2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125229A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed characteristic.) It follows that the adjoint of a finitely supported ideal I of order a has order sup(a+1-d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, the adjoint of I is also finitely supported. Also: if this I is a normal ideal, then its reduction number is the least integer > d(1-1/a).12 pages. Main results and proofs similar to previous version, but with numerous enhancementsCommutative AlgebraAlgebraic Geometry13H05; 13C99; 14F17text