Mixed Artin–Tate motives over number rings

This paper studies Artin–Tate motives over bases S ⊂ Spec OF , for a number field F . As a subcategory of motives over S, the triangulated category of Artin–TatemotivesDATM(S) is generated by motives φ∗1(n), where φ is any finite map. After establishing the stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin–Tate motives (MATM(S)) is two, and there is an equivalence DATM(S) ∼= Db(MATM(S)). © 2010 Elsevier B.V. All rights reserved. Geometric motives, as developed by Hanamura [5], Levine [8] and Voevodsky [14], are established as a valuable tool in understanding geometric and arithmetic aspects of algebraic varieties over fields. However, the stupefying ambiance inherent tomotives, exemplified by Grothendieck’s motivic proof idea of theWeil conjectures, remains largely conjectural— especially what concerns the existence of mixed motives MM(K) over some field K . That category should be the heart of the so-called motivic t-structure on DMgm(K), the category of geometric motives. Much the same way as the cohomology groups of a variety X over K , e.g. Hét(X×KK , Ql), l-adic cohomology for l ≠ char K are commonly realized as cohomology groups of a complex, e.g. RΓl(X, Ql), there should bemixedmotives hn(X) that are obtained by applying truncation functors belonging to the t-structure to M(X), the motive of X . However, progress on mixed motives has proved hard to come by. To date, such a formalism has been developed for motives of zeroand one-dimensional varieties, only. This is due to Levine [7], Voevodsky [14], Orgogozo [9] and Wildeshaus [16]. Building upon Voevodsky’swork, Ivorra [6] and recently Cisinski andDéglise [3] developed a theory of geometricmotives DMgm(S) over more general bases. The purpose of this work is to join the ideas of Beilinson et al. on perverse sheaves [2] with the ones on Artin–Tate motives over fields to obtain a workable category of mixed Tate and Artin–Tate motives over bases S which are open subschemes of Spec OF , the ring of integers in a number field F . As over a field, this provides some evidence for the existence and properties of the conjectural category of mixed motives over S. The triangulated category DTM(S) (DATM(S)) of Tate (Artin–Tate) motives is defined 2.2 to be the triangulated subcategory ofDMgm(S) (with rational coefficients) generated by direct summands of 1(n) and i∗1(n) (φ∗1(n), respectively). Here, 1 is a shorthand for the motive of the base scheme, (n) denotes the Tate twist, i : Spec Fp → S is a closed point, φ : V → S is any finite map and φ∗ : DMgm(V ) → DMgm(S) etc. denotes the pushforward functor on geometric motives. In case S is a finite disjoint union of Spec Fp, the usual definition of (Artin–)Tate motives over S is recalled in Definition 2.1. The following theorem and its ‘‘proof’’ is an overview of the paper. Theorem 0.1. The categories DTM(S) and DATM(S) are stable under standard functoriality operations such as i, j∗ etc. for open and closed embeddings j and i, respectively. Both categories enjoy a non-degenerate t-structure called motivic t-structure. Its heart is denoted MTM(S) or MATM(S), respectively and called category of mixed (Artin–)Tate motives. The functors i, j∗ etc. feature exactness properties familiar from the corresponding situation of perverse sheaves. For example, i is left-exact, and j∗ is exact with respect to the motivic t-structure. E-mail address: [email protected]. 0022-4049/$ – see front matter© 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2010.11.019 J. Scholbach / Journal of Pure and Applied Algebra 215 (2011) 2106–2118 2107 The cohomological dimension ofMTM(S) andMATM(S) is one and two, respectively. We have an equivalence of categories Db(MATM(S)) ∼= DATM(S) and likewise for Tate motives. The ‘‘site’’ of mixed Artin–Tate motives over S has enough points in the sense that a mixed Artin–Tate motive over S is zero if and only if its restrictions to all closed points of S vanish. Proof. The first statement is Theorem2.4. It is proven using the localization, purity and base-change properties of geometric motives. We will write T (S) for