## Nagoya Mathematical Journal

### On the Steinitz module and capitulation of ideals

#### Abstract

Let $L$ be a finite extension of a number field $K$ with ring of integers ${\cal O}_L$ and ${\cal O}_K$ respectively. One can consider ${\cal O}_L$ as a projective module over ${\cal O}_K$. The highest exterior power of ${\cal O}_L$ as an ${\cal O}_K$ module gives an element of the class group of ${\cal O}_K$, called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover $Y$ of an algebraic curve $X$, the Steinitz module as an element of the Picard group of $X$ is the sum of the line bundles on $X$ which become trivial when pulled back to $Y$. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in ${\cal O}_K$ is said to capitulate in $L$ if its extension to ${\cal O}_L$ is a principal ideal.)

#### Article information

Source
Nagoya Math. J., Volume 160 (2000), 1-15.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631498

Mathematical Reviews number (MathSciNet)
MR1804137

Zentralblatt MATH identifier
0971.11055

Subjects
Primary: 11R37: Class field theory
Secondary: 11R29 14C22: Picard groups

#### Citation

Khare, Chandrashekhar; Prasad, Dipendra. On the Steinitz module and capitulation of ideals. Nagoya Math. J. 160 (2000), 1--15. https://projecteuclid.org/euclid.nmj/1114631498

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