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Let $R$ be a discrete valuation ring, $m$ the maximal ideal and $f:X \to \mathrm{Spec}(R)$ be a flat, proper morphism of relative dimension $1$. Assume further that $X$ is regular. For any $n>0$, denote by $X_n:=X \times_{\mathrm{Spec}(R)} \mathrm{Spec}(R/m^n)$ and by $i_n:X_n \hookrightarrow X$ the natural closed immersion.

Let $\mathcal{L}$ be an invertible sheaf on $X$. Suppose $\mathcal{L}'$ is another invertible sheaf on $X$ satisfying the property: for all $n>0$, $i_n^*\mathcal{L} \cong i_n^*\mathcal{L}'$. Is $\mathcal{L} \cong \mathcal{L}'$? Any reference which deals with similar questions will be also very welcome.

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  • $\begingroup$ Doesn't this follow from the formal function theorem? Have you looked at Hartshorne? $\endgroup$
    – Mohan
    Oct 14, 2015 at 21:59
  • $\begingroup$ You'd better ask that the various isomorphisms are compatible with respect to pullback by the transition maps. Once you do that, there's a (more general) theorem of Grothendieck that gives an equivalence of categories between formal coherent sheaves and coherent sheaves on $X$. (I remember this from arxiv.org/pdf/1006.0497v2.pdf, thm 6.48, but there are certainly better references, for example FGA explained chapter 8, that we give there) $\endgroup$
    – Mattia Talpo
    Oct 15, 2015 at 3:42
  • $\begingroup$ @MattiaTalpo All the results that I am seeing assumes $R$ is complete under certain valuation. Is there a reference where $R$ is not necessarily complete? $\endgroup$
    – Jana
    Oct 16, 2015 at 0:04
  • $\begingroup$ @Mohan: Could you please elaborate a little more on how to use the formal function theorem. $\endgroup$
    – Jana
    Oct 16, 2015 at 0:06
  • $\begingroup$ Replacing $\mathcal{L},\mathcal{L}'$ with $\mathcal{L}\otimes\mathcal{L}'^{-1}$, suffices to show that if $\mathcal{L}$ restricted to the $n$th thickening of the closed point of $R$ is trivial for all $n$, then $\mathcal{L}$ is trivial. By formal function theorem, we see that the completion of $f_*\mathcal{L}$ is the completion of $R$ and then it is standard to show that $f_*\mathcal{L}$ is $R$ and the the map $R\to f_*\mathcal{L}\otimes R/m^n$ is a surjection for all $n$. So, this section will give a trivialization of $\mathcal{L}$. $\endgroup$
    – Mohan
    Oct 16, 2015 at 2:57

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