For a number field $F$ and an odd prime number $p,$ let $\tilde{ F}$ be the compositum of all $\mathbb{ Z}_p$-extensions of $F$ and $\tilde{ \Lambda}$ the associated Iwasawa algebra. Let $G_{ S}(\tilde{ F})$ be the Galois group over $\tilde{ F}$ of the maximal extension which is unramified outside $p$-adic and infinite places. In this paper we study the $\tilde{ \Lambda}$-module $\mathfrak{ X}_{ S}^{ (-i)}(\tilde{ F}):=H_1(G_S(\tilde{ F}), \mathbb{ Z}_p(-i))$ and its relationship with $X(\tilde{ F}(\mu_p))(i-1)^\Delta,$ the $\Delta:=\mathrm{ Gal}(\tilde{ F}(\mu_{ p})/\tilde{ F})$-invariant of the Galois group over $\tilde{ F}(\mu_{ p})$ of the maximal abelian unramified pro-$p$-extension of $\tilde{ F}(\mu_{ p}).$ More precisely, we show that under a decomposition condition, the pseudo-nullity of the $\tilde{ \Lambda}$-module $X(\tilde{ F}(\mu_p))(i-1)^\Delta$ is implied by the existence of a $\mathbb{ Z}_{ p}^d$-extension $L$ with $\mathfrak{ X}_{ S}^{ (-i)}(L):=H_1(G_S(L), \mathbb{ Z}_p(-i))$ being without torsion over the Iwasawa algebra associated to $L,$ and which contains a $\mathbb{ Z}_{ p}$-extension $F_{ \infty}$ satisfying $H^2(G_{ S}(F_{ \infty}),\mathbb{ Q}_{ p}/\mathbb{ Z}_p(i))=0.$ As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer $i\equiv 1 \mod{ [F(\mu_p):F]}.$ This existence is fulfilled for $(p, i)$-regular fields.
Categories: math.NT