On the Invertibility of Motives of Affine Quadrics

Tom Bachmann

Received: January 13, 2017

Communicated by Nikita Karpenko

Abstract. We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k, ℤ[1/e]) (where k is a perfect field of exponential characteristic e). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) DM(k, ℤ[1/e]), the construction of which constitutes the main part of this work.

2010 Mathematics Subject Classification: Primary 14C15; Secondary 11E04, 14C25 Keywords and Phrases: Quadric, motive, invertible

1 Introduction

For a perfect field k, Voevodsky has constructed a triangulated category DM(k) containing the classical category Chow(k) of Chow motives [20]. Like Chow(k), DM(k) is a tensor category. We denote the tensor product by ⊗ = ⊗_{DM(k)} and the unit by 𝟙 = 𝟙_{DM(k)}. As in any tensor category, we have the notion of invertible objects: an object E ∈ DM(k) is called invertible if there exists an object F ∈ DM(k) and an isomorphism E ⊗ F ≈ 𝟙. The set of isomorphism classes of invertible objects forms an abelian group under ⊗ and is called the Picard group. We denote it P ic(DM(k)). Po Hu [12] was the first to construct interesting elements in P ic(DM(k)), related to certain low-dimensional quadrics. In this direction we prove the following result (see Theorem 33 in Section 5).

Theorem. Let k be a perfect field of exponential characteristic e not two, ϕ(t_{1}, …, t_{n}) a non-degenerate quadratic form over k and a ∈ k^{×}. Write X_{ϕ}^{a} for the affine quadric defined by the equation ϕ(t_{1}, …, t_{n}) = a. Then the reduced motive M̃(X_{ϕ}^{a}) ∈ DM(k, ℤ[1/e]) is invertible.

This result has a number of predecessors. Work of Voevodsky can be used to show that reduced versions of the Rost motives [19] are invertible. As observed by Hu-Kriz [13, Proposition 5.5], the reduced Rost motives are reduced motives of affine Pfister quadrics. They go further and explore analogies with the Hopf invariant one problem. In [12] this culminates in certain conjectures about smash products of affine Pfister quadrics implying their invertibility. Moreover the conjectures are proven in low dimensions. The best method the author knows of attacking the study of Picard groups of tensor categories (to the extent that it even deserves the name “method”) is to construct “realisation functors” F : DM(k) → C. If F is a tensor functor, it induces a homomorphism P ic(DM(k)) → P ic(C). If F is sufficiently nice, and C sufficiently simple, one may hope to compute P ic(C) and relate it to P ic(DM(k)). We mention in passing that a good test for the “niceness” of F seems to be conservativity (i.e. the property that F detects isomorphisms). This will be illustrated later. There are well known realisation functors out of DM(k), but none of them seem helpful to our problem. If k ⊂ ℂ there is the Hodge realisation, but this factors through the natural functor DM(k) → DM(ℂ) and hence provides no interesting information about quadrics (which over ℂ are distinguished by only their dimension). There is also étale realisation, but this factors through DM(k) → DM_{et}(k). In DM_{et}(k) our problem turns out to be very simple and not indicative of the complexity encountered in DM(k) (i.e. in the Nisnevich topology). What we propose in this work is to construct purpose-built realisation functors DM(k) → C into big but easy to understand categories. (Actually we do not quite achieve this; limitations will be explained later.) To motivate our constructions, we explain two analogous but simpler problems obtained by replacing DM(k) by another category. First let G be a finite group. There exists the stable G-equivariant homotopy category SH(G) [15]. Its objects (called genuine G-spectra) are roughly pointed G-spaces, where maps inducing weak equivalences on all fixed point sets have been turned into isomorphisms, and all representation spheres are invertible objects. If H ≤ G is a subgroup, the set of cosets G/H can naturally be turned into a pointed G-space (adding a base point ∗ with trivial action). We denote the associated spectrum by Σ^{∞}G/H_{+}. The objects Σ^{∞}G/H_{+} generate SH(G). There is a functor, called geometric fixed points functor, and denoted Φ = Φ^{G} : SH(G) → SH (where SH = SH({e}) is the classical stable homotopy category) which turns out to be very useful. It is a tensor functor with the property that Φ^{G}(Σ^{∞}G/G_{+}) = S (the sphere spectrum), whereas Φ^{G}(Σ^{∞}G/H_{+}) = 0 for any proper subgroup H < G. There are also natural functors SH(G) → SH(H) (treating G-spaces as H-spaces) allowing us to construct the more general geometric fixed points functors Φ^{H} : SH(G) → SH(H) → SH. As it turns out the collection {Φ^{H}}_{H} (with H ranging over all subgroups of G) is as nice as one may ask (in particular conservative). Consequently these functors were used in [10] to study P ic(SH(G)). We now come to a second, more algebraic, example. Let R be a (commutative unital) ring. Suppose we want to study P ic(D(R)), the Picard group of the derived category of R-modules. Let m be a maximal ideal of R. Recalling that D(R) can be identified with a subcategory of K(P (R)), the homotopy category of chain complexes of projective R-modules, it is easy to construct a functor Φ^{m} : D(R) → D(R/m) with the propetry that Φ^{m}(R[0]) = R/m[0]. (This is just ⊗_{R}^{L}R/m.) It turns out that the collection {Φ^{m}}_{m} (where m ranges over all maximal ideals) is as nice as one needs (at least when restricted to subcategories of sufficiently small objects in D(R)). Moreover the categories D(R/m) are easy to understand. Consequently, these functors have implicitly been used by Fausk in his study of the Picard group of derived categories [9]. Our construction for DM^{gm}(k) uses a conglomerate of these ideas. The technical notion of weight structures is the glue that holds our constructions together. We proceed roughly as follows. Recall that DM^{gm}(k) is generated as a triangulated category by the Chow motives. Let S be the triangulated subcategory generated by those Chow motives not affording a (non-vanishing) Tate summand. The basic idea is to consider the (Verdier Quotient) functor φ_{0}^{k} : DM^{gm}(k) → DM^{gm}(k)/S. The right hand side does not seem initially easier to understand, but it is at least clear that it is generated by the images of Tate motives. Using weight structure theory one obtains a functor t : DM^{gm}(k)/S → K^{b}(T ate), where T ate is the category of (pure) Tate motives, and K^{b} means bounded chain homotopy category.^{1} Combined with base change to arbitrary fields, we thus obtain a collection of functors Φ^{l} : DM^{gm}(k) → DM^{gm}(l) → DM^{gm}(l)/S → K^{b}(T ate). We not that if T ∈ T ate is a Tate motive then Φ^{k}(T) = T. If instead M ∈ Chow affords no (non-zero) Tate summands, then Φ^{k}(M) = 0. This is rather similar to the geometric fixed points functor Φ^{G} from stable equivariant homotopy theory. Since the general Φ^{l} are obtained from Φ^{k} by base change, just as Φ^{H} is obtained from the Φ^{G} construction by base change (restriction to a subgroup), we will call the functors Φ^{l} “generalized geometric fixed points functors.” A natural question is when these functors have good properties. For our purposes we definitely need tensor functors, which is to say we need S to be a tensor ideal. This is just not true in general. However, if instead of looking at the full DM^{gm}(k) we look at the subcategory DQM^{gm}(k) generated by the (products of) smooth projective quadrics, and use coefficients modulo two, then we can show that S is a tensor ideal. Moreover, using more properties of weight structures, we prove the collection of generalized fixed points functors to be conservative and Pic-injective (i.e. inducing an injection on Picard groups; see Theorem 31 in Section 5):

