nLab
Nullstellensatz

Contents

Context

Cohesive Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Geometry

Contents

Idea

Nullstelle means zero locus. Hilbert’s Nullstellensatz (theorem about zero loci) characterizes the joint zero loci of an ideal of functions in a polynomial ring. This is a foundational result in algebraic geometry at the heart of the duality by which rings are dually interpreted as spaces (varieties).

Classical formulation in algebraic geometry

In classical algebraic geometry over an algebraically closed field kk, the Nullstellensatz concerns the fixed points of a standard Galois connection between ideals II of the polynomial ring k[x 1,,x n]k[x_1, \ldots, x_n] and subsets Vk nV \subseteq k^n. The Galois connection is induced by a relation IVI \perp V iff f(x)=0f(x) = 0 for all fI,xVf \in I, x \in V. Accordingly, letting Idl(k[x 1,,x n])Idl(k[x_1, \ldots, x_n]) be the set of ideals ordered by inclusion and P(k n)P(k^n) the set of subsets of k nk^n ordered by inclusion, there are contravariant maps Ideal:P(k n)Idl(k[x 1,,x n])Ideal: P(k^n) \to Idl(k[x_1, \ldots, x_n]) and Var:Idl(k[x 1,,x n])P(k n)Var: Idl(k[x_1, \ldots, x_n]) \to P(k^n) defined by

Ideal(S)={fk[x 1,,x n]: xxSf(x)=0},Var(I)={xk n: ffIf(x)=0}Ideal(S) = \{f \in k[x_1, \ldots, x_n]: \forall_x x \in S \Rightarrow f(x) = 0\}, \qquad Var(I) = \{x \in k^n: \forall_f f \in I \Rightarrow f(x) = 0\}

which defines the Galois connection.

The fixed points of VarIdeal:P(k n)P(k n)Var \circ Ideal: P(k^n) \to P(k^n) are, by definition, the closed sets of the Zariski topology on the affine space k nk^n (“affine space” here in the sense of algebraic geometry). These are the closed subvarieties of k nk^n.

On the other side, the fixed points of IdealVarIdeal \circ Var is what the classical Nullstellensatz is about:

Theorem

(Strong Nullstellensatz) The fixed points of IdealVarIdeal \circ Var are precisely the radical ideals II of k[x 1,,x n]k[x_1, \ldots, x_n], i.e., ideals II such that for any rr \in \mathbb{N}, the condition f rIf^r \in I implies fIf \in I.

Thus there is a Galois correspondence between closed subvarieties and radical ideals.

Typically the way that the strong Nullstellensatz is proved is by reduction to the so-called “weak Nullstellensatz” by means of the “Rabinowitsch trick”. (The “weak” here may be misleading, as the weak Nullstellensatz may be considered the core result and the strong Nullstellensatz a corollary.) We now turn to these.

A weak Nullstellensatz: Existence of points

One weak version of the Nullstellensatz says the following (e.g. theorem 3.99.1 here).

For kk an algebraically closed field and II a proper ideal in the polynomial ring k[X 1,,X n]k[X_1, \cdots, X_n], the set V(I)V(I) (of nn-tuples x=(x i)k nx = (x_i) \in k^n such that all polynomials in II vanish when evaluated on these xx) is an inhabited set.

Before giving a proof, we remark (in view of more abstract formulations to come later) that an element of V(I)V(I) is just a kk-algebra homomorphism of the form

kk[X 1,,X n]/I. k \longleftarrow k[X_1, \cdots, X_n]/I \,.

Dually this is a morphism of affine schemes (ring spectra) of the form

Spec(k)Spec(k[X 1,,X n]/I). Spec(k) \longrightarrow Spec(k[X_1, \cdots, X_n]/I) \,.

Moreover, since Spec(k)Spec(k) is the terminal object in this context, such a map is the same as a “point”, a global element of Spec(k[X 1,,X n]/I)Spec(k[X_1, \cdots, X_n]/I).

Hence in this form the Nullstellensatz simply says that (for kk algebraically closed) affine schemes have points. This formulation of the Nullstellensatz leads one to a more general abstract formulation.

