  A 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).
