dimension of affine variety

Definition

If $X$ is a topological space, we define the dimension of $X$ (denoted $\dim X$ ) to be the supremum of all integers $n$ such that there exists a chain $Z_0\subset Z_1\subset \dots \subset Z_n$ of distinct irreducible closed subsets of $X$.

We define the dimension of an affine or quasi-affine variety to be its dimension as a topological space.

Example. The dimension of ${\mathbf{A}}^1$ is $1$. Indeed, the only irreducible closed subsets of ${\mathbf{A}}^1$ are the whole space and single points.

Definition

In a ring $A$, the height of a prime ideal $\mathfrak{p}$ is the supremum of all integers $n$ such that there exists a chain ${\mathfrak{p}}_0\subset {\mathfrak{p}}_1\subset \dots \subset {\mathfrak{p}}_n=\mathfrak{p}$ of distinct prime ideals. We define the dimension (or Krull dimension) of $A$ to be the supremum of the heights of all prime ideals.

Proposition

If $Y$ is an affine algebraic set, then the dimension of $Y$ is equal to the dimension of its affine coordinate ring $A(Y)$.

\begin{proof} If $Y$ is an affine algebraic set in ${\mathbf{A}}^n$, then the closed irreducible subsets of $Y$ correspond to prime ideals of $A=k\left[x_1,\dots ,x_n\right]$ containing $I(Y)$. These in turn correspond to prime ideals of $A(Y)$. Hence $\dim Y$ is the length of the longest chain of prime ideals in $A(Y)$, which is its dimension.