(Weil)
Let \(X\) be a noetherian integral separated scheme regular in codimension one, and let \(K(X)\) be its function field. A prime divisor is a codimension-one integral closed subscheme of \(X\). A Weil divisor is a finite formal \(\ZZ\)-linear combination of prime divisors; these form the group \(\operatorname{Div}(X)\). For a prime divisor \(Y\) with generic point \(\eta_Y\), let \(v_Y\) be the normalized valuation of the DVR \(\OO_{X,\eta_Y}\). For \(f\in K(X)^\times\), the principal Weil divisor is \[ \operatorname{div}_X(f)\coloneqq\sum_{\operatorname{codim}(Y,X)=1}v_Y(f)Y. \] Only finitely many coefficients are nonzero [@Har10a, Lemma II.6.1]. The divisor class group \(\Cl(X)\) is \(\operatorname{Div}(X)\) modulo principal Weil divisors.
(Cartier)
Let \(X\) be a scheme, and let \(\mck\) be its sheaf of total quotient rings. Explicitly, for an open set \(U\), let \(S(U)\subseteq\Gamma(U,\OO_X)\) consist of sections whose germ at every point of \(U\) is a non-zero-divisor; \(\mck\) is the sheafification of \(U\mapsto S(U)^{-1}\Gamma(U,\OO_X)\). A Cartier divisor is a global section of \(\mck^\times/\OO_X^\times\). It is represented by an open cover \((U_i)\) and elements \(f_i\in\mck^\times(U_i)\) such that \(f_i/f_j\) is a unit on each overlap. For integral \(X\), the sheaf \(\mck\) is the constant sheaf of its function field and the \(f_i\) are nonzero rational functions [@Har10a, Chapter II, §6].
(Principal, effective, support)
A Cartier divisor \(D\) is principal if it is represented by a single element of \(\Gamma(X,\mck^\times)\). The Cartier divisor class group \(\CaCl(X)\) is the group of Cartier divisors modulo principal Cartier divisors. The divisor \(D\) is effective if it has local equations \(f_i\in\OO_X(U_i)\), necessarily non-zero-divisors in their stalks. Its support is the complement of the largest open subset on which all its local equations are units [@Har10a, Chapter II, §6].
For a Cartier divisor \(D\) on a scheme \(X\), the sheaf \(\OO_X(D)\subseteq\mck\) is locally generated by \(f_i^{-1}\) when \(f_i\) is a local equation of \(D\). The rule \([D]\mapsto[\OO_X(D)]\) gives an injective homomorphism \(\CaCl(X)\to\Pic(X)\). It is an isomorphism when \(X\) is integral [@Har10a, Corollary II.6.14 and Proposition II.6.15]. If \(D\) is effective, \(\OO_X(-D)\) is the ideal sheaf locally generated by the \(f_i\) [@Har10a, Proposition II.6.18].
(Cartier and Weil comparison)
Let \(X\) be a noetherian integral separated normal scheme. Sending a Cartier divisor with local equations \(f_i\) to the Weil divisor with coefficient \(v_Y(f_i)\) at a prime divisor \(Y\) meeting \(U_i\) identifies Cartier divisors with locally principal Weil divisors. It induces an injection \(\CaCl(X)\to\Cl(X)\). For such \(X\), the support of a Cartier divisor is the union of the prime divisors with nonzero coefficient in its associated Weil divisor [@Har10a, Remark II.6.11.2 and the proof of Proposition II.6.11]. If \(X\) is locally factorial, the comparison is an isomorphism, and \(\Cl(X)\cong\Pic(X)\) [@Har10a, Proposition II.6.11 and Corollary II.6.16]. In particular, \(\CaCl(X)\cong\Cl(X)\cong\Pic(X)\) for every regular integral separated noetherian scheme \(X\), since regular local rings are factorial [@Har10a, Remark II.6.11.1A].
(A Weil divisor that is not Cartier)
Let \(k\) be a field, put \(A=k[x,y,z]/(xy-z^2)\), and let \(X=\Spec A\). This is a normal domain: the map \(x\mapsto s^2\), \(y\mapsto t^2\), \(z\mapsto st\) identifies \(A\) with the second Veronese subring of \(k[s,t]\), which is normal by the graded-domain argument in P-AGH2514PROJNORM. The identification is injective because \(A=k[x,y]\oplus z k[x,y]\), whose two summands map to distinct even-even and odd-odd monomials in \(s,t\). The ruling \(D=V(x,z)\) is a prime divisor, and its class generates \(\Cl(X)\cong\ZZ/2\ZZ\) [@Har10a, Example II.6.5.2].
At the vertex with maximal ideal \(\mathfrak m=(x,y,z)\), the ideal of \(D\) is \((x,z)A_{\mathfrak m}\). Its image in \(\mathfrak m A_{\mathfrak m}/\mathfrak m^2A_{\mathfrak m}\) has dimension two: the defining relation is quadratic, so the classes of \(x,y,z\) are linearly independent. A principal ideal contained in \(\mathfrak m A_{\mathfrak m}\) has image of dimension at most one there. Thus the ideal of \(D\) is not principal at the vertex, and \(D\) is not Cartier. Since every nonzero divisor class is represented by \(D\), the Cartier-to-Weil injection has zero image, so \(\Pic(X)=0\).
On a smooth integral curve over a field, the local ring at every closed point is a DVR and the generic local ring is its function field. The curve is regular and locally factorial, so its Weil and Cartier divisor classes both identify with its Picard group.
Dependencies
D-QJ5M9Normal domains and schemes, and their relation to regularityuses
Backlinks
D-VARFANOFano, Calabi--Yau, and general typerelated-toPR-ET5PQ$H^1$ classifies line bundles and extensionsusesPR-DIVLBCartier divisors, invertible sheaves, and the Picard groupusesD-DIVEQUIVLinear, algebraic, and numerical equivalenceusesD-DIVREDMULTReduced divisors, multiplicity at a point, and pullback of divisorsusesFE-ADOKKThe Picard group of a nodal and of a cuspidal curveusesFE-DIVPNThe class group of projective space and of a hypersurface complementusesPR-Y5S7VThe degree of a divisor on a projective curveusesT-MOVLEMThe moving lemmausesD-DIVODThe sheaf $\OO_X(D)$ and effective divisors as sectionsusesPR-DIVZEROPOLEZeros and poles of a rational functionuses