G-topology

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

- S. Bosch, U. Güntzer, Reinhold Remmert, section 9.1 of
*Non-Archimedean Analysis – A systematic approach to rigid analytic geometry*, 1984 (pdf)

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 $p$-adic Analysis and Applications*, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

In theorem 8.14 of

- Oren Ben-Bassat, Kobi Kremnizer,
*Non-Archimedean analytic geometry as relative algebraic geometry*(arXiv:1312.0338)

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.