All Questions
8 questions
20
votes
1
answer
2k
views
Are flat morphisms of analytic spaces open?
Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The ...
- 19.6k
9
votes
0
answers
577
views
What lies between algebraic geometry and analytic geometry?
Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to ...
- 64k
7
votes
0
answers
2k
views
Generalized GAGA
So, I have heard GAGA works for Rigid Analytic spaces. I know next to nothing about this, but it made me curious as to whether there are any other contexts in which GAGA "works". Of course, this is a ...
- 373
4
votes
1
answer
421
views
Representability of relative Hilbert and Picard functors over analytic spaces
Let $f:X \to S$ be a morphism of complex analytic spaces. Then, just like in the case of schemes, we can define the relative Hilbert and Picard functors. For instance, if $\text{An}_{/S}$ denotes de ...
user96873
4
votes
0
answers
321
views
Can we see the completion of a scheme along a subscheme as an adic space?
Classically, formal schemes were invented to study completions of schemes along closed subschemes. Eventually, people started using them for more arithmetical reasons. (I.e., to study non-archimedean ...
- 773
3
votes
1
answer
307
views
How to show analytification functor commutes with forgetful functor?
Let $k$ be a field complete with respect to a non-trivial non-archimedean
absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$.
Denote $X\rightsquigarrow X^{...
- 452
1
vote
1
answer
209
views
Reference request: Gruson's theorem on the tensor product of Banach spaces over a non-Archimedean field
I am looking for a reference for theorem 3.21 of these notes: https://web.math.princeton.edu/~takumim/Berkovich.pdf
The theorem states that if $k$ is a non-Archimedean field and $X$ and $Y$ are $k$-...
- 63
1
vote
0
answers
183
views
Contractibility of the quotient of an analytification of a smooth variety by a finite group (if the field is trivially valued)
Let $k$ be a field and $X$ be a smooth irreducible $k$-variety with an action of a finite group $G$. I consider $k$ as a trivially valued field.
It is known from results of Berkovich ("Smooth p-...
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