Affine variety

Last updated January 27, 2022

Definition. A algebraic subset $A\subset\mathbb{A}^n$ is called an affine variety if it is irreducible.

Definition. The dimension of an affine variety $X$ is defined as $$ \dim(X)=\max{k\in\mathbb{N}\mid X\supsetneq X_1\supsetneq \cdots\supsetneq X_k} $$ where each $X_i$ is a nonempty affine variety.

Quasi-varieties

Definition. An open subset of an affine variety (wrt to the subspace topology induced from the Zariski topology) is called a quasi-affine variety.

Note that every affine variety is a quasi-affine variety, since the whole space is open (and closed).

Normal

For $X$ an affine variety, we say that it is normal if $K[X]$ is integrally closed in its field of fractions.

Properties

1. If $X$ is an affine variety, then $I(X)$ is prime.
2. Let $X$, $Y$ be affine varieties and $A(X)$, $A(Y)$ their coordinate rings. Then $X\cong Y$ iff $A(X)\cong A(Y)$.
   1. Hence $A(-)$ is a (contravariant) functor witnessing that the category of affine varieties and the category of f.g. integral domains over $k$ are equivalent.
3. Not every quasi-variety is a variety, e.g. $\mathbb{A}^2\setminus (0,0)$.