). This generalization builds on the original lemma over fields by using linear combinations of generators to construct the polynomial subring, but requires the base ring to satisfy additional properties for stronger conclusions to hold globally. The Cohen-Seidenberg theorems provide essential context for this extension, guaranteeing that integrality is preserved under base change from R to rings R/\mathfrak{p} for primes \mathfrak{p} \subset R. This allows reduction to the field case over residue fields, ensuring that the relative quasi-finiteness corresponds to the Tor-dimension of A over R being at most d, meaning \Tor_i^R(A, k(\mathfrak{p})) = 0 for i > d and generic primes \mathfrak{p}. In particular, if A is projective as an R- or the extension has finite Tor-dimension, the R[f_1, \dots, f_d] embeds injectively into A. For more general Noetherian base rings, the full finiteness may fail, as demonstrated by counterexamples where the morphism to the affine space is quasi-finite but not finite. However, when R is a complete or an , the lemma strengthens: the fibers over generic points are geometrically or , and the morphism \mathrm{Spec}(A) \to \mathrm{Aff}^d_R is finite with geometrically reduced fibers, ensuring flatness in the generic fiber. rings, which include all complete s and rings over fields, satisfy the necessary regularity conditions for these properties to hold uniformly. This generalized form was developed in works building on Noether's original result, notably by Nagata in his studies of local rings and by Matsumura in systematic treatments of dimension theory for Noetherian rings.

Applications

Generic Freeness

Generic freeness is a significant application of the Noether normalization lemma in the study of over , highlighting how finite modules become after a suitable generic specialization. Consider a k and a finitely generated M over the R = k[x_1, \dots, x_d]. There exists a nonzero element f \in R such that the localized M_f is as an R_f- of rank equal to the of M \otimes_K K over the fraction K of R. Geometrically, this implies the existence of a Zariski-open U \subset \mathbb{A}^d_k such that the \widetilde{M} associated to M is locally over U, and since U is affine, it is over the coordinate ring of U. The proof proceeds by applying Noether normalization to embed R into a larger via generic linear changes of variables, reducing the problem to showing freeness over this , and then extending via localization. Specifically, Noether normalization provides a finite injective from a k[y_1, \dots, y_d] to R, where the y_i are generic linear combinations of the x_j, ensuring the extension is without zero divisors outside a thin set. For the M, a by submodules with cyclic quotients allows induction on the length, using the normalization to control the generic dimension and localize to make each step . This generic choice avoids rank drops by ensuring that relations in the presentation do not degenerate. Central to this is the rank function \mathrm{rk}(M), which equals \dim_K(M \otimes_R K) and remains constant on a dense open subset of \mathrm{Spec}(R). The loci of lower rank are closed sets defined by the vanishing of determinant ideals, specifically the Fitting ideals generated by the (r+1) \times (r+1) minors of a presentation matrix of M, where r = \mathrm{rk}(M); these ideals are proper since the generic rank is achieved over K, so their zero sets do not cover \mathrm{Spec}(R). Thus, outside this closed set, the presentation matrix has full generic rank, implying local freeness, which globalizes to freeness after localization. A representative way to achieve this freeness is by selecting generic linear combinations for the variables, such as y_i = \sum a_{ij} x_j with coefficients a_{ij} \in k in , which perturbs the relations in M to prevent torsion and ensure the module is projective (hence ) over the new . This property simplifies computations in elimination theory by allowing reductions to modules over generic hypersurfaces, thereby facilitating the extraction of eliminants without rank deficiencies. In the context of Gröbner bases, generic freeness enables effective algorithms for modules over Noetherian rings by making presentations after inverting elements, aiding in the computation of syzygies and resolutions.