either DTM(S) or DATM(S). The existence of the motivic t-structure on T (S) is proven in three steps. The first ingredient is the well-known motivic t-structure on Artin–Tate motives over finite fields (Lemma 3.6). The second step is the study of a subcategory T̃ (S) ⊂ T (S) generated by φ∗1(n), where φ is finite and étale (Artin–Tate motives), or just by 1(n) (Tatemotives). This category is first equippedwith an auxiliary t-structure. Using the cohomology functor for the auxiliary t-structure, amotivic t-structure on T̃ (S) is defined in Section 3. This statement uses (and its proof imitates) the corresponding situation for Artin–Tate motives over number fields due to Levine and Wildeshaus. Thirdly, the t-structure on T̃ (S) is glued with the one over finite fields, using the general gluing procedure of t-structures of [2], see Theorem 3.8. Much the sameway as with perverse sheaves, there are shifts accounting for dim S = 1, that is to say, i∗1(n) and 1(n)[1] are mixed Tate motives. Beyond the formalism of geometric motives, the only non-formal ingredient of the motivic t-structure are vanishing properties of the algebraic K -theory of number rings, number fields and finite fields due to Quillen, Borel and Soulé. The exactness statements are shown in Theorem4.2. This theoremgives some content to the exactness axioms for general mixedmotives over S [11, Section 4]. The key step stone is the following: for any immersion of a closed point i : Spec Fp → S, the functor i maps the heart T 0(S) of T (S) to T [−1,0](Spec Fp), that is, the category of (Artin–)Tate motives over Fp whose only nonzero cohomology terms are in degrees −1 and 0. The proof is a careful reduction to basic calculations relying on facts gathered in Section 3 about the heart of T̃ (S). The cohomological dimensions are calculated in Proposition 4.4. The Artin–Tate case is a special (but non-conjectural) case of a similar fact for general mixed motives over S. The difference in the Tate case is because the generators of DTM(S) have a good reduction at all places. By an argument of Wildeshaus, the identity on T 0(S) extends to a functor Db(T 0(S)) → T (S) (Theorem 4.5). While it is an equivalence in the case of Tate motives for formal reasons, the Artin–Tate case requires some localization arguments. The last statement is Proposition 4.6. It might be seen as a first step into motivic sheaves. Deligne and Goncharov define a category of mixed Tate motives over rings OS of S-integers of a number field F [4, 1.4., 1.7.]. Unlike themixed Tatemotiveswe study, their category is a subcategory ofmixed Tatemotives over F , consisting of motives subject to certain non-ramification constraints, akin to Scholl’s notion of mixed motives over OF [12]. This paper is an outgrowth of part of my thesis. I owe many thanks to Annette Huber for her advice during that time. I am also grateful to Denis-Charles Cisinski and Frédéric Déglise for teaching me their work on motives over general bases. 1. Geometric motives This section briefly recalls some properties of the triangulated categories of geometric motives DMgm(X), where X is either a number field F or an open or closed subscheme of Spec OF . All of this is due to Cisinski and Déglise [3]. In this section, all references in brackets refer to op. cit., e.g. [Section 14.1]. Let X be any of the afore-mentioned bases. There is the triangulated category DM(X) of Beilinson motives and its subcategory DMgm(X) of compact objects.1 Objects of the latter category will be referred to as geometric motives. The categories are related by adjoint functors f ∗ : DM(X) DM(Y ) : f∗, (1) where f : Y → X is any map [13.2.11, 1.1.11]. If f is separated and of finite type this adjunction restricts to an adjunction between the subcategories of compact objects [14.1.5, 14.1.26] and there is an adjunction [13.2.11, 2.4.2] f! : DMgm(Y ) DMgm(X) : f . (2) If f is smooth in addition, f ∗ : DMgm(X) → DMgm(Y ) also has a left adjoint f♯ [13.2.11, 1.1.2]. These five functors respect composition of morphisms in the sense that there are natural isomorphisms f∗ ◦ g∗ = (f ◦ g)∗, f ∗ ◦ g = (g ◦ f ) etc. (3) for any two composable maps f and g [Section 1.1, 2.4.21]. The category DMgm(X) enjoys inner Hom’s, denoted Hom, and a tensor structure such that pullback functors f ∗ are monoidal [13.2.11, 1.1.28]. The unit of the tensor structure is denoted 1. 