Well-known example: Consider the function $$f(x,y)=\left\{\begin{array}{c} \frac{x^2y}{x^4+y^2} & \text{if }(x,y)\neq(0,0)\\ 0 & \text{if }(x,y)=(0,0) \end{array}\right.$$ …

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 …

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 …

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 a …

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 …

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, …