# Questions tagged [valuative-criteria]

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9
questions

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### Morphisms satisfying the valuative criterion of properness

Is there an example of a morphism of schemes $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ that satisfies the valuative criterion (as stated in the point (3) here) but is not of finite type?

**5**

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**2**answers

605 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**

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**1**answer

339 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 ...

**21**

votes

**3**answers

701 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 ...

**13**

votes

**1**answer

595 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**

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**1**answer

952 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 $K$-...

**7**

votes

**1**answer

743 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

**2**answers

845 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)→...

**7**

votes

**1**answer

1k 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) ...