The Universal Solution

This is the first in a series of posts taken from notes I took in a two-semester course on Algebraic Geometry taught by Arthur Ogus.  Mostly these are things that seemed to me particular to Ogus.

The first few weeks were about the following.  Any map of R-algebras $\ R[x_1,\dotsc, x_n] \to A$ is determined by an n-tuple with values in $\ A$ (the images of the $\ x_i$).  Given a set of equations $\ E \subset R[x_1,\dotsc, x_n]$, we can ask whether these equations are solved in $\ A$, i.e. if $\ E \mapsto 0$ in $\ A$.  When this is true, we in fact have a morphism $\ R' = R[x_1, \dotsc, x_n]/I \to A$.

The Ogus way…

You have a functor $\ \mathbb{A}^m$ sending an R-algebra $\ A$ to the set of m-tuples with values in $\ A$.  Given a set of equations $\ E \subset R[x_1,\dotsc, x_n]$ you have a subfunctor $\ Z_E$ sending $\ A$ to

$\ Z_E(A) = \{a. \in \mathbb{A}^n| f(a.) = 0, f \in E\}$.

In other words, $\ Z_E(A)$ consists of solutions to $\ E$ with values in $\ A$.  The claim is that there is a universal solution: the images of the $\ x_i$ in $\ R'$.  This mean for every solution $\ a. \in Z_E(A)$ there is a unique morphism $\ \theta R' \to A$ such that $\ a. = \theta(\bar{x}_i)$.  Actually, Ogus proved a more general statement about $\ R[X] := R\mathbb{N}^{(X)}$ in the context of free monoids and associated R-algebra they generated and appealing to a map $\ X \to R[X]$ with a universal property, etc. etc.

A corollary: TFAE

1. $\ E$ generates a proper ideal of $\ R[X]$ iff $\ R' \ne 0$
2. These exists $\ A$ such that $\ Z_E(A) \ne \emptyset$
3. There exists a field k such that $\ Z_E(k) \ne \emptyset$