Are Weil divisors defined over any algebraic scheme?

Question

I notice that:

In fulton's book "Intersection Theory", $r$-cycles are defined in any algebraic scheme (see page 10). In particular, Weil divisors (i.e. cycles of codimension 1) are well defined. But In Hartshorne's book "Algebraic Geometry" says that Weil divisors are defined only over some noetherian integral schemes (see page 129). In fact Hartshorne define Weil divisors only on a noetherian integral separated scheme which is regular in codimension 1 (see page 130).

Which is correct?

Answer 1

Let's examine the definitions first. Here's Fulton's definitions:

B.1.1. An algebraic scheme over a field $K$ is a scheme $X$, together with a morphism of finite type from $X$ to $\operatorname{Spec} K$.

B.1.2. A variety is a reduced and irreducible (integral) algebraic scheme. A subvariety $V$ of a scheme $X$ is a reduced and irreducible closed subscheme of $X$.

(p.10, section 1.3) Let $X$ be an algebraic scheme. A $k$-cycle on $X$ is a finite formal sum $$\sum n_i[V_i]$$ where the $V_i$ are $k$-dimensional subvarieties of $X$ and the $n_i$ are integers.

(p.29, section 2.1) Let $X$ be an $n$-dimensional variety. A Weil divisor on $X$ is an $(n-1)$ cycle on $X$.

And here's Hartshorne's:

Let $X$ be a noetherian integral separated scheme which is regular in codimension one. A prime divisor on $X$ is a closed integral subscheme of codimension one. A Weil divisor is an element of the free abelian group $\operatorname{Div} X$ generated by the prime divisors.

As you can see, the definitions are relatively close:

Trait	Fulton	Hartshorne
$X$ noetherian	Yes	Yes
$X$ finite type over a field	Yes	Not necessarily
$X$ integral	Yes	Yes
$X$ separated	Not necessarily	Yes
$X$ regular in codim 1	Not necessarily	Yes
Elements are formal finite sums of integral closed subschemes	Yes	Yes

The elements they have in common are some of the key features: working in a noetherian integral scheme and considering formal sums of integral closed subschemes.

If you compare the definitions to other sources like the Stacks Project or Vakil, you can see that these key features are basically present in everyone's definition (Stacks generalizes a little to the locally noetherian situation, but adds the requirement that the sums be locally finite). The general idea remains the same, but some of the implementation details are a little different - that doesn't make any one of the definitions correct or wrong, just slightly better suited for different work.