Cycle Maps on Cohomology Theories for Dg-Categories and Their Applications

Cycle maps on cohomology theories for dg-categories and their applications

Satoshi Mochizuki

Abstract In this article, we propose noncommutative versions of Tate conjecture and Hodge conjecture. If we consider these conjectures for a dg-category of perfect complexes over a certain schemes X, then they are equivalent to the classical Tate and Hodge conjectures for X respectively. We also propose a strategy of how to prove these conjectures by utilizing a version of motivic Bass conjecture.

Introduction

For a projective and smooth variety X over a field k, the i-th Chow group CHi(X) is generated by cycles of codimension i on X modulo rational equivalence. (i) If k = C the field of complex numbers, then we can define the cycle class morphism from the i-th Chow i i 2i group into the 2i-th Betti cohomology group with rational coefficients ρ (X)Q : CH (X) ⊗ Q → H (X,Q). Since X is a Kahler¨ manifold, there exists a decomposition of its cohomology with complex coefficients

k M p,q H (X,C) = H (X) p+q=k

where Hp,q(X) is the subgroups of cohomology classes which are represented by harmonic form of type (p,q). i 2i i,i 2i We set Hdg (X) := H (X,Q) ∩ H (X)(⊂ H (X,C)) and we call it the group of Hodge classes of degree 2i i i on X. We can show that Imρ (X)Q ⊂ Hdg (X). In [Hod50], Hodge conjectured the following: i i Conjecture 1 (Hodge conjecture). For each i, we have an equality Imρ (X)Q = Hdg (X). (ii) If a prime l is invertible in k, we can deﬁne the cycle class morphism from the i-th Chow group into the i i 2i 2i-th l-adic etale´ cohomology group with i-th Tate twist coefﬁcient ρ (X)Q : CH (X) ⊗ Q → Het´ (X,Ql(i)). i i 2k It can extends to Qi-linear morphism ρ (X)Ql : CH (X) ⊗ Ql → Het´ (X,Ql(i)). In [Tat65], Tate conjectured following:

i Conjecture 2 (Tate conjecture). For any i, the image of the cycle class morphism ρ (X ×k Speck¯) in the Ql arXiv:2002.04373v1 [math.AG] 11 Feb 2020 2i ¯ cohomology group He´t(X ×k Speck,Qi(i)) is exactly the union of ﬁxed parts of open subgroups of the Galois group Gal(k¯/k).

m (iii) If k is a finite field Fq of characteristic p with q = p , then there exits a p-adic version of Tate conjecture [Mil07]. Let W(k) be the associated ring of p-typical Witt vectors and we set K := W(k)[1/p] the fraction ∗ ∗ field of W(k) and Hcrys(X) := Hcrys(X/W(k)) ⊗W(k) K the crystalline cohomology groups of X.

∗ 2∗ Frp Conjecture 3 (p-adic Tate conjecture). The cycle class map CH (X) ⊗ Qp → Hcrys(X)(∗) with values in the Qp-vector subspace of those elements which are fixed by the crystalline Frobenius Frp is surjective. In this article, we propose noncommutative versions of Tate conjecture 6 and Hodge conjecture 10. If we consider these conjectures for a dg-category of perfect complexes over a certain schemes X, then they are equivalent to the classical Tate and Hodge conjectures for X respectively. For Tate conjectures, in [Tab18], Tabuada already formulated noncommutative Tate conjecture and p-adic Tate conjecture for saturated dg- categories over a finite field of characteristic p. His modi operandi are founded upon Thomason’s result in [Tho89] and calculations of topological cyclic homology respectively (see § 2.3 and § 2.4). The key ingredi- ent of our method in this article is the l-adic Chern character established in [TV19]. For Hodge conjecture,

1 in [KKP08], Katzarkov, Kontsevich and Pantev discussed the Hodge theory of algebraic noncommutative spaces and Kontsevich stated a version of noncommutative Hodge conjecture in [Kon08]. Our handiwork in this article is different from it and is based upon the topological K-theory and the Deligne-Beilinson cohomology theory for dg-categories over C studied in [Bla15]. We also propose a strategy of how to prove these conjectures by utilizing a version of motivic Bass conjecture.

Conventions. p (i) For a subring A ⊂ R and non-negative integer p, we set A(p) := (2πi) A(⊂ R).