1 DM(X) is denoted DMB(X) in [3, Sections 13.2, 14.1]. 2108 J. Scholbach / Journal of Pure and Applied Algebra 215 (2011) 2106–2118 In particular f 1X = 1Y for f : Y → X . The motive of any separated scheme f : Y → X of finite type is defined as f!f 1 and denoted M(Y ). (For f smooth, [Section 1.1.] puts M(Y ) := f♯f 1. The two agree, see Lemma 1.2.) The tensor structure in DMgm(X) is such that M(Y )⊗M(Y ) = M(Y×XY ) (4) for any two smooth schemes Y and Y ′ over X [1.1.35]. There is a distinguished object 1(1) such that M(PX ) = 1 ⊕ 1(1)[2]. Tensoring with 1(1) is an equivalence on DMgm(X) [2.1.5], and 1(n) is defined in the usual way in terms of tensor powers of 1(1).We exclusivelyworkwith rational coefficients, i.e., all morphism groups areQ-vector spaces. If X is regular,morphisms in DMgm(X) are given by HomDMgm(X)(1, 1(q)[p]) ∼= K2q−p(X) (q) Q , (5) the q-th Adams eigenspace in algebraic K -theory of X , tensored with Q [Section 13.2]. Having rational coefficients (or coefficients in a bigger number field) is vital when it comes to vanishing properties of Hom-groups in DMgm(X). (With integral coefficients, the existence of a t-structure is unclear even in the case of Artin motives over a field.) For any closed immersion i : Z → X with open complement jwe have the following functorial distinguished localization triangles in DMgm(X) [2.2.14, 2.3.3]: j!j∗ → id → i∗i∗. (6) Moreover i∗i∗ = id [2.3.1], so that i∗j! = 0, (7) and i∗ is fully faithful. There is an isomorphism of functors f! ∼= −→ f∗ (8) for any proper map f [2.2.14, 2.2.16]. For smooth and quasi-projective maps f of constant relative dimension d there is a relative purity isomorphism [Theorem 1, p. 5] f ! ∼= f (d)[2d]. (9) Moreover, when i : Z → X is a closed immersion of constant relative codimension c and Z and X are regular, we have an isomorphism i1 ∼= i1(−c)[−2c]. (10) This is called absolute purity [Sections 2.4, 13.4]. Finally, for f : Y → X , g : X ′ → X , f ′ : Y ′ := X ′×XY → X ′ and g ′ : Y ′ → Y , there is a natural base-change isomorphism of functors [Section 2.2] f ∗g! ∼= g ′ ! f . (11) The Verdier dual functor DX : DMgm(X) → DMgm(X) is defined by DX (M) := Hom(M, π 1(1)[2]) for anyM ∈ DMgm(X), where π : X → Spec Z denotes the structural map. Lemma 1.1. For an open subscheme X of Spec OF we have DX (−) = Hom(−, 1(1)[2]). Secondly, we have DSpec Fq(−) = Hom(−, 1). Proof. The structural map π : X → Spec Z factors as X j → Spec OF i → AZ p → Spec Z, where j is an open immersion, i is a closed immersion and p is the projection. Thus we have π 1 = π1 by absolute purity (10), applied to i, and relative purity (9), applied to j and p. Using (10) we get the second statement. The Verdier dual functor exchanges ‘‘!’’ and ‘‘∗’’, that is, there are natural isomorphisms [Section 14.3] D(f M) ∼= f D(M), f!D(M) ∼= D(f∗M). (12) For example, the Verdier dual of (6) yields a distinguished triangle i∗i! → id → j∗j∗. (13) Lemma 1.2. For f : X → Y smooth, we have a natural isomorphism f!f 1 = f♯f 1. Proof. This is well known.We can assume f is of constant relative dimension d. Then the claim follows from the adjunctions f♯ f ∗ (9) = f (−d)[−2d] and f!(d)[2d] f (−d)[−2d]. Let X = Spec OF . The colimit over the triangles (13) over increasingly small open subschemes j : U ⊂ X is still a distinguished triangle. For any geometric motiveM over X we get the following distinguished triangle in DM(X): ⊕pip∗i ! pM → M → η∗η M, (14) where η : Spec F → Spec OF is the generic point, the sum runs over all closed points p ∈ X , ip is the closed immersion. Indeed colimj∗j∗M = η∗η∗M for anyM ∈ DMgm(X) [Section 14.2]. J. Scholbach / Journal of Pure and Applied Algebra 215 (2011) 2106–2118 2109 2. Triangulated Artin–Tate motives Recall the following classical definition. We apply it to a number field or a finite field: Definition 2.1. Let K be a field. The category of Tate motives DTM(K) over K is by definition the