Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Abstract

In this article we further the study of noncommutative numerical motives. By
exploring the change-of-coefficients mechanism, we start by improving some of
our previous main results. Then, making use of the notion of Schur-finiteness,
we prove that the category NNum(k)_F of noncommutative numerical motives is
(neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not
Tannakian. In order to solve this problem we promote periodic cyclic homology
to a well-defined symmetric monoidal functor HP on the category of
noncommutative Chow motives. This allows us to introduce the correct
noncommutative analogues C_NC and D_NC of Grothendieck's standard conjectures C
and D. Assuming C_NC, we prove that NNum(k)_F can be made into a Tannakian
category NNum'(k)_F by modifying its symmetry isomorphism constraints. By
further assuming D_NC, we neutralize the Tannakian category NNum'(k)_F using
HP. Via the (super-)Tannakian formalism, we then obtain well-defined
noncommutative motivic (super-)Galois groups. Finally, making use of
Deligne-Milne's theory of Tate triples, we construct explicit homomorphisms
relating these new noncommutative motivic (super-)Galois groups with the
classical ones.

Full text

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arXiv:1110.2438v1 [math.KT] 11 Oct 2011
NONCOMMUTATIVE NUMERICAL MOTIVES,
TANNAKIAN STRUCTURES, AND
MOTIVIC GALOIS GROUPS
MATILDE MARCOLLI AND GONC ¸ALO TABUADA
Abstract. In this article we further the study of noncommutative numerical
motives, initiated in [29, 30]. By exploring the change-of-coefficients mecha-
nism, we start by improving some of the main results of [29]. Then, making
use of the notion of Schur-finiteness, we prove that the category NNum(k)F
of noncommutative numerical motives is (neutral) super-Tannakian.
the commutative world, NNum(k)F is not Tannakian. In order to solve this
problem we promote periodic cyclic homology to a well-defined symmetric
monoidal functor HP∗ on the category of noncommutative Chow motives.
This allows us to introduce the correct noncommutative analogues CNC and
DNC of Grothendieck’s standard conjectures C and D. Assuming CNC, we
prove that NNum(k)Fcan be made into a Tannakian category NNum†(k)Fby
modifying its symmetry isomorphism constraints. By further assuming DNC,
we neutralize the Tannakian category Num†(k)F using HP∗. Via the (super-
)Tannakian formalism, we then obtain well-defined noncommutative motivic
(super-)Galois groups. Finally, making use of Deligne-Milne’s theory of Tate
triples, we construct explicit homomorphisms relating these new noncommu-
tative motivic (super-)Galois groups with the classical ones.
As in
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Appendix A.
Appendix B.
References
Introduction
Differential graded categories
Noncommutative pure motives
Change of coefficients
Schur-finiteness
Super-Tannakian structure
Periodic cyclic homology
Noncommutative standard conjecture CNC
Tannakian structure
Noncommutative homological motives
Motivic Galois groups
Tannakian formalism
Orbit categories
2
5
5
6
8
9
10
13
14
17
20
25
27
29
Date: October 12, 2011.
2000 Mathematics Subject Classification. 14F40, 18G55, 19D55.
Key words and phrases. Noncommutative motives, periodic cyclic homology, Tannakian for-
malism, motivic Galois groups.
The first named author was supported by the NSF grants DMS-0901221 and DMS-1007207.
The second named author was supported by the J.H. and E.V. Wade award.
1

