This is a short introduction to Scheme Theory, as modeled on the article by Brian Lawrence.
A variety here is a zero set that can be covered by a finite number of affine varieties. Hence, a morphism between varieties can be considered to be a bunch of affine morphisms, as long as they agree on the intersections.
We need a shift in perspective. What this means is that we need to start thinking about the coordinate ring rather than the points themselves.
Now let us think about the following example: the coordinate ring of in is . However, the coordinate ring of is also ; it is not . The reasons for this can be worked out easily. Hence, the variety in this case is not accurately recovered from the coordinate ring. We started off with the variety , and got back . We need a new concept, which would allow us to accurately get back the variety from the coordinate ring- something that would allow nilpotents.
An affine scheme, written as , is the data of a ring . A morphism of affine schemes , is a morphism of rings . An affine scheme over a field is a scheme where is equipped with a -Algebra structure.
Why are morphisms defined backwards here? In other words, why is defined as ? This is because are the coordinate rings. Let be the variety corresponding to the coordinate ring . Then a map defines a map , and vice-versa. Maybe is a formal representation of . It is at least easy to remember which way the arrow goes this way.
How do we recover points from coordinate rings? Hilbert’s Nullstellensatz tells us that we can recover them using maximal ideals. Hence, our aim right now is to take an affine morphism, and construct a morphism between varieties. Hence, if the affine morphism is , we want to construct a map .
Given a ring homomorphism , for any prime ideal , is also prime. This is an elementary exercise in ring theory. It is however, not true in general that the inverse image of a maximal ideal is also maximal. For example, consider the map defined by inclusion. Then the only maximal ideal of is , the inverse of which is also just . It is easy to see that is not a maximal ideal in . For instance, , and for any .
We define the points of the affine scheme to be prime ideals. Why? Let us work this through. We have a scheme morphsim , where both and are coordinate rings. Now let us take a prime ideal in . From the discussion above, we know that is a prime ideal in . Hence, if prime ideals were points, we have taken a point in , and mapped it to . In a way, we have constructed a map from to .
However, this is a little weird. Points correspond to maximal ideals, and not prime ideals. All maximal ideals are prime, but the converse is not true. Do we really have a map from ? No. At least not in the traditional sense. What we have is a map from some “stuff” in , which includes points, to “stuff” in , which too includes points (possibly not all). Hence, something that’s not a point in may map to a point in , and a point in may map to something that is not a point in . We’re gonna call this “stuff” generic points. Hence, generic points in go to generic points in . This is a classic example of formulating new definitions to suit our world-view.
Now that we have the concept of “generic” points, we also need a name for “actual points”. This name is “classical points”. Hence, we’ll refer to maximal ideals in as classical points.
So what exactly is a scheme? A scheme is a coordinate ring, whose prime ideals are its points. Simple. It generalizes the notion of a variety. How? A variety has a set of points and an associated coordinate ring. A scheme has a larger set of points, and an associated coordinate ring. Hence the generalization is in the set of points; at least in this instance. Also, as discussed before, although and correspond to different coordinate rings but the same variety, they correspond to different schemes. Why? If a scheme was the data of its “points” (read generic points), then the points of are different from those of (the cosets look different, for starters). Hence, we now allow for distinguishing between multiplicities.