Hilbert's Nullstellensatz

The Nullstellensatz (which is German for "zeroes theorem") is one of the cornerstones of algebraic geometry. It gives us a correspondence between geometric objects and algebraic objects. This allows us to use algebra to study geometry.

Before we state the theorem some preliminaries. Fix k an algebraically closed field, for example one might take the complex numbers k=C. Now consider n-space consisting of all n-tuples

kⁿ={ (a₁,...,a₁) : each a_i is in k }.

This is a geometric object, an an n-tuple a=(a₁,...,a_n) in kⁿ is a point in n-space.

The corresponding algebraic object is the polynomial ring R=k[x₁,...,x_n]. For each point a in kⁿ consider the ideal of R defined by

m_a=(x₁-a₁)R + ... + (x_n-a_n)R

Lemma

1. Each m_a is a maximal ideal of k[x₁,...,x_n].
2. If a and b are distinct points then the ideals m_a and m_b are distinct

Proof: 1. Given any polynomial f=f(x)=f(x₁,...,x_n) we can evaluate it at a point a=(a₁,...,a_n) to form f(a)=f((a₁,...,a_n). Thus, we have a function e_a:R-->k defined by e_a(f)=f(a). Clearly this is a surjective ring homomorphism and its kernel contains m_a. By the first isomorphism theorem we see that the kernel is a maximal ideal. It follows from this that m_a is proper. We will show in a minute that it coincides with this kernel.

Next notice that there is an automorphism of rings of R defined by mapping each polynomial f(x₁,...,x_n) to f(x₁-a₁,...,x_n-a_n). This automorphism takes the ideal m₀ into m_a so WLOG we only need to prove the result in the case a=0=(0,...,0). Now consider some polynomial f=f(x₁,...,x_n). Suppose that f is not in m₀. f can be written as f=c + g, the sum of a constant term c in k plus a term g which is a sum of monomials all of which have total degree at least one. Thus g is in m₀. We cannot have c=0 for then f would be in m₀ after all. Thus c is nonzero and any ideal containing m₀ and f must also contain c and hence be all of R.

2. If f is in m_a then, from the above f(a)=0. Thus to show that m_a and m_a are distinct we just have to find f in m_a that doesn't vanish at b. Since a and b are distinct it must be that for some i we have a_i is different to b_i. Consider then the polynomial x_i-a_i which is in m_a. This polynomial takes the nonzero value b_i-a_i at the point b and so we see that it is not in m_b.

Hilbert's Nullstellensatz Every maximal ideal of k[x₁,...,x_n] has the form m_a for some point a in kⁿ.

Here is a proof of Hilbert's Nullstellensatz. This result has an immediate corollary which is also sometimes called the Nullstellensatz.

Corollary There is a bijection between the set of points of kⁿ and the set of maximal ideals of k[x₁,...,x_n]. If a is a point the corresponding maximal ideal is m_a.

This corollary opens up lots of interesting questions, such as, can we generalise it to apply not just to points but to appropriate higher dimensional subsets of n-space? For example, if we consider the line in k² consisting of all points (a,b) of the form 3a+2b=4 might this correspond to the ideal (not maximal this time) (3x+2y-4)k[x,y] in k[x,y]? The answer is yes, see the correspondence between closed sets for the Zariski topology and radical ideals in the polynomial ring.

..

I'd like to develop hamster bong's excellent writeup above more deeply within the context of algebraic geometry, by stating and proving a version that with some minimal effort follows from the above formulation in terms of maximal ideals of polynomial rings. First, then, some motivation.

Varieties and Ideals

For a ground field k, an affine algebraic variety is, roughly speaking, a set of points (essentially, N-tuples) Y ⊂ k^N (I'll denote this A^N from now on) such that there exists a finite collection of polynomials F₁...F_r drawn from the polynomial ring k[X₁,...,X_N] such that a point x=(x₁,...,x_N) is in Y if and only if F₁(x)=...=F_r(x)=0. This seems like a useful construct for a 'zero places theorem'!

In other words, we seek to describe geometric objects (regions in affine space) in terms of polynomials, to which we can then apply the full force of algebra. But an interesting problem (that I've touched on before) involves the opposite process- given a collection of polynomials, where, if anywhere, do they all vanish?

It turns out that a good way to approach this is in terms of ideals, since if a collection of points evaluate at zero on some finite set of polynomials, then they'll also give a zero for the most likely infinite assortment of ways of combining those polynomials via addition, multiplication from the ring, and so on. There are hence two processes at work:

Definition: Given an affine algebraic variety Y⊂A^N, the ideal I(Y) ⊂ k[X₁,...,X_N] is given by F∈I iff F(x)=0 ∀x∈Y.

Definition: Given an ideal I of k[X₁,...,X_N], the corresponding variety V(I) is the set {x∈A^N | F(x)=0 ∀F∈I}.
Note that I initially defined a variety in terms of a finite set of polynomials, to make the conversion from geometry to algebra more palatable. Fortunately, this is still the case here- whatever awkward and messily infinite ideal I you might offer, by Hilbert's basis theorem the polynomial ring is Noetherian and thus I will be finitely generated. That in turn ensures the existence of finitely many polynomials F₁,...,F_r such that I = <F₁...F_r> and thus V(I)= { x∈A^N | F₁(x)=...=F_r(x)=0}.