triangulated subcategory of DMgm(K) generated by 1(n) where n ∈ Z. The smallest full triangulated subcategory DATM(K) stable under tensoring with 1(n) and containing direct summands of motives f∗1, where f : K ′ → K is any finite map, is called a category of Artin–Tate motives over K . For a scheme S of the form S = ⊔Spec Ki, a finite disjoint union of spectra of fields, we put DATM(S) := ⊕iDATM(Ki) and likewise for DTM. This section gives a generalization of that definition to bases S which are open subschemes of Spec OF based on the idea that Artin–Tate motives over S should be compatible with the ones over F and Fp under standard functoriality. Definition 2.2. The categories DTM(S) ⊂ DMgm(S) of Tate motives and DATM(S) ⊂ DMgm(S) of Artin–Tate motives over S are the triangulated subcategories generated by the direct summands of 1(n), i∗1(n) (Tate motives) and φ∗1(n), (Artin–Tate motives) respectively, where n ∈ Z, φ : V → S is any finite map (including those that factor over a closed point) and i : Spec Fp → S is the immersion of any closed point of S. Remark 2.3. • We can assume by localization (see (6), (13)) that the domain of φ is a reduced scheme. • The category of Tate motives DTM(S) agrees with the triangulated category generated by the above generators (without taking direct summands). Indeed, by (5), the endomorphism rings End(1(n)), End(i∗1(n)) identify with K0(S) (0) Q and K0(Fp) (0) Q , respectively, which are both one-dimensional over Q. Hence these objects do not have any proper direct summands. For brevity, we write T (S) or T for DATM(S) or DTM(S) in the sequel. In most proofs, we will only spell out the case of Artin–Tate motives. Theorem 2.4. Let j : S ′ → S be any open immersion, i : Z → S be any closed immersion and f : V → S any finite map such that V is regular. Let η : Spec F → S be the generic point. Then the functors f∗ (8) = f!, f ∗ and f ! preserve Artin–Tate motives. Similar statements hold for Artin–Tate and Tate motives for j and i. Moreover, η, the Verdier dual functor D and the tensor product on DMgm(S) respect the subcategories of (Artin–)Tate motives. The functor η∗ does not respect Artin–Tate motives: we will see in Proposition 4.6 that any Artin–Tate motive M of the formM = η∗Mη , whereMη is a geometric motive over F , necessarily satisfiesM = 0. Proof. The stability of (Artin–)Tate motives under j, η, i∗ and i, f ∗ and — for Artin–Tate motives, under f∗ — is immediate from the definition, (8), and (11). For example, iφ∗1(n) = φ ∗1(n). Here φ : S ′ → S is any finite map and φ : Z ′ → Z is its pullback along i. Let i : Z ′ → S ′ be the pullback of i. For the stability under i we use iφ∗1 (11) = φ ∗ i1. We can assume S ′ is reduced and, since the zero-dimensional case is easy, one-dimensional. Let n : S ′′ → S ′ be the normalization map; let v : Y ′ ⊂ S ′ be the ‘‘exceptional divisor’’, i.e., the smallest (zero-dimensional) closed reduced subscheme such that n−1(S \Y ) → S \Y ′ is an isomorphism. Moreover, put z : Y ′′ := Y ′×S′S ′′ → S ′′ → S . Consider the the distinguished triangle 1S′ → v∗1Y ′ ⊕ n∗1S′′ → z∗1Y ′′ . It is a special case of [3, Theorem 4, p. 5] or can alternatively be derived from localization. Note that in∗1S′′ (11) = n ∗ i1S′′ (10) = n ∗ 1(−1)[−2] by the regularity of S . Here, again, n and i denote the pullback maps. Similar considerations for iv∗1Y ′ and iz∗1Y ′′ show that i1S′ is an Artin–Tate motive. For the stability under j∗ it is sufficient to show j∗φ′ ∗1 is an Artin–Tate motive over S for any finite flat map φ ′ : V ′ → S . Choose some finite flat (possibly non-regular) model φ : V → S of φ, i.e., V×SS ′ = V , so that jφ∗1 = φ ∗1 is an Artin–Tate motive over S . The localization triangle (13) i∗i!φ∗1 → φ∗1 → j∗j∗φ∗1 and the above steps show that j∗φ′ ∗1 is an Artin–Tate motive over S. To see the stability under the Verdier dual functor D, it is enough to see that D(φ∗φ∗1) (12) = φ!