Deformation rings and base change (Q1961088)

Let \(\rho\) be a linear representation of the absolute Galois group of \({\mathbb Q}\) in \(\text{GL}_n({\mathbb F})\), where \({\mathbb F}\) is a finite field of characteristic \(p\). On the Galois group of the maximal extension of \({\mathbb Q}\) which is unramified outside some fixed finite set \(S\) of places there exists a universal deformation \(\rho^{\mathfrak univ} \) of \(\rho\) in \(\text{GL}_n(R_{\mathbb Q})\) where \(R_{\mathbb Q}\) is a local noetherian \(W({\mathbb F})\)-algebra with residue field \({\mathbb F}\). If \(F\) is a finite extension of \({\mathbb Q}\), there is an analog universal deformation of the restriction \(\rho_F\) of \(\rho\) to the absolute Galois group of \(F\), which takes values in some \(\text{GL}_n(R_F)\). Due to universality there is a local ring homomorphism \(\alpha:R_F \rightarrow R_{\mathbb Q}\) which is the main objective of the present paper. The first half of this is devoted to studying \(\alpha\) in the case that \(p\) does not divide \([F:{\mathbb Q}]\). If \(I\) denotes the kernel of \(\alpha\), then one gets a split extension \(R_F/I^2 \rightarrow R_{\mathbb Q}.\) It remains open whether \(\alpha\) itself is a split extension. In the second half of the paper it is shown that for \(n=2\) this splitting would imply the existence of a \(p\)-adic lifting of \(\rho.\) A condition is given under which \(p\) is not a zero divisor in \(R_{\mathbb Q}.\)