Cyclotomic units in z p -extensions

Cohomology groups of the class groups over a Zpextension

“…Applying the isomorphism α 2 • α 1 to the cohomology sequence induced from the short exact sequence 0 → C n → U n → U n /C n → 0 and using H 2 (H, C n ) = 0 (Lemma 2.1) one obtains H 1 (H, U n /C n ) = 0, H 0 (H, U n ) ≃ H 0 (H, U n /C n ). (9) It follows that β 1 in (8) with L = K n is an isomorphism and by Lemma 3.4 the composite map β 3 •β 2 is also an isomorphism. Thus β 2 is injective and β 3 is surjective.…”

On the group of modular units and the ideal class group

“…Let A n be the p-primary part of the ideal class group of K n , E n the unit group of K n , C n the cyclotomic unit group of K n and B n the p-primary part of E n /C n . J.-M. Kim, S. Bae and I.-S. Lee proved the following remarkable result in [8]:…”

“…Moreover, B has no non-trivial Z p -torsion element because it is finitely generated over Z p and the natural mapping B n → B m is injective for all m > n (cf. [8,Theorem 1], [11,Lemma 1]). Similarly, we can see that A χ is a cyclic Λ χ -module.…”

Fitting ideals of class groups in a $ℤ_p$-extension

“…For a result similar to Proposition 3, see Theorem 2 of [10]. We now want to show that A and C have the same characteristic ideal.…”