(ii) For an abelian group H and a unital, associative and commutative ring B, we write HB for H⊗ZB. For a category C enriched over the category of abelian groups, we write C B for the category such that

ObC B = ObC and HomC B (x,y) = HomC (x,y)B for any x and y ∈ ObC B. p (iii) For a family of objects {E }p∈I in an additive category indexed by a subset I of Z the ring of integers, ∗ M p p we set E := E . For a family of objects {E (q)}(p,q)∈J in an additive category indexed by a subset p∈I 2∗ M 2p J of Z×Z the set of ordered pair of integers, we set E (∗) := E (p). p∈Z (2p,p)∈J (iv) For a cochain complex E = E∗ in an additive category closed under inﬁnite products, we adopt the notation E∗[u±] := ∏E∗[−2i]. i∈Z 1 Images of cycle maps

Let A be a commutative associative unital ring and we set S = SpecA. By a dg-category (over A), we mean a small category enriched over the symmetric monoidal category Ch(A) of chain complexes of A-modules. We denote the category of small A-linear dg-categories over A and A-linear dg-functors by dgCatA and the stable ∞-category of spectra by Sp. A dg-functor f : A → B is a Morita equivalence if it induces an equivalence of triangulated categories L f! : D(A ) → D(B) on derived categories. We denote the ∞-category of the localization of dgCatA with respect to Morita equivalences by dgCatA.

1.1 Localizing invariants

For a stable presentable ∞-category D, an ∞-functor E : dgCatA → D is called a localizing invariant if it preserves filtered colimits and satisfies localization. We explain the last condition more precisely. A sequence i p of triangulated categories T → T 0 → T 00 is exact if pi ' 0 and if i is fully-faithful and if every objects 00 0 in T is a direct summand of some object in T /T . We say that a sequence A → B → C in dgCatA is called exact if the induced sequence of triangulated categories of derived categories of compact objects c c c D A → D B → D C is exact. We say that a ∞-functor E : dgCatA → D satisfies localization if E sends an exact sequence of dg-categories A → B → C to a cofiber sequence E(A ) → E(B) → E(C ) in D. We denote the ∞-category of localizing invariant to D by Funloc(dgCatA,D). In [CT11] and [CT12], Cisinski and Tabuada constructed the stable presentable ∞-category MotA and a localizing invariant U : dgCatA → MotA which satisfies the following universality: For a stable presentable ∞-category D, we have induced equivalence of ∞-categories

∗ L U : Fun (MotA,D) ' Funloc(dgCatA,D) where the left-hand side denotes the ∞-category of colimit-preserving functors. In the literature [CT12, 7.5], universal property is written by the language of derivators. But by utilizing [Coh13], we can translate universal property of MotA by the language of ∞-categories as in [BGT13]. Typical examples of localizing invariants dgCatA → Sp are K, HK, HN and HP the non-connective K- theory, the homotopy K-theory, the negative cyclic homology theory and the periodic cyclic homology theory respectively. There exists natural maps K → HK, ch: K → HN and HN → HP.

2 (i) We let l be a prime number invertible in A. We denote the ∞-category of constructible Ql-complexes on the etale´ site Set´ of S by L et´ (Set´ ,l). It is a symmetric monoidal ∞-category and we denote by S hA,Ql := M ⊗n −1 Ind(L et´ (Set´ ,l)). We let T := Ql[2](1) and we consider the E∞-ring object Ql(β) := T (= Ql[β,β ]) in n∈Z h . We denote the ∞-category of ( )-modules by Mod ( h ). In [BRTV18], we construct the S A,Ql Ql β Ql (β) S A,Ql ∞-functors r : dgCat → Mod ( h ) which is called the l-adic realization functor and the geometric l A Ql (β) S A,Ql realization functor | − |: S hA,Ql → Sp. Moreover we define a natural transformation chl : HK(−) → |rl(−)| from the homotopy invariant K-theory HK to |rl(−)| which we call l-adic Chern character (see [TV19, 2.3.1]). (ii) If A = , in [Bla15], Blanc constructed a localizing invariant top : dgCat → Sp the topological K- C K C top ch top ch theory and showed that the composition K → HN → HP factors through K → K → HP. We define top top ch Deligne-Beilinson cohomology HBD : dgCatA → Sp by the pull-back of the diagram K → HP ← HN in Funloc(dgCatA,Sp) (compare [Bla15, 4.33]). By definition, there exists the natural maps chBD : K → HBD top and pr top :H → . K BD K

