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On the Steinitz module and capitulation of ideals

Published online by Cambridge University Press:  22 January 2016

Chandrashekhar Khare  and
Dipendra Prasad
Chandrashekhar Khare
Affiliation:
Tata Institute of Fundamental Research, Colaba, Bombay-400005, Indiashekhar@math.tifr.res.in
Dipendra Prasad
Affiliation:
Tata Institute of Fundamental Research, Colaba, Bombay-400005, Indiashekhar@math.tifr.res.in
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Abstract

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Let L be a finite extension of a number field K with ring of integers and respectively. One can consider as a projective module over . The highest exterior power of as an module gives an element of the class group of , 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 is said to capitulate in L if its extension to is a principal ideal.)

Keywords

11R3711R2912A3516A26
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 160 , 2000 , pp. 1 - 15
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

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