Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, the functor $$\mathcal{F}\longmapsto (\mathcal{F}(C\setminus\{x\}),\mathcal{F}_x,\iota_x)$$ from the category of coherent sheaves of $\mathcal{O}_C$-modules to the category of triplets $(M,N,\iota)$ where $M$ is a finitely generated $\mathcal{O}_C(C\setminus\{x\})$, $N$ is a finitely generated $\mathcal{O}_{C,x}$-module and $\iota_x:M\otimes_{\mathcal{O}_C(C\setminus\{x\})}k(C)\to N\otimes_{\mathcal{O}_{C,x}}k(C)$ is an isomorphism of vector spaces over $k(C)$ (which corresponds to the gluing data in the latter case) is an equivalence of categories.

Let $R$ be a Dedekind domain which is also a $k$-algebra. I want to know whether the same machinery applies for $C_R:=C\times_k \operatorname{Spec} R$ with the subscheme $\{x\}\times_k \operatorname{Spec}R$. Namely, given a triplet $(M,N,\iota)$ where $M$ is a finitely generated $\mathcal{O}_C(C\setminus\{x\})\otimes R$-module, $N$ is a finitely generated $\mathcal{O}_{C,x}\otimes R$ and $\iota$ is a $Q:=\operatorname{Quot}(k(C)\otimes_k R)$-linear isomorphism from $M\otimes_{\mathcal{O}_{C}(C\setminus \{x\})\otimes_k R} Q$ to $N\otimes_{\mathcal{O}_{C,x}\otimes_k R} Q$, can I get a corresponding coherent sheaf on $C_R$?

Sorry for the heavy notations. Any "elementary" answer is very welcome!

Many thanks!


The Beauville-Laszlo theorem holds in much greater generality - see Tag 0BNI on the Stacks Project.

Let $A$ be any ring and let $f\in A$ be a non-zero divisor. Then the category of $f$-torsion free $A$-modules $M$ is equivalent to the category of triples $(M_1, M_2, \varphi)$ where $M_1$ is an $A_f$-module, $M_2$ is an $f$-torsion free module over the $f$-adic completion $\widehat{A}$ of $A$, and $\phi \colon M_1 \otimes_{A_f} (\widehat{A})_f \xrightarrow{\sim} M_2 \otimes_{\widehat{A}} (\widehat{A})_f$ is an isomorphism.

The equivalence sends an $A$-module $M$ to $(M_f, \varprojlim_n M/f^n M, \phi)$, where $\phi$ is the natural isomorphism.

Furthermore, this equivalence respects the full subcategories of finitely generated, flat, and finite projective modules on either side.

This has the following consequence: let $X$ be a smooth relative curve with geometrically connected fibers over any scheme $S$. Take any section $x \in X(S)$ and let $\Gamma_x \hookrightarrow X$ be the image of $x$. Define $U$ to be the open subset $X - \Gamma_x$ and $\widehat{X}$ as the formal completion of $X$ along $\Gamma_x$. (In particular, we could take a smooth geometrically connected curve $X_0$ over some field $k$ and let $X = X_0 \times_k S$, $x \in X_0(S) = X(S)$.)

Then the natural restriction maps give an equivalence of categories from the category of finitely generated quasi coherent sheaves on $X_S$ without sections vanishing along $\Gamma_x$ (resp. flat f.g. quasicoherent sheaves, resp vector bundles) and the category of triples of such sheaves on $U$, such sheaves on $\widehat{X}$, and an isomorphism on $\widehat{X} - \Gamma_x$.

This is just because a section of a relative smooth curve is a Cartier divisor.

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  • $\begingroup$ This is crucial when studying the geometry of the moduli stack $\mathrm{Bun}_G(X)$ of $G$-bundles on a curve $X$. $\endgroup$ – dorebell Jul 16 '19 at 5:53
  • $\begingroup$ Thank you very much ! This is clearer now $\endgroup$ – Stabilo Jul 17 '19 at 11:19

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