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Some Ring-Theoretic Properties of \(\mathbf {A}_{{{\mathrm{inf}\,}}}\)

  • Kiran S. KedlayaEmail author
Conference paper
  • 51 Downloads
Part of the Simons Symposia book series (SISY)

Abstract

The ring of Witt vectors over a perfect valuation ring of characteristic p, often denoted \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), plays a pivotal role in p-adic Hodge theory; for instance, Bhatt–Morrow–Scholze have recently reinterpreted and refined the crystalline comparison isomorphism by relating it to a certain \(\mathbf {A}_{{{\mathrm{inf}\,}}}\)-valued cohomology theory. We address some basic ring-theoretic questions about \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), motivated by analogies with two-dimensional regular local rings. For example, we show that in most cases \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), which is manifestly not noetherian, is also not coherent. On the other hand, it does have the property that vector bundles over the complement of the closed point in \({{\,\mathrm{Spec}\,}}\mathbf {A}_{{{\mathrm{inf}\,}}}\) do extend uniquely over the puncture; moreover, a similar statement holds in Huber’s category of adic spaces.

Keywords

Witt vectors Perfectoid rings 

Notes

Acknowledgements

The author was supported by NSF (grant DMS-1501214, DMS-1802161), UC San Diego (Warschawski Professorship), Guggenheim Fellowship (fall 2015), and IAS (Visiting Professorship 2018–2019). Some of this work was carried out at MSRI during the fall 2014 research program “New geometric methods in number theory and automorphic forms” supported by NSF grant DMS-0932078. Thanks to Jaclyn Lang, Judith Ludwig, and Peter Scholze for additional feedback.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSan Diego, La JollaUSA

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