Why do we need a Nullstellensatz?

The above definitions look similar almost to the point of being circular. So, one might reasonably expect that I(V(J))=J or V(I(Y))=Y; that is, these processes are mutual inverses. Sadly, that isn't quite true. Considering the set-theoretic notion of an inverse image, this isn't too surprising- properties such as injectivity and surjectivity guide us as to when images and inverses behave nicely when layered. The following examples show that moving between ideals and varieties (formulae and points) we can gain objects (more formulae that evaluate to zero, more points that disappear on the set of polynomials).

Example 1: Y≠V(I(Y))

For simplicity, take the ground field to be C the complex numbers and restrict our attention to a single dimension (N=1). Then the integers Z are a perfectly good subset of C; so we'll let them be our variety Y.

So I(Y) is the set of polynomials that vanish for all integers- that is, every integer is a root of such polynomials. Sounds like a tall order? There's only one such 'polynomial', the zero polynomial. So I(Y)={0}.

You may have spotted the problem now. If not, consider V(I(Y)). This, by the above, is V({0})- the set of points in C such that, when evaluated at that point, the zero polynomial gives zero. That's all of them, i.e., V(I(Z))=C.

In general therefore Y≠V(I(Y)), although Y⊂V(I(Y)).

Example 2: J≠I(V(J))

Sticking with k=C and N=1, we consider the ideal J generated by the polynomial X². That is, J=<X²>. So V(J) is all the complex numbers c such that c²=0; there's only one of those, 0. So V(J)={0}.

Now we look at I(V(J))=I({0}). That's the collection of polynomials which evaluate to zero at zero. Certainly J is contained in that set; but there are more- any linear polynomial of the form aX will also satisfy this condition. So now I(V(J))=<X>≠<X²>.

Again we can only conclude J⊂I(V(J)), not equality.

These simple examples aren't quite pathological- there was no particular reason I needed the set {0} in each case, it's just the most striking case. Factors that do matter are those such as algebraic closure, nilpotent elements, and reducibility. So having hopefully convinced you of the need for a Nullstellensatz, I'd best supply one.

Hilbert's Nullstellensatz for varieties and ideals

If k is algebraically closed and I is an ideal of k[X₁,...,X_N] then:

• V(I)=∅ ⇒ I=k[X₁,...,X_N]
• I(V(I)) is the radical ideal of I, the set of all f in k[X₁,...,X_N] such that some power of f is in I.

Proof

For the first part we argue by contraposition- that any ideal I which isn't the whole ring generates a non-empty variety V(I). So let I be such an ideal. The polynomial ring k[X₁,...,X_N] is Noetherian, so there is some maximal ideal M containing I (possibly I itself, but we don't care). By hamster bong's formulation of the Nullstellensatz above, there is a point a=(a₁,...,a_N) from A^N such that M=<X₁-a₁,...X_N-a_N>, that is, M={f | f(a)=0}. So, a is in V(M). But any point which vanishes on a set M must vanish on any subset of M, such as I- that is, a is in V(I). Hence, V(I) is non-empty and the first part of the theorem holds.

Now for the second part, pick some I and ideal of k[X₁,...,X_N] and an f from I(V(I)) (forgive the notation!).

We construct a larger ring in N+1 variables, k[X₁,...,X_N,Y], and take the ideal J=<I, 1-Yf(x)>. Suppose there is some (a₁,...,a_N,b) in V(J). Then (a₁,...,a_N) is in V(I) and 1-bf(a₁,...,a_N)=0 by construction. Since f is from I(V(I)), it vanishes on V(I) and in particular at (a₁,...,a_N) so 1=0. This is clearly absurd in any ring, so there cannot be such a point (a₁,...,a_N,b). In other words, we have shown V(J)=∅ and thus, by the first part, J=k[X₁,...,X_N,Y].

In particular, therefore, 1 is in J. So we can chose some q_i from k[X₁,...,X_N] and q,p_i from k[X₁,...,X_N,Y] so that 1 = (Σp_iq_i)+q(1-Yf), where the q_i are generators for I.

We introduced Y as an arbitrary variable; now we fix Y to be 1/f(x) to obtain 1 = Σp_i(x,1/f(x))q_i(x)

Set r=max{deg_Y{p_i(x,Y)}} and multiply each side by f(x)^r to clear denominators: f(x)^r = Σf(x)^rp_i(x,1/f(x))q_i(x). Observe that the RHS is an element of I, so the LHS is. That is, f is in the radical of I.

Finally, suppose f is in the radical of I. Then f^r is in I and hence in I(V(I)). But a polynomial ring of a field is an integral domain, so if f^r(p)=0, then f(p)=0, i.e. f is in I(V(I)). Hence, I(V(I))= the radical of I and the proof is complete.