  In the present paper, it is proved that any complete local domain of mixed
characteristic has a weakly almost Cohen-Macaulay algebra in the sense that
some system of parameters is a weakly almost regular sequence, which is a
notion defined via a valuation. The central idea of this result originates from
the main statement obtained by Heitmann to prove the Monomial Conjecture in
dimension 3. In fact, A weakly almost Cohen-Macaulay algebra is constructed
over the absolute integral closure of a complete local domain by applying the
methods of Fontaine rings and Witt vectors. A connection of the main theorem
with the Monomial Conjecture is also discussed.
