Noether Normalization · Invariance: The Mathematics of Emmy Noether

← Invariance: The Mathematics of Emmy Noether

By Part 12 a quotient $A = k[x_1, \ldots, x_n]/I$ is the coordinate ring of a variety cut out by finitely many equations. Noether normalization (1926) finds inside it a polynomial ring over which it is finite:

\text{there are algebraically independent } y_1, \ldots, y_d \in A \text{ with } A \text{ a finite } k[y_1, \ldots, y_d]\text{-module.}

Finite means finitely many $f_1, \ldots, f_r \in A$ such that every element is $p_1 f_1 + \cdots + p_r f_r$ with each $p_i \in k[y_1, \ldots, y_d]$ . Geometrically the inclusion is a projection from the variety to $d$ -space in which every point has finitely many preimages, and none escape to infinity. Here $d \le n$ , with equality only for $I = 0$ , and $d$ is the dimension of the variety.

Two sheets over a line

On the elliptic curve $A = k[x, y]/(y^2 - x^3 + x)$ , the relation $y^2 = x^3 - x$ lowers any power of $y$ , so every element is $a(x) + b(x)\, y$ . $A$ is generated as a $k[x]$ -module by $\{1, y\}$ , and the projection $(x, y) \mapsto x$ has two sheets.

x = 1.50

Figure 1. Left: the real curve $y^2 = x^3 - x$ and the fiber over $x$ . Right: the same fiber in the complex $y$ -plane, the two roots $\pm\sqrt{x^3 - x}$ . The real count jumps between 0 and 2; the complex count is 2 everywhere, the roots meeting only at the branch points $x = -1, 0, 1$ , where they turn from real to imaginary.

Two sheets over a plane

The quadric cone $A = k[x, y, z]/(z^2 - xy)$ works the same way one dimension up. Every element is $a(x, y) + b(x, y)\, z$ , and forgetting $z$ is a two-sheeted projection onto the plane, branched over $xy = 0$ . (The cone's axis is the line $x = y,\ z = 0$ : substituting $x = u + v,\ y = u - v$ gives $z^2 + v^2 = u^2$ .)

Figure 2. The base plane of the cone, sampled on a 41 by 41 grid and shaded by the number of real points $z = \pm\sqrt{xy}$ above each sample: two in the quadrants where $xy > 0$ , none where $xy < 0$ , one on the axes. Drag the point, or focus the plot and use the arrow keys, to read its fiber. The census counts complex points too.

Why a generic linear change works

Not every projection is finite. On the hyperbola $xy = 1$ , projecting to $x$ gives one point over each $x \ne 0$ and none over $x = 0$ : the point runs off to infinity, and $y$ satisfies no monic equation over $k[x]$ . The proof repairs this with a change of variables

x_i \;\mapsto\; x_i + c_i\, x_n \quad (i = 1, \ldots, n - 1)

For generic $c_i$ a polynomial $f \in I$ acquires a pure power $c\, x_n^{\deg f}$ with $c \ne 0$ , so dividing by $c$ gives a monic equation for $x_n$ over the other variables and $A$ is finite over $k[x_1', \ldots, x_{n-1}']$ . If those still satisfy a relation, repeat; each round removes a variable, until the rest are algebraically independent. Generic constants need an infinite field; over a finite field the proof substitutes powers instead, such as $x_i \mapsto x_i + x_n^{r^i}$ for a large integer $r$ .

c = 0.25 t = 0.50

Figure 3. The hyperbola $xy = 1$ projected to $t = x - c\,y$ , which is the substitution $x \mapsto x + c\,y$ . The fiber over $t$ solves $c\,y^2 + t\,y - 1 = 0$ . Left: the lines $x - c\,y = t$ and the real fiber points. Right: $|y|$ of both complex fiber points against $t$ , on a log scale, with $c = 0$ in red for comparison. The strip below counts fiber points with multiplicity at 601 sampled $t$ . Any $c \ne 0$ keeps both points bounded; at $c = 0$ one of them escapes.

The rank depends on the subring

The number of sheets is the rank of $A$ over the subring, and it depends on the subring chosen. The curve $(t^3, t^4, t^5)$ has coordinate ring $k[t^3, t^4, t^5]$ , spanned by the powers $t^m$ with $m$ a sum of 3s, 4s and 5s. Over $k[t^3]$ , sort the exponents by their residue mod 3: each residue class starts at some least exponent $w_r$ and then contains every $w_r + 3q$ , so $t^{w_0}, t^{w_1}, t^{w_2}$ generate a free module of rank 3.

curve over

Figure 4. Exponents of the coordinate ring of a monomial curve, one row per residue class mod $a$ for the chosen subring $k[t^a]$ . Filled cells are exponents in the ring, empty cells are the gaps. The first filled cell of each row (outlined) is a module generator, and everything after it follows by multiplying by $t^a$ . Change the subring and the number of rows, the rank, changes with it; the curve and its dimension do not.

Counting the dimension

The $d$ of a normalization is the Krull dimension of $A$ , the length of the longest chain of prime ideals, and equally the transcendence degree of its fraction field over $k$ . Normalization is what makes these agree, and it makes dimension computable. With a graded Gröbner basis from Part 12, count the monomials of degree at most $s$ outside the leading-term ideal. That count $\mathrm{HF}(s)$ is the dimension of the polynomials of degree at most $s$ modulo $I$ , and for large $s$ it is a polynomial in $s$ of degree $d$ , which is what $d$ free variables contribute.

Figure 5. Six coordinate rings, each run through Buchberger's algorithm in the page (graded reverse lex, exact rationals). The chart plots $\mathrm{HF}(s)$ on log-log axes, where a slope of $d$ means growth like $s^d$ ; the table reads $d$ off exactly as the first $d$ with a constant $d$ -th difference. The fiber column adds the subring's equations at five seeded random rational base points and counts the points of each fiber. Select a row to trace it.

Together with the basis theorem, this closes Act III. Ideals of polynomials are finitely generated, so their quotients are the coordinate rings of varieties, and each is a finite cover of a polynomial ring whose number of variables is the dimension of the variety.