On algebraic geometry
The name of the game is spot the error ! I'm tired, work doesn't, and all is jellyfish. As my son's finally asleep i'm ready to make mistakes. Let's do that ambitionally and see what i can absorb / convey about modern algebraic geometry. In an hour. I do find motives pretty sexy because i have no clue what they are. But i fear that would be aiming too high for now. So this is going to be a series of posts.
As you know, a circle can be described by the equation x^2 + y^2 = r^2 with x and y variables and r being a constant (also called radius in this case). So the circle is all points in the x,y space where the term x^2 + y^2 - r^2 is zero (the vanishing locus). That's an example from classic algebraic geometry. I trust modern math distinguishes itself by vastly generalizing classical ideas. How can we do that?
For one, i didn't specify what kind of objects x, y and r are. Classically we'd first use the real numbers. Another popular choice is the complex numbers C, as they offer quite a lot of structures, some of which are as tame as it can get. On the other hand we have the more broken structures like Galois extensions. Those are for systematic exploration for what can be done between the rationals and the reals (rather: algebraic numbers). The above are all considered classical and will serve as examples to motivate modern notions.
Suppose we're stuck with the rational numbers Q (fractions). We'll find that the polynomial x^2 - 2 has no zeros (solutions, roots). One might suggest sqrt(2), but that's not a fraction (it's not in Q).
The field extension Q(sqrt(2)) adds sqrt(2) as an element to the rational numbers. A single new element won't do because we'll want arithmetic operations in Q(sqrt(2)) to always yield elements from that set (field). So we also have to add 2*sqrt(2), 1/3 sqrt(2), 5/3 * sqrt(2) and so on (the closure). And voila, here's a subset of R containing all points expressible by fractions and sqrt(2).
Wait, a subset of R? Some nomenclature: They call Q(sqrt(2)) a field extension . Q(sqrt(2))/Q should mean that each of Q(sqrt(2)) can be constructed as the zero of a polynomial over Q. That's an algebraic extension . Algebraic basically means polynomial. A Galois extension is one that is interesting and simple:
For example, when extending Q with the cube root of 2, it's closure contains complex roots which aren't on the real line R. Also, since there's only one real root, we don't get a splitting field as required (see the wikipedia page for the definition). So the extension R/Q(2^(1/3)) is not considered a Galois extension, but C/Q(2^(1/3)) is!
You see, the Galois extension is all about polynomials, as is our circle example. Polynomials are going to be a central object for our studies. They're also the silver bullet to enter a whole universe of new functions. Remember, we're about to generalize as hell...
Here's the plan: What considerations are important in classic algebraic geometry (may i say "applications")? Such that we can ask: What are the basic requirements (axioms) so we can make sense of these problems? And to have a chance to attack them. What useful concepts emerge?
I'll be trying to get at schemes , sheaves , stacks , and much of what follows in future posts. That is, i'll be trying to find applications which, when generalized, will gently lead to the concept.
Looking at the related wikipedia entries yielded a positive surprise. I tried to approach the topic some years ago (whilst ignoring anything french and behind paywalls) and had to give up because the situation was much too barren.
Good intro to the classical part but does give connections to the modern approaches in part 4:
Please note the inclusions near the end of the introduction, before the toc:
One interesting resource (there are many) is Lieven Lebruyn's Neverendingbooks . Here's what got me to add an image of David Mumford: