Principal parts of an invertible sheaf on a curve

Assume the Axiom of Choice as inherited from the curve and sheaf suppliers (The Axiom of Choice); AC implies Dependent Choice by AC implies DC implies countable choice, which supplies the choice hypothesis of the finite-support lemma below. Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field), with generic point $\eta$ (Generic points of irreducible closed subsets) and function field $K={\mathcal{O}}_{C,\eta }=k(C)$ (Function field of an integral finite-type scheme). Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_C$-module (Invertible sheaves, Modules on a ringed space).

Sheaf of meromorphic sections. Write ${\mathcal{L}}_{\eta }$ also for the constant sheaf with value the generic stalk ${\mathcal{L}}_{\eta }$, a one-dimensional $K$-vector space. This is the sheaf of meromorphic sections of $\mathcal{L}$; its global sections include zero. In the convention of Rational section line bundle, a rational section is a nonzero element of this space (The constant sheaf is the sheaf of locally constant functions, A sheaf on a topological space).

The natural map $\mathcal{L}\to {\mathcal{L}}_{\eta }$ is injective: on an affine trivializing open $U=\mathrm{Spec}A$ it is the map $A\to K$ in a frame, and $A$ is a domain because $C$ is integral (Invertible sheaves, Curves over a field).

Sheaf of principal parts. Define $\mathcal{P}(\mathcal{L}):=\mathrm{coker}(\mathcal{L}⟶{\mathcal{L}}_{\eta })={\mathcal{L}}_{\eta }\mathrm{/}\mathcal{L}\mathrm{.}$ The stalk sequence is exact by A sequence of abelian sheaves is exact exactly when it is exact on every stalk. At the generic point the two stalks are both ${\mathcal{L}}_{\eta }$, so $\mathcal{P}(\mathcal{L}{)}_{\eta }=0$. At a closed point $p$, exactness gives $\mathcal{P}(\mathcal{L}{)}_p={\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p\mathrm{.}$ Trivializing ${\mathcal{L}}_p=A_p{e}_{\mathcal{L}}$ identifies this quotient with $K\mathrm{/}{A}_p$, where $A_p={\mathcal{O}}_{C,p}$ is a DVR with uniformizer $t_p$ (Local rings at closed points of smooth curves are discrete valuation rings). It is a torsion $A_p$-module: if a class is represented by $a\mathrm{/}b$ with $a,b\in A_p$ and $b\mathrm{\ne }0$, then multiplication by $b$ kills it. By the published DVR normal-form theorem Every nonzero fraction is a unit times a power of a uniformiser, every class belongs to some submodule $t_p^{-n}{A}_p\mathrm{/}{A}_p$ with $n\ge 0$: if its representative has valuation $m<0$, take $n=-m$; if $m\ge 0$, its class is zero. Multiplication by $t_p^n$ identifies that submodule with $A_p\mathrm{/}(t_p^n)$, which has length $n$ by Length and valuation in a DVR. Thus $K\mathrm{/}{A}_p$ is the union of these finite-length torsion submodules. A local principal part at $p$ is an element of ${\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$.

Quasi-coherence. On an affine trivializing open $U=\mathrm{Spec}A$ with fraction field $K$, the restriction of ${\mathcal{L}}_{\eta }$ is the associated sheaf $\tilde{K}$: on a distinguished open $D(f)$ with $f\mathrm{\ne }0$, the constant sheaf has sections $K$ and $\tilde{K}(D(f))=K_f=K$; on $D(0)=\mathrm{\varnothing }$ both have zero sections. The restriction of $\mathcal{L}$ is $\tilde{A}$. Therefore both are quasi-coherent, and their quotient is quasi-coherent by the published Kernels and cokernels of quasi-coherent modules. The principal-parts sheaf is torsion, not coherent; its stalk $K\mathrm{/}{A}_p$ is not finitely generated over the DVR. (Module sheaf on an affine scheme, Sections and restrictions on distinguished opens of an affine scheme, Quasi-coherent module on a scheme)

Finite support of rational sections. First verify the hypotheses of A meromorphic unit has locally finite nonzero order support. The affine coordinate rings of $C$ are finite-type algebras over the Noetherian field $k$ (Field), hence Noetherian by Every algebra of finite type over a Noetherian ring is a Noetherian ring; thus $C$ is locally Noetherian. It is quasi-compact because its structure morphism is proper (Proper morphisms), so $C$ is Noetherian in the sense of Locally Noetherian and Noetherian schemes. Every point is either the generic point or closed, since $C$ is integral of dimension one. The generic local ring is the field $K$; every closed-point local ring is a DVR by Local rings at closed points of smooth curves are discrete valuation rings. A field is integrally closed; a DVR is a valuation ring by Discrete valuation rings and therefore integrally closed by Valuation rings are integrally closed. Thus all local rings are integrally closed domains, so $C$ is normal in the sense required by Weil divisor normal noetherian scheme. In particular, each closed point is a prime divisor.

