Chapter I — The Language of Schemes

§7. Rational Maps

Translation status. Skeleton with definitions and principal statements; full proofs reference .

7.1. Rational maps and rational functions

(7.1.1) Two $S$-morphisms $f_1:U_1\to Y$, $f_2:U_2\to Y$ from dense open subsets $U_1,U_2\subset X$ are equivalent if they agree on a dense open of $U_1\cap U_2$. This is an equivalence relation among morphisms from dense opens; an equivalence class is a rational $S$-map $X⇢Y$.

Definition (7.1.2). A rational $S$-map from $X$ to $Y$ (or $S$-rational map) is an equivalence class as above. A rational function on $X$ is a rational map $X⇢\mathrm{Spec}(\mathbb{Z}[T])$ — equivalently, an element of the ring of rational functions of $X$ (defined below).

Proposition (7.1.5). When $X$ is irreducible with generic point $\eta$, every rational map $X⇢Y$ is determined by a morphism $\mathrm{Spec}(\kappa (\eta ))\to Y$ of $S$-preschemes (i.e., by a $\kappa (\eta )$-rational point of $Y$).

Proposition (7.1.7). For $X$ irreducible, rational $S$-maps $X⇢Y$ correspond bijectively to morphisms $\mathrm{Spec}({\mathcal{O}}_{X,\eta })\to Y$ of $S$-preschemes.

Corollary (7.1.8). For $X$ integral, the ring of rational functions on $X$ $R(X)$ is the field of fractions of ${\mathcal{O}}_{X,\eta }$ (the local ring at the generic point).

Corollary (7.1.9). For $X$ integral with function field $R(X)=K$, rational maps $X⇢Y$ correspond bijectively to $K$-rational points of $Y$ (when $Y\to S$ is appropriate).

Proposition (7.1.11). For $X$ having finitely many irreducible components $X_i$ with generic points ${\eta }_i$, the ring of rational functions is $R(X)={\prod }_i{\mathcal{O}}_{X,{\eta }_i}$ (product of local rings at generic points).

Corollaries (7.1.12)–(7.1.16). Functoriality and base-change properties of $R(X)$; rational maps compose when defined; the field/ring $R(X)$ is the ring of rational sections of ${\mathcal{O}}_X$ along the generic-fiber subprescheme.

7.2. Domain of definition of a rational map

Definition (7.2.1). The domain of definition of a rational map $f:X⇢Y$ is the union of all open $U\subset X$ on which $f$ is represented by an honest morphism $U\to Y$. A rational map is defined at $x\in X$ if $x$ is in its domain.

Proposition (7.2.2). The domain of definition is open.

Corollaries (7.2.3)–(7.2.7). Properties of domains of definition: stability under composition; restriction; behavior under base change.

(7.2.8) A rational map $f:X⇢\mathrm{Spec}(B)$ to an affine target is defined at $x$ iff each generator $b\in B$ "extends" to a section of ${\mathcal{O}}_X$ in a neighborhood of $x$.

Proposition (7.2.9). For $X$ integral and $Y$ separated, a rational map $f:X⇢Y$ is determined by its restriction to any nonempty open of the domain of definition.

7.3. Sheaf of rational functions

(7.3.1) Define the sheaf of rational functions ${\mathcal{K}}_X$ on $X$ as the sheaf associated with the presheaf U ↦ R(U) = (ring of rational functions on U).

Definition (7.3.2). A meromorphic function on $X$ is a section of ${\mathcal{K}}_X$. The sheaf ${\mathcal{K}}_X$ is an ${\mathcal{O}}_X$-module containing ${\mathcal{O}}_X$ as a subsheaf.

Proposition (7.3.3). When $X$ is integral, ${\mathcal{K}}_X$ is the constant sheaf with stalks $R(X)$.

Corollaries (7.3.4)–(7.3.6). Behavior of ${\mathcal{K}}_X$ under open immersions, sums, and reduction.

Proposition (7.3.7). For $X$ locally Noetherian, the total ring of fractions sheaf ${\mathcal{K}}_X$ is the sheafification of $U\mapsto S_U^{-1}\Gamma (U,{\mathcal{O}}_X)$ where S_U is the set of non-zero-divisors in $\Gamma (U,{\mathcal{O}}_X)$.

7.4. Torsion and torsion-free sheaves

(7.4.1) For an ${\mathcal{O}}_X$-module $\mathcal{F}$, the torsion subsheaf $t(\mathcal{F})\subset \mathcal{F}$ is the kernel of $\mathcal{F}\to \mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{K}}_X$. $\mathcal{F}$ is torsion-free if $t(\mathcal{F})=0$, torsion if $t(\mathcal{F})=\mathcal{F}$.

Proposition (7.4.2). Torsion is functorial; $t(\mathcal{F})$ is the largest torsion subsheaf.

(7.4.2) For $\mathcal{F}$ torsion-free of finite rank, the rank is the rank of $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{K}}_X$ as an ${\mathcal{K}}_X$-module.

Corollary (7.4.3). On an integral prescheme, the rank of a torsion-free coherent sheaf is well-defined (constant).

Proposition (7.4.5)–(7.4.6). Torsion behavior under direct sums, tensor products, and pullback.