Act I: Where polynomials take over
It is a large theme in commutative algebra that the only thing standing in the way of two rings is what polynomials in these rings look like. Think about this briefly: the only difference between and is that, in the latter ring, the polynomial is always the zero function, while in the former it is never the zero function.
This idea motivates a lot of the definitions in ring theory, usually phrased in language like (though often not identical to) “a ring is said to have property if the roots of polynomials with property always are in .” Fine examples of this include:
- The definition of an integral domain: we say that is a domain if, for all nonzero, the function has no roots in ;
- The definition of a field: we say that is a field if, whenever is nonzero, the function has a root in ;
- The definition of being algebraically closed: we say that a field is algebraically closed if every polynomial in has a root in ;
- The definition of a separable field extension: we say an extension of fields is separable if every element of has a separable minimal polynomial in ;
- The definition of a normal field extension: we say an extension of fields is normal if is the splitting field of a family of polynomials in ;
- The definition of an integrally closed domain: we say that an integral domain with field of fractions is integrally closed if, any root in of a monic polynomial in is actually an element of .
And so on. Roots of polynomials say an enormous amount about the rings they take their coefficients in! Hence they are more than sensible things to study — they are, in a very real sense, the only thing to study.
Act II: Where geometry arises
So we have now become interested in sets of the form
A natural generalization of this is to allow polynomials in several variables, so that we start looking at sets of the form
Sets of this form are called affine varieties, and they are the first step in the study of classical algebraic geometry. Rather conveniently, affine varieties happen to satisfy the axioms of closed sets for a topology: the empty set and the whole space are both affine varieties, and the collection of affine varieties is closed under arbitrary intersection and finite union. This observation defines the (classical) Zariski topology on .
Act III: A formulation using ideals
Classically it was entirely common to work with , which has the desirable property that (since is algebraically closed) affine varieties tend to be non-empty (they are only empty if contains a constant, non-zero function).
Working with algebraically closed fields one is eventually lead to Hilbert’s Nullstellensatz, which one can use to formulate a theory of varieties entirely in terms of maximal ideals in . In this formulation, one defines
and then one creates a topology — still called the Zariski topology — on where a set is closed if it is of the form
When is algebraically closed, this topology on is homeomorphic to the classical Zariski topology on by the map (this is a consequence of the Nullstellensatz). Suddenly we have a formulation of the Zariski topology entirely in the language of ideals.
Act IV: Functoriality leads to
There are some obvious ways to generalize the work in Act III. First, there’s no reason in particular to only look at polynomial rings like ; we may as well consider arbitrary rings and look at their , which we can define and topologize in the same way.
Once we’ve done this, we have a mapping, where rings give us topological spaces. In the modern world it is natural to ask this mapping to be functorial — that is, if rings are going to give us topological spaces, we may as well ask that ring homomorphisms give us continuous maps.
Let’s set aside for a moment and simply consider what this functor might look like. Our experience with suggests that it is profitable to have a topological space whose points are ideals of , so we’ll stick with this idea, trying to dial it in to something more precise.
First we need to ask ourselves: should our functor be co- or contravariant? As it happens, if we want our topological spaces to have points corresponding to ideals, then our hand is forced: we need only look at the initial and terminal objects in Ring and Top.
- The terminal object in Ring is the zero ring, which has no proper ideals. It should therefore correspond to the empty topological space, which is the initial object in Top.
- If is a field, then has only one ideal (the zero ideal), hence should be sent to a topological space with only one point (the final object in Top).
Thus the map needs to correspond to the map , which is a contravariant relationship. Thus our functor is going to be contravariant.
Luckily, there is a good, contravariant way for ring homomorphisms to move around ideals: if is a ring homomorphism, and is an ideal, then is also an ideal (often called the contraction of ).
With this in mind we now return to . If is a ring homomorphism, it is sadly the case that the contraction of a maximal ideal in is not likely to be a maximal ideal in (that is, the map is not likely to be a function ). The standard example of this is to look at the inclusion map , noting that the unique maximal ideal of does not pull back to a maximal ideal in .
It is true, however, that if is a maximal ideal, then will be a prime ideal. In fact, this is true even if is just a mere prime ideal itself (not necessarily maximal). In particular, if we allow ourselves to expand a little, generalizing to the set
then contraction does induce a function . Furthermore, if we endow with a topology in exactly the same way we did with , then contraction will be continuous. We’ve thus developed a contravariant functor from Ring to Top!
Act V: Curtains, and on to schemes
And so the journey to schemes begins. I won’t define schemes here; my point was to motivate, not elucidate. I’ll just say what’s missing.
In the classical theory of varieties, one is lead to consider functions from a variety in back down to . There’s a reasonable definition for what it means for a function to be “well behaved” enough to be worth looking at — continuity is a part of it, but obviously it wouldn’t be algebra unless there was an algebraic condition as well.
This definition, after a good deal of modernization, lends itself quite well to generalization. One defines what the class of “good functions” on should be, in such a way that both the topology of and the algebra of are incorporated, then creates a new structure on which carries around this data. This combination — and the extra structure that describes the “good functions” — are what we call an affine scheme. A scheme is a topological space (again with some structure that describes the “good functions”) which is locally isomorphic to affine schemes.
The handwaving above suggests why it is so time-consuming to define schemes: not only do we need to define , but we also need to introduce the extra structure for tracking the “good functions,” as well as a characterization of these functions (so that the structure can track it!)
But once one has managed through this process (experience suggests this is where most would-be algebraic geometers give up and decide to study something else), one is left with a powerful theory indeed — in no small part due to the functor from Act IV.