# Glueing modules over $\{x\}\times \operatorname{Spec} R$

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.

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