In this series of posts we will walk you through the main paradigm shifts that took place in algebraic geometry over the last century. The short story is : the use of (higher) categories led to (more) flexible geometric objects..

That is, if varieties are 0-categorical geometric objects, then schemes are 1-categorical and stacks 2-categorical objects. The longer story starts here.

Algebraic geometry addresses a very natural problem. Say, you have a (possibly infinite) system of polynomial equations in a number of complex variables

what can you say about its set of solutions? The solutions to just one of these equations form an hypersurface in n-dimensional affine space , so we want to describe the intersection of these (possibly infinite) hypersurfaces.

Linear algebra tells us the answer in the easy case when all the polynomials are linear forms,

and in this case we have to consider the intersections of the corresponding hyperplanes in . A quick recap of the main facts :

• (basis) : we can reduce to a finite set of linear forms. Take a basis for the subspace in the -dimensional space of all linear forms in n variables.
• (existence) : there is a solution to the linear system of equations iff the constant function . In fact, is the space of all linear forms vanishing on solutions.
• (size) : if solutions exist, they form an affine subspace of of dimension .

Perhaps surprisingly, David Hilbert and Emmy Noether
were able to extend these facts to arbitrary systems of polynomial equations!

Their main idea was to translate the problem into algebra, whence algebraic geometry. In the polynomial algebra we can consider the ideal generated by all the polynomials . Clearly, if in a point all the polynomials evaluate to zero, then so does every polynomial contained in the ideal . That is, all zero sets are in fact zero-sets of ideals in .

The Hilbert basis theorem asserts that the polynomial algebra is Noetherian, that is, every ideal is finitely generated. In particular, our ideal is of the form for a finite set of polynomials . Phrased differently, all zero sets are zero sets of finite systems of polynomial equations, similar to the situation of linear equations. The difference between the two is that the new polynomials cannot necessarily be chosen among the original poynomials .

An existence criterium for solutions is given by Hilbert’s weak Nullstellensatz. It states that there exist solutions to the system of polynomial equations whenever the associated ideal is proper, that is, does not contain the constant function . Compare this to the existence criterium for systems of linear equations.

The easy part of the proof is that whenever is a proper ideal, it is contained in a maximal ideal determining a quotient map to a field

If we could now show that , we would be done as then the point would be a zero for all polynomials in and hence a common zero to all polynomials .

However, showing that turns out to be somewhat harder than expected and is a consequence of the Noether Normalization lemma. This says that an affine commutative -algebra, that any quotient algebra of the form for an ideal , is always a finitely generated module over a polynomial subalgebra . The proof can be viewed as the polynomial version of solving a linear system by substitution of variables.

The number of remaining variables in the formulation of the Noether normalization lemma is then the dimension or the size of the zero set of , that is, the number of free parameters on which it depends. The ‘size’ refers here to the biggest chunk of the solution set. In general, a zero set can be decomposed in its irreducible components, that is zero-sets which cannot be written as the union of two strictly smaller sets of zeroes.

In our dictionary between the geometry of zero-sets and the algebra of ideals in the polynomial algebra, irreducible zero-sets correspond to prime ideals, that is ideals for which imples that or belongs to . The geometric picture of writing the zero-set of as the finite union of its irreducible components corresponds to the algebraic picture of determining the finite number of prime ideals, minimal over the ideal .

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Proofs of these results can be found in the book Undergraduate Algebraic Geometry by Miles Reid. More details (and, at times, more accurate proofs) are to be found in his other Student Textbook Undergraduate Commutative Algebra.

Summarizing, Hilbert and Noether gave polynomial versions of basic facts from linear algebra : we can reduce to finite systems of equations, we can calculate whether the system has a solution and can determine the size of the set of all solutions. Further, we have translated the geometric problems on zero-sets into algebraic problems concerning ideals of polynomial algebras.

However, this dictionary is not perfect as there may be distinct ideals in having the same set of zeroes. For example, it is clear that zeroes of a polynomial are also zeroes of any power . The full-blown version of the Hilbert Nullstellensatz assures us that this is the only problem that remains, that is, two ideals and have the same set of zeroes precisely when their radicals coincide . Recall that for some and that it is the intersection of all prime ideals containing , which fits nicely with the above decomposition of zero-sets as the union of their irreducible components.

So, taking as our geometric objects the zero sets in we are only considering radical ideals in . Next time we will extend our class of geometric objects beyond zero-sets so that we can attach distinct objects to distinct ideals, and we will explain why this may be of use.