  R. Heitmann's proof of the Direct Summand Conjecture has opened a new
approach to the study of homological conjectures in mixed characteristic.
Inspired by his work and by the methods of almost ring theory, we discuss a
normalized length for certain torsion modules, which was introduced by G.
Faltings. Using the normalized length and the Frobenius map, we prove some
results of local cohomology for local rings in mixed characteristic, which has
an immediate implication for the subject of splinters studied by A. Singh.
