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Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations. First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...

Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely $$D=\sum_{\mathfrak p\;\...

The background to my question, in a nutshell, is: If $k$ is a field and $X$ a $k$-variety, i.e. an integral, separated, finite type $k$-scheme, which discrete rank $1$ valuations on $k(X)$ come from ...

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...