On a projective variety over an algebraically closed field \(k\), the ring of global regular functions is \(k\); the sheaf \(\OO_X\) records the regular functions on every open subset.
The sheaf of regular functions on a variety (Tag D-VARREG)
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].
\(A(X)\) consists of the functions regular on all of \(X\), \(\OO_{X,p}\) of germs of functions regular near \(p\), and \(k(X)\) of rational functions regular on some nonempty open set.
Affine, quasi-affine, projective, and quasi-projective varieties
Affine, quasi-affine, projective, and quasi-projective varieties; the affine charts of $\PP^n$ (Tag D-VARQAP)
An affine variety is an irreducible closed subset of \(\AA^n\); a quasi-affine variety is an open subset of one. A projective variety is an irreducible closed subset of \(\PP^n\); a quasi-projective variety is an open subset of one. Affine, quasi-affine, and projective varieties are quasi-projective.
(Standard charts)
\(U_i \da \ts{x_i \neq 0} \subseteq \PP^n\) is open, the map \[ \phi_i: U_i \to \AA^n, \qquad \tv{x_0 : \cdots : x_n} \mapsto \qty{ x_0/x_i, \ldots, \widehat{x_i/x_i}, \ldots, x_n/x_i } \] is an isomorphism of varieties, and \(\PP^n = \union_{i=0}^n U_i\).
For \(Y \subseteq \PP^n\) closed, \(\phi_i(Y\cap U_i)\) is the affine variety cut out by the dehomogenizations \(f(x_0,\ldots,1,\ldots,x_n)\), with \(1\) in position \(i\), for \(f \in I(Y)\). For \(Z\subseteq\AA^n\cong U_0\) closed, the projective closure of \(Z\) is cut out by the homogenizations of all elements of \(I(Z)\).
Homogenizing only a generating set can give a larger set: for the twisted cubic \(Z=V(y-x^2,z-x^3)\), the homogenizations \(yw-x^2\) and \(zw^2-x^3\) both vanish on the line \(V(w,x)\) at infinity, which is not contained in the projective closure of \(Z\).
\(\AA^2 \sm \ts{0}\) is quasi-affine and not affine: a function regular off a point of \(\AA^2\) extends to \(\AA^2\), so its ring of global regular functions is \(k[x,y]\), and the inclusion \(\AA^2\sm\ts0\to\AA^2=\Spec k[x,y]\) is not an isomorphism.
Two regular functions that agree on a dense open set agree, so a rational function is determined by its restriction to any nonempty open set on which it is regular.
Two morphisms agreeing on a dense open agree (Tag PR-QX0VL)
Let \(f : X \to Y\) be separated, let \(Z\) be a reduced scheme over \(Y\), and let \(g, h : Z \to X\) be \(Y\)-morphisms agreeing on a dense open \(U \subseteq Z\). Then \(g = h\).
This is the good property to name, and the proof shows why both hypotheses are there. The pair \((g,h)\) gives \(Z \to \fiberprod{X}{Y}{X}\), and the locus where \(g\) and \(h\) agree is the preimage of the diagonal. Separatedness makes that preimage closed; it contains the dense \(U\), so it is all of \(Z\) topologically. Reducedness upgrades the topological statement to an equality of morphisms, because a closed subscheme of a reduced scheme with the whole space as support is the whole scheme.
So the analogue for quasi-separated is weaker in exactly the way the diagonal is: quasicompactness of the diagonal gives a retrocompact agreement locus, not a closed one, and the conclusion fails. Quasi-separatedness is what one assumes to make cohomology and direct images behave, not to make morphisms equal.