Now choose a finite affine cover by opens trivializing $\mathcal{L}$ for a nonzero global rational section $s$. On each such open $U_i$, write $s{\mathrm{\mid }}_{U_i}=f_i{e}_i$ with $f_i\in K^{\times }$, viewed as a global section of the constant sheaf of meromorphic functions ${\mathcal{K}}_C$ (Sheaf total quotient rings, The constant sheaf is the sheaf of locally constant functions). Every closed point of the smooth integral curve is a prime divisor. The support where ${\mathrm{ord}}_p(f_i)\mathrm{\ne }0$ is locally finite by A meromorphic unit has locally finite nonzero order support, hence finite on $C$ by quasi-compactness. Taking the finite union over the cover shows that $s$ is regular at all but finitely many closed points; indeed, outside this union each $f_i$ is a unit in the local DVR, so $s$ is a local generator. The zero meromorphic section is regular everywhere. This uses only the local coefficient of a rational section and the stated principal-Weil-divisor supplier; it does not require a Cartier-divisor identification.

The direct sum of skyscrapers. For each closed point $p$, let ${\mathcal{S}}_p$ be the skyscraper sheaf with value ${\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$ at $p$ and zero stalks elsewhere, with ${\mathcal{O}}_C$ acting through ${\mathcal{O}}_{C,p}$ at $p$ (A skyscraper sheaf of abelian groups at a point, Modules on a ringed space). Put $\mathcal{S}:={⨁}_{p\in C_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}}{\mathcal{S}}_p\mathrm{.}$ By Stalks, coproducts and right exactness of the abelian sheaf tensor product, this coproduct is the sheafification of the presheaf direct sum and its stalk at any point is the direct sum of the summand stalks. By the sheafification property Sheafification of a presheaf, a section of this sheafification is locally represented by finite-support families, so its support is locally finite.

For any open $U$ and $q\in \mathcal{P}(\mathcal{L})(U)$, the quotient map locally lifts $q$ to a meromorphic section $r$ (Kernel sheaves are objectwise, while cokernels and images are sheafified, Sheafification of a presheaf). Around each $x\in U$, shrink to an affine trivializing neighborhood on which a lift exists and write $r=fe$ (or $r=0$); here $f\in K$ gives a global rational coefficient, so the preceding finite-support argument applies. The nonzero stalks of $q$ there are contained in the finite pole support of $r$. Hence the family of germs $(q_p{)}_{p\in U\cap C_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}}$ is locally finite and defines a section of $\mathcal{S}(U)$. This assignment gives a natural ${\mathcal{O}}_C$-linear sheaf morphism $\gamma :\mathcal{P}(\mathcal{L})⟶\mathcal{S}\mathrm{.}$ At a closed point $p$, the stalk map is the identity on ${\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$: the $p$-summand is the only summand with nonzero stalk there. At the generic point both stalks vanish, since every closed point has a neighborhood of $\eta$ avoiding it and coproduct stalks are direct sums. Thus $\gamma$ is an isomorphism by the published A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk. This proves that $\mathcal{P}(\mathcal{L})$ is the direct sum of the closed-point skyscraper sheaves with the stated fibers.

Because $C$ is quasi-compact, a locally finite support subset of $C$ is finite. Consequently $H^0(C,\mathcal{P}(\mathcal{L}))={⨁}_{p\in C_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}}{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$ is the vector space of finite-support families of local principal parts. For every global meromorphic section $s\in {\mathcal{L}}_{\eta }$, its diagonal family $(s+{\mathcal{L}}_p{)}_p$ has finite support by the preceding argument, so the diagonal map ${\mathcal{L}}_{\eta }⟶H^0(C,\mathcal{P}(\mathcal{L})),s⟼(s+{\mathcal{L}}_p{)}_p$ is well-defined. Its image consists of the principal parts of global meromorphic sections. A global regular section maps to zero because each of its germs lies in ${\mathcal{L}}_p$. The quotient of finite-support principal parts by this diagonal image is the cohomology space computed in the next item of this development.