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2MATILDE MARCOLLI AND GONC ¸ALO TABUADA
1. Introduction
Noncommutative motives. Over the past two decades Bondal, Drinfeld, Kaledin,
Kapranov, Kontsevich, Orlov, Van den Bergh, and others, have been promoting a
broad noncommutative (algebraic) geometry program where “geometry” is per-
formed directly on dg categories; consult [5, 6, 7, 14, 15, 19, 23, 24, 25, 26]. In this
vein, Kontsevich introduced the category NChow(k)F of noncommutative Chow
motives (over a base commutative ring k and with coefficients in a field F); see
§3.1. Recently, making use of Hochschild homology, the authors introduced the
category NNum(k)F of noncommutative numerical motives; see §3.2. Under mild
assumptions on k and F (see Theorem 4.6) the category NNum(k)F is abelian
semi-simple. The precise relation between NChow(k)Q, NNum(k)Q, and the clas-
sical categories of Chow and numerical motives (with rational coefficients) can be
depicted as follows
(1.1)
Chow(k)Q
?????????????
Num(k)Q
Chow(k)Q/−⊗Q(1)
????????????
Num(k)Q/−⊗Q(1)
RN
τ??
τ??
R
??NChow(k)Q
???????????
??NNum(k)Q
.
Here, NChow(k)Q/−⊗Q(1)and Num(k)Q/−⊗Q(1)are the orbit categories associated
to the auto-equivalence −⊗Q(1) (see Appendix B) and R and RN are fully-faithful
functors; consult [29] (or the survey [34]) for further details.
Motivating questions. In the commutative world, the category Num(k)F of nu-
merical motives is known to be not only abelian semi-simple but also (neutral)
super-Tannakian. Moreover, assuming the standard conjecture C (or even the sign
conjecture C+), Num(k)F can be made into a Tannakian category Num†(k)F by
modifying its symmetry isomorphism constraints; see Jannsen [18]. By further as-
suming the standard conjecture D, the classical Weil cohomologies can be used to
neutralize the Tannakian category Num†(k)F; see [2, §6]. As explained in Appen-
dix A, the (super-)Tannakian formalism provides us then with well-defined motivic
(super-)Galois groups sGal(Num(k)F) and Gal(Num†(k)F) encoding deep arith-
metic/geometric properties of smooth projective k-schemes. This circle of results
and conjectures in the commutative world leads us naturally to the following ques-
tions in the noncommutative world:
Question I: Is the category NNum(k)F (neutral) super-Tannakian ?
Question II: Does the standard conjecture C (or the sign conjecture C+) admits
a noncommutative analogue CNC ? Does CNC allows us to make NNum(k)F into
a Tannakian category NNum†(k)F ?
Question III: Does the standard conjecture D admits a noncommutative ana-
logue DNC ? Does DNC allows us to neutralize NNum†(k)F ?
Question IV: Assuming that Questions I, II and III have affirmative answers,
how do the noncommutative motivic (super-)Galois groups hence obtained relate
with sGal(Num(k)F) and Gal(Num†(k)F) ?

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NONCOMMUTATIVE MOTIVES, TANNAKIAN STRUCTURES, AND GALOIS GROUPS3
Statement of results. By exploring the change-of-coefficients mechanism we start
by improving the main results of [29] concerning the semi-simplicity of the category
NNum(k)F and the relation between the commutative and the noncommutative
world; see Proposition 4.4 and Theorem 4.6. Then, making use of the notion of
Schur-finiteness (see §5), we answer affirmatively Question I.
Theorem 1.2. Assume that F is a field of characteristic zero and that k and F
are as in Theorem 4.6. Then, the category NNum(k)F is super-Tannakian. If F is
moreover algebraically closed, then NNum(k)F is neutral super-Tannakian.
Theorem 1.2 (with F algebraically closed) combined with the super-Tannakian
formalism gives rise to a super-Galois group scheme sGal(NNum(k)F), which we will
call the noncommutative motivic super-Galois group. Among other consequences,
NNum(k)F is ⊗-equivalent to the category of finite dimensional F-valued super-
representations of sGal(NNum(k)F).
The category NNum(k)F is not Tannakian since the rank of each one of its
objects is not necessarily a non-negative integer. In order to solve this problem, we
start by promoting periodic cyclic homology to a well-defined symmetric monoidal
functor HP∗on the category of noncommutative Chow motives; see Theorem 7.2.
Then, given a smooth and proper dg category A in the sense of Kontsevich (see
§2.1), we formulate the following conjecture:
Noncommutative standard conjecture CNC(A): The K¨ unneth projectors
π±
A: HP∗(A) ։ HP∗
±(A) ֒→ HP∗(A)
are algebraic, i.e. they can be written as π±
correspondences.
As in the commutative world, the noncommutative standard conjecture CNC
is stable under tensor products, i.e. CNC(A) + CNC(B) ⇒ CNC(A ⊗kB); see
Proposition 8.2. Its relation with the sign conjecture C+(see §8) is the following:
given a quasi-compact and quasi-separated k-scheme Z, it is well-known that the
derived category Dperf(Z) of perfect complexes of OZ-modules admits a natural dg
enhancement Ddg
perf(Z); consult Lunts-Orlov [28] (or [10, Example 4.5]). When Z
is moreover smooth and proper, the dg category Ddg
the sense of Kontsevich.
A= HP∗(π±
A), with π±
Anoncommutative
perf(Z) is smooth and proper in
Theorem 1.3. Let k and F be fields of characteristic zero with k a field extension
of F. Then, C+(Z) ⇒ CNC(Ddg
perf(Z)).
Intuitively speaking, Theorem 1.3 shows us that when restricted to the com-
mutative world, the noncommutative standard conjecture CNC is more likely to
hold than the sign conjecture C+(and therefore than the standard conjecture C).
Hence, it answers affirmatively to the first part of Question II. Our answer to the
second part is the following:
Theorem 1.4. Assume that k is a field of characteristic zero, and that F is a field
extension of k or vice-versa. Then, if the noncommutative standard conjecture CNC
holds, the category NNum(k)F can be made into a Tannakian category NNum†(k)F
by modifying its symmetry isomorphism constraints.
In order to answer Question III, we start by observing that the F-linearized
Grothendieck group K0(A)F of every smooth and proper dg category A is endowed

