Hilbert’s Nullstellensatz

Hilbert’s Nullstellensatz is one of the cornerstones of Algebraic Geometry. Due to its widespread use, many versions of the statement float about, causing confusion and frustration. Instead of getting bogged down by the exact statement, we will look at the idea and spirit of the statement.

Mathematics makes great strides forward when connections between two seemingly unrelated concepts are made, especially if one of them is theoretic and the other practical. (To mathematicians “practical” often means “easy-to-do-calculations-with”.)

Hilbert’s Nullstellensatz is one such connection, linking the worlds of algebra and geometry. The setting in which this bridge exists, is when working with an algebraically closed field K and examining polynomial rings K[x1,x2,…,xn].

Much like in high school, we are interested in finding the zeros of functions, i.e. solutions to the equation f(x)=0, where f is an element of K[x1,x2,…,xn].
But we needn’t focus on the solutions to just one equation. We could look for the solutions to many equations, i.e. find all the solutions to fi(x)=0, where fi lie in K[x1,x2,…,xn].
The set of solutions to a system of polynomial equations was originally called an algebraic variety. Due to conventions, it is now known as an algebraic set, with the word algebraic variety being reserved for irreducible algebraic sets.

Despite the name change, we still use V to denote the following function, which sends ideals of K[x1,x2,…,xn] to algebraic sets:

Function V copy

Something that mathematicians are fond of is going backwards once they’ve reached a place. In particular, they are curious about inverse functions.

The great thing is, there is a simple function which does go the other way! It takes any subset of Kn (e.g. algebraic sets), sends it to an ideal of K[x1,x2,…,xn] and is therefore known as I.

Function I copy

With these definitions in mind, one can succinctly state Hilbert’s Nullstellensatz:

Hilberts Nullstellensatz copy

Note that equality statements such as V(I(J))=r(J) are proven by showing that both sides are subsets of the other. Typically, one direction is easier to prove than the other. The same is true here.

Hilbert Partial Proof copy

The other direction requires more substantial work and will be omitted. A plethora of proofs can be found via Google.

Knowing that I(V(r(J)))=r(J) for all ideals J of K[x1,x2,…,xn], we might wonder what V(I(S)) is for any subset SKn. The answer is V(I(S))=cl(S), i.e. the closure of S with respect to the Zariski topology. Since algebraic sets are closed, V(I(S))=S for all algebraic sets S.

Having defined two functions, I and V, which are almost mutual inverses, we try to adjust their domains, in the attempt to find a proper one-to-one correspondence.

Noting that r(r(J))=r(J), we deduce that I(V(r(J)))=r(J). This piques our interest and we restrict V to the set of radical ideals of K[x1,x2,…,xn], i.e. V:{radical ideals of K[x1,x2,…,xn]} → {algebraic sets of Kn}.

What about I? At this point, I will admit to dishonesty. When defining I, we could’ve noted that r(I(S))=I(S) for all SKn, i.e. I(S) is always a radical ideal. Restricting I to the algebraic sets of Kn won’t change the fact that I maps into the set of radical ideals of K[x1,x2,…,xn], i.e. I:{algebraic sets of Kn} → {radical ideals of K[x1,x2,…,xn]}.

 

Quick Summary:Hilberts Nullstellensatz Diagram copy

  1. V:{radical ideals of K[x1,x2,…,xn]} → {algebraic sets of Kn}
  2. I:{algebraic sets of Kn} → {radical ideals of K[x1,x2,…,xn]}
  3. I(V(r(J)))=r(J) for all ideals J of K[x1,x2,…,xn]
  4. V(I(S))=S for all algebraic sets S of Kn

Conclusion: There is a one-to-one correspondence between the radical ideals of K[x1,x2,…,xn] and the algebraic sets of Kn, induced by the functions I and V.

This establishes a link between radical ideals (algebraic objects which aren’t too difficult to manipulate) and algebraic sets (geometric objects, despite their name, which can be tricky to work with). A bridge has been built!

 

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Leave a Reply

Your email address will not be published. Required fields are marked *