Theory of noncommutative motives

Abstract

The theory of motives was originally conceived by Alexander Grothendieck as a universal cohomology theory for algebraic varieties. In the decades since it was first introduced, it has become a vast and profoundly sophisticated subject systematically developed in many directions spanning algebraic and arithmetic geometry, homotopy theory and higher category theory. The quest for a fully developed theory of motives as envisioned by Grothendieck drove a great deal of fundamental research in the aforementioned disciplines, while delivering fantastic and long-promised results and settling classical questions as it reached maturity in the past decades. This quest is arguably not complete, since the abelian category of mixed motives, originally established by Grothendieck himself as the ultimate desideratum of a satisfactory theory of motives, has proven elusive. However, ideas of motivic nature as a programmatic approach to cohomology theories and invariants have proven extremely useful in a variety of other contexts. Noncommutative algebraic geometry is precisely one of these contexts. Following ideas of Maxim Kontsevich, Goncalo Tabuada and Marco Robalo independently developed theories of "noncommutative" motives which fully encompasses the classical theory of motives and helps assemble so-called additive invariants such as Algebraic K-Theory, Hochschild Homology and Topological Cyclic Homology into a motivic formalism in the very precise sense of the word. In this expository work, we will review the fundamental concepts at work, which will inevitably involve a foray into the formalism of enhanced and higher categories. We will then discuss Kontsevich's notion of a noncommutative space, sharpened and made precise over the years by Toen, Tabuada, Robalo and others and introduce noncommutative motives as "universal additive invariants" of noncommutative spaces. We will conclude by offering a brief sketch of Robalo's construction of the noncommutative stable homotopy category.