## 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 $$X_ S$$.

### MSC:

 11R23 Iwasawa theory 11R34 Galois cohomology

Zbl 0721.00006