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.
A second approach is due to Bertrand Toën, Michel Vaquié, Gabriele Vezzosi. They construct a motivic stable homotopy theory for noncommutative spaces (in the sense of Kontsevich).
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.
Write
(stable) $Cat_\infty^{stab}$ for the (∞,1)-category of stable (∞,1)-categories;
(small with linear maps) $Cat_\infty^{ex}$ for the $(\infty,1)$-category of small stable $\infty$-categories with finite (infinity,1)-limit/colimit-preserving functors between them;
(idem-complete) $Cat_\infty^{perf} \hookrightarrow Cat_\infty^{ex}$ for the further full sub-(infinity,1)-category of the small stable $\infty$-categories on the idempotent complete (infinity,1)-categories.
(Blumberg-Gepner-Tabuada 10, def. 2.12 and above def 2.14)
The inclusion from def. 1 is a reflective sub-(infinity,1)-category
the reflector $Idem$ being idempotent completion.
Passing to ind-objects yields an equivalence of (infinity,1)-categories
with the stable compactly generated (∞,1)-categories.
Say that a morphism in $Cat_\inty^{ex}$ is a Morita equivalence if it is an $Idem$-equivalence, hence if it becomes an equivalence of (∞,1)-categories under idempotent completion, prop. 1.
Say that a sequence in $Cat_\infty^{ex}$ is (split-)exact if it is an exact sequence (…see section 5…) under idempotent completion, prop. 1
A functor $Cat_\infty^{ex} \to \mathcal{D}$ to a stable presentable (∞,1)-category is called a localizing invariant (additive invariant) if it
inverts Morita equivalences, def. 2;
preserves filtered (∞,1)-colimits;
sends (split-) exact sequences, def. 3, to (split) cofiber sequences (…see section 5…).
The $(\infty,1)$-category $Mot_{add}$ or $Mot_{loc}$, respectively, of noncommutative motives is the universal localizing/additive invariant, def. 4
(Blumberg-Gepner-Tabuada 10, theorem 1.1, section 8)
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).
For $\mathcal{A}, \mathcal{B} \in Cat_\infty^{stab}$ with $\mathcal{B}$ smooth and proper, hence a compact object, then the hom-spectrum in $Mot_{loc}$ between $\mathcal{A}$ and $\mathcal{B}$ is given by the non-connective algebraic K-theory $\mathbb{K}$ of the tensor product in that there is a natural equivalence
(Blumberg-Gepner-Tabuada 10, theorem 9.36)
By (Blumberg-Gepner-Tabuada 10, theorem 9.36), the morphisms of noncommutative motives from $\mathcal{A}$ to $\mathcal{B}$ for $\mathcal{B}$ suitably dualizable/compact are given by
hence by the non-connective algebraic K-theory of the Deligne tensor product of the two categories.
Thinking of these as categories of quasicoherent sheaves on some spaces (by definition in noncommutative algebraic geometry), this are $\mathbb{K}$-cocycles on the product correspondence space.
(…)
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
which is given by sending a C*-algebra to the dg-category of perfect complexes over (the unitalization of) its underlying associative algebra (Mahanta 13).
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).
Bernardara and Tabuada have used noncommutative motives to compute the rational Chow groups of certain complete intersections of curves. See (Bernardara-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 $Mot_{add}$ | noncommutative motives $Mot_{loc}$ | 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
Maxim Kontsevich, Noncommutative motives, talk at the conference on Pierre Deligne’s 61st birthday (2005) (pdf of part of the talk, notes by Zoran Skoda)
Maxim Kontsevich, Geometry and Arithmetic - Non-commutative motives, talk at Institute for Advanced Study October 20, 2005 (video)
The following article has the treatment of $A_\infty$-categories representing smooth, proper, separated etc. noncommutative varieties, notions which are used in Kontsevich’s approach to motives in the above talks.
An abstract characterization of noncommutative motives in dg-category theory and higher algebraic K-theory is in
Denis-Charles Cisinski, Gonçalo Tabuada, Non connective K-theory via universal invariants. Compositio Mathematica 147 (2011), 1281–1320 (arXiv:0903.3717)
Denis-Charles Cisinski, Gonçalo Tabuada, Symmetric monoidal structure on Non-commutative motives, (arxiv/1001.0228)
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
See also
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;
Goncalo Tabuada, Chow motives versus non-commutative motives (arxiv/1103.0200
Goncalo Tabuada, Galois descent of additive invariants, arxiv/1301.1928
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
Ivo Dell'Ambrogio, Gonçalo Tabuada, Tensor triangular geometry of non-commutative motives, arxiv/1104.2761
For the approach of Bertrand Toën-Michel Vaquié-Gabriele Vezzosi, see
and the Ph.D. thesis of Marco Robalo, under the supervision of Bertrand Toën:
Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)
Marco Robalo, Noncommutative Motives II: K-Theory and Noncommutative Motives, June 2013, (arxiv:1306.3795)
Also the lectures notes:
Another survey article is
Discussion of how the derived category of a scheme determines its commutative and noncommutative Chow motive is in
In
it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ 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.
Goncalo Tabuada, Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives, (arXiv).
Marcello Bernardara, Goncalo Tabuada. Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives. (arXiv)