Theorem. Let k be a perfect field of exponential characteristic e not two, and Φ^{l} : DQM^{gm}(k, 𝔽_{2}) → T ate(𝔽_{2}) the functors constructed above. Then {Φ^{l}}_{l}, as l ranges over finitely generated extensions of k, forms a conservative, Pic-injective family of tensor triangulated functors.

1Actually for this to be true we need to consider DM^{gm}(k, 𝔽) where 𝔽 is a field. This is not really a problem.

Documenta Mathematica 22 (2017) 363–395

It is then not hard to use general properties of base change and change of coefficients for DM to build a conservative and Pic-injective family for DQM^{gm}(k, ℤ[1/e]). It turns out that one additional functor Ψ : DQM(k, ℤ[1/e]) → T ate(ℤ[1/e]) suffices. (It is related to geometric base change.) In more detail, the paper is organised as follows. In Section 2 we introduce our notations regarding Chow motives and collect some results. The main idea is to use the absence of degree one zero-cycles in a variety to conclude that it is free of Tate summands in a strong sense. This observation is what will allow us in a later section to establish that our “geometric fixed points functors” Φ^{l} are tensor. In Section 3 we review in some detail the categories DM(k, A) (triangulated motives over the perfect field k with coefficients in the commutative ring A) and their behaviour under change of coefficients and base. All the material is well known, but sometimes hard to source. We then construct a convenient conservative and Pic-injective collection on DM(k, A). The targets are always DM(k′, A′) with either k simplified (e.g. k′ separably closed) or A simplified (e.g. A′ a field). Section 4 constitutes the technical heart of our work. We first rapidly review Bondarko’s theory of weight structures. After that we carry out the programme outlined above, of constructing a conservative and Pic-injective family of functors {Φ^{l}}_{l} : DQM^{gm}(k, 𝔽_{2}) → K^{b}(T ate(𝔽_{2})). The remaining sections contain applications. In Section 5 we prove that all affine quadrics have invertible motives. This is rather satisfying, since affine quadrics are fairly natural “generalised spheres.” Also the result has been known in the étale topology for a long time. Compare the beginning of this introduction for a history of this problem. Section 6 contains the second set of applications. In [12, Conjecture 1.4] Po Hu has stated certain conjectures about the motivic spectra of affine Pfister quadrics, namely certain formulas they should satisfy under smash product. We establish the analogues (or “images”) of these formulas in DM(k) by an easy computation involving our fixed points functors. The list of applications of our methods does not end here, but the amount of material we want to stuff into one article does. As directions of future work, let us mention the following possibilities. The structure of P ic(DQM(k)) can be investigated. One may replace the set of projective quadrics by projective homogeneous varieties for a fixed group G. Also using (almost) the same methods, it is possible to study DATM(k), the subcategory of DM(k) generated by M(Spec(l)){i} for l/k finite separable and i ∈ ℤ, i.e. Artin-Tate motives. This will be treated in forthcoming work. We also note that our results for DM have applications to the study of the stable motivic homotopy category SH(k). In forthcoming work [1] we show that if k is a field of finite 2-étale cohomological dimension, then the functor SH(k) → DM(k) is conservative and Pic-injective, when restricted to compact spectra. Consequently the suspension spectral of affine quadrics are also invertible, and the Hu conjectures hold for spectra (over such fields). Whenever we talk about quadrics or quadratic forms, we shall assume that the base field has characteristic different from two. This will be restated with the most important theorems. Our results are stated over perfect base fields, because this is when DM(k) is best understood. However actually everything goes through over arbitrary base fields, using [7]. Throughout this text, we will omit brackets around the arguments of functors whenever convenient. For example MX means the same thing as M(X). The author wishes to thank Fabien Morel for suggesting this topic of investigation and for providing many helpful insights, and Mikhail Bondarko for comments on a draft of this paper. He also wishes to thank an anonymous referee for many helpful comments and suggestions.

