  Part of these notes was written as the author's 2013 master thesis. For
proper flat schemes over a complete discrete valuation ring of mixed
characteristic, we construct an isomorphism of certain subgroups of the Picard
group and the first cohomology group of the structure sheaf. When the Picard
scheme is available and smooth, it recovers the isomorphism coming from its
formal completion. A reinterpretation of an old theorem of Mattuck is given.