φ D(1) 1.1 = φ∗φ!1(1)[2] 2110 J. Scholbach / Journal of Pure and Applied Algebra 215 (2011) 2106–2118 is an Artin–Tate motive for any finite map φ : V → S with reduced domain (Remark 2.3). If V is zero-dimensional, this follows from purity (10), (9) and the regularity of S. If not, there is an open (non-empty) immersion j : S ′ → S such that V ′ := V×SS ′ is regular (for example, take S ′ such that V /S ′ is étale). Let i be the complement of j. We apply the localization triangle (13) to φ∗φ1. By base-change (11) we obtain i∗φ′′ ∗φ i1 → φ∗φ!1 → j∗φ′ ∗φ j1. Here φ and φ is the pullback of φ along i and j, respectively. By the regularity of S and purity we have i1 = 1(−1)[−2], so the left hand term is an Artin–Tate motive. The right one also is by purity. This shows the claim for D. The stability under f , i, and j! now follow for duality reasons. As for the stability under tensor products we note that φ∗1⊗φ ∗1 (4) = (φ×φ)∗1 if φ and φ are (finite and) smooth, cf. (4). Using the localization triangle (6), it is easy to reduce the general case of merely finite maps φ, φ to this case. Remark 2.5. Theorem 2.4 also holds for a similarly defined category of Artin–Tate motives over open subschemes S of a smooth curve over a field. Proposition 2.6. Let M ∈ DATM(S) be any Artin–Tate motive. Then there is a finite map f : V → S such that f M ∈ DTM(S) ⊂ DATM(S). We describe this by saying that f splitsM. Proof. As f ∗ is triangulated, this statement is stable under triangles (with respect to M), and also under direct sums and summands. Therefore, we only have to check the generators, i.e., M = φ∗1(n) with φ : S ′ → S a finite map with reduced domain. The corresponding splitting statement for Artin–Tate motives over finite fields is well-known. Therefore, by localization (6), (13), it is sufficient to find a splitting map f after replacing S by a suitable small open subscheme, so we may assume φ étale. We first assume that φ is moreover Galois of degree d, i.e., S ′×SS ′ ∼= S ′⊔d, a disjoint union of d copies of S . In that case one has φφ∗1 = 1⊕d by base-change (11), so the claim is clear. In general φ need not be Galois, so let S ′′ be the normalization of S in some normal closure of the function field extension k(S )/k(S). Both μ : S ′′ → S and ψ : S ′′ → S ′ are generically Galois. By shrinking S we may assume both are Galois. From Hom(1S′ , ψ∗1S′′) = Hom(1S′′ , 1S′′) = Q and Hom(ψ∗1S′′ , 1S′) = Hom(1S′′ , ψ 1S′) = Hom(1S′′ , 1S′′) = Qwe see that 1S′ is a direct summand ofψ∗1S′′ . Thereforeμφ∗1S′ is a summand of μφ∗ψ∗1S′′ = μμ∗1S′′ = 1 deg S /S , a Tate motive. 3. The motivic t-structure In this section, we establish the motivic t-structure on the category of Artin–Tate motives over S (Theorem 3.8). It is obtained by the standard gluing procedure, applied to the t-structures on Artin–Tate motives over finite fields and on a subcategory T̃ (S ) ⊂ T (S ) for open subschemes S ′ ⊂ S. Under the analogy of mixed (Artin–Tate) motives with perverse sheaves, the objects in the heart of the t-structure on T̃ (S ) correspond to sheaves that are locally constant, i.e., have good reduction. We refer to [2, Section 1.3.] for generalities on t-structures. Definition 3.1 (Compare [7, Def. 1.1]). For −∞ ≤ a ≤ b ≤ ∞, let T̃[a,b] denote the smallest triangulated subcategory of T (S) containing direct factors of φ∗1(n), a ≤ −2n ≤ b, where φ : S ′ → S is a finite étalemap. For Tatemotives, φ is required to be the identity map. (We will not specify this restriction expressis verbis in the sequel.) Furthermore, T̃[a,a] and T̃[−∞,∞] are denoted T̃a and T̃ . If it is necessary to specify the base, we write T̃[a,b](S) etc. We need the following vanishing properties of the K -theory of number fields, related Dedekind rings and finite fields up to torsion. In order to weigh the material appropriately, it should be said that the content of the theorem below is the only non-formal part of the proofs in this paper, and all complexity occurring with Artin–Tate motives ultimately lies in these computations. Theorem 3.2 (Borel, Quillen, Soulé). Let φ : S ′ → S and ψ : V → S be two finite maps with zero-dimensional domains. HomS(φ∗1, ψ∗1(n)[m]) =  finite-dimensional n = m = 0 0 else. Now let φ : S ′ → S and ψ : V → S be two finite étale maps over S. Then HomS(φ∗1, ψ∗1(n)[m]) =  finite-dimensional n = m = 0 finite-dimensional m = 1, n odd and positive 0 else.