On Greenberg's generalized conjecture

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