Abstract
We revisit Huber’s theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber’s theory compatible with Berkovich’s construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.
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Acknowledgments
The author was supported by NSF (grant DMS-1101343) and UC San Diego (Warschawski Professorship), and by MSRI during fall 2014 (via NSF grant DMS-0932078). Thanks to Antoine Ducros, Roland Huber, Thomas Scanlon, and Michael Temkin for helpful discussions.
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Kedlaya, K.S. Reified valuations and adic spectra. Res. number theory 1, 20 (2015). https://doi.org/10.1007/s40993-015-0021-7
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DOI: https://doi.org/10.1007/s40993-015-0021-7