The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu)$ and an extension $\omega$ of $\nu$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $\omega$ is essentially finitely generated over the valuation ring of $\nu$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

中文翻译：

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就经典不变量而言，估值环的基本有限生成

本文的主要目标是研究经典不变量的估值扩展的一些性质。更具体地说，我们考虑一个值域 $(K,\nu)$ 和 $\nu$ 的扩展 $\omega$ 到 $K$ 的有限扩展 $L$。然后我们研究 $\omega$ 的估值环何时在 $\nu$ 的估值环上本质上是有限生成的。我们根据 Hagen Knaf 的扩展的经典不变量提出了一个必要条件，并表明在某些特定情况下，这个条件也是充分的。我们还研究了分级代数的相应扩展何时有限生成。对于这个问题，我们提出了一个等价的条件（比有限代估值环的条件弱）。