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## Equations for formal toric degenerations

Bernard H Teissier
• Function : Author
• PersonId : 1067524

#### Abstract

Let $R$ be a complete equicharacteristic noetherian local domain and $\nu$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $\nu$ on $R\setminus \{0\}$ is not finitely generated in general. We produce equations in an appropriate generalized power series ring for the algebra encoding the degeneration of $R$ to the toric graded algebra ${\rm gr}_\nu R$ associated to the filtration defined by $\nu$. We apply this to represent $\nu$ as the limit of a sequence of Abhyankar semivaluations (valuations on quotients) of $R$ with finitely generated semigroups.

#### Domains

Mathematics [math] Algebraic Geometry [math.AG]

### Dates and versions

hal-03954995 , version 1 (24-01-2023)

### Identifiers

• HAL Id : hal-03954995 , version 1
• ARXIV :

### Cite

Bernard H Teissier. Equations for formal toric degenerations. 2023. ⟨hal-03954995⟩

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