2 Some Results about Chow Motives

We begin with some notation. We take for granted the notion of an additive category. An additive category C is called Karoubi-closed if every idempotent endomorphism of an object of C corresponds to a direct sum decomposition. By a tensor category we mean an additive category provided with a suitably compatible symmetric monoidal structure [8, Section 1]. In particular this means that the monoidal operation is bi-additive. We shall always denote the monoidal operation by ⊗ = ⊗_{C} and call it tensor product. The tensor unit is generically denoted 𝟙 = 𝟙_{C}. Now our conventions regarding Chow motives. By SmP roj(k) we denote the category of smooth projective varieties over the field k. It is a symmetric monoidal category using cartesian product as monoidal product. We shall assume understood the existence and functoriality properties of the Chow ring A^{∗}(X). Grading is by codimension and the equivalence relation we use is rational equivalence. Lower index means grading by dimension. For convenience if 𝔽 is any coefficient ring, we put A^{∗}(X, 𝔽) = A^{∗}(X) ⊗_{ℤ} 𝔽. It is then possible to construct a Karoubi-closed tensor category Chow(k, 𝔽) together with a covariant symmetric monoidal functor M = M_{𝔽} : SmP roj(k) → Chow(k, 𝔽) which has the following properties. The unit object is 𝟙_{Chow(k,𝔽)} = 𝟙 = M(Spec(k)). There exists an object 𝟙{1} such that M(ℙ^{1}) ≈ 𝟙 ⊕ 𝟙{1}. We call 𝟙{1} the Lefschetz motive. It is invertible. For any n ∈ ℤ and M ∈ Chow(k, 𝔽) we write M{n} := M ⊗ 𝟙{1}^{⊗n}. For any X, Y ∈ SmP roj(k) and i, j ∈ ℤ we have

Hom_{Chow(k,𝔽)}(M(X){i}, M(Y){j}) = A_{dim X+i−j}(X × Y).

In particular we have Hom(MX, 𝟙{i}) = A^{i}(X, 𝔽) and Hom(𝟙{i}, MX) = A_{i}(X, 𝔽). Composition is by the usual push-pull convolution. In the remainder of this section we collect some results about Chow motives which we will need throughout the article. None of them are hard, so probably most of this is well known.

Recall first that if l/k is a field extension then the functor SmP roj(k) → SmP roj(l), X ↦ X_{l} induces a functor Chow(k, 𝔽) → Chow(l, 𝔽) called base change and denoted M ↦ M_{l}. We need to know something about this in the inseparable case.

Lemma 1. Let l/k be a purely inseparable extension of fields of characteristic p and 𝔽 a coefficient ring in which (the image of) p is invertible. Then the base change Chow(k, 𝔽) → Chow(l, 𝔽) is fully faithful.

Proof. It suffices to prove that for X ∈ SmP roj(k) we have A_{∗}(X, 𝔽) = A_{∗}(X_{l}, 𝔽). By the definition of rational equivalence as in [11, Section 1.6] it is enough to show that Z_{∗}(X, 𝔽) → Z_{∗}(X_{l}, 𝔽) is an isomorphism for all X. Let Z ⊂ X be a reduced closed subscheme and |Z_{l}| the reduced closed subscheme underlying Z_{l}. Then the image of [Z] under Z_{∗}(X, 𝔽) → Z_{∗}(X_{l}, 𝔽) is n[|Z_{l}|], where n is the multiplicity of Z_{l}. This is easily seen to be a power of p, whence Z_{∗}(X, 𝔽) → Z_{∗}(X_{l}, 𝔽) is injective. It is also surjective since X_{l} → X is a homeomerphism on underlying spaces. This concludes the proof.

We now investigate “Tate summands”. Denote by T ate(k, 𝔽) ⊂ Chow(k, 𝔽) the smallest (strictly) full Karoubi-closed additive subcategory containing 𝟙{i} for all i. This is independent up to equivalence of k and we will just write T ate(𝔽) if no confusion can arise. (It is a tensor category.) We say M ∈ Chow(k, 𝔽) is Tate-free if whenever M ≈ T ⊕ M′ with T ∈ T ate(k, 𝔽), then T ≈ 0. The next proposition holds in much greater generality, but this version is all we need.

Proposition 2. Let 𝔽 be a finite ring and M ∈ Chow(k, 𝔽). Then there exist T ∈ T ate(𝔽) and M′ ∈ Chow(k, 𝔽) with M′ Tate-free and M ≈ T ⊕ M′.

Proof. Splitting off Tate summands inductively, the only problem which could occur is that M might afford arbitrarily large Tate summands. The impossibility of this follows (for example) from the finiteness of étale cohomology of complete varieties [17, Corollary VI.2.8].