1.2 Relation between the classical and noncommutative motive theories In this subsection, we recall relationship between the classical and noncommutative motive theories. First recall that Kontsevich introduce the triangulated category NMMA of noncommutative mixed motives over A in [Kon05], [Kon06], [Kon09] and [Kon10]. We say that a triangulated full subcategory of a triangulated category is thick if it is closed under direct summands. This notion is equivalent to the notion of epaisse´ subcategories of triangulated categories in the sense of Verdier [Ver77] by Rickard’s criterion [Ric89, 1.3]. loc There exists a natural fully faithful embedding NMMA into MotA whose essential image is the thick triangulated subcategory spanned by U (A ) for all saturated dg-categories A (see [CT12, 8.5]). Here recall that a dg-category A is saturated if for any two objects x and y in A , the complex A (x,y) is a perfect op op complex of A-modules and the right dg A ⊗L A -module A (−,−): A ⊗L A → Chdg(A) is a perfect op dg A ⊗L A -module where Chdg(A) is the dg-category of chain complexes of A-modules. We denote the category of projective and smooth varieties over A by ProjSmA. For any projective and smooth scheme X over A, let PerfX be the dg-category of perfect complexes of quasi-coherent sheaves on X which is fibrant with respect to the model structure on Ch(QcohX ) the category of chain complexes of quasi-coherent sheaves on X whose cofibrations are monomorphisms and whose weak-equivalences are quasi-isomorphisms. Then by the work of Toen¨ and Vaquie´ [TV07, 3.27], the dg-category PerfX is saturated. In particular U (PerfX ) is in NMMR. If a category which we concern admits both structures of a symmetric monoidal category and of a triangulated category, then May proposed suitable compatibility axioms of both structures in [May01, §4]. For example, since Ho(MotA) is a homotopy category of a certain symmetric monoidal model category, as illus- trated in [May01, §6] we can show that the symmetric monoidal triangulated category Ho(MotA) satisfies the compatibility axioms (TC1), (TC2) and (TC3) in [May01, p.47-49]. Since NMMA is a thick subcategory of Ho(MotA), NMMA also satisfies these axioms. Next we review the notion of dualizable objects in symmetric monoidal categories. An object x in a symmetric monoidal category R is called dualizable if there exists an object x∨ in R and morphisms ev: x ⊗ ∼ id⊗ ev⊗id ∼ ∼ x∨ → 1 and δ : 1 → x∨ ⊗ x such that compositions x → x ⊗ 1 →δ x ⊗ x∨ ⊗ x → 1 ⊗ x → x and x∨ → ⊗ ⊗ ∼ 1 ⊗ x∨ δ→id x∨ ⊗ x ⊗ x∨ id→ev x∨ ⊗ 1 → x∨ are identities. The object x∨ is called the dual of x. A typical example of dualizing objects is a non-commutative motive U (A ) in Ho(MotA) associated with a saturated dg-category A over A by [CT12, 4.8]. In particular by lemma 4 below, all objects in NMMA are dualizable. Lemma 4. Let T be an idempotent complete triangulated category. Then (i) If x ⊕ y is dualizable, then x and y are also dualizable. Moreover assume that T satisfies May’s axioms (TC1), (TC2) and (TC3) in [May01, p.47-49]. Then (ii) ([Voe10, 8.3]). In a distinguished triangle x → y → z → Σx in T , if two of x, y and z are dualizable, then third one is also.

(iii) In particular, If x is dualizable, then Σnx is dualizable for any integer n.

3 ∨ Thus we obtain the functor NM: ProjSmA → NMMA which sends an object X to U (Perfdg X) . For a projective and smooth scheme X over A, we call NM(X) the noncommutative motive associated with X. Next we define K-motive functor. Let SH A be a ∞-category of Morel-Voevodsky motivic stable homotopy category over SpecA. We denote the homotopy K-theory spectrum in SH A by KGL and we write ModKGL(SH A) for the ∞-category of KGL-modules. We define KM: ProjSmA → Ho(ModKGL(SH A)) ∞ to be a functor by sending a scheme X in ProjSmA to Σ X+ ∧ KGL in Ho(ModKGL(SH A)) the homotopy category of KGL-modules. For a projective and smooth scheme X over A, we call KM(X) the K-motive associated with X. In [Tab14], Tabuada constructed the comparison functor between KM and NM. We denote the small- est triangulated category which contains NMMA and closed under infinite direct sums in Ho(MotA) by ⊕ NMMA . If A = k is a perfect field, then he construct the fully faithful symmetric monoidal triangulated ⊕ functor Φ: Ho(ModKGL(SH k)) → NMM which makes the diagram below commutative Q A,Q

ProjSmA

KMQ NMQ w $ ⊕ Ho(ModKGL(SH k)) / NMMA, . Q Φ Q

As a formal consequence, we obtain the following Proposition.

Proposition 5. Let k be a perfect ﬁeld and let E : dgCatk → Sp be a localizing invariant and let chE : K → E be a map of localizing invariants. Assume that there exists a motivic spectrum L in SH k such that 2∗ (i) L (∗) admits a KGLQ-module structure, 2∗ (ii) Restriction of E on NMMk is represented by L (∗). Namely we have an equality 2∗ E ' Hom ⊕ (−,Φ(L (∗)). NMMk

2∗ (iii) Restriction of chE on Ho(ModKGL(SH k)) corresponds to the structure map chL : KGLQ → L (∗). Then for any projective and smooth variety X, the following diagram is commutative

