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Arithmetic on Curves of Genus 1 III. The Tate‐Šafarevič and Selmer Groups
Author(s) -
Cassels J. W. S.
Publication year - 1963
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-13.1.768-s
Subject(s) - mathematics , prime power , lemma (botany) , prime (order theory) , isomorphism (crystallography) , invariant (physics) , combinatorics , pure mathematics , discrete mathematics , biology , ecology , chemistry , poaceae , crystal structure , mathematical physics , crystallography
I AM afraid that an oversimplification has slipped into §5 (p. 277, ‘Simplifications in the prime‐power case’). The restriction maps (5.2), (5.3), and (5.4) are not isomorphisms as stated, only injections. More precisely, let A be a Γ‐module and let Γ″ be a subgroup of Γ of finite index prime to the prime number q . Then H j (Γ″, A ) ( j ⩾0) can be injected naturally into a Γ‐module, and the restriction map H j (Γ, A )→ H j (Γ″, A ) gives an isomorphism between the q ‐primary part of H j (Γ″, A ) and that portion of the q ‐primary part of H j (Γ″, A ) which is invariant under Γ. (Cf. A. Douady, Séminaire Bourbaki 12 (1959–60), Exposé 189. The result stated is trivial for j = 0 and can be deduced for j > 0 by ‘ddvissage’.) As we wish to prove a natural duality (Theorem 1.1) the facts stated above show that it is indeed permissible to make the simplifications set out in Lemma 5.1.