Lemma 3. Let 𝔽 be a field. Then if M, N ∈ Chow(k, 𝔽) are Tate-free so is M ⊕ N.

Proof. A motive with 𝔽-coefficients is Tate-free if and only if it affords no summand of the form 𝟙{n} for any n. Let i : 𝟙{n} → M ⊕ N and p : M ⊕ N → 𝟙{n} be inclusion of and projection to a summand, for M, N arbitrary. Write i = (i_{M}, i_{N})^{T} and p = (p_{M}, p_{N}). Then id = pi = p_{M}i_{M} + p_{N}i_{N}. Since Hom(𝟙{n}, 𝟙{n}) = 𝔽 ≠ 0 we must have p_{M}i_{M} ≠ 0 or p_{N}i_{N} ≠ 0. Suppose the former holds. Then since 𝔽 is a field we may replace i_{M} by a multiple ci_{M} such that p_{M}(ci_{M}) = 1. Thus 𝟙{n} is a summand of M. Similarly in the other case. This establishes the contrapositive of the lemma.

Lemma 4. Let 𝔽 be a field. Then any morphism in T ate(k, 𝔽) factoring through a Tate-free object is zero.

Proof. Since 𝔽 is a field any Tate motive is a sum of 𝟙{n} for various n, so it suffices to consider a morphism 𝟙{n} → 𝟙{m} factoring through a Tatefree object. Since Hom(𝟙{n}, 𝟙{m}) = 0 for n ≠ m we may assume n = m. Consider a ∈ Hom(𝟙{n}, M) and b ∈ Hom(M, 𝟙{n}). If ba ≠ 0 then there exists c ∈ 𝔽 such that (cb)a = id. It follows that (cb), a present 𝟙{n} as a summand of M. This establishes the contrapositive.

We need tools to recognise Tate-free motives. To do so, we introduce some more notation. For X ∈ SmP roj(k) there exists the degree map deg : A_{0}(X, 𝔽) → 𝔽 (corresponding to pushforward along the structure map Hom(𝟙, MX) → Hom(𝟙, 𝟙)). Write I_{𝔽}(X) = deg(A_{0}(X, 𝔽)) for the image of the degree map. This is the ideal inside 𝔽 generated by the degrees of closed points. The utility of this notion is as follows.

Lemma 5. Let 𝔽 be a field and suppose I_{𝔽}(X) ≠ 𝔽. Then MX is Tate-free.

Proof. As before MX is Tate-free if and only if it affords no summand 𝟙{N} for any N. Given i ∈ Hom(𝟙{N}, MX) = A_{N}(X, 𝔽) and p ∈ Hom(MX, 𝟙{N}) = A^{N}(X, 𝔽), the composite pi ∈ Hom(𝟙{N}, 𝟙{N}) = 𝔽 is obtained by push-pull convolution. In this case it is just deg(p ∩ i) and so is contained in I_{𝔽}(X). Thus pi ≠ 1 and (p, i) is not a presentation of 𝟙{N} as a summand of X.

Lemma 6. Let X, Y ∈ SmP roj(k). Then I_{𝔽}(X × Y) ⊂ I_{𝔽}(X) ∩ I_{𝔽}(Y).

Proof. We recall that I_{𝔽}(X ×Y) is just the ideal generated by degrees of closed points. So let z ∈ X × Y be a closed point. Then z → X × Y corresponds to morphisms z → X and z → Y. It follows that deg(z) ∈ I_{𝔽}(X) and similarly deg(z) ∈ I_{𝔽}(Y). This implies the result.

Suppose S ⊂ SmP roj(k) is a set of smooth projective varieties. We write ⟨S⟩_{Chow(k,𝔽)}^{⊗,T} for the smallest strictly full, additive, Karoubi-closed, tensor subcategory of Chow(k, 𝔽) containing all Tate motives and also MX for each X ∈ S. Assuming 𝔽 is a field, this means that a general object of ⟨S⟩_{Chow(k,𝔽)}^{⊗,T} is (isomorphic to) a summand of

T ⊕ M(X_{1}^{(1)} × ⋯ × X_{n_{1}}^{(1)}){i_{1}} ⊕ ⋯ ⊕ M(X_{1}^{(m)} × ⋯ × X_{n_{m}}^{(m)}){i_{m}},

with T ∈ T ate(𝔽) and X_{i}^{(j)} ∈ S, i_{r} ∈ ℤ. The following proposition (or rather its failure to generalise) is the basic reason why in the construction of fixed point functors we will need to restrict to subcategories.

Proposition 7. Let 𝔽 be a finite field and S ⊂ SmP roj(k) be such that I_{𝔽}(X) = 0 for all X ∈ S (i.e. such that all closed points of X have degree divisible by the characteristic of 𝔽). Then any object M ∈ ⟨S⟩_{Chow(k,𝔽)}^{⊗,T} can be written as T ⊕ M′, where T ∈ T ate(𝔽) and M′ is (isomorphic to) a summand of

M(X_{1}^{(1)} × ⋯ × X_{n_{1}}^{(1)}){i_{1}} ⊕ ⋯ ⊕ M(X_{1}^{(m)} × ⋯ × X_{n_{m}}^{(m)}){i_{m}},

for some X_{i}^{(j)} ∈ S, i_{r} ∈ ℤ. Moreover any such M′ is Tate-free.