π0(clE (Perfdg X)) (π0(K(Perfdg X)) / π0(E(Perfdg X)) (1) Q Q

o o CH∗(X) / L2∗(X,∗) . Q Q clL

1.3 Noncommutative Tate conjecture Let k be a countable perfect ﬁeld and let l be a prime number invertible in k. Then for any projective and 2∗ smooth variety X over k, we have isomorphisms π0(|rl(Perfdg X)|) ' Het´ (X,Ql(∗)) (see [TV19, §2.2, §2.3]) and HK(Perfdg X) ' K(X) (see for example [Wei89, p.477] or [Cis13, 2.3]). On the other hand, there exists p,q p ¯ a motivic spectrum E et´ ,l in SH k such that E et´ ,l (X) ' Het´ (X ×k k,Ql(q)) for any projective smooth variety 2∗ over k (see [CD12, §2.1.5, §3.3]). E et´ ,l(∗) admits a unique KGLQ-module structure (see [TV19, §2.2]). From these observations and Proposition 5, we propose the following conjecture: Conjecture 6 (Noncommutative Tate conjecture). Let k be a perfect ﬁeld and let l be a prime number invertible in k and let A be a saturated dg-category over k. Then the image of

chl ⊗kk¯ : HK (A ⊗kk¯) → π (|rl(A ⊗kk¯)|) 0 Ql 0 is just the Gal(k¯/k)-invariant part of π0(|rl(A ⊗kk¯)|).

4 Argument above implies the following. Corollary 7. Let X be a projective and smooth variety over a countable perfect ﬁeld k. Let l be a prime number invertible in k. Then Tate conjecture for X and l is equivalent to Tate conjecture for Perfdg X and l.

1.4 Noncommutative Hodge conjecture

Let X be a projective and smooth variety over C of dimension d. We have an isomorphism π0 HBD(Perfdg X)) = 2∗ HBD(X,Q(∗)) (see [Bla15, 4.36]). On the other hand, there exists a motivic spectrum H BD in SH such p,q p C that H (X) ' H (X, (q)) and H 2∗(∗) admits a KGL-module structure (see [HS15, 3.6]). BD Q BD Q Q i i− 2i−1 i 2i−1 For an integer 0 ≤ i ≤ d, we set Jac (X) := H2 1(X,C)/(H (X,Z(i)) + F H (X,C)) and call it the i-th intermediate Jacobian of X. There exists a short exact sequence (see [EV88, 7.9]). 0 → Jac∗(X) → H2∗ (X, (∗)) → Hdg∗(X) → 0 (2) Q BD Q top Recall that there exists a natural map pr top :H → . K BD K Deﬁnition 8 (Noncommutative Jacobian, Noncommutative Hodge class group). For a saturated dg-category , we set Jac( ) := Kerπ (pr top ( )) and Hdg( ) := Imπ (pr top ( )) and call them the Jacobian of A A 0 K A A 0 K A A and the Hodge class group of A respectively. By deﬁnition, there exists a short exact sequence

0 → Jac(A ) → HBD(A ) → Hdg(A ) → 0. (3) The terminologies above are justiﬁed by the following lemma. Lemma 9. Let X be a projective and smooth variety over C. Then there exists canonical isomorphisms ∗ ∗ Jac(Perfdg X) ' Jac (X) and Hdg(Perfdg X) ' Hdg (X) which makes diagram below commutative: Q Q Q Q

0 0 / Jac(Perfdg X) / H (Perfdg X) / Hdg(Perfdg X) / 0 (4) Q BD Q Q

o o o 0 / Jac∗(X) / H2∗ (X, (∗)) / Hdg∗(X) / 0. Q BD Q

Proof. We recall that i-th Beilinson-Deligne complex of sheaves Q(i)BD(X) is given by the following formula i ∗ ∗ Q(i)BD(X) = Cone(Q(i)X ⊕ F ΩX → ΩX ). Thus there exists a commutative diagram of distinguished triangles

∗ i ∗ Q(i)BD(X) / QX / ΩX /F ΩX / ΣQ(i)D (X)

o i ∗ ∗ ∗ i ∗ i ∗ F ΩX / ΩX / ΩX /F ΩX / ΣF ΩX

−∗ in the derived category of Q-sheaves on X. We denote HP∗(X), HN∗(X) and HBetti(X,Q) for the complex of Z-modules which calculates the periodic cyclic homology and the negative K-theory and rational Betti −∗ ∗ ± cohomology complex of X respectively. There exists canonical isomorphisms HP∗(X) ' H (X,ΩX )[u ] and −∗ i ∗ ∗ ∗ HN∗(X) ' ∏H (X,F ΩX )[−2i] (see [Wei97, 2.7, 3.3]). We set HBD(X,Q) := H (X,∏Q(i)BD(X)[2i]). i≤0 i≥0 By taking the hyper-cohomology on X, applying the shift [−2i] and taking the product on all integer i, we obtain the commutative diagram of distinguished triangles

