Non-commutative Motives and the K-theory of Derived Azumaya Algebras

This paper investigates the interplay between non-commutative geometry and motivic theory in the context of derived algebraic geometry. We introduce the notion of a derived Azumaya algebra over a quasi-compact, quasi-separated derived scheme and study its algebraic K-theory. Our central contribution is the construction of a non-commutative motive associated with such an algebra. We establish a fundamental relationship between the K-theory of the derived Azumaya algebra and the K-theory of its base scheme, twisted by its class in the derived Brauer group. This is achieved by leveraging the theory of localizing invariants, particularly the work of Tabuada on non-commutative motives, and extending it to the framework of derived stacks. We prove a dévissage theorem for the K-theory of perfect modules over a derived Azumaya algebra, relating it to the motive of the underlying scheme. Furthermore, we construct a localization sequence that connects the K-theory of the algebra, its center, and a suitably defined quotient. These results provide a new perspective on the structure of Brauer groups in derived settings and offer computational tools for K-theoretic invariants of non-commutative spaces arising in string theory and mirror symmetry.