Proof. By Lemma 6 we know that I_{𝔽}(X_{1}^{(j)} ×… X_{n_{j}}^{(j)}) = 0 and thus by Lemmas 5 and 3 we conclude that any M′ as displayed is indeed Tate-free. So it suffices to establish the first part. By definition we may write

M ⊕ M′′ ≈ T ⊕ M(X_{1}^{(1)} ⊗ ⋯ ⊗ X_{n_{1}}^{(1)}){i_{1}} ⊕ ⋯ ⊕ M(X_{1}^{(m)} ⊗ ⋯ ⊗ X_{n_{m}}^{(m)}){i_{m}},

with T ∈ T ate(𝔽) and X_{i}^{(j)} ∈ S. Using Proposition 2 we write M ⊕ M′′ ≈ M′ ⊕ M′′′ ⊕ T′, where M′, M′′ are maximal Tate-free summands in M, M′′ respectively and T′ is Tate. Writing out the inverse isomorphisms M′ ⊕ M′′′ ⊕ T′ ⇆ T ⊕ M(X_{1}^{(1)} …) ⊕ … in matrix form and using Lemma 4 we conclude that T′ ≈ T via the induced map. The Lemma below yields that M′ ⊕ M′′ ≈ M(X_{1}^{(1)} …) ⊕ . . .. This finishes the proof.

Lemma 8. Let C be an additive category and let U, T, X, T′ ∈ C be four objects. Suppose we are given an isomorphism ϕ : U ⊕ T → X ⊕ T′ such that the component T → T′ is also an isomorphism. Then there is an isomorphism ϕ̃: U → X.

Proof. Let us write

ϕ = (\begin{matrix}α & a \\ b & f \end{matrix}) ψ = (\begin{matrix}β & a′ \\ b′ & g \end{matrix}) ,

where ψ is the inverse of ϕ. By assumption f is an isomorphism. Writing out ϕψ = id_{X⊕T} and ψϕ = id_{U⊕T′} one obtains

bβ = −fb′ βa = −a′f αβ + ab′ = id_{U} βα + a′b = id_{X} .

Put α̃ = α − af^{−1}b : U → X. Then the above relations imply that α̃ is an isomorphism with inverse β.

3 Review of Voevodsky Motives

In this section we collect some facts about DM(k, A) that we will need for the applications. Most of this is well-known, but in some cases we were unable to locate adequate references. We will assume throughout that k is a perfect field, and re-state this assumption with each theorem. Fix a ring A. We follow the construction of DM(k, A) and DM^{eff}(k, A) in [5]. Write Sm(k) for the symmetric monoidal category of smooth schemes over k (monoidal operation being cartesian product) and Cor(k) for the symmetric monoidal category with same objects as Sm(k) but morphisms given by finite correspondences. There is a natural monoidal functor Sm(k) → Cor(k). Write Shv^{tr}(k, A) for the abelian category of Nisnevich sheaves of A-modules on Cor(k), i.e. (additive) presheaves Cor(k)^{op} → A-Mod such that the restriction Sm(k)^{op} → Cor(k) → A-Mod is a sheaf in the Nisnevich topology. There is a functor A_{tr}• : Sm(k) → Shv^{tr}(k, A) sending X ∈ Sm(k) to the presheaf with transfers it represents (which turns out to be a sheaf). The category Shv^{tr}(k, A) carries a right exact tensor structure making A_{tr} a monoidal functor. The category DM^{eff}(k, A) is then the 𝔸^{1}-local derived category of Shv^{tr}(k, A) [5, Example 3.15]. We write L_{𝔸^{1}} : D(Shv^{tr}(k, A)) → DM^{eff}(k, A) ⊂ D(Shv^{tr}(k, A)) for the localisation functor and denote the composite Sm(k) ⟶^{A_{tr}} Shv^{tr}(k, A) → DM^{eff}(k, A) by M_{A}^{eff} or M^{eff} if no confusion can arise. The category DM^{eff}(k, A) is compactly generated, and the subcategory of compact objects DM^{eff,gm}(k, A) is the thick subcategory generated by M_{A}^{eff}(X) for X ∈ Sm(k) [5, Example 5.5]. The category DM^{eff}(k, A) carries a symmetric monoidal structure making M_{A}^{eff} : Sm(k) → DM^{eff}(k, A) a monoidal functor [5, Example 2.4]. If X ∈ Sm(k) then we write M̃^{eff}X for the homotopy fibre of M^{eff}X → M^{eff}Spec(k) = 𝟙. This is the reduced (effective) motive of X. If (X, x) is a pointed scheme then there is a canonical isomorphism M^{eff}X ≃ M̃^{eff}X ⊕ M^{eff}x = M̃^{eff}X ⊕ 𝟙. In this situation we will write M^{eff}(X, x) for M̃^{eff}X, with this direct sum decomposition understood. Throughout this text, we write 𝔾_{m} for the pointed scheme (𝔸^{1} \ 0, 1). The next step is to stabilise DM^{eff}(k, A) by inverting M^{eff}𝔾_{m} in the monoidal structure. Here we depart slightly from the notation of [5] and write Sp(Shv^{tr}(k, A)) for the abelian category of symmetric spectra in Shv^{tr}(k, A) [5, Section 6]. This is in keeping with the notation in [6, Section 5.3]. Then DM(k, A) is the Ω − 𝔸^{1}-local derived category of Sp(Shv^{tr}(k, A)) [5, Example 6.25]. There is an adjunction Σ^{∞} : Shv^{tr}(k, A) ⇆ Sp(Shv^{tr}(k, A)) extending to an adjunction

Σ^{∞} : DM^{eff}(k, A) ⇆ DM(k, A) : Ω^{∞}.

