Some results on simple complete ideals having one characteristic pair
<p>Let <em>α</em> be a regular local two-dimensional ring, and let<span style="text-decoration: underline;"> </span><em><span style="text-decoration: underline;">m </span>= (x, y) </em>be its maximal ideal. Let <em>m >...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Università degli Studi di Catania
2003-05-01
|
| Series: | Le Matematiche |
| Subjects: | |
| Online Access: | http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/178 |
| Summary: | <p>Let <em>α</em> be a regular local two-dimensional ring, and let<span style="text-decoration: underline;"> </span><em><span style="text-decoration: underline;">m </span>= (x, y) </em>be its maximal ideal. Let <em>m > n > 1</em> be coprime integers, and let<span style="text-decoration: underline;"><em> p</em></span> be the integral closure of the ideal<em> (x^m , y^n )</em>. Then<span style="text-decoration: underline;"> <em>p</em></span> is a simple complete <span style="text-decoration: underline;"><em>m</em></span>-primary ideal, and its value semigroup is generated by <em>m, n</em>.</p><p>We construct a minimal system of generators <em>{z_0 , . . . , z_n }</em> of <span style="text-decoration: underline;"><em>p</em></span>, and from this we get a minimal system of generators of the polar ideal <span style="text-decoration: underline;"><em>p'</em></span> of<span style="text-decoration: underline;"> <em>p</em></span>, consisting of<em> n = θ</em> elements. In particular, we show that <span style="text-decoration: underline;"><em>p</em></span> and <span style="text-decoration: underline;"><em>p'</em></span> are monomial ideals. When <em>α = κ[ [ x, y ] ]</em>, a ring of formal power series over an algebraically closed field <em>κ</em> of characteristic zero, this implies the existence of some relevant property.</p> |
|---|---|
| ISSN: | 0373-3505 2037-5298 |