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Hi,

let $V$ be a complete DVR with uniformizer $\pi$. Let $m$ be a NON zero integer, $a\in V[[u,v]]/(uv-\pi)^{\times}$ and $f=\pi^{m}a$. Consider $F$ as the kernel of the diagram

$$ V[[u,v]]/(uv-\pi)[\frac{1}{\pi}]\oplus \widehat{V[[u,v]]/(uv-\pi)}\stackrel{\rightarrow}{\rightarrow} \widehat{V[[u,v]]/(uv-\pi)}[\frac{1}{\pi}] $$

where the upper arrow is a map from the first factor induced by multiplication by $f$ and the bottom is the canonical one on the second factor. If I understand correctly the theorem of Beauville-Laszlo

http://math1.unice.fr/~beauvill/pubs/descente.pdf

$F$ is finitely generated projective $V[[u,v]]/(uv-\pi)$-module.

Let now $\mathfrak{m}=(u,v,\pi)$ and $k$ be the residue field at $\mathfrak{m}$.

What can I say about the fiber $F(k)$?

Can I find a diagram (using different localizations and morphisms) such that $F(k) \cong \mathfrak{n}$, where $\mathfrak{n}=(u,v)$ in $V/(\pi)[[u,v]]$?

Improve this question
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    $\begingroup$ In the noetherian case, BL is a special case of usual faithfully flat descent. For any noetherian ring $R$ (such as $V[[u,v]]/(uv-\pi)$) and any $\pi \in R$, an element of $R$ divisible by $\pi$ in the $\pi$-adic completion $R'$ of $R$ is divisible by $\pi$ in $R$ because the natural map $R/\pi R \rightarrow R'/\pi R'$ is injective (even an isomorphism), and the diagonal map $R \rightarrow R[1/\pi] \times R'$ is injective (by faithful flatness considerations locally along the zeros of $\pi$ in Spec($R$)), so the $F$ you ask about is always the original noetherian ring $R$. $\endgroup$
    – user28172
    Commented Feb 25, 2013 at 12:30

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