  In this paper we construct new derived invariants with integral coefficients
using the theory of motifs, and give several applications. Specifically, we
obtain the following results: For complex algebraic surfaces, we prove that
certain torsion in the abelianized fundamental group is a derived invariant. We
prove that the collection of Hodge-Witt cohomology groups is a derived
invariant. In particular, Hodge-Witt reduction and ordinary reduction are
preserved by derived equivalence when the characteristic is sufficiently large.
Finally, using the techniques of non-commutative algebraic geometry, we prove
that Serre's ordinary density conjecture is true for cubic $4$-folds which
contain a $\mathbb{P}^2$.
