On the Steinitz module and capitulation of ideals

Khare, Chandrashekhar ; Prasad, Dipendra (2000) On the Steinitz module and capitulation of ideals Nagoya Mathematical Journal, 160 . pp. 1-15. ISSN 0027-7630

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

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