Discreet evaluation ring

In the mathematical subfield of commutative algebra , discrete evaluation rings are special local rings with particularly good properties.

definition

A discrete evaluation ring is a main local ideal ring that is not a body .

A producer of the maximal ideal is uniformisierendes element or shortly Uniformisierendes . It is also known as DVR (for discrete valuation ring ) or DBR.

properties

• A discrete evaluation ring is a dedekind ring, in particular a regular local integrity ring.
• The spectrum Spec of a discrete evaluation ring consists of exactly two points: ${\displaystyle R}$ ${\displaystyle R}$
  • A closed point, the special point , belonging to the maximum ideal (if the uniformizing element is) ${\displaystyle (\pi )}$ ${\displaystyle \pi }$
  • and an uncompleted (but open) point, the generic point . ${\displaystyle (0)}$
• For a review discrete ring is a discrete evaluation on the quotient field defined (if for in ). This rating has as a rating ring. ${\displaystyle R}$ ${\displaystyle Quot(R)\rightarrow \mathbb {Z} ;{\frac {a}{b}}\mapsto v_{\pi }(a)-v_{\pi }(b)}$ ${\displaystyle v_{\pi }(a)=n}$ ${\displaystyle (a)=(\pi )^{n}}$ ${\displaystyle R}$ ${\displaystyle R}$
• If one assigns its evaluation ring to a discretely evaluated body and applies the above construction to it, one obtains a discretely evaluated body that is isomorphic . In other words: these constructions induce an equivalence of categories between discrete valued bodies and discrete valuation rings . ${\displaystyle F}$ ${\displaystyle {\mathcal {O}}_{F}}$ ${\displaystyle F}$

Examples

• The ring of p-adic integers for each prime number . is tight in . ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} _{p}}$
• The ring of rational numbers that are p -adic integer for a prime number . It is and is tight in . ${\displaystyle p}$
           ${\displaystyle \mathbb {Z} _{(p)}=\left.\left\{{\frac {z}{n}}\,\right|z,n\in \mathbb {Z} ,p\nmid n\right\}}$
  ${\displaystyle \mathbb {Z} _{(p)}=\mathbb {Z} _{p}\cap \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} _{(p)}}$
• The ring of formal power series in an indeterminate over a body . ${\displaystyle k[[T]]}$ ${\displaystyle k}$
• The ring of convergent power series

${\displaystyle \mathbb {C} \{T\}=\left.\left\{\sum _{i=0}^{\infty }a_{i}T^{i}\,\right|\exists r>0\colon \sum _{i=0}^{\infty }a_{i}z^{i}\ \mathrm {konvergiert\ f{\ddot {u}}r} \ |z|<r\right\}\subset \mathbb {C} [[T]].}$

• The local ring to a smooth point on an algebraic curve .

literature

• MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Chapter 9, ISBN 0-201-00361-9