We turn now to the proof of the weak Nullstellensatz.

Lemma

If kk is an uncountable algebraically closed field, then for any proper ideal Ik[X 1,,X n]I \subset k[X_1, \ldots, X_n] there is (a 1,,a n)k n(a_1, \ldots, a_n) \in k^n such that f(a 1,,a n)=0f(a_1, \ldots, a_n) = 0 for all fIf \in I.

Proof

By the axiom of choice, II is contained in a maximal ideal MM. So it suffices to prove it in the case where I=MI = M is maximal, where we prove there is an isomorphism ϕ:k[X 1,,X n]/Mk\phi: k[X_1, \ldots, X_n]/M \cong k. Then the composite

k[X 1,,X n]quotk[X 1,,X n]/Mϕkk[X_1, \ldots, X_n] \stackrel{quot}{\twoheadrightarrow} k[X_1, \ldots, X_n]/M \stackrel{\phi}{\to} k

sends each X iX_i to a value a ia_i, so we have an inclusion of maximal ideals

(X 1a 1,,X na n)M(X_1 - a_1, \ldots, X_n - a_n) \subseteq M

which must be an equality. In that case (a 1,,a n)(a_1, \ldots, a_n) is the desired point.

Since kk is algebraically closed, it suffices to prove that the extension field F=k[X 1,,X n]/MF = k[X_1, \ldots, X_n]/M over kk is algebraic over kk. Clearly FF as a vector space over kk has countable dimension. Supposing to the contrary there is an element tFt \in F that is transcendental over kk, we have a subquotient vector space of FF,

k(t)k[t] akk[1ta]k,\frac{k(t)}{k[t]} \cong \bigoplus_{a \in k} \frac{k[\frac1{t - a}]}{k},

where the isomorphism is by partial fraction decomposition (ultimately by the Chinese remainder theorem?), but the vector space on the right clearly has uncountable dimension because there are uncountably many aka \in k, and this leads to a contradiction.

Now we reduce to the case where kk is uncountable by employing a model-theoretic argument:

Theorem

(Weak Nullstellensatz) If kk is an algebraically closed field, then for any proper ideal Ik[X 1,,X n]I \subset k[X_1, \ldots, X_n] there is (a 1,,a n)k n(a_1, \ldots, a_n) \in k^n such that f(a 1,,a n)=0f(a_1, \ldots, a_n) = 0 for all fIf \in I.

Proof

Let UU be a non-principal ultrafilter on \mathbb{N}, and let KK be the ultrapower k /Uk^\mathbb{N}/U. By Łoś's ultraproduct theorem?, KK is an algebraically closed field. Also KK is uncountable (Lemma below).

First we claim K kIK \otimes_k I is a proper ideal of K kk[X 1,,X n]K[X 1,,X n]K \otimes_k k[X_1, \ldots, X_n] \cong K[X_1,\ldots, X_n]. For suppose we have 1= i=1 sf ip i1 = \sum_{i = 1}^s f_i p_i for p 1,,p sIp_1, \ldots, p_s \in I and f iK[X 1,,X n]f_i \in K[X_1, \ldots, X_n]. Writing the f if_i as KK-linear combinations of monomials X αX^\alpha, there are finitely many q 1,,q tIq_1, \ldots, q_t \in I, each of the form X αp iX^\alpha p_i, so as to enable us to write 1= i=1 ta iq i1 = \sum_{i=1}^t a_i q_i for a i=[(a i,m)]Ka_i = [(a_{i, m})] \in K and q iIq_i \in I. By Łoś‘s theorem,

{m:1= i=1 ta i,mq iink[X 1,,X n]}U;\{m \in \mathbb{N}: 1 = \sum_{i=1}^t a_{i, m} q_i\; in\; k[X_1, \ldots, X_n]\} \in U;

this set is nonempty, and so there is at least one mm rendering the equation true. This contradicts 1I1 \notin I.