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4 MATILDE MARCOLLI AND GONC ¸ALO TABUADA
with two well-defined equivalence relations: one associated to periodic cyclic homol-
ogy (∼hom) and another one associated to numerical equivalence (∼num); consult
§10 for details. This motivates the following conjecture:
Noncommutative standard conjecture DNC(A): The following equality holds
K0(A)F/∼hom= K0(A)F/∼num .
Its relation with the standard conjecture D (see §10) is the following.
Theorem 1.5. Let k and F be fields of characteristic zero with k a field extension
of F. Then, D(Z) ⇒ DNC(Ddg
perf(Z)).
Similarly to Theorem 1.3, Theorem 1.5 shows us that when restricted to the
commutative world, the noncommutative standard conjecture DNC is more likely
to hold than the standard conjecture D. Hence, it answers affirmatively to the first
part of Question III. Our answer to the second part is the following:
Theorem 1.6. Let k be a field of characteristic zero and F a field extension of k.
If the noncommutative standard conjectures CNCand DNChold, then NNum†(k)F
is neutral Tannakian category.Moreover, an explicit fiber functor neutralizing
NNum†(k)F is given by periodic cyclic homology.
Theorem 1.6 combined with the Tannakian formalism gives rise to a Galois
group scheme Gal(NNum†(k)F), which we will call the noncommutative motivic
Galois group. Since by Theorem 4.6 the category NNum†(k) is not only abelian
but moreover semi-simple, the Galois group Gal(NNum†(k)F) is pro-reductive, i.e.
its unipotent radical is trivial; see [2, §2.3.2]. Similarly to the super-Tannakian
case, the category NNum†(k)F is ⊗-equivalent to the category of finite dimensional
F-valued representations of Gal(NNum†(k)F).
Finally, making use of all the above results as well as of the Deligne-Milne’s
theory of Tate triples, we answer Question IV as follows:
Theorem 1.7. Let k be a field of characteristic zero. If the standard conjectures
C and D as well as the noncommutative standard conjectures CNCand DNChold,
then there exist well-defined surjective group homomorphisms
(1.8)sGal(NNum(k)k) ։ Ker(t : sGal(Num(k)k) ։ Gm)
(1.9)Gal(NNum†(k)k) ։ Ker(t : Gal(Num†(k)k) ։ Gm).
Here, Gm denotes the multiplicative group, t is induced by the inclusion of the
category of Tate motives, and the (super-)Galois groups are computed with respect
to periodic cyclic homology and de Rham cohomology.
Theorem 1.7 was sketched by Kontsevich in [23]. Intuitively speaking, it shows
us that the ⊗-symmetries of the commutative world which can be lifted to the
noncommutative world are precisely those that become trivial when restricted to
Tate motives. The difficulty of replacing k by a more general field of coefficients F
relies on the lack of an appropriate “noncommutative ´ etale/Betti cohomology”.
In Appendix A we collect the main results of the (super-)Tannakian formalism
and in Appendix B (which is of independent interest) we recall the notion of orbit
category and describe its behavior with respect to four distinct operations.
Acknowledgments: The authors are very grateful Eric Friedlander, Dmitry Kaledin
and Yuri Manin for useful discussions.