−∗ −∗ ± −∗ ∗ i ∗ −∗ HBD(X,Q) / HBetti(X,Q)[u ] / ∏i H (X,ΩX /F ΩX )[−2i] / ΣHBD(X,Q)

o −∗ ∗ i ∗ ∗ HN∗(X) / HP∗(X) / ∏i H (X,ΩX /F ΩX )[−2i] / ΣHN∗(X)

5 in the derived category of Q-vector spaces. By applying EM: D(Ab) → Ho(Sp) the Eilenberg-Maclane top functor (see [SS03, B.1]) to the top distinguished triangle above and by the isomorphism K (X)∧S EMQ ' −∗ ± EM(HBetti(X,Q)[u ]) (see in the proof of Proposition 4.36 in [Bla15]), we obtain the distinguished triangle

pr top ∧EMQ K top −∗ ∗ i ∗ HBD(Perfdg X)∧EMQ → K (Perfdg X)∧EMQ→EM(∏i H (X,ΩX /F ΩX )[−2i])→ΣHBD(Perfdg X)∧EMQ. ∗ By taking π0, we obtain the canonical isomorphism Hdg(Perfdg X) ' Hdg (X) which makes the diagram Q Q below commutative

π0 HBD(Perfdg X) / Hdg(Perfdg X) Q Q

o o H2∗ (X, (∗)) / Hdg(X) . BD Q Q ∗ Thus by short exact sequences (2) and (3), we obtain the desired isomorphism Jac(Perfdg X) ' Jac (X) . Q Q

From observations above and Proposition 5, we propose the following conjecture: Conjecture 10 (Noncommutative Hodge conjecture). For a saturated dg-category A over C, the cycle map K ( ) → Hdg( ) is surjective. 0 A Q A Q By argument above, we obtain the following: Corollary 11. Let X be a projective and smooth variety over C. Then Hodge conjecture for X is equivalent to Hodge conjecture for Perfdg X.

2 Strategy of how to prove conjectures

In the article [MY20], we give a proof of the following theorem (compare [RS19]). It is a version of motivic Bass conjecture (see [MY20]). For a dg-algebra B over A, We write B for the dg-category with the one object ∗ and such that B(∗,∗) = B. We denote U (A) in Ho(MotA) the homotopy category of MotA by 1. Theorem 12. For any saturated dg-category A over A, there exists a non-negative integer n such that U (A ) ⊕n is a direct summand of 1 in Ho(MotA) the homotopy category of MotA. Theorem 12 enable us that questions for saturated dg-categories over A or proper and smooth schemes over SpecA reduce to questions for nonconnective K-theory of the base ring A. In this section, we will illustrate how to reduce several such questions to the questions for the base ring A. Proof of Conjecture 6 and Conjecture 10. Since for Conjecture 6 and Conjecture 10, questions are closed under direct sums and direct summands, we shall assume A = k where k is a base ﬁeld and in this case conjectures are well-known. In particular, by Corollary 7 and Corollary 11, we obtain the following corollaries. Corollary 13 (Hodge conjecture). Hodge conjecture is true. Corollary 14 (Tate conjecture). For a projective and smooth variety X over a countable perfect ﬁeld k and for a prime number l invertible in k, Tate conjecture is true.

2.1 Lattice conjecture The following conjecture is called the lattice conjecture (see [Kal11, 8.6] and [Bla15, 4.25]). Corollary 15 (Lattice conjecture). Let A be a saturated dg-category. Then the map top top ch ∧S EMC: K (A ) ∧S EMC → HP(A ) is an equivalence where EMC stands for the Eilenberg-Maclane spectrum associated to C. top Proof. Since K and HP factor through MotC and since this question is closed under direct sums and direct summands, we shall assume that A = C. In this case, the result is known (see remark after Conjecture 7.25 in [Bla15]).

6 2.2 Noncommutative Parshin conjecture In [Gei08], [Gei15], Geisser proved that Tate conjecture over finite fields implies Parshin conjecture. Kahn also gives a similar remark about logical connections between Bass conjecture and Parshin conjecture for proper and smooth varieties over finite fields and Beilinson-Soule´ vanishing conjecture [Sou85] (see for example [Kah05, Theorem 39 and p.396]). In this subsection, by using (a version of) motivic Bass conjecture, we will also show the following noncommutative Parshin conjecture which is proposed by Tabuada in [Tab18]. By virtue of Tabuada’s result [Tab18, 1.3], it implies the original Parshin conjecture (more precise statement see below).

Corollary 16 (Noncommutative Parshin conjecture). Let A be a saturated dg-category over a ﬁnite ﬁeld Fq. Then Kn(A )Q = 0 for every n 6= 0 where Kn(A ) stands for the n-th non-connective algebraic K-theory of A .

