Cartier divisors, invertible sheaves, and the Picard group

Cartier divisors are line bundles with rational sections; on locally factorial schemes, Cartier, Weil, and Picard classes coincide.Suggest a slogan

$D \mapsto \OO_X(D)$ is an injection $\CaCl(X) \injects \Pic(X)$, and it is a bijection onto the invertible subsheaves of $\mck$ [@Har10a, Proposition II.6.13]. When $X$ is integral every invertible sheaf is such a subsheaf, so $\CaCl(X) \cong \Pic(X)$ [@Har10a, Proposition II.6.15].

$\Pic(X) \cong H^1(X, \unitsof{\OO_X})$ (P-AGH345PICH1), and the sequence \[ 0 \to \unitsof{\OO_X} \to \unitsof{\mck} \to \unitsof{\mck}/\unitsof{\OO_X} \to 0 \] has cohomology sequence $\unitsof{\mck(X)} \to \Div_{\mathrm{Ca}}(X) \to \Pic(X) \to H^1(X, \unitsof{\mck})$.

(Divisors, line bundles with sections, and class groups)

Let $X$ be an integral separated Noetherian scheme with function field $K$, and write $\mathcal{P}(X)$ for the group of isomorphism classes of pairs $(\mathcal{L}, s)$ with $\mathcal{L}$ invertible and $s$ a nonzero rational section of $\mathcal{L}$, under tensor product.

$\Div_{\mathrm{Ca}}(X) \to \mathcal{P}(X)$, $D = \{(U_i, f_i)\} \mapsto (\OO_X(D), 1)$, is an isomorphism, with inverse $(\mathcal{L}, s) \mapsto \{(U_i, \phi_i(s)^{-1})\}$ for local trivializations $\phi_i$. For an effective $D$ with invertible ideal sheaf $\mci_D$, $\OO_X(D) = \mci_D^{\vee}$ and $1$ is the canonical section vanishing on $D$.
If $X$ is regular in codimension one, $\Div_{\mathrm{Ca}}(X) \to \Div(X)$, $\{(U_i, f_i)\} \mapsto \sum_Y v_Y(f_i) Y$ with $U_i \cap Y \neq \emptyset$, is a homomorphism, and $(\mathcal{L}, s) \mapsto \div s$ is the composite with the inverse of part 1.
Under these maps, $t \in \unitsof{K}$ goes to the principal Cartier divisor $\{(X, t)\}$, the pair $(\OO_X, t)$, and the principal Weil divisor $\div t$, so they descend to $\CaCl(X) \to \Cl(X)$ and $\CaCl(X) \cong \Pic(X)$.
If every local ring $\OO_{X,x}$ is a unique factorization domain, then $\CaCl(X) \cong \Cl(X) \cong \Pic(X)$, with inverse $D \mapsto (\OO_X(D), 1)$ [@Har10a, Proposition II.6.11].

Both halves are the same observation: gluing data for a line bundle is a cocycle $g_{ij} \in \unitsof{\OO}(U_{ij})$, and Cartier data $f_i$ produces the cocycle $g_{ij} = f_i/f_j$. Saying “a Cartier divisor is a line bundle together with a choice of rational section” is the cleanest one-line answer, and it explains the failure of surjectivity for non-integral $X$: a line bundle with no rational section is not a divisor.

Combined with the injection $\CaCl \injects \Cl$, the chain to recite is \[ \Pic(X) \cong \CaCl(X) \injects \Cl(X) , \] an isomorphism throughout when $X$ is locally factorial. The gap at the second arrow is measured by the singularities.

What is the correspondence between Cartier divisors and invertible sheaves?

Why is $\Pic(X) = H^1(X, \unitsof{\OO_X})$?

Which line bundles come from divisors?

Why does $\Pic(X) \cong H^1(X, \OO_X^\times)$ need the comparison of Čech and derived-functor $H^1$?

Describe the maps between Cartier divisors, Weil divisors and line bundles with rational sections.

Dependencies

D-DIVODThe sheaf $\OO_X(D)$ and effective divisors as sectionsuses
D-5PQ5WWeil and Cartier divisors, and when they agreeuses

Backlinks

P-AGXMISCCALCULATIONSOral calculations on projective space, sections and divisorsrelated-to
D-TORSORTorsors under a sheaf of groupsrelated-to

What links to this

Line bundles and linear systems