We write M = M_{A} : Sm(k) → DM(k, A) for the evident composite, and call MX for X ∈ Sm(k) the motive of X. Similarly for the reduced motive M̃X. As before, DM(k, A) is a compactly generated tensor triangulated category. The subcategory DM^{gm}(k, A) of compact objects is the thick triangulated subcategory generated by the objects of the form MX ⊗ (M𝔾_{m})^{⊗i} for X ∈ Sm(k) and i ∈ ℤ. So far all of this is very formal and the base k did not really enter. Since k is perfect, we have Voevodsky’s remarkable results at hand. Firstly, a bounded above complex in D(Shv^{tr}(k, A)) is 𝔸^{1}-local if and only if its homology sheaves are [16, Proposition 14.8]. Moreover a model for L_{𝔸^{1}} is given by the 𝔸^{1}-chain complex C_{∗} [16, Corollary 14.9]. Next there is the cancellation theorem: for E, F ∈ DM^{eff}(k, A) we have Hom(E ⊗ M𝔾_{m}, F ⊗ M𝔾_{m}) = Hom(E, F) [21, Corollary 4.10]. (Voevodsky only states this for E, F bounded above, but the general case follows using compact generation and taking limits: Let C_{1} ⊂ DM^{eff}(k, A) be the class of objects F such that Hom(MX[i], F) = Hom(MX[i] ⊗ M𝔾_{m}, F ⊗ M𝔾_{m}) for all X ∈ Sm(k). Then C_{1} is closed under cones (by the five lemma), shifts, isomorphisms and arbitrary sums (because the MX are compact) and contains all bounded above complexes, hence C_{1} = DM^{eff}(k, A). Next let C_{2} ⊂ DM^{eff}(k, A) be the class of objects E such that Hom(E, F) = Hom(E ⊗ M𝔾_{m}, F ⊗ M𝔾_{m}) for all F ∈ DM^{eff}(k, A). Then C_{2} is closed under shifts, cones (by the five lemma) isomorphisms and arbitrary sums, and contains DM^{gm}(k, A) (since C_{1} = DM^{eff}(k, A)) and hence C_{2} = DM^{eff}(k, A).) This implies that Σ^{∞} : DM^{eff}(k, A) → DM(k, A) is fully faithful. Indeed if E ∈ D(Shv^{tr}(k, A)) is 𝔸^{1}-local then Σ^{∞}E ∈ D(Sp(Shv^{tr}(k, A))) is an Ω-spectrum by cancellation (i.e. E ≃ R\underline{Hom}(M𝔾_{m}, E ⊗ M𝔾_{m})). Finally there are the homotopy t-structures: the category DM^{eff}(k, A) has a non-degenerate t-structure with heart the category of homotopy invariant sheaves with transfers, and DM(k, A) affords a non-degenerate t-structure with heart the category of homotopy modules with transfers. The functor Ω^{∞} is texact and hence Σ^{∞} is right-t-exact. We will now discuss functoriality of DM(k, A) in k and A. The tool to do this is [5, Proposition 3.11], which implies that if f^{∗} : Shv^{tr}(k, A) → Shv^{tr}(l, B) is a functor which preserves colimits, coverings, and multiplication by 𝔸^{1} and 𝔾_{m}, then there are induced adjunctions

Lf^{∗} : DM^{eff}(k, A) ⇆ DM^{eff}(l, B) : Rf_{∗}
Lf^{∗} : DM(k, A) ⇆ DM(l, B) : Rf_{∗}.

Here Lf^{∗} commutes with Σ^{∞} and Rf_{∗} commutes with Ω^{∞}. If the functor f^{∗} we started with was monoidal then so is the extended functor Lf^{∗}. We now specialise to base change. For this, let f : Spec(l) → Spec(k) be an extension of (perfect) fields. Then f^{∗} : Shv^{tr}(k, A) → Shv^{tr}(l, A) satisfies the requirements outlined above and so we get Lf^{∗} : DM(k, A) ⇆ DM(l, A) : Rf_{∗}, and similarly for DM^{eff}. If f is finite separable then f_{#} : Sm(l) → Sm(k) induces f_{#} : Shv^{tr}(l, A) → Shv^{tr}(k, A) and then Lf_{#} : DM(l, A) ⇆ DM(k, A) : Rf^{∗}. See also [5, Example 6.25] again. In this situation we have Rf^{∗} = f^{∗} = Lf^{∗}. The following result is surely well-known, but we could not find a reference, so include the easy proof.

Proposition 9. Let f : Spec(l) → Spec(k) be an algebraic (separable) extension of the perfect field k. Then Lf^{∗} : DM(k, A) → DM(l, A) is t-exact. Suppose that A is a ring such that for each finite subextension l/l′/k, the (image of the) integer [l′ : k] is a unit in A. Then Lf^{∗} : DM(k, A) → DM(l, A) is conservative and t-exact.

Thus we shall write Lf^{∗} = f^{∗} also in this situation.

