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Let $A\in GL_n(\mathbb{C}((t)))$, i.e. some invertible matrix over the ring of Laurent series. It is known that there are $P,Q\in GL_n(\mathbb{C}[[t]])$, such that $PAQ$ is diagonal. This is just Smith normal form over the PID $\mathbb{C}[[t]]$.

Now, replace $\mathbb{C}$ with some $\mathbb{C}$-algebra $R$. I wonder if every $A\in GL_n(R((t)))$ still has a Smith normal form Zariski-locally on $R$. Namely, if there are $f_1,...,f_n\in R$ that generate the unit ideal and such that for each $i=1,...,n$, there are $P_i,Q_i\in GL_n(R_{f_i}[[t]])$ such that $P_i A Q_i$ is diagonal.

It may be better to start with the question whether there is some non-empty Zariski open set over which $A$ has a Smith normal form. I tried to follow the algorithm for finding Smith normal form and see if it can be done over $R[[t]]$ when I am allowed to invert some elements, but I got confused. Note that we can easily reduce to the case that $R$ is noetherian.

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  • $\begingroup$ You ask if every vector bundle on some sort of formal $\widehat{\mathbb{P}^1_{R}}$ is a direct sum of line bundles. $\endgroup$ – Martin Brandenburg Oct 13 '13 at 0:05

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