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While p-adic numbers and other non-archimedean fields may be complete with respect to the norm topology induced from their non-archimedean valuation, this topology is generally not a useful topology on the affinoid domains over these spaces. For instance analytic continuation does not in general exist for these topologies.

To remedy this (Tate 71) introduced a Grothendieck topology for such analytic spaces with respect to which there is a good analytic geometry, namely what is called rigid analytic geometry.

This is simply the Grothendieck topology whose coverings are the covers by affinoid domain embeddings. It should probably be called the Tate topology. But in (Bosch-Güntzer-Remmert 84,section 9.1.4) it is called the G-topology (where actually “G” is short for “Grothendieck”…) and since then that name stuck.


The concept was introduced in

  • John Tate, Rigid analytic spaces, Invent. Math. 12:257–289, 1971

A standard textbook account is in section 9 of

Lectures notes in the context of Berkovich analytic spaces include

  • Vladimir Berkovich, section 3.2 of Non-archimedean analytic spaces, lectures at the Advanced School on pp-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

In theorem 8.14 of

the G-topology is re-derived from more general abstract grounds.

Last revised on May 28, 2021 at 08:22:49. See the history of this page for a list of all contributions to it.