I have begun to study some algebraic geometry. I think I understand at an abstract, high level the purpose of generic points in scheme theory. However, my current knowledge is a superficial history with a serious shortage of illustrative examples. This post will describe a secondhand, concise summary of one motivation of introducing schemes and ask
- whether the history is basically correct,
- if it is correct, can you please give examples or exercises that illustrate some of the points made.
Here is the pseudohistory/rationalization for schemes.
When we do geometry with affine and projective varieties with their Zariski topology, sheaves are not strictly necessary at the introductory level, but they can be a useful tool, as in n many areas in geometry. For example, a continuous map of varieties should induce a natural transformation between the sheaves of regular functions. In particular there are many arguments that involve localization, and we must reason about what happens to a function on the space as we descend lower through the lattice of opens and study as a function as its domain converges to finer and finer approximations to a point. Conversely, given a germ in the stalk at a point we often have to reason about whether a property of the germ is already attained at the level of some finite open containing the point. Many arguments in early commutative algebra and algebraic geometry involved arguments of the form "Take a sufficiently generic point", i.e. a point which belongs to many open dense sets, and thus will escape annoyances or flaws in the argument that hold only on "small" closed sets, like a matrix being non-invertible on the closed set where its determinant is zero. The exact choice generic point was often irrelevant, and only introduced to argue a broader point about the space. Sometimes it was easy to prove that sufficiently generic points existed, but other times there were very many open sets being intersected and it was not at all clear that the intersection would be nonempty in general. Krull and Noether both proved major theorems guaranteeing the existence of "sufficiently general points" in varieties over various kinds of fields under different collections open sets, which allowed for more sophisticated forms of this reasoning. Such theorems are analogous to the Baire Category theorem in analysis.
However, there are sometimes families of opens which just don't have any intersection at all. These generate a descending filter in the lattice of opens whose intersection is empty, and yet it makes sense to take the "stalk" at this nonexistent point because we still have the possibility of taking the filtered colimit of the sheaf along the filter of opens. I conjecture that there were historical problems in geometry that naturally gave rise to these filters with no intersection and the bizarre "localization" of a sheaf at this filter with no underlying point; the addition of new points to the space for each irreducible closed set in the process of "soberification" allows us to make sense of these filters by framing them precisely as the filter of all opens containing a point, and the "stalk" as a true stalk of a sheaf at a point. Perhaps it is even strictly unnecessary for an argument about the variety for the generic point to exist, as one could develop a circumlocution which did not need the underlying point to actually exist, but it was just convenient and mentally relieving to talk about the stalk at a point. In some sense the filter "wants" to have a point in its intersection, and very well looks like it might, but due to the poor separation properties of the underlying space it doesn't - nevertheless we can take the stalk along it. Replacing a variety with the associated scheme guarantees that every completely prime filter in the lattice of opens is actually the completely prime filter of opens containing a certain point, which allows us much more facility in making arguments based on "generic points".
My question is: Is my conjecture basically correct? I understand there is an easy justification for adding the primes to the maximal ideals in a variety, i.e. the pullback of a maximal ideal along a ring homomorphism will be prime in general if we go outside the category of f.g. reduced $k$-algebras, but I would be pleased if there was any evidence for my conjecture because it would be a more geometric than algebraic justification for schemes and "schematic points."
In particular I would appreciate if you could supply specific examples or exercises for study that use the technique of localization at generic points to prove some result of independent geometric interest. I find the abstract justification above convincing but I will not really understand unless I see the technique in action. I am looking for examples both in the old sense, where one localized at generic points that 'really did' exist (i.e. maximal primes, closed points in the variety) and in the new sense where one is forced to localize at the prime ideals.
Thank you!
