Schemes in algebraic geometry I : the affine plane
I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today. My turn now. I recommend Igor Dolgacev’s lectures which is one of my favorite ways of speaking of schemes.
Interesting mathematics come up when algebraic varieties (things defined by several polynomial equations) are no longer defined as mere sets (sets of tuples of numbers satisfying the equation) but mope complex mathematical objects. Differential geometry, for example, gives the structure of a complex manifold to algebraic varieties in , which is still an efficient way of proving theorems. However, during the 20th century, a lot of mathematicians tried to develop a new structure which would avoid the use of analysis to concentrate on the algebraic aspects (I don’t know exaclty who, but expect Hilbert, Zariski, Chevalley, Grothendieck to have played a role). Grothendieck approach using category theory and functors of points is now widely used and is a very impressive way of tying together intuition, commutative algebra, and geometry.
There are many ways of reverse engineering Grothendieck’s definition of a scheme (see EGA1 if you want to know how this is related to Chevalley’s definition of a scheme). The first thing to say is probably what properties and notions are needed for schemes:
- there should be a line, a plane, and more generally affine spaces for any dimension n
- there should be a notion of subscheme (subvariety) defined by a set of equations (or an ideal of equations, to take into account possible algebraic combinations of equations)
- we should be able to define maps between schemes, for example to parameterize a scheme by another
- it would be nice to have a simple way of using schemes to recover the other standard structures on algebraic varieties
- it would be also nice to have a topology
- it would be very nice to have a notion of cohomology, but that’s a very long story. I hope Andreas from Motivic Stuff will tell about this
An important choice at the beginning is to allow working with rings instead of fields. This means the coordinates of our “points” are not always something you can divide by, as it is the case with rational, real or complex numbers. For example, points with integer coordinates have the property of having a modulo p reduction, which is often used in arithmetic.
Let’s begin with the affine plane: the traditional definition of the affine plane is the set of tuples of numbers. Keep this definition, but now we do not specify what sort of numbers they are. It was already traditional in the 19th century to freely switch between real and complex numbers when working on the real affine plane. We also need to define what are equations: in algebraic geometry, equations are polynomials, so there is a natural object : the set of polynomials with integral coefficients, which describe the equations we allow. With the choice of as coefficients, satisfying an equation has a sense for arbitrary pairs of coordinates in any ring (thus arbitrary points). A point gives a value for any equation : this value is an element of the ring R where a and b were picked, and this evaluation map is a ring morphism .
When , the ring morphism allows to recover the point: we just have to test whether or evaluates to zero in order to find the coordinates. Of course, this does not work for points with arbitrary real numbers as coordinates: for this would have to replace by , and we would say we work with the real affine plane, but now coordinates would only have a sense in rings where equations with real coefficients have a value: is such a ring, and rings where real numbers have a sense are called -algebras. So the general philosophy is the following: to find the location of a point corresponding to an evaluation map , look at the kernel of this map (the set of polynomials evaluating to zero).
The “functor of points” point of view would say pairs of coordinates in are in one-to-one correspondance with ring homomorphisms (tihs is a well-known adjunction of functors). So there are two equivalent ways of speaking of the abstract affine plane: either use the (intuitive) “pairs of coordinates” functor, which associates to each ring R the set , or use the (algebraic) ring . In a similar way, the affine plane for real numbers is either the functor mapping -algebras A to the set or the ring .
The title of the next post in this series will be prime spectra and generic points.