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 $F(k)$?

Is it true that $F \cong \mathfrak{n}$, where $\mathfrak{n}=(u,v)$ in $V/(\pi)[[u,v]]$?

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 $V/(\pi)[[u,v]]$?