My Brush With Algebraic Geometry

Sometime back in the mid-Ougties I decided that I would try to learn some Algebraic Geometry.  I did what anyone would do: I bought Hartshorne's Algebraic Geometry and Eisenbud's Commutative Algebra from Amazon and started reading. Some many months later, when I realised I had no idea what "divisors" really were, I ran across a simple explanation in Shaferavtich's book and bought that. Apparently many people have problems learning from Hartshorne's comprehensive but very abstract book. (A divisor, by the way, is a generalisation to higher dimensions of the idea of the roots of a polynomial.)

Modern algebraic geometry is not about conics and envelopes of curves and anything else you may recognise from A-levels. In the same way that Georg Cantor had to develop the theory of infinite sets to cope with zeros of Fourier series, Alexander Grothendieck developed vast swathes of category theory to cope with problems in algebraic geometry and frame it in a general setting. Of this approach, its high priest, Robin Hartshorne, says: "the person who works with schemes has to carry a considerable load of technical baggage... sheaves, abelian categories, cohomology, spectral sequences and so forth". Not to mention a hefty lump of commutative algebra. 

Why algebraic geometry? Isn't algebra one thing and geometry another. Well, here's how it works.  Descartes taught us how to take a curve and describe it with an equation such as y = 3x+4. This can be written in the form f(x,y) = 0 (y - 3x+4 = 0 in the example). In other words, a curve or a finite set of points is the solution to an equation: it's the set of points (x,y) such that f(x,y) = 0. Now let's take the ring of real-valued functions of two variables with real coefficients, R(x,y), and consider the ideal (f) generated by f(x,y). Factor R(x,y) by the ideal (f) to get the quotient ring R(x,y)/(f). Each one of these steps generates an unique object, so we can take a curve and associate it with a quotient ring. We can also associate the curve with a field of rational functions by looking at all the real-valued rational functions defined everywhere on the curve: this is the function field. It turns out that (roughly) the curves are equivalent if their function fields are isomorphic. So we can learn a lot about curves from these algebraic objects. But notice how quickly it went from something you did in school to something you can do an entire Maths degree and still avoid (commutative algebra).

The subject is cluttered with a lot of what's known in the trade as "machinery", which is one of those terms that mathematicians get and is difficult to explain. "Machinery" is needed for a number of things: one is using equivalence classes to mod out issues with specific co-ordinate representations of spaces or functions; another is to ensure that local representations of a part of a space or function are "glued together" in a coherent way; and yet another is representing a space or object by a set of ideals or filters.  The first piece of real machinery most mathematics undergraduates come across is a differentiable manifold or a fibre bundle.

The abstract spaces of algebraic geometry, called schemes, use all the machinery ever devised and then some more. A scheme is a locally ringed space such that every point has an open neighbourhood that is isomorphic to the spectrum of some ring. A locally ringed space is a topological space with a gadget called a sheaf that assigns a ring of functions (think of them as polynomials) to each open set. The spectrum of a ring is its set of prime ideals with the Zariski topology and a sheaf of rings defined in a manner so horrible not even Shaferavitch could simplify what's on page 70 of Hartshorne. When I first tried getting to grips with the definition I thought of schemes as like manifolds made up of locally patched Stone spaces (very "sort of").  As an example, the spectrum of the integers consists of: a) ideals generated by the prime numbers, b) co-finite subsets of the (ideals generated by the) primes and c) a set of polynomials which don't have a particular prime as a zero.

It's very good for stretching the brain. But in the end, I'm a logician and just not that interested in curves and numbers and other such things. I found Goldblatt's Topoi: Categorical Analysis of Logic recently and am reading that almost for light relief. It uses categories called topoi to construct alternative models of set theory. Toposes were invented by Grothendieck to solve problems arising in algebraic geometry. Hence my interest: toposes are used in mathematical logic and set theory to do all sorts of clever things, and second because algebraic geometry doesn't fit with the conventional foundational / axiomatic approach to mathematics. I will go on dipping into the books, but there are some things you really do need to study full-time, and algebraic geometry is one of them.