Proof. Since f^{∗} : Shv^{tr}(k, A) → Shv^{tr}(l, A) preserves homotopy invariant sheaves, it follows that Lf^{∗} : D(Sp(Shv^{tr}(k, A))) → D(Sp(Shv^{tr}(k, A))) preserves 𝔸^{1}-local objects. Since f^{∗} : Shv^{tr}(k, A) → Shv^{tr}(l, A) preserves contractions, Lf^{∗} : D(Sp(Shv^{tr}(k, A))) → D(Sp(Shv^{tr}(l, A))) preserves 𝔸^{1} − Ω-local objects. Thus t-exactness of Lf^{∗} : D(Sp(Shv^{tr}(k, A))) → D(Sp(Shv^{tr}(l, A))) implies t-exactness of Lf^{∗} : DM(k, A) → DM(l, A). It thus remains to show: if F ∈ Shv^{tr}(k, A) and f^{∗}F = 0, then F = 0. Let X ∈ Sm(k), x ∈ F (X). It suffices to show that x = 0. By [17, Proposition II.2.2 and Lemma II.3.3] we have 0 = f^{∗}x ∈ F (X ⊗_{k}l) = colim_{l/l′/k} F (X ⊗_{k}l′), where the colimit is over finite subextensions. Thus there exists a finite subextension l/l′/k with (l′/k)^{∗}(x) = 0. But then by a transfer argument one finds that [l′ : k]x = 0, whence x = 0 since [l′ : k] is a unit in A by assumption.

Next we consider change of coefficients. The construction and basic properties must be well known, but again we could not find convenient references. Let α : A → B be a ring homomorphism. There is a natural adjoint functor pair

α_{#} : Shv^{tr}(k, A) ⇄ Shv^{tr}(k, B) : α^{∗}.

Here α_{#}F is the sheaf associated to X ↦ F (X) ⊗_{A} B and α^{∗}F (X) = F (X), viewed as an A-module. In particular α_{#} is monoidal and preserves colimits, so [5, Proposition 3.11] applies to give us adjunctions

Lα_{#} : DM^{eff}(k, A) → DM^{eff}(k, B) : Rα_{#}
Lα_{#} : DM(k, A) → DM(k, B) : Rα_{#}.

In order to manipulate these functors efficiently, we need a standard result.

Lemma 10. Any object E ∈ DM(k, A) is a filtered homotopy colimit of objects of the form E′ ⊗ 𝔾_{m}^{⊗i} with E′ ∈ DM^{eff}(k, A) bounded above and i ∈ ℤ.

Proof. Let DM^{eff}(k, A) be a model for DM(k, A). Then we may alternatively model DM(k, A) via Spt^{Σ}(DM^{eff}(k, A), M𝔾_{m}), i.e. via (symmetric) M𝔾_{m}- spectra in DM^{eff}(k, A). Let E ∈ DM(k, A) = Ho(Spt^{Σ}(DM^{eff}(k, A), M𝔾_{m})) have fibrant replacement (E_{1}, E_{2}, …). Then E ≃ hocolim_{i}(E_{i})_{≥−i} ⊗ 𝔾_{m}^{⊗−i}, as one sees immediately by computing the homotopy sheaves on both sides.

In order to use this, recall that any left adjoint functor of triangulated categories (assumed to have all countable coproducts) commutes with filtered homotopy colimits (by filtered we always mean ω-filtered, i.e. with a countable indexing set) and if it additionally preserves a compact generating set, then its right adjoint also commutes with filtered homotopy colimits. Thus essentially all our functors commute with filtered homotopy colimits. In particular Rα^{∗} commutes with filtered homotopy colimits.

Lemma 11. Let k be perfect and α : A → B a ring homomorphism. Then the functors Rα^{∗} commute with Σ^{∞}. In fact for E ∈ DM^{eff}(k, B) we have Rα^{∗}(E ⊗ 𝔾_{m}) ≃ Rα^{∗}(E) ⊗ 𝔾_{m}.

Of course Lα_{#} always commutes with Σ^{∞}, for formal reasons.

Proof. Let E ∈ DM^{eff}(k, B). Then Σ^{∞}E ≃ (E, E ⊗ 𝔾_{m}, E ⊗ 𝔾_{m}^{⊗2}, …). (Here by E ⊗ 𝔾_{m} we mean the derived tensor product in DM^{eff}, in particular this notation implies an 𝔸^{1}-local object.) By the cancellation theorem, this is an Ω-spectrum. It follows that Rα^{∗}Σ^{∞}E = (Rα^{∗}E, Rα^{∗}(E ⊗ 𝔾_{m}), …). It is thus enough to show that Rα^{∗}(E ⊗ 𝔾_{m}) ≃ Rα^{∗}(E) ⊗ 𝔾_{m}. Since Lα_{#} is symmetric monoidal Rα^{∗} is lax symmetric monoidal and there is a natural comparison map. Since DM^{eff}(k, B) is generated as a localising subcategory by MX for X ∈ Sm(k) and ⊗, Rα^{∗} commute with arbitrary sums, we may assume E = MX. In this case a fibrant model of E ⊗ 𝔾_{m} is given by C_{∗}B_{tr}(X_{+} ∧ 𝔾_{m}). Resolving B freely as an A-module, it follows that Rα^{∗}E ∈ DM^{eff}(k, A)⊗𝔾_{m}. A calculation using adjunction and the cancellation theorem allows us to conclude by the Yoneda lemma.

Proposition 12. Let k be perfect, α : A → B be flat, E ∈ DM^{gm}(k, A) and F ∈ DM(k, A). Then

Hom(E, F) ⊗_{A} B ≈ Hom(Lα_{#}E, Lα_{#}F).

