The notion of motives in algebraic geometry can be adapted to derived noncommutative geometry. The idea and the first version has been developed by Maxim Kontsevich. There is a remarkable observation that the category of Chow motives (after localizing at the Lefschetz motive) can be embedded into the category of Kontsevich’s noncommutative motives. More recently this direction has been systematically studied by Cisinski and Tabuada.
There is another approach by Arne Ostvaer.
In noncommutative geometry à la Alain Connes, Connes and Matilde Marcolli have also introduced some motivic ideas. Marcolli also has a recent collaboration with Tabuada on the algebraic side, see her webpage.
The definition in (Blumberg-Gepner-Tabuada 10) is the following.
the reflector being idempotent completion.
Say that a sequence in is (split-)exact if it is an exact sequence (…see section 5…) under idempotent completion, prop. 1
A functor to a stable presentable (∞,1)-category is called a localizing invariant (additive invariant) if it
The localization property here (be additive, invert Morita, preserve split sequences) is of the same form as that which defines the localization of C*-algebras to KK-theory in noncommutative stable homotopy theory. See at KK-theory – Universal characterization. See also (Blumberg-Gepner-Tabuada 10, paragraph 1.5).
(Blumberg-Gepner-Tabuada 10, theorem 9.36)
By (Blumberg-Gepner-Tabuada 10, theorem 9.36), the morphisms of noncommutative motives from to for suitably dualizable/compact are given by
The category of ordinary Chow motives, after factoizing out the action of the Tate motive? essentially sits inside that of noncommutative Chow motives. This is recalled as (Tabuada 11, theorem 4.6). For more see (Tabuada 11 ChowNCG).
This relation is best understood as being exhibited by K-motives, see there.
Noncommutative motives receive a universal functor from KK-theory
Tabuada has used noncommutative motives to compute the cyclic homology of twisted projective homogeneous varieties?. Also, he showed that the noncommutative motive of such a variety is trivial if and only if the Brauer classes? of the associated central simple algebras are trivial. See (Tabuada 13).
|geometric context||universal additive bivariant (preserves split exact sequences)||universal localizing bivariant (preserves all exact sequences in the middle)||universal additive invariant||universal localizing invariant|
|noncommutative algebraic geometry||noncommutative motives||noncommutative motives||algebraic K-theory||non-connective algebraic K-theory|
|noncommutative topology||KK-theory||E-theory||operator K-theory||…|
A survey is in
Discussion of Maxim Kontsevich’s definition of noncommutative motives include
The following article has the treatment of -categories representing smooth, proper, separated etc. noncommutative varieties, notions which are used in Kontsevich’s approach to motives in the above talks.
Gonçalo Tabuada, K-theory via universal invariants, Duke Math. J. 145 (2008), no.1, 121–206.
and a further lift of this to (∞,1)-category theory is in
with discussion of the corresponding cyclotomic trace in
Bivariant cyclic cohomology and Connes’ bilinear pairings in Non-commutative motives, arxiv/1005.2336;
Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives, arxiv/1101.0731;
Matilde Marcolli, Goncalo Tabuada, Kontsevich’s noncommutative numerical motives, arxiv/1108.3785; Noncommutative motives, numerical equivalence, and semi-simplicity, arxiv/1105.2950; Noncommutative numerical motives, Tannakian structures, and motivic Galois groups, arxiv/1110.2438
Also the lectures notes:
Another survey article is
it is shown that there is a universal functor from KK-theory to the category of noncommutative motives, which is the category of dg-categories and dg-profunctors up to homotopy between them. This is given by sending a C*-algebra to the dg-category of perfect complexes of (the unitalization of) its underlying associative algebra.