Connection to Hilbert's Nullstellensatz

The Noether normalization lemma plays a crucial role in proving the weak form of , which states that if k is an and \mathfrak{m} is a in the k[x_1, \dots, x_n], then \mathfrak{m} = (x_1 - a_1, \dots, x_n - a_n) for some a = (a_1, \dots, a_n) \in k^n. To establish this, consider the A = k[x_1, \dots, x_n]/\mathfrak{m}, which is a of k and finitely generated as a k-. By the Noether normalization lemma, there exist algebraically elements y_1, \dots, y_r \in A such that A is a finite (hence integral) extension of the polynomial subring k[y_1, \dots, y_r]. Since A is a field, the subring k[y_1, \dots, y_r] must also be a field, as integral extensions of domains are domains and the only field among polynomial rings over a is the constant ring itself (i.e., r = 0). Thus, A is a finite extension of k. If k is algebraically closed, then A = k by properties of algebraic closures. This implies that the maximal ideal \mathfrak{m} corresponds precisely to at a point in k^n, establishing the between maximal ideals and points in . The argument proceeds by contradiction: suppose r > 0, so the transcendence degree of A over k is positive. Then k[y_1, \dots, y_r] contains non-constant elements, and since A is over it, the maximality of \mathfrak{m} would be violated, as A could not be a field unless r = 0. If A \neq k, the transcendence degree exceeds 0, leading to zero-divisors or non-field structure in the extension, contradicting the assumption that A is a . This forces A = k, confirming that maximal ideals are of the specified form. The strong form of Hilbert's Nullstellensatz, which asserts that for any ideal \mathfrak{a} \subseteq k[x_1, \dots, x_n], the \sqrt{\mathfrak{a}} = I(V(\mathfrak{a})), follows by combining the weak form with properties of radical ideals. Specifically, the weak Nullstellensatz identifies the maximal ideals containing \mathfrak{a} as corresponding to points in the V(\mathfrak{a}), and the radical is the of those maximals. Noether underpins this by ensuring that finitely generated algebras over k have the required integral structure to control radicals via maximal ideals.

Examples and Illustrations

Basic Polynomial Examples

A simpler case occurs with field extensions viewed as k-algebras. For the rational function field L = k(t), the transcendence degree over k is 1. Here, the subfield k(t) itself serves as the purely transcendental extension, and L is a finite extension of degree 1 over it, satisfying the lemma trivially since the two coincide. This computation aligns the algebraic dimension with the transcendence degree, highlighting the lemma's role in normalizing such extensions.

Quotient Ring Examples

One illustrative example of Noether's normalization lemma applied to quotient rings is the coordinate ring of a quadric surface, given by A = k[x, y, z] / (x^2 + y^2 - z^2), where k is a field of characteristic not equal to 2. This ring is an integral domain of Krull dimension 2, so the lemma guarantees a polynomial subring of rank 2 over which A is finite. Consider the subring k[x, z] \subseteq A; the images of x and z are algebraically independent, and the image of y satisfies the monic equation t^2 = z^2 - x^2 over k[x, z], making A integral over this subring. Thus, A is a free module of rank 2 over k[x, z] with basis \{1, \overline{y}\}, where the bar denotes the image in the quotient. A similar computation applies to the quadric cone R = k[x, y, z] / (y^2 - x z), also of 2. Here, the k[x, z] \subseteq R serves as a Noether , with \overline{x} and \overline{z} algebraically independent, and \overline{y} satisfying the t^2 - x z = 0. The structure is again of rank 2 with basis \{1, \overline{y}\}, demonstrating the finiteness. These examples highlight how imposing a quadratic relation reduces the effective number of variables from 3 to 2 while preserving the extension property. For hypersurfaces, consider B = k[x, y] / (f(x, y)), where f is an of positive degree over a k. The of B is 1, so the lemma yields a subring isomorphic to k over which B is finite. By a suitable linear , one can assume f is monic in y, say f(x, y) = y^d + a_{d-1}(x) y^{d-1} + \cdots + a_0(x); then k \subseteq B, and the image of y satisfies this monic equation over k. Consequently, B is a free module of rank d over k with basis \{1, \overline{y}, \dots, \overline{y}^{d-1}\}. This construction shows how the relation imposed by the irreducible hypersurface drops the dimension from 2 to 1, with the extension remaining finite.