2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163082Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G. Faltings that was used by Taylor and Wiles in \cite{TW} for a simplification of the proof of the minimal deformation problem.Commutative Algebratext