The Nonarchimedean Scottish Book

Revision as of 05:48, 18 December 2015

The original Scottish Book was a compilation of problems, primarily concerning functional analysis, assembled at the Scottish Cafe in Lvov during the 1930s and 1940s. This web site is an analogous compilation of problems concerning nonarchimedean functional analysis and related topics, including analytic geometry (especially adic spaces) and perfectoid rings, fields, and spaces.

An ideal entry in this list includes the name of the proposer, the date of the initial post, a detailed statement of the problem, and attributed comments (including references as appropriate; these can be listed at the bottom of the page).

For security reasons, editing this wiki requires logging into the server. I have created a communal login credential for editing this page; please contact me for the details. Alternatively, I am wiling to accept entries by email and post them manually. (If someone with more free time than me wants to reimplementing this using better technology, I would support this.)

Problem 1

Proposed by Kiran S. Kedlaya, 17 December 2015.

Statement: Let K be a nonarchimedean commutative Banach ring whose underlying ring is a field. Suppose in addition that K is uniform, i.e., its norm is equivalent to a power-multiplicative norm. Is K necessarily a nonarchimedean field, that is, is the topology on K defined by some multiplicative norm?

Proposer's comment: This question acquires a negative answer if the uniform condition is omitted. Details available upon request.

Problem 2

Proposed by Kiran S. Kedlaya, 17 December 2015.

Statement: Let (A,A^+) be an adic Banach ring which is Tate (i.e., A contains a topologically nilpotent unit). Suppose that Spa(A,A^+) is a perfectoid space (for some prime p). Is A necessarily a perfectoid algebra?

Proposer's comment: This is true if A is of characteristic p.

Problem 3

Proposed by David Hansen, 17 December 2015.

Statement: Let A be a perfectoid Tate ring with an action of a finite group G. Is the fixed subring A^G perfectoid?

Proposer's comment: This is true if A is of characteristic p, by an easy argument. This is also true if A is a Q_p-algebra, by a more subtle argument of Kedlaya.

Problem 4

Proposed by David Hansen, 18 December 2015.

Statement: Let A be a sheafy Tate ring, and suppose some Zariski-open subset of some Spa(A,A^+) is a perfectoid space. Is A perfectoid?

Proposer's comment: This is a stronger version of Problem 2.

Problem 5

Proposed by David Hansen, 18 December 2015.

Statement: Let A be a stably uniform Tate ring over Q_p, and let R^+ \subset R be a perfectoid Tate ring in characteristic p. Consider the Tate ring (W(R^+) \widehat{\otimes} A^\circ)[1/p], where the completion is taken for the p-adic topology. Is this ring stably uniform?

Proposer's comment: The answer is yes when A is finite etale over some Q_p<X_1,...,X_n>.

Problem 6

Proposed by David Hansen, 18 December 2015.

Statement: Call a Tate ring A "sousperfectoid" if there exists a perfectoid Tate ring B and a continuous A-algebra map A-->B which admits a continuous A-Banach module splitting. This class includes perfectoid Tate rings, and any sousperfectoid ring is stably uniform and hence sheafy. If R is sousperfectoid, then any rational localization of R is sousperfectoid, and so are R<X>, R<X^1/p^infty> and R' for any finite etale R'/R.

Is there an example of a stably uniform Tate ring which is not sousperfectoid?

References

To be added later.