arXiv:2105.05230math.AG
Line bundles on perfectoid covers: case of good reduction
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Abstract
We study Picard groups and Picard functors of perfectoid spaces which are limits of rigid spaces. For sufficiently large covers that are limits of rigid spaces of good reduction, we show that the Picard functor can be represented by the special fibre. We use our results to answer several open questions about Picard groups of perfectoid spaces from the literature, for example we show that these are not always $p$-divisible. Along the way, we construct a "Hodge--Tate spectral sequence for $\mathbb G_m$" of independent interest.
I solve 034. By [1, Theorem 4.1], the Neron-Severi rank of the perfectoid cover is the same as the Neron-Severi rank of the reduction. For a product E x E' of elliptic curves, it is well known that NS(E x E') = NS(E) + NS(E') + Hom(E,E'); see [2, Prop. 2.3]. Since E = E' here…
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