(Regular functions)
Let \(X\) be a variety and \(U \subseteq X\) open. A function \(\phi: U \to k\) is regular if for every \(p \in U\) there is an open \(U_p \ni p\) and polynomials \(f, g\) with \(g\) nowhere zero on \(U_p\) and \(\restrictionof{\phi}{U_p} = f/g\). These form a sheaf of rings \(\OO_X\), with restriction of functions: regularity is a local condition, and a function that is zero on each set of an open cover is zero. The stalk \(\OO_{X,p}\) is the local ring at \(p\), with maximal ideal \(\mfm_p\) the regular functions vanishing at \(p\), and the stalk at the generic point is the function field \(k(X)\).
(Sections on an affine variety)
For \(X\) affine with coordinate ring \(A(X) = k[X] = k[x_1,\ldots,x_n]/I(X)\) and \(D(f) \da X \sm V(f)\), \[ \OO_X(D(f)) = A(X)\invert{f}, \qquad \OO_X(X) = A(X), \qquad \OO_{X,p} = A(X)_{\mfm_p}, \qquad k(X) = \operatorname{Frac} A(X) . \]
(Sections as an intersection in the function field)
Let \(X\) be a variety. For every nonempty open \(U \subseteq X\), restriction identifies \(\OO_X(U)\) with a subring of \(k(X)\), and if \(U = \bigcup_i U_i\) with each \(U_i\) nonempty and open, then \[ \OO_X(U) = \bigcap_i \OO_X(U_i) \quad \text{inside } k(X) . \] In particular, for an ideal \(\mfa \neq 0\) of \(k[x_1,\ldots,x_n]\), the open set \(\AA^n \sm V(\mfa) = \bigcup_{0 \neq f \in \mfa} D(f)\) has \[ \OO_{\AA^n}(\AA^n \sm V(\mfa)) = \bigcap_{0 \neq f \in \mfa} k[x_1,\ldots,x_n]\invert{f} \subseteq k(x_1,\ldots,x_n) . \] [@Har10a, §I.3]
(Rational and birational maps)
A rational map \(\varphi \colon X \dashrightarrow Y\) of varieties is an equivalence class of pairs \((U, \varphi_U)\) with \(U \subseteq X\) nonempty open and \(\varphi_U \colon U \to Y\) a morphism, where \((U, \varphi_U) \sim (V, \varphi_V)\) if \(\varphi_U\) and \(\varphi_V\) agree on \(U \cap V\). It is dominant if some, hence every, \(\varphi_U\) has dense image. A birational map is a rational map with a rational inverse, and \(X\) and \(Y\) are birational if one exists.
(Rational maps and function fields)
Over an algebraically closed field \(k\), sending \(X\) to \(k(X)\) gives an arrow-reversing equivalence between varieties with dominant rational maps and finitely generated field extensions of \(k\) with \(k\)-homomorphisms. In particular \(X\) and \(Y\) are birational if and only if \(k(X) \cong k(Y)\) over \(k\). [@Har10a]
Regularity requires a quotient presentation only locally, not globally. On \(\PP^n\) every regular function is locally a ratio of forms of equal degree although there are no nonconstant global regular functions; on \(X=V(xw-yz)\subseteq\AA^4\), the function \(x/y=z/w\) is regular on a union of two opens without being represented there by one global quotient.
On \(\Spec A\), the structure sheaf is defined in the same way: its sections over \(U\) are the functions \(s\colon U\to\coprod_{\mfp\in U}A_\mfp\) that are locally of the form \(a/f\) with \(a,f\in A\) [@Har10a, §II.2].
Dependencies
PR-7OT2ZThe ideal-variety correspondenceuses