Questions tagged [valuation-rings]
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9
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On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
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The Zariski Riemann Space, but with Local Rings
The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would ...
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Is there a good notion of higher-rank archimedean norm?
Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
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Frobenius and mixed characteristic valuation rings
Let $R$ be an $\mathbf{F}_p$-algebra. Kunz's theorem says that if $R$ is Noetherian, then the Frobenius of $R$ is flat iff $R$ is regular. Following the philosophy that valuation rings often behave ...
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On the closed subring of valuation rings
Let $(K,v)$ be a (not necessarily discrete) valued field of characteristic $p$ with valuation ring $\mathcal{O}$ and residue field $k$. We endow $K$ and $\mathcal{O}$ with the valuation topology. ...
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References on topological ringed spaces
This is a follow up to this question of mine.
First of all, let me fix some terminologies, which may or may not be standard:
Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)...
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Derivation for genus-degree formula from algebraic functions field theory
This is a copy of my question from math.stackexchange: https://math.stackexchange.com/questions/4517289/derivation-for-genus-degree-formula-from-algebraic-functions-field-theory. I didn't get any ...
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Approximating a scheme with irreducible fibers
Let $R$ be a (non-trivial) valuation ring of an algebraically closed field. Let $V$ be an integral affine scheme over $S=\mathrm{Spec} R$ and assume that $V(R)\neq \emptyset$ (so e.g. $V$ is ...
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Does separability of residue fields implies separability of $L/K$?
Let $A$ be discrete valuation domain, and $K$ be quotient field of $A$. Let $L$ be a finite extension of $K$ and $B$ be the integral closure of $A$ in $L$.Does separability of residue fields implies ...