In rigid analytic geometry, some sources refer to "rigid spaces", where others refer to "rigid analytic varieties". Do these two terms stand for the same thing, or is there a difference between them?
More details:
Comparing the definition of a rigid analytic variety in [BGR, Definition 9.3.1/4] with the definition of a rigid space in [FvdP, Definition 4.3.1] (for the strong topology), there seem to be two differences in the definitions:
- A rigid space is a type of G-ringed space, whereas a rigid analytic variety is a type of locally G-ringed space.
- The G-topology on a rigid analytic variety $X$ is required to satisfy condition (G2), namely that if $U\subseteq X$ is admissible open and $\mathfrak U=(U_i)_{i\in I}$ is a covering of $U$ by admissible opens, then $\mathfrak U$ is an admissible covering if and only if it has an admissible refinement. There is no such requirement in the definition of a rigid space in [FvdP].
Of these differences, I don't think the first causes any problems: rigid spaces are automatically locally G-ringed since affinoids are.
The second difference seems like it could be more problematic. If we take $X$ an affinoid space, then I think we can find a G-subtopology $T$ of the strong topology for which $(X,T,\mathcal O_X|_T)$ satisfies all the conditions of a rigid analytic variety except (G2). I'd suggest something like taking an strong-admissible covering $(X_i)_{i\in I}$, declaring the $T$-admissible sets in $X$ to be the strong-admissible ones, and declaring the $T$-admissible coverings of a $T$-admissible $U\subseteq X$ to be the trivial covering $\{U\}$ together with all strong-admissible coverings which refine the covering $(U\cap X_i)_{i\in I}$. This seems to satisfy all the properties required of a rigid space (in particular $(X_i)_{i\in I}$ is a $T$-covering of $X$ by affinoids equipped with the strong topology), but does not satisfy (G2) in general.
I'm tempted to suggest based on this that the definitions of a rigid analytic variety in [BGR] and a rigid space in [FvdP] should be the same, but that the definition in [FvdP] accidentally omits the condition (G2). Is this a reasonable conclusion?
References
[BGR] Siegfried Bosch, Ulrich Güntzer and Reinhold Remmert, Non-Archimedean Analysis. A Series of Comprehensive Studies in Mathematics, volume 261. Springer 1984.
[FvdP] Jean Fresnel and Marius van der Put, Rigid Analytic Geometry and Its Applications. Progress in Mathematics, volume 218. Birkhäuser 2004.
