# Tagged Questions

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### Noetherianity of valuation ring and valuation being discrete

I need a hint for left to right part of the following: Let $K$ be a valued field with $\nu$ and $\mathcal{O}_\nu$ be its valuation ring. Then, $\mathcal{O}_\nu$ is Noetherian if and only ...
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### Help in this proof on DVRs

I'm trying to understand this proof: Anyone could clarify the converse please? I really need help. Thanks a lot
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### A question on valuation overrings of a PID

Let $A$ be a PID and let $K$ be its quotient field. Let $V$ be a valuation ring of $K$ containing $A$ and assume $V\neq K$. Show that $V$ is a local ring $A_{(p)}$ for some prime element $p$. I ...
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### Localizations of the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$

Consider the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$. Let $A_M$ denote the localization of $A$ with respect to maximal ideal $M$. My question is: Is $A_N$ a DVR, where $N$ is the maximal ideal ...
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### Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
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Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
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### Extending discrete valuation to a function field

Consider a field extension $Q$ over $L$, not necessarily finite. Let $R$ be a valuation ring in $Q$ and $A$ a DVR (discrete valuation ring) in $L$ such that $A \to R$ is local. Let $x \in Q$ be ...
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### Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let ...
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### Some question on localization of a polynomial ring and DVR

Let $A$ be a ring, $P$ be a prime ideal of the polynomial ring $A[x]$ and let $Q=P \cap A$. There are two questions... (1) $A[x]_P \cong A_Q[x]_{m_Q}$? (2) If $A_Q$ is a DVR then $A_Q[x]_{m_Q}$ is ...