Let \(k\) be a number field, \(k^c\) an algebraic closure of \(k\) and \(\Sigma\) a finite set of places of \(k\) containing all Archimedean places \({\Sigma}_{\infty}\) and places above a fixed prime \(p.\) For a finite extension \(L\) of \(k\) let \({\Sigma}_L\) be the set of places of \(L\) above \({\Sigma}.\) Let \({\mathcal O}_{L,{\Sigma}}\) be the subring of \(L\) consisting of elements that are integral outside \({\Sigma}_L\) and \({\mathrm{Cl}}_{\Sigma}(L)\) the ideal class group of \({\mathcal O}_{L,{\Sigma}}.\) Assume that \(k_{\Sigma}\) is the maximal extension of \(k\) in \(k^c\) unramified outside \(\Sigma ,\) \(G_{k,{\Sigma}}=\text{Gal}(k_{\Sigma}/k)\) and \({\rho}: G_{k,{\Sigma}}\rightarrow {\mathrm{Aut}}_{{\mathbb Z}_p}(T)\) is a continuous representation of \(G_{k,{\Sigma}}\) on a finitely generated free \({\mathbb Z}_p\)-module \(T.\) Let further \(H^i({\mathcal O}_{L,{\Sigma}}, T):=H^i({\mathrm{Spec}}({\mathcal O}_{L,{\Sigma}})_{\mathrm{{\acute{e}}t}}, T) \) be the \(p\)-adic étale cohomology groups and \(H^i_c({\mathcal O}_{L,{\Sigma}}, T):=H^i_c({\mathrm{Spec}}({\mathcal O}_{L,{\Sigma}})_{\mathrm{{\acute{e}}t}}, T) \) be the compactly supported étale cohomology groups. The main result of the paper is the following:
Theorem. Let \(F/E\) be a Galois extension of fields with \(k\subseteq E\subseteq F\subset k^c ,\) \(F/k\) finite and \(F/E\) unramified outside \({\Sigma}_E.\) Then in each degree \(i\) there are decompositions of \({\mathbb Z}_p[G_{F/E}]\)-modules \[ H^i({\mathcal O}_{F,{\Sigma}}, T)=P^i_{F/E,T}\oplus R^i_{F/E,T}\] and \[ H^i_c({\mathcal O}_{F,{\Sigma}}, T)=P^i_{F/E,T,c}\oplus R^i_{F/E,T,c}\] where the modules \(P^i_{F/E,T}\) and \(P^i_{F/E,T,c}\) are projective and one has \[ {\mathrm{{rk}_p}}(R^i_{F/E,T})\leq m^i_{F/E,T}\cdot {\mathrm{{rk}}}(T) \quad \mathrm{and} \quad {\mathrm{{rk}_p}}(R^i_{F/E,T,c})\leq m^i_{F/E,T,c}\cdot {\mathrm{{rk}}}(T) \] for integers \(m^i_{F/E,T}\) and \(m^i_{F/E,T,c}\) depending only on \(i\), \([F:E],\) \({\mathrm{{rk}_p}}({\mathrm{Cl}}_{\Sigma}(F_T)\) and \(\# {\Sigma}_{f,F}.\)
In the above theorem \(F_T:=k_TF,\) where the field \(k_T\) is a finite extension of \(k({\mu}_p)\) that corresponds to the action of \(G_{k({\mu}_p),{\Sigma}}\) on \(T/p\) and \({\Sigma}_{f,F}={\Sigma}_F\backslash{\Sigma}_{\infty}.\) The constants \(m^i_{F/E,T}\) and \(m^i_{F/E,T,c}\) can be given explicitly. As an application od this result the author obtains new bounds on the changes in ranks of Selmer groups over extensions of number fields. The author also discusses the consequences of his theorem to the Galois structure of some natural arithmetic modules, e.g., unit groups, higher algebraic \(K\)-groups and ray class groups.