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NONCOMMUTATIVE MOTIVES, TANNAKIAN STRUCTURES, AND GALOIS GROUPS5
Conventions: Throughout the article, we will reserve the letter k for the base
commutative ring and the letter F for the field of coefficients. The pseudo-abelian
envelope construction will be denoted by (−)♮.
2. Differential graded categories
In this section we collect the notions and results concerning dg categories which
are used throughout the article. For further details we invite the reader to consult
Keller’s ICM address [21]. Let C(k) the category of (unbounded) cochain complexes
of k-modules. A differential graded (=dg) category A is a category enriched over
C(k). Concretely, the morphisms sets A(x,y) are complexes of k-modules and the
composition operation fulfills the Leibniz rule: d(f◦g) = d(f)◦g+(−1)deg(f)f◦d(g).
A dg functor is a functor which preserves the differential graded structure. The
category of small dg categories will be denoted by dgcat(k).
Let A be a (fixed) dg category. Its opposite dg category Aophas the same objects
and complexes of morphisms given by Aop(x,y) := A(y,x). A right dg A-module (or
simply an A-module) is a dg functor Aop→ Cdg(k) with values in the dg category
of complexes of k-modules. We will denote by C(A) the category of A-modules
and by D(A) the derived category of A, i.e. the localization of C(A) with respect to
the class of quasi-isomorphisms; consult [21, §3] for details. The full triangulated
subcategory of D(A) formed by its compact objects [33, Def. 4.2.7] will be denoted
by Dc(A).
As proved in [36, Thm. 5.3], the category dgcat(k) carries a Quillen model struc-
ture whose weak equivalence are the derived Morita equivalences, i.e. the dg func-
tors A → B which induce an equivalence D(A)
categories. The homotopy category hence obtained will be denoted by Hmo(k).
The tensor product of k-algebras extends naturally to dg categories, giving rise to
a symmetric monoidal structure −⊗k− on dgcat(k). The ⊗-unit is the dg category
k with one object and with k as the dg algebra of endomorphisms (concentrated
in degree zero). As explained in [21, §4.3], the tensor product of dg categories can
be naturally derived −⊗L− giving thus rise to a symmetric monoidal structure on
Hmo(k).
Let A and B be two dg categories. A A-B-bimodule is a dg functor A ⊗L
Cdg(k), or in other words a (Aop⊗L
subcategory of D(Aop⊗L
kB) spanned by the (cofibrant) A-B-bimodules X such that
for every object x ∈ A the associated B-module X(x,−) belongs to Dc(B).
∼
→ D(B) on the associated derived
kBop→
kB)-module. Let rep(A,B) be the full triangulated
2.1. Smooth and proper dg categories. Following Kontsevich [23, 24], a dg
category A is called smooth if the A-A-bimodule
A(−,−) : A ⊗L
kAop→ Cdg(k)(x,y) ?→ A(x,y)
belongs to Dc(Aop⊗L
the complex A(x,y) of k-modules belongs to Dc(k). As proved in [10, Thm. 4.8], the
smooth and proper dg categories can be conceptually characterized as being pre-
cisely the dualizable (or rigid) objects of the symmetric monoidal category Hmo(k).
kA). It is called proper if for each ordered pair of objects (x,y),
3. Noncommutative pure motives
In this section we recall the construction of the categories of noncommutative
Chow and numerical motives.