By Lemma , there exists (a 1,,a n)K n(a_1, \ldots, a_n) \in K^n belonging to Var(K kI)Var(K \otimes_k I). By Hilbert's basis theorem, Ik[X 1,,X n]I \subset k[X_1, \ldots, X_n] is generated by finitely many polynomials p 1,,p sp_1, \ldots, p_s, so Var(K kI)K nVar(K \otimes_k I) \subseteq K^n is defined by finitely many formulas p 1(X)=0,,p s(X)=0p_1(X) = 0, \ldots, p_s(X) = 0 and thus by Łoś‘s theorem again,

{m:p i(a 1,m,a 2,m,,a n,m)=0fori=1,,s}U\{m \in \mathbb{N}: p_i(a_{1, m}, a_{2, m}, \ldots, a_{n, m}) = 0\; for\; i = 1, \ldots, s\} \in U

and so this set is nonempty and therefore there is a point (a 1,m,,a n,m)k n(a_{1, m}, \ldots, a_{n, m}) \in k^n that belongs to Var(I)Var(I).

To prove KK uncountable, we use the fact that an algebraically closed field is at least infinite. Then the missing piece is supplied by the following result.

Lemma

If XX is an infinite set and UU is a non-principal ultrafilter on \mathbb{N}, then the ultrapower X /UX^\mathbb{N}/U has cardinality |X| 0{|X|}^{\aleph_0} (so at least 2 02^{\aleph_0}).

Proof

For jj \in \mathbb{N}, put [j]={i:i<j}[j] = \{i \in \mathbb{N}: i \lt j\}. If XX is infinite, then there is a mono i j:X [j]Xi_j: X^{[j]} \to X for each j0j \geq 0. For any h:Xh: \mathbb{N} \to X, let h| [j]h|_{[j]} denote the composite [j]hX[j] \hookrightarrow \mathbb{N} \stackrel{h}{\to} X, and define h:Xh': \mathbb{N} \to X by h(j)=i j(h| [j])h'(j) = i_j(h|_{[j]}). Let [h] U[h']_U denote the value of hh' under the quotient map X X /UX^\mathbb{N} \to X^\mathbb{N}/U.

Claim: The map X X /UX^\mathbb{N} \to X^\mathbb{N}/U that takes hh to [h] U[h']_U is injective. For, if g,h:Xg, h: \mathbb{N} \to X are distinct, then they differ at some ii \in \mathbb{N}. Then g| [j]h| [j]g|_{[j]} \neq h|_{[j]} whenever i<ji \lt j, which in turn implies i j(g| [j])i j(h| [j])i_j(g|_{[j]}) \neq i_j(h|_{[j]}) i.e. g(j)h(j)g'(j) \neq h'(j) for all jj such that j>ij \gt i. (That is, “for almost all jj”, belonging to a cofinite set that belongs to UU.) Thus [g] U[h] U[g']_U \neq [h']_U, which establishes the claim. The lemma follows immediately from the claim.

Rabinowitsch’s “trick”

Rabinowitsch, who later renamed himself Rainich (see the story here), showed how to deduce the Strong Nullstellensatz from the Weak Nullstellensatz. Often this is described as Rabinowitsch’s “trick” (i.e., something involving a stroke of cleverness), but see the remark below which places the trick conceptually within the general method of localization.

Let kk be algebraically closed and let Ik[x 1,,x n]I \subseteq k[x_1, \ldots, x_n] be an ideal. To show that Ideal(Var(I))=IIdeal(Var(I)) = \sqrt{I}, suppose that fk[x 1,,x n]f \in k[x_1, \ldots, x_n] vanishes at any point where all of a given set of polynomials f 1,,f sk[x 1,,x n]f_1, \ldots, f_s \in k[x_1, \ldots, x_n] vanish. It suffices to show that for some rr we have that f rf^r belongs to the ideal generated by the f if_i.

