## Tropical Geometry, part 3.

So we have discussed the following idea. Given a geometric object $X$, we can study it by studying the functions that are defined on $X$, which we will write as $\mathcal{O}_X$ (I’m not actually sure what the $\mathcal{O}$ stands for, but this is in a certain I’m-slightly-lying-to-you way the standard way of writing this).

Now, functions are objects that we can add together ($f(x) + g(x)$), we can multiply them together ($f(x)g(x)$), and perhaps if we feel like it, we can also scale them by multiplying them by a real (or even complex) number ($\lambda \cdot f(x)$). They are, to use mathematical terminology, a ring or an algebra. So restated, as above, we can associate to every geometric thingy $X$ its associated ring/algebra $\mathcal{O}_X$.

One of the great shifts in the 20th century is that you can actually do the reverse to this as well. That is, to every ring $R$, there is a canonically associated geometric object (a scheme) which we denote as $\mathrm{Spec}\, R$. Moreover, these associations are inverse to each other. That is, we have (in a certain sense)

$\mathrm{Spec}\, \mathcal{O}_X = X$

and

$\mathcal{O}_{\mathrm{Spec}\, R} = R$.

(I should really stress again that I am slightly lying to you here. There is a context in which this is 100% true, but there are some subtleties to what I am saying. Caveat lector.)

Let’s go over a few examples just to ground ourself here. The simplest non-trivial example in some sense is the following. If we write the ring of polynomials in one variable as $\mathbb{C}[x] = \{f(x) = a_0 + a_1x + \cdots + a_nx^n \mid a_i \in \mathbb{C}\}$ then this is certainly a ring (in fact, as algebra, because you can multiply polynomials by real or complex numbers) since you can add and multiply polynomials together. So what is the corresponding geometric object? It is just the complex plane! The rough idea is that a polynomial is determined by its roots, and so we identify a polynomial $f(x)$ with its zero set. That is,

$f(x) \leftrightarrow \{z \in \mathbb{C} \mid f(z) = 0\}$

For another similar example, if let consider polynomials in two variables (for example, $f(x, y) = 4y^2 - 2xy + 11xy^2 - \pi x$) and let the ring/algebra of all of these be written as $\mathbb{C}[x,y]$, then we have that

$\mathrm{Spec}\, \mathbb{C}[x,y] = \mathbb{C}^2$

and you may be able to guess how this generalizes.

Finally, to tie ourselves into the previous post, consider the following example. Suppose that we define the ring $R$ to be the collection of all two-variable polynomials $f(x, y)$ where we identify any two of them if their difference is a multiple of $h(x, y) = x^2 + y^2 - 1$. You can check that this makes sense as a definition, but given that, then we have that $\mathrm{Spec}\, R$ iscaveat lector, again  the circle!

So the tl;dr version of this post: up to some finicky details that can be dealt with, algebraic things like rings and algebras are the same as geometric things. This is a powerful, powerful tool.