Twisting a coherent sheaf by an invertible sheaf on an integral proper curve

Given: a field $k$, an integral proper $k$-scheme $X$ of dimension one, an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$, a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the generic point $\xi$ of $X$, and the Axiom of Choice (The Axiom of Choice).

Set-up: since $X$ is proper over the field $k$ it is of finite type over $k$, and its affine charts are spectra of Noetherian rings, so $X$ is locally Noetherian and quasi-compact, hence Noetherian (Proper morphisms, Locally Noetherian and Noetherian schemes, Every algebra of finite type over a Noetherian ring is a Noetherian ring). On the locally Noetherian scheme $X$ a quasi-coherent module is coherent if and only if it is of finite type (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves), and $\chi (X,\mathcal{G})$ is defined for every coherent $\mathcal{G}$ (Euler characteristic of a coherent sheaf).

Generic point and rank: because $X$ is integral it has a unique generic point $\xi$, and $\kappa (\xi )={\mathcal{O}}_{X,\xi }$ is its function field; the stalk of a coherent module at $\xi$ is a finite-dimensional $\kappa (\xi )$-vector space, so $r={\dim }_{\kappa (\xi )}{\mathcal{F}}_{\xi }$ is a finite integer (Integral schemes, Chain dimension and the empty-space convention). The degree is defined by ${\mathrm{deg}}_X(\mathcal{L})=\chi (X,\mathcal{L})-\chi (X,{\mathcal{O}}_X)$ (Degree of an invertible sheaf on a proper one-dimensional scheme).

Exactness of twisting: an invertible module is locally free of rank one (Invertible sheaves, Locally free sheaves of finite rank), and tensoring with a locally free module is exact: exactness of a sequence of sheaves is stalkwise (A sequence of abelian sheaves is exact exactly when it is exact on every stalk), at each point the stalk of $\mathcal{L}$ is free of rank one, and tensoring modules over a ring by a free module preserves kernels and cokernels. Tensor products of quasi-coherent modules are quasi-coherent (Tensor product preserves quasi-coherence), and the tensor product of an invertible module with a coherent module is coherent: the question is local, and on an affine open trivialising $\mathcal{L}$ the unit isomorphism of The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$ identifies $\mathcal{L}\otimes \mathcal{G}$ with $\mathcal{G}$ for a coherent $\mathcal{G}$, while coherence is a local condition (Coherent module sheaves, Coherent sheaves on a locally Noetherian scheme).

Euler characteristic is additive in short exact sequences of coherent modules on the proper $k$-scheme $X$: $\chi (X,\mathcal{F})=\chi (X,{\mathcal{F}}^{\mathrm{\prime }})+\chi (X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ (Euler characteristic is additive in short exact sequences).

For a short exact sequence $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of coherent sheaves, the stalk sequence at $\xi$ is a short exact sequence of $\kappa (\xi )$-vector spaces (A sequence of abelian sheaves is exact exactly when it is exact on every stalk, Localisation of modules is exact), and dimension is additive on short exact sequences (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$); hence $r(\mathcal{F})=r({\mathcal{F}}^{\mathrm{\prime }})+r({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$.

Noetherian devissage (Noetherian devissage for coherent proper pushforward): let $\mathcal{P}$ be a property of coherent ${\mathcal{O}}_X$-modules which holds for the zero module and satisfies the two-of-three property in every short exact sequence of coherent modules. Suppose that for every integral closed subscheme $Z\subseteq X$ with generic point $\eta$ there is a coherent ${\mathcal{O}}_X$-module $\mathcal{G}$ with $\mathrm{Supp}(\mathcal{G})\subseteq Z$, whose stalk ${\mathcal{G}}_{\eta }$ is annihilated by ${\mathfrak{m}}_{\eta }$ and is one-dimensional over $\kappa (\eta )$, and for which $\mathcal{P}(\mathcal{G})$ holds. Then $\mathcal{P}$ holds for every coherent ${\mathcal{O}}_X$-module.

The closed points: for a closed point $p\in X$ the canonical morphism $i:\mathrm{Spec}\kappa (p)\to X$ is a closed immersion, $i_{\ast }\kappa (p)$ is a coherent ${\mathcal{O}}_X$-module with support $\{p\}$, and $\mathcal{L}\otimes i_{\ast }\kappa (p)\cong i_{\ast }\kappa (p)$; moreover $\chi (X,i_{\ast }\kappa (p))=[\kappa (p):k]$ (Euler characteristic of a closed point, and invariance under an invertible twist, Support of a module sheaf).

The integral closed subschemes of the one-dimensional Noetherian space $X$ are $X$ itself and the closed points: a proper irreducible closed subset of the integral one-dimensional Noetherian scheme $X$ has dimension zero (otherwise a length-one chain inside it extends to a length-two chain in $X$), and an integral zero-dimensional scheme consists of its single generic point, which is then closed (Chain dimension and the empty-space convention, Integral schemes).

The Axiom of Choice enters through the Euler-characteristic, additivity and devissage suppliers [L1]–[L4]; the tensor and stalk computations below make no selection.