Introduce a fresh variable x 0x_0, and observe that there is no point (a 0,a 1,,a n)k n+1(a_0, a_1, \ldots, a_n) \in k^{n+1} where the polynomials 1x 0f,f 1,,f s1 - x_0 f, f_1, \ldots, f_s simultaneously vanish. By the Weak Nullstellensatz (Theorem ), these polynomials must generate the unit ideal, say

1=p 0(x)(1x 0f)+p 1(x)f 1++p s(x)f s1 = p_0(x)(1 - x_0 f) + p_1(x) f_1 + \ldots + p_s(x) f_s

where x=(x 0,x 1,,x n)x = (x_0, x_1, \ldots, x_n). There is a kk-algebra map k[x 0,x 1,,x n]k(x 1,,x n)k[x_0, x_1, \ldots, x_n] \to k(x_1, \ldots, x_n) sending x 01/fx_0 \mapsto 1/f and x ix ix_i \mapsto x_i for 1in1 \leq i \leq n, where the above identity yields

1= i=1 sp i(1/f,x 1,,x n)f i1 = \sum_{i=1}^s p_i(1/f, x_1, \ldots, x_n) f_i

which, if rr is the highest degree of x 0x_0 appearing in a p ip_i, may be rewritten in the form f r= i=1 sg if if^r = \sum_{i=1}^s g_i f_i in k[x 1,,x n]k[x_1, \ldots, x_n] by clearing denominators.

Remark

Proving that fIf \in \sqrt{I} means we prove that ff is nilpotent in k[x 1,,x n]/Ik[x_1, \ldots, x_n]/I, in other words that the localization of k[x 1,,x n]/Ik[x_1, \ldots, x_n]/I with respect to the (image of the) multiplicative system {1,f,,f n,}\{1, f, \ldots, f^n, \ldots\} vanishes. Clearly we localize by inverting (the image of) ff, by constructing

(k[x 1,,x n]/I)[f 1]=k[x 0,x 1,,x n]/(I,1x 0f).\left(k[x_1, \ldots, x_n]/I\right)[f^{-1}] = k[x_0, x_1, \ldots, x_n]/(I, 1 - x_0 f).

Clearly Var(I,1x 0f)=Var(I, 1 - x_0 f) = \emptyset, and so by the weak Nullstellensatz the ideal (I,1x 0f)(I, 1 - x_0 f) is improper, i.e., the localization vanishes.

General abstract formulations

By the discussion above at Existence of points it follows that phrased in terms of the sheaf topos over the Zariski site, the weak Nullstellensatz says that “objects have global points”.

Motivated by this, in (Lawvere 07, def. 2) is a proposal for a general abstract formulation of what constitutes a Nullstellensatz, formalized in the context of cohesion.

A cohesive topos H\mathbf{H} over some base topos S\mathbf{S} is a relative topos such that the terminal geometric morphism extended to an adjoint quadruple (ΠDiscΓcoDisc)(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) with Disc,coDisc:SHDisc, coDisc \colon \mathbf{S}\hookrightarrow \mathbf{H} full and faithful and Π\Pi preserving finite products. The top adjoint triple here induces a canonical natural transformation

ptp:ΓΠ ptp \;\colon\; \Gamma \longrightarrow \Pi

which deserves to be called the points-to-pieces transformation.

Consider the condition that ptpptp is on every object XX of H\mathbf{H} an epimorphism. In terms of the geometric interpretation of cohesion this means that “all pieces of XX have at least one point”. (See at cohesive topos – Pieces have points) This statement is what in Lawvere 07, def. 2 c) is also referred to as Nullstellensatz. More comments on this are in (Lawvere 11). (In (Johnstone 11) this condition is called “punctual local connectedness”.)

For a detailed analysis of how this general abstract concept indeed captures the traditional meaning of Nullstellensatz, see (Lawvere 15, Menni), also (Tholen).

References

Traditional

The model theoretic perspective

  • M. Bunge, On the transfer of an abstract Nullstellensatz , Comm. Alg. 10 (1982) pp.1891-1906.

  • W. A. MacCaull, Hilbert’s Nullstellensatz revisited , JPAA 54 (1988) pp.289-297.

  • V. Weispfenning, Nullstellensätze - a model theoretic framework , Z. Math. Logik Grundl. Math. 23 (1977) pp. 539-545.

In terms of cohesion

Discussion of the Nullstellensatz in terms of cohesion:

Last revised on April 9, 2019 at 06:46:18. See the history of this page for a list of all contributions to it.