Definition of Sheaf of Rational Functions on Integral Scheme?

Question

Given an integral (i.e. irreducible and reduced) scheme $X$ of finite type over an algebraically closed field $k$. We can consider $X$ as a variety. On a cover $\{U_i\}$ of affine varieties, $U_i \simeq V(I_i)$ for $I_i \subset k[x_1,\dots ,x_n]$, the field of rational functions $K(U_i)$ is given as the quotient field of the coordinate ring $k[x_1,\dots,x_n]/I_i$. We may consider the sheaf of rational functions on $U_i$, which just gives the field of rational functions on every open. This should define a sheaf on all of $X$.

Given an arbitrary scheme $X$. In the section on Cartier Divisors in Hartshorne, there is a very complicated definition for the sheaf of rational functions on $X$ given. The text states: "On an arbitrary scheme, the sheaf $\mathcal{K}$ replaces the concept of function field of an integral scheme."

What is a good definition for the sheaf of rational functions on an arbitrary integral scheme? That is, a definition not as specific as the first one, but not as general as the second one.

The question arose when studying the chapter on line bundles on curves in Gathmann's lecture notes. There the sheaf of rational functions on a curve $X$ is used without prior introduction. Does the word curve imply that the scheme is a variety? I always thought of a curve as any one-dimensional scheme.

Answer 1

If $X$ is an integral scheme, then $X$ has a unique generic point - the point associated to the zero ideal in any open affine neighbourhood. We can define the function field of $X$ to be the local ring at the generic point.

The sheaf of rational functions $\mathcal K_X$ on $X$ is a constant sheaf. The group of sections over any non-empty open set $U$ is the function field of $X$.

For practical purposes, this information is of little use unless we are able to identify $\mathcal O_X$ as a subsheaf of $\mathcal K_X$. We can do this as follows. First, we cover $X$ with a collection of open affines. In any open affine subset ${\rm Spec } A \subset X$, the generic point is associated to the zero ideal $(0) \subset {\rm Spec } A$, and the function field is the localisation $A_{(0)}$. Given a point in ${\rm Spec}A$, represented by a prime ideal $\mathfrak{p} \subset A$, there is a natural inclusion morphism $i_{\mathfrak p} : A_{\mathfrak p} \to A_{(0)}$. For any open set $U \subset {\rm Spec}A$, we can define $\mathcal O_X (U) \subset \mathcal K_X(U)$ as the subring of $A_{(0)}$ consisting of all elements in $A_{(0)}$ that are in the image of $i_{\mathfrak p}$ for all $\mathfrak p \subset U$. If $U$ intersects more than one open affine in our affine cover, then $\mathcal O_X(U)$ is the subring of the function field consisting of elements that obey this criterion in each open affine that $U$ intersects.