6
$\begingroup$

I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature.

Let us fix a non-archimedean complete valuation field $k$ (the valuation is not necessarily non-trivial) The first definition is as in TEMKIN: An analytic space over $k$ consists of a locally Hausdorff topological space $X$, a net $\tau$ on $X$(regarded as a category), and a functor from $\tau$ to the category of $k$-affinoid spaces sending morphisms to affinoid domain embeddings, such that the underlying topological spaces of $U\subset X$ and $\tau(U)$ are naturally identified.

The second definition I found is in Berkovich's original book: A $k$-analytic space is a locally ringed space $X$ with an equivalence class of collections of pairs $(U_i,\phi_i)$, where $U_i\subset X$ is open, $\phi_i$ is an open immersion of $U_i$ into a $k$-affinoid space subject to the usual compatibility conditions as in the definition of a mainfold.

I'm not pretty sure what is the connection between these two definitions, I guess that the first definition axiomatizes analytic domains in the second definition, but I haven't been able to make this precise.

$\endgroup$
  • 6
    $\begingroup$ The connection between the two is explained in 1.5 of Berkovich's big IHES paper on etale cohomology for non-archimedean analytic spaces (where he introduces the more general notion you mention at the start). What Berkovich had overlooked when he wrote the original little book is that the construction in the book is not general enough to account for all reasonable (e.g., quasi-compact and separated) rigid-analytic spaces (see 1.6 in the IHES paper). And there are much more than 2 definitions of non-archimedean analytic spaces in the literature: also Tate, Raynaud, Huber, Fujiwara-Kato,... $\endgroup$ – nfdc23 Jun 9 '16 at 13:52
5
$\begingroup$

Let me give more a few more details than in nfdc23's comment. The most general definition of a analytic space is the one that you find in Berkovich's IHES paper. One requirement is that for every point $x$ and every open subset $U$ containing $x$, there exist finitely many affinoid domains $V_1,\dotsc,V_n$ of $U$ that all contain $x$ and whose union is a neighborhood of $x$. (The way I picture it is that of a brick wall, where some points lie on the edges of several bricks.)

You could require something stronger, namely that every point has a basis of affinoid neighborhoods (i.e. you can always take $n=1$ above). Such a space is called a good space and this is the class of spaces that is defined in the red book.

To see a typical example of a non-good space, consider the bi-disk $\mathcal{M}(k\{T,U\})$ and, inside it, the space $X$ that is the union of the two affinoid domains defined by $\{|T|=1\}$ and $\{|U|=1\}$. One can prove that the Gauss point has no affinoid neighborhood in $X$. On the other hand, it is important to have these kinds of spaces in the theory. For instance, this one appear as the generic fiber of the (non-affine) formal scheme $Spf(k^\circ\{T,U\})\setminus \{(0,0)\}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.