  Let k be a base commutative ring, R a commutative ring of coefficients, X a
quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of
rank r, and Hmo(R) the category of noncommutative motives with R-coefficients.
Assume that 1/r belongs to R. Under this assumption, we prove that the
noncommutative motives with R-coefficients of X and A are isomorphic. As an
application, we show that all the R-linear additive invariants of X and A are
exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic
homology (and all its variants), topological Hochschild homology, etc. Making
use of these isomorphisms, we then computer the R-linear additive invariants of
differential operators in positive characteristic, of cubic fourfolds
containing a plane, of Severi-Brauer varieties, of Clifford algebras, of
quadrics, and of finite dimensional k-algebras of finite global dimension.
Along the way we establish two results of independent interest. The first one
asserts that every element of the Grothendieck group of X which has rank r
becomes invertible in the R-linearized Grothendieck group, and the second one
that every additive invariant of finite dimensional algebras of finite global
dimension is unaffected under nilpotent extensions.
