Questions tagged [valuative-criteria]
The valuative-criteria tag has no usage guidance.
10
questions
2
votes
0
answers
89
views
What are étale coverings of the spectrum of a discrete valuation ring?
This question comes when I try the valuative criterion on properness of the moduli space of stable sheaves. Let $X$ be a projective scheme over $\Bbbk$ with an ample line bundle $\mathcal L$. Let $P(t)...
2
votes
0
answers
120
views
Direct proof that for $X$ an integral scheme with every valuation on $K(X)$ having unique center, the same is true for all closed integral subschemes
Suppose $X$ is an integral scheme of finite type over a field $k$, and assume that every valuation on $K(X)/k$ has a unique center. Is there a direct proof that the same is true for every integral ...
5
votes
2
answers
1k
views
When are valuative criteria useful?
We have valuative criteria for properness and universal closedness of morphisms of schemes. I would agree that these criteria shed some light on the geometric nature of these morphisms. However, are ...
6
votes
1
answer
532
views
Valuative criterion to extend morphism of schemes
Let $k$ be an algebraically closed field, $X, Y$ integral $k$-schemes and $Y$ proper over $k$. Let $U$ be a non-empty open subset $U \subset X$
and $f:U \to Y$ a morphism of finite-type. Suppose ...
22
votes
3
answers
825
views
Can continuity of a function be checked by restricting to smooth curves?
Well-known example: Consider the function
$$f(x,y)=\left\{\begin{array}{c}
\frac{x^2y}{x^4+y^2} & \text{if }(x,y)\neq(0,0)\\
0 & \text{if }(x,y)=(0,0)
\end{array}\right.$$
When restricted to ...
14
votes
1
answer
839
views
Can a single DVR witness all specializations on a variety?
If $X$ is a noetherian scheme with points $x$ and $\xi$ so that $x$ is in the closure of $\{\xi\}$, then there exists a discrete valuation ring $V$ and a map $Spec(V)\to X$ sending the generic point ...
7
votes
1
answer
1k
views
Valuative criterion for properness
Let $f : X \rightarrow Y$ be a finite type morphism of Noetherian schemes. The valuative criterion for properness runs as follows. Suppose that for any DVR $R$ with fraction field $K$ that any $K$-...
9
votes
1
answer
1k
views
Example where you *need* non-DVRs in the valuative criteria
The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it ...
5
votes
3
answers
1k
views
Can the valuative criteria be checked "on a dense open"?
The valuative criterion for separatedness (resp. properness) says that a noetherian scheme X is separated (resp. proper) if and only if
for any DVR R, with fraction field K,
any map Spec(K)→...
10
votes
1
answer
2k
views
Can the valuative criteria for separatedness/properness be checked "formally"?
Suppose f:X→Y is a morphism of finite type between locally noetherian schemes. The valuative criterion for separatedness (resp. properness) says roughly that f is a separated (resp. proper) ...