Definition of codimension of variety

Question

Let $X$ be a variety over field $k$.
A Weil divisor on $X$ is an integral linear combination of irreducible subvarieties of $X$ of codimension $1$.
So I want to know the definition of codimension of a subvariety.

Let $Y$ be a subvariety of $X$ , where subvariety means a closed subscheme of an open subscheme of $X$ to be a variety.

What is the definition of codimension of $Y$ in $X$ $??$

Answer 1

Just assume $Y$ is irreducible here.

Let's do the affine case first. If $S = \text{Spec}(A)$ is an irreducible affine variety (with $A$ a finitely-generated $k$-algebra) and $Z = V(\mathfrak p)$ is an irreducible closed subvariety with $\mathfrak p\subset A$ a prime ideal, then the codimension of $Z$ in $S$ is $\dim A - \dim A/\mathfrak p = \text{ht}(\mathfrak p)$. (This equality is a fact of integral $k$-algebras and other catenary rings.) In other words, it is the length $\ell$ of the longest chain of irreducible closed subvarieties $Z\subsetneq Z_1 \subsetneq \dots \subsetneq Z_\ell = S$.

Now suppose $Y$ is an irreducible closed subvariety of the open irreducible affine subvariety $U$ of $X$. Let $V\subset U$ be an open affine which intersects $Y$. Now $Y_V = Y\cap V$ is a closed irreducible subvariety of the irreducible affine variety $V$. The codimension of $Y$ in $X$ is the codimension of $Y_V$ in $V$. You can check that this is independent of the various choices.