On Greenberg?s generalized conjecture

For a number field F and an odd prime number p, let F similar to be the compositum of all Zp-extensions of F and ?similar to the associated F similar to) be the Galois group over F similar to of Iwasawa algebra. Let GS( the maximal extension which is unramified outside p-adic and infinite places. In this paper we study the ?similar to-module X(-i) S ( F similar to ) := H1(GS(F similar to),Zp(-i)) and its relationship with X(F similar to(mu p))(i - 1)Delta, the Delta := Gal( F similar to(mu p)/ F similar to)-invariant of the Galois group over F similar to(mu p) of the maximal abelian unramified pro -p-extension of F similar to(mu p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the ?similar to-module X(F similar to(mu p))(i -1)Delta is implied by the existence of a Zdp-extension L with X(-i) S(L) := H1(GS(L), Zp(-i)) being without torsion over the Iwasawa algebra associated to L, and which contains a Zp-extension F infinity satisfying H2(GS(F infinity),Qp/Zp(i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i equivalent to 1 mod [F(mu p) : F]. This existence is fulfilled for (p, i)-regular fields. (c) 2022 Elsevier Inc. All rights reserved.