Proof. Let A be a saturated dg-category over a finite field Fq. Since the question is closed under direct sums and direct summands, by Theorem 12, we shall assume that A = Fq. Then by calculation of algebraic K-theory of finite fields by Quillen [Qui72], we obtain the result. In particular we will obtain the following original Parshin conjecture.

Corollary 17 (Parshin conjecture). Let X be a proper and smooth variety over a ﬁnite ﬁeld. Then i-th algebraic K-group K (X) of X and j-th motivic cohomology group H j (X, (k)) are trivial for all positive i Q M Q integer i > 0 and a pair of integers j 6= 2k. Similarly we obtain the following:

Corollary 18. Assume that R is a regular ring. Then for any saturated dg-category A over R, the −n-th non-connective K-group K−n(A ) of A is trivial for any n > 0. Proof. As in the proof of non-connective Parshin conjecture, we shall assume that A = R. Then the result is well-known (see for example [TT90, Proposition 6.8 (b)] or [Sch06, Example 9.5]).

2.3 Noncommutative Thomason-Tate conjecture Thomason tried to show that Bass conjecture for projective and smooth schemes over finite fields implies Tate conjecture for varieties over finite fields and as its working hypothesis, in the article [Tho89], he stated the equivalent statement of Tate conjecture over finite fields. More precisely Thomason showed that the following statement equivalent to the Tate conjecture on X for all i. We say that an abelian group A is l-reducible if the group Hom(Z(l∞),A) is trivial where Z(l∞) stands for the the Prufer¨ l-group, namely A has no non-trivial l-divisible elements.

n Conjecture 19 (Thomason conjecture). For all n, the group π−1LKU K(X ×Fq SpecFq ) is l-reducible where n LKU K(X ×Fq SpecFq ) stands for the Bousﬁeld localization of the algebraic K-theory spectrum of X ×Fq SpecFqn with respect to topological complex K-theory KU. Here is a noncommutative version of Thomason conjecture which is proposed by Tabuada in [Tab18].

Tabuada showed that for a proper and smooth variety X over a ﬁnite ﬁeld Fq, Tate conjecture for X ×Fq SpecFq is equivalent to noncommutative Tabuada-Thomason-Tate conjecture below for PerfX (see [Tab18, 1.3]).

Corollary 20 (Tabuada-Thomason-Tate conjecture). Let Fq be a finite field and l be a prime number which n is prime to q and let A be a saturated dg-category over Fq. Then π−1LKU K(A ⊗Fq Fq ) is l-reducible for n n all n where LKU K(A ⊗Fq Fq ) stands for the Bousfield localization of the algebraic K-spectrum if A ⊗Fq Fq with respect to topological complex K-theory KU.

Proof of Corollary 20. Since for each positive integer n and a saturated dg-category A over Fq, a dg-category n n n n A ⊗Fq Fq is a saturated dg-category over Fq , by replacing k = Fq with Fq and A with A ⊗Fq Fq , we shall only give a proof for n = 1. Since for any spectra X, we have an equivalence of spectra ([Bou79, 4.3]) ∞ 0 LKU X ' X ∧ LKU (Σ S ), LKU preserves distinguished triangles and direct sums. Thus by Theorem 12, we reduced to the case for B = Fq. Since Tate conjecture for SpecFq is true, π−1LKU K(Fq) is l-reducible by Thomason’s theorem [Tho89]. Hence we obtain the result.

7 2.4 Noncommutative p-adic Tate conjecture Tabuada proposed a noncommutative p-adic Tate conjecture and proved that it implies the original p-adic version of Tate conjecture in [Tab18, 1.3]. Recall the notations from introduction. Let k be a finite field m Fq of characteristic p with q = p , and let W(k) be the associated ring of p-typical Witt vectors and we set K := W(k)[1/p] the fraction field of W(k). Let A be a saturated dg-category over k. Then Tabuada constructed the p-adic noncommutative cycle map

ϕm ρA : K0(A ) ⊗ K → TP0(A )1/p (5)

