Prove the following theorem of Kleiman: if \(X\) is a noetherian, integral, separated, locally factorial scheme, then every coherent sheaf on \(X\) is a quotient of a locally free sheaf (of finite rank).
First show that open sets of the form \(X_s\), for various \(s \in \Gamma(X, \mcl)\), and various invertible sheaves \(\mcl\) on \(X\), form a base for the topology of \(X\).
Now use (II, 5.14) to show that any coherent sheaf is a quotient of a finite direct sum \(\bigoplus_i \mcl_i^{n_i}\) for various invertible sheaves \(\mcl_i\) and various integers \(n_i\).
Hint
For (a), given a closed point \(x \in X\) and an open neighborhood \(U\) of \(x\), to show there is an \(\mcl, s\) such that \(x \in X_s \subseteq U\), first reduce to the case that \(Z=X-U\) is irreducible. Then let \(\zeta\) be the generic point of \(Z\). Let \(f \in K(X)\) be a rational function with \(f \in \mco_x\), \(f \notin \mco_\zeta\). Let \(D=(f)_{\infty}\), and let \(\mcl=\mcl(D)\), \(s \in \Gamma(X, \mcl(D))\) correspond to \(D\) (II, §6).
Solution
Write \(K=K(X)\) and view every local ring of the integral scheme \(X\) as a subring of \(K\). For an invertible sheaf \(L\) and section \(s\), let \(X_s\) be the open subset where \(s\) is a local frame. Local factoriality implies normality and makes every Weil divisor Cartier, by the Cartier--Weil comparison.
1. If \(x\notin\overline{\{z\}}\), then there is \(f\in\OO_{X,x}\) with \(f\notin\OO_{X,z}\).
Suppose instead that \(\OO_{X,x}\subseteq\OO_{X,z}\) inside \(K\). The induced morphism \(\Spec\OO_{X,z}\to\Spec\OO_{X,x}\to X\) agrees at the generic point with the canonical morphism \(\Spec\OO_{X,z}\to X\). They must agree everywhere: their equalizer is a closed subscheme because \(X\) is separated, contains the generic point, and is therefore the whole reduced integral source, as in P-AGH242AGREEDENSE.
The image of \(\Spec\OO_{X,x}\to X\) consists of the generalizations of \(x\). This follows on an affine neighborhood \(\Spec A\) from the correspondence between primes of \(A_{\mathfrak p_x}\) and primes of \(A\) contained in \(\mathfrak p_x\). Equality of the two morphisms would therefore make \(z\) a generalization of \(x\), meaning \(x\in\overline{\{z\}}\), a contradiction. The asserted noninclusion of local rings supplies \(f\), which is necessarily nonzero.
2. For every point \(x\) and every open neighborhood \(U\), there is an effective Cartier divisor \(D\) with \(x\notin\supp D\) and \(X\setminus U\subseteq\supp D\).
First suppose \(Z=X\setminus U\) is irreducible and nonempty, with generic point \(z\). Step 1 gives \(f\in\OO_{X,x}\setminus\OO_{X,z}\). Define its pole divisor \[ D=(f)_\infty=\sum_H\max\{0,-v_H(f)\}H, \] where \(H\) ranges over prime divisors. This is a finite effective Weil divisor [@Har10a, Lemma II.6.1], and is an effective Cartier divisor because \(X\) is locally factorial [@Har10a, Proposition II.6.11].
No component of \(D\) contains \(x\): at a height-one point generalizing \(x\), the regular germ \(f\in\OO_{X,x}\) remains regular after localization and has nonnegative valuation. On the other hand, \(z\) belongs to \(\supp D\). Indeed, express \(f\) as a relatively prime fraction in the UFD \(\OO_{X,z}\). Since \(f\) is not in that ring, its denominator has an irreducible factor not cancelled by the numerator. The corresponding height-one prime gives a codimension-one point of \(X\) generalizing \(z\), with negative valuation of \(f\). The closure of that point is a component of \(D\) containing \(z\). As the support is closed, it contains all of \(Z=\overline{\{z\}}\).
In general, \(X\setminus U\) has finitely many irreducible components \(Z_1,\ldots,Z_t\) because \(X\) is noetherian. Apply the preceding construction to each \(Z_i\) and take the sum of the resulting effective Cartier divisors. Their supports all avoid \(x\), and their union contains \(X\setminus U\), proving the assertion. If the complement is empty, take \(D=0\).
3. The opens in (a) form a basis.
Given \(x\in U\), choose \(D\) from step 2 and put \(L=\OO_X(D)\). The effective divisor gives a canonical global section \(s\) of \(L\), namely the rational function \(1\) in \(\OO_X(D)\subseteq K\). If \(a\) is a local equation for \(D\), the frame of \(L\) is \(a^{-1}\) and the coefficient of \(s\) in this frame is \(a\). Thus \(s\) is a frame exactly away from \(D\), and \[ x\in X_s=X\setminus\supp D\subseteq U. \] Every point and neighborhood have such a refinement, proving the basis assertion, including at nonclosed points.
4. Every coherent sheaf \(F\) is a quotient of a finite direct sum of invertible sheaves.
For each \(x\), choose an affine neighborhood \(U_x\) and a finite family of sections generating \(F|_{U_x}\). By step 3, choose \(W_x=X_{s_x}\subseteq U_x\) containing \(x\), with \(s_x\in\Gamma(X,L_x)\) and \(L_x\) invertible. Quasi-compactness of \(X\) gives a finite cover by such opens, which we label \(W_i=X_{s_i}\) for \(1\le i\le t\). Let \(a_{ij}\in\Gamma(W_i,F)\) be the restrictions of the finite generating families just chosen.
By the section-extension lemma [@Har10a, Lemma II.5.14], for each pair \((i,j)\) there are \(m_{ij}\ge0\) and \[ b_{ij}\in\Gamma(X,F\otimes L_i^{\otimes m_{ij}}) \quad\text{with}\quad b_{ij}|_{W_i}=a_{ij}\otimes s_i^{\otimes m_{ij}}. \] The lemma applies because \(X\) is noetherian and \(F\) is quasi-coherent; it does not require \(W_i\) to be affine. Each \(b_{ij}\) defines a morphism \(L_i^{\otimes(-m_{ij})}\to F\). On \(W_i\), use \(s_i\) to trivialize \(L_i\); the image of the resulting frame is precisely \(a_{ij}\). Hence the sum of these maps is a surjection \[ \boxed{\bigoplus_{i,j}L_i^{\otimes(-m_{ij})}\twoheadrightarrow F.} \] There are finitely many pairs, and the displayed source is locally free of finite rank. This proves (b) and the theorem.
5. Q.E.D.
Steps <1>1–<1>3 prove (a), including the existence of the rational function used in the hint, and step 4 proves (b).