For proper flat schemes over complete discrete valuation rings 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 functor is representable and smooth, our construction recovers and gives finer information to the isomorphism coming from its formal completion. An alternative proof of an old theorem of Mattuck is given.
"Notes on a $p$-adic Exponential Map for the Picard Group." Tokyo J. Math. 42 (1) 35 - 49, June 2019. https://doi.org/10.3836/tjm/1502179262