ϕm where TP0(A )1/p stands for K-linear subspace of TP0(A )1/p = TP0(A ) ⊗W(k) K the 0-th topological periodic cyclic homology of A consisting of those elements which are fixed by the K-linear endomorphism ϕm the m-times cyclotomic Frobenius (for more detail, see below). Here is a noncommutative p-adic Tate conjecture and we will show it by utilizing motivic Bass conjecture. In particular we obtain the p-adic Tate conjecture. Corollary 21 (Noncommutative p-adic Tate conjecture). For a saturated dg-category A over k, the cycle map (5) is surjective. Before giving our proof of Corollary 21, we briefly recall some notations and fundamental results of topological periodic cyclic homology from [BM17], [HM97], [Tab18] and so on. We write THH for the topological Hochschild homology theory. Since THH is a localizing invariant on the category of small dg- loc categories (over Z) by [BM12, 1.2, 7.1], we can regard it as a functor on Motk to Ho(Spt) the category of spectra by [CT12, 7.2]. Topological periodical cyclic homology TP is defined as the Tate cohomology of the 1 circle group action on topological Hochschild homology. Namely TP(A ) = THH(A )tS for any dg-category loc A over k. We can also regard TP as the symmetric monoidal functor from Motk to Ho(Spt) by [CT12, 7.5] and [BM17, Theorem A] (see also [AMN18]). By the work of Hesselholt and Madsen [HM97, §5], we obtain the following computation results. In particular for a saturated dg-category over k, TP0(A )1/p := TP0(A ) ⊗W(k) K has a structure of K-vector space. Lemma 22. As a graded W(k)-algebra, we have an isomorphism ∼ ±1 TP∗(k) → SW(k){v } (6) for some v ∈ TP−2(k). Here SW(k) stands for the symmetric algebra over W(k).

We obtain the cycle class map ρ : K0(−)K(= K0(−) ⊗ K) → TP0(−)1/p by universality of K-theory [CT12, 8.7] (see also around Lemma 3.7 in [Tab18]). We write ρX for ρPerfX for a scheme X over k and we denote ρSpecA by ρA for a k-algebra A. In [Tab18], by utilizing [NS18, II 4.2], Tabuada constructed the cyclotomic Frobenius morphism

ϕ : TP0(A )1/p → TP0(A )1/p for a saturated dg-category A over k which satisﬁes the following conditions (see around Lemma 3.7 in [Tab18]). Lemma 23. Let A be a saturated dg-category over k. Then m (i) ϕ is a K-linear map where m = logp #k where #k stands for the number of elements in k.

ϕm (ii) The image of the cycle map ρA : K0(A )K → TP0(A )1/p is in TP0(A )1/p the K-linear subspace of m TP0(A )1/p of those elements which are ﬁxed by the K-linear endomorphism ϕ . m (iii) ϕ : TP0(k)1/p → TP0(k)1/p is the identity morphism.

ϕm (iv) ρk : K0(k)K → TP0(k)1/p is just the identity map of K via the canonical isomorphisms K0(k)K ' K and ϕm TP0(k)1/p ' K.

Proof of Corollary 21. The question is closed under direct summand, thus by Theorem 12, we reduce to the problem for B = k. For this case, assertion is true by Lemma 23 (iv).

8 2.5 Noncommutative standard conjectures In this subsection, We consider the noncommutative Beilinson and Voevodsky nilpotence conjectures which were proposed by Tabuada respectively Bernardara, Marcolli and Tabuada in [Tab18] and [BMT18] respectively. Let A be a saturated dg-category over a ﬁeld k. Then there exists a pairing χ : K0(A ) × K0(A ) → Z i which sends a pair ([M],[N]) to an integer ∑(−1) dimk HomDc(A )(M,n[−i]) where Dc(A ) stands for the i full subcategory of all compact objects in D(A ) the derived category of A . We say that an element x in K0(A ) is numerically equivalent to zero if for any element y in K0(A ), χ(x,y) = 0. We denote the quotient group of K ( ) by subgroup of all elements which are numerically equivalent to zero by K ( ) . Notice 0 A Q 0 A num that we have the canonical isomorphism K0(A ) ' HomNMMk (1,U (A )) and therefore we can regard an element of K0(A ) as a morphism 1 → U (A ). An element x in K0(A ) is ⊗-nilpotent if there exists an integer ⊗n n > 0 such that x = 0 where ⊗ is the symmetric monoidal structure on NMMk. We write K0(A )nil for the quotient group of K0(A )Q by the subgroup of all ⊗-nilpotent elements. Then we will show the followings.

Corollary 24 (Noncommutative Beilinson conjecture). If k is a ﬁnite ﬁeld, then the equality K0(A )num = K ( ) holds. 0 A Q

Corollary 25 (Noncommutative Voevodsky nilpotence conjecture). The equality K0(A )nil = K0(A )num holds.

As in [Tat94, 2.9], the Beilinson conjecture and the (p-adic) Tate conjecture implies the following strong Tate conjecture. Corollary 26 (Strong Tate conjecture). Let X be a projective and smooth scheme X over a finite field k. Then the order of the pole of the Hasse-Weil zeta function ζ(x,s) of X at −i is equal to the dimension of i CH (X)Q,num for 0 ≤ i ≤ dimX. By the work of Bernardara, Marcolli and Tabuada [BMT18, 1.1], noncommutative nilpotence conjecture implies the original nilpotence conjecture and as in [Voe95, p.194], nilpotence conjecture implies the standard conjecture D and it might be well-known that the standard conjecture D implies the standard conjecture B (see for example [Kle94, 5.1]). Thus finally we obtain the following: Corollary 27 (Standard conjecture). Grothendieck’s standard conjectures B and D are true.

