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Iwasawa modules up to isomorphism. (English) Zbl 0732.11061
Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values L-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 171-207 (1989).
[For the entire collection see Zbl 0721.00006.]
In this interesting paper, the author proposes some methods for the study of finitely generated modules over the completed group ring \(\Lambda ={\mathbb{Z}}_ p[[G]]\) of a compact p-adic Lie group G (not necessarily commutative). A basic tool is the “homotopy theory” for such \(\Lambda\)- modules X, which amounts to considering them up to projective factors. Analogues of homotopy groups are certain \(\Lambda\)-modules \(E^ i(X)=Ext^ i_{\Lambda}(X,\Lambda)\). In the case \(G\simeq {\mathbb{Z}}_ p\), the \(\Lambda\)-module X is determined up to isomorphism by \(E^ 0(x)\simeq \Lambda^ r\) (where \(r=rank_{\Lambda}X)\), \(E^ 1(X)\), \(E^ 2(X)\), and a certain class in \(Ext^ 2_{\Lambda}(E^ 2(X),E^ 1(X))\). Essentially two arithmetical applications are given:
1) If k is a finite extension of \({\mathbb{Q}}_ p\), K/k a \({\mathbb{Z}}_ p\)- extension, \(G=Gal(K/k)\), M the maximal abelian pro-p-extension of K, then \(X=Gal(M/K)\) is determined by \(\mu_ K(p)\) \((=the\) group of p-power roots of 1 in K) and a canonical class \(\chi \in H^ 2(G,\mu_ K(p)^{\vee})\) (where \(^{\vee}\) denotes the Pontryagin dual).
2) If k is a finite extension of \({\mathbb{Q}}\), S a finite set of primes in k containing the primes above p.\(\infty\), K/k an S-ramified \({\mathbb{Z}}_ p\)- extension, \(K^ S\) (resp. \(M^ S)\) the maximal (resp. maximal abelian) S-ramified pro-p-extension of K, then \(X_ S=Gal(M^ S/K)\) is determined by \(W_ S=E_ 2(p)^{Gal(K^ S/K)}\) (where \(E_ 2(p)\) is the dualizing module of \(Gal(K^ S/k)\), in the sense of Galois cohomology) and a canonical class \(\chi \in H^ 2(G,W_ S^{\vee})\). Note that \(W_ S^{\vee}\) is related in a precise way to the \(\Lambda\)- torsion of \(X_ S\).

11R23 Iwasawa theory
11R34 Galois cohomology
Zbl 0721.00006