Contents

# Contents

## Idea

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.

## Definition

### As the universal additive/localizing invariant

The definition in (Blumberg-Gepner-Tabuada 10) is the following.

Write

###### Proposition

The inclusion from def. is a reflective sub-(infinity,1)-category

$Cat^{perf} \stackrel{\overset{Idem}{\longleftarrow}}{\hookrightarrow} Cat^{ex}$

the reflector $Idem$ being idempotent completion.

###### Remark

Passing to ind-objects yields an equivalence of (infinity,1)-categories

$Ind \colon Cat_\infty^{per} \stackrel{\simeq}{\longrightarrow} Pr_{St,cg}^{L} \hookrightarrow Pr_{St}^{L}$
###### Definition

Say that a morphism in $Cat_\infty^{ex}$ is a Morita equivalence if it is an $Idem$-equivalence, hence if it becomes an equivalence of (∞,1)-categories under idempotent completion, prop. .

###### Definition

Say that a sequence in $Cat_\infty^{ex}$ is (split-)exact if it is an exact sequence (…see section 5…) under idempotent completion, prop.

###### Definition

A functor $Cat_\infty^{ex} \to \mathcal{D}$ to a stable presentable (∞,1)-category is called a localizing invariant (additive invariant) if it

1. inverts Morita equivalences, def. ;

2. preserves filtered (∞,1)-colimits;

3. sends (split-) exact sequences, def. , to (split) cofiber sequences (…see section 5…).

###### Definition

The $(\infty,1)$-category $Mot_{add}$ or $Mot_{loc}$, respectively, of noncommutative motives is the universal localizing/additive invariant, def.

$\mathcal{U}_{loc} \colon Cat_\infty^{ex} \to Mot_{loc} \,.$
$\mathcal{U}_{add} \colon Cat_\infty^{ex} \to Mot_{add} \,.$
###### Remark

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).

## Properties

### Relation to algebraic K-theory

###### Theorem

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

$Hom_{Mot_{loc}}(\mathcal{U}_{loc}(\mathcal{B}), \mathcal{U}_{loc}(\mathcal{A})) \simeq \mathbb{K}(\mathcal{B}^{op}\widehat \otimes \mathcal{A}) \,.$

### Relation to correspondences equipped with cocycles

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

$Maps(\mathcal{U}_{loc}(\mathcal{B}), \mathcal{U}_{loc}(\mathcal{A})) \simeq \mathbb{K}(\mathcal{B}^{op}\otimes\mathcal{A}) \,,$

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.

(…)

### Relation to Chow motives

The category of ordinary Chow motives, after factorizing 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.

### Relation to KK-theory

Noncommutative motives receive a universal functor from KK-theory

$KK \longrightarrow NCC_{dg}$

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).

## Applications

geometric contextuniversal additive bivariant (preserves split exact sequences)universal localizing bivariant (preserves all exact sequences in the middle)universal additive invariantuniversal localizing invariant
noncommutative algebraic geometrynoncommutative motives $Mot_{add}$noncommutative motives $Mot_{loc}$algebraic K-theorynon-connective algebraic K-theory
noncommutative topologyKK-theoryE-theoryoperator K-theory

## References

A survey is in

• Goncalo Tabuada, A guided tour through the garden of noncommutative motives, in Guillermo Cortinas, Topics in Noncommutative Geometry Clay Mathematics Proceedings Vol 16, 2012 (arxiv1108.3787);

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

and a further lift of this to (∞,1)-category theory is in

with discussion of the corresponding cyclotomic trace in

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:

Also the lectures notes:

• Marco Robalo, Noncommutative motives and K-theory, talk at [Higher Categories and Topological Quantum Field Theories,]

Vienna, 2013](http://www.mat.univie.ac.at/~favero/Workshops/Higher.html), notes

Another survey article is

Discussion of how the derived category of a scheme determines its commutative and noncommutative Chow motive is in

• Adeel Khan, On derived categories and noncommutative motives of varieties, arXiv.

In

• Snigdhayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality, (pdf, talk notes)

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.

Last revised on August 20, 2020 at 03:55:13. See the history of this page for a list of all contributions to it.