Sintlattice

Sintlattice - Math 676. The lattice of S-integers Let K be...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 676. The lattice of S-integers Let K be a number field, and let S be a finite set of places of K containing the set S of archimedean places. Recall that we define the ring O K,S of S-integers in K to be the set of a K such that a is v-integral for all (necessarily non-archimedean!) v 6 S ; that is, || a || v 1 for all v 6 S . For a O K,S , we have a O K,S if and only if a 6 = 0 and a, 1 /a K each lie in O K,S . That is, a O K,S if and only if a K and || a || v , || 1 /a || v 1 for all v 6 S . This final condition says || a || v = 1 for all v 6 S . Hence, O K,S = { x K ||| x || v = 1 for all v 6 S } . 1. Preliminaries We first wish to show that O K,S can be concretely constructed from O K and knowledge of the class number. For each non-archimedean place of K , we let p v denote the corresponding prime ideal of O K . Since the class group is killed by the class number, for all non-archimedean v the ideal p h ( K ) v in O K is principal. Hence, the finite product Q v S- S p h ( K ) v has the form a S O K , so 1 /a S K is non-integral at precisely those non-archimedean v that lie in S (if S = S then this product is empty and we may interpret the product over the empty set S- S to be the unit ideal O K ; a S is an element of O K in this case). Having constructed one such element, we now show that any such element allows us to construct O K,S as a localization of O K : Lemma 1.1. For a O K- { } , we have O K,S = O K [1 /a ] if and only if the finite set of non-archimedean v for which || 1 /a || v > 1 is exactly the set S- S . ( Equivalently, the condition is that the prime factors of a O K are exactly the primes p v for v S- S ) . Proof. If O K,S = O K [1 /a ] then 1 /a is v-integral for all v 6 S , so || 1 /a || v 1 for all v 6 S . We wish to show that || 1 /a || v > 1 for the other non-archimedean places, namely those v S- S . Suppose otherwise, so || 1 /a || v 1 for some v S- S . That is, assume 1 /a is v-integral for some non-archimedean v S . Since all elements of O K are also v-integral, it follows that all elements of O K,S = O K [1 /a ] are v-integral. However, this is not true: by finiteness of the class group we have p h ( K ) v = a O K for some a O K- { } , and clearly 1 /a O K,S (since a is even a local unit at all places not in S ) yet 1 /a is not v-integral for the place v S (as || 1 /a || v > 1 due to the prime factorization of a O K )....
View Full Document

Page1 / 3

Sintlattice - Math 676. The lattice of S-integers Let K be...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online