Before showing corollaries 24 and 25, we will recall a general theory of ideals of ringoids which we will use to prove corollaries. Let R be a ringoid, in other words, a category enriched over the category of abelian groups. An ideal I of R is a family of subgroups I (x,y) of HomR(x,y) for all ordered pair (x,y) ∈ Ob(R ×R) which satisfies the condition that for any objects x, y, z and u of R and any pair of morphisms f : x → y and g: z → u, we have an inclusion gI (y,z) f ⊂ I (x,y). In this situation, we define R /I to be a ringoid by setting ObR /I := ObR and HomR /I (x,y) := HomR(x,y)/I (x,y) for any pair of objects x and y in R. Then the compositions in R induces a composition of R /I . We call the category R /I the quotient category of R by I . From now on, we assume R admits a unital symmetric monoidal structure (⊗,1). An ⊗-ideal I of R is an ideal of I such that for all triple of objects x, y and z and a morphism f : x → y in I (x,y), idz ⊗ f and f ⊗ idz are also in I (z ⊗ x,z ⊗ y) and I (x ⊗ z,y ⊗ z) respectively. We will illustrate typical two examples of ⊗-ideals. The first one is NilR which is defined as follows. A morphism f : x → y in R is ⊗-nilpotent if there exits an integer n > 0 such that f ⊗n = 0. For a pair of objects x and y in R, we denote the set of all ⊗-nilpotent morphisms from x to y by NilR(x,y). Then we can show that the family {NilR(x,y)}(x,y)Ob(R ×R) is a ⊗-ideal. Assume that an object x in R is dualizable. Then for a morphism f : x → x, we define Tr( f ) to be a f ⊗id ∨ ev morphism 1 → 1 by compositions 1 →δ x∨ ⊗x ' x⊗x∨ →x x⊗x∨ → 1 and we call it the (categorical) trace of f . We say that a morphism f : x → y is numerically equivalent to zero if for any morphism g: y → x, Tr(g f ) = 0 and we denote the set of all morphisms from x to y which are numerically equivalent to zero by NumR(x,y). Moreover if we assume that all objects in R are dualizable, then the family {NumR(x,y)}(x,y)∈Ob(R ×R) forms a ⊗-ideal by [AKO02, 7.1.1]. If HomR(1,1) is a field, then we have an inclusion NilR(x,y) ⊂ NumR(x,y) for any pair of objects x and y in R by [AKO02, 7.1.4]. We write NMMA,nil and NMMA,num for the quotient categories (NMMA) /Nil(NMM ) and (NMMA) /Num(NMM ) respectively. These notions are compatible Q A Q Q A Q

9 with the notions in introduction. Namely for a saturated dg-categories A over a ﬁeld k, we have the following isomorphisms

K0(A )# ' HomNMMk,# (1,U (A )) (7) for # ∈ {nil,num} by [MT14, 4.4]. Moreover NMMk,# for # ∈ {nil,num} are symmetric monoidal triangulated categories by Lemma 28 below.

Lemma 28. (i) Assume that R is additive. Then for any pair of families of objects {xi}i∈I and {y j} j∈J indexed by non-empty ﬁnite sets I and J, the canonical isomorphism

M M ∼ M HomR( xi, y j) → HomR(xi,y j) i∈I j∈J (i, j)∈I×J

induces an isomorphism M M ∼ M I ( xi, y j) → I (xi,y j). (8) i∈I j∈J (i, j)∈I×J In particular R /I is an additive category. (ii) Moreover assume that R is a triangulated category. Then we can make R /I into a triangulated category where a triangle in R /I is distinguished if and only if it is isomorphic to a distinguished triangle in R. (iii) ([AKO02, 6.1.2]). If R is a symmetric monoidal category and I is a ⊗-ideal, then the symmetric monoidal structure on R makes R /I into a symmetric monoidal category.

Proof of Corollaries 24 and 25. Since Corollaries 24 and 25 are equivalent to assertions that

Num(1,U (A ))Q = 0 and Nil(1,U (A ))Q = Num(1,U (A ))Q respectively, if assertions are true for A , then assertions are also true for direct summands of U (A ) by the equality (8). Thus by Theorem 12, we shall assume that A = k a ﬁnite dg-cell over k. Since K0(k) = Z, assertions for A = k are true.

Acknowledgement The author wish to express my deep gratitude to Seidai Yasuda for giving several com- ments and he greatly appreciate Kanetomo Sato, Kento Yamamoto, Takashi Suzuki, Masana Harada and Kazuya Kato for stimulative discussions in the early stage of the works.

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SATOSHI MOCHIZUKI DEPARTMENT OF MATHEMATICS, CHUO UNIVERSITY, BUNKYO-KU, TOKYO, JAPAN. e-mail: [email protected]