Proof. By Lemma 10 (and twisting E), we may assume that F ∈ DM^{eff}(k, A) and is bounded above. Then by the cancellation theorem we may assume that E ∈ DM^{gm,eff}(k, A) as well. There is a natural map from the left hand side to the right hand side. Using the 5-lemma and the fact that DM^{gm,eff}(k, A) is generated by MX for X ∈ Sm(k), we may reduce to E = MX (shifting F if necessary). In this case Hom(MX, F) is given by the hypercohomology H^{0}(X, F^{•}). Since ⊗_{A}B is exact it commutes with hypercohomology and preserves 𝔸^{1}-invariance of cohomology sheaves, so we have H^{0}(X, F^{•}) ⊗_{A} B = H^{0}(X, F^{•} ⊗_{A} B) = H^{0}(X, (Lα_{#}F)^{•}).

Proposition 13. Let k be perfect, A a ring, a ∈ A a non zero divisor and α : A → A/(a) the natural map. Then for E ∈ DM(k, A) there is a natural distinguished triangle

E ⟶^{·a} E → Rα^{∗}Lα_{#}E.

This triangle yields the typical Bockstein sequences one expects for reduction of coefficients.

Proof. By Lemmas 10 and 11 we may assume that E ∈ DM^{eff}(k, A) and is bounded above. In this case Rα^{∗}Lα_{#}E is computed by resolving E by a complex of representable sheaves C^{•} and then C^{•}/(a) is a model for Rα^{∗}Lα_{#}E. (Note that since C^{•} has homotopy invariant cohomology, so does α_{#}C^{•} = C^{•}/(a), by considering the (ordinary) Bockstein sequence. Hence we may apply α^{∗} immediately to α_{#}C^{•} instead of having to 𝔸^{1}-localise first.) Since a is not a zero divisor the sequence 0 → C^{•} → C^{•} → C^{•}/(a) → 0 is exact and yields the desired triangle.

With this preparation out of the way, we can prove our conservativity and Pic-injectivity theorem. Recall that Hom_{DM(k,A)}(𝟙, 𝟙[i]) = A if i = 0 and = 0 else.

Theorem 14. Let k be a perfect field and A a PID of characteristic zero. Let f : Spec(k^{s}) → Spec(k) be a separable closure. The collection of functors {f^{∗}} ∪ {Lα_{π#}}_{π} is conservative. If A has primes of arbitrary large characteristic, the collection is also Pic-injective (both on DM(k, A)). Here α_{π} : A → A/(π) runs through the primes of A.

We could prove essentially the same theorem with A replaced by a Dedekind domain (of characteristic zero) with only slightly more work.

Proof. We first show conservativity. Let E ∈ DM(k, A) with Lα_{π#}E = 0 for all π and f^{∗}E = 0. We must show that E = 0. Let T ∈ DM^{gm}(k, A). It suffices to prove that Hom(T, E) = 0. Now by Proposition 13 we have the triangle E ⟶^{π} E → Rα_{π}^{∗}Lα_{π#}E = 0. Thus multiplication by π is an isomorphism on Hom(T, E). Let K = F rac(A). Since π was arbitrary it follows that Hom(T, E) is a K-vector space. Since K ⊗_{A} K ≠ 0 we concude that Hom(T, E) = 0 provided that Hom(T, E) ⊗_{A} K = 0. Let α_{0} : A → K be the (flat) localisation. By proposition 12 we know that Hom(T, E) ⊗_{A} K = Hom(Lα_{0#}T, Lα_{0#}E), so it suffices to show that Lα_{0#}E = 0. But K is of characteristic zero, so by proposition 9 it is enough to show that f^{∗}Lα_{0#}E = 0. Since Lα_{0#} and f^{∗} “commute”, this follows from the assumption that f^{∗}E = 0. Now we prove Pic-injectivity. Let E ∈ DM(k, A) be such that f^{∗}E ≈ 𝟙_{k^{s}} and Lα_{π#}E ≈ 𝟙_{A/(π)}. As a first step, I claim that there exists a finite extension k ⊂ l ⊂ k^{s} such that g^{∗}E ≈ 𝟙_{l}, where g : Spec(l) → Spec(k). As in the proof of Proposition 9 we find that Hom(𝟙_{k^{s}}, f^{∗}E) = colim_{k⊂l⊂k^{s}} Hom(𝟙_{l}, (l/k)^{∗}E), where the colimit is over finite subextensions. Hence there exist l and an element t ∈ Hom(𝟙_{l}, g^{∗}E) such that (k^{s}/l)^{∗}(t) is an isomorphism. The commutative diagram

Hom(𝟙_{l}, g^{∗}E)  ⟶  Hom(𝟙_{k^{s}}, f^{∗}E) ≈ A
↓  ↓
Hom(𝟙_{l,A/(π)}, Lα_{π#}g^{∗}E) ⟶^{≈} Hom(𝟙_{k^{s},A/(π)}, Lα_{π#}f^{∗}E) ≈ A/(π)

shows that Lα_{π#}(t) is an isomorphism. Thus by the first part (conservativity), t is an isomorphism.

Now we consider Hom(𝟙_{k}, E). From the Bockstein triangles and the assumption Lα_{π#} ≈ 𝟙_{A/(π)} we get the exact sequences

Hom(𝟙_{A/(π)}, Lα_{π#}E[−1]) = 0 → Hom(𝟙_{k}, E) ⟶^{π} Hom(𝟙_{k}, E)

→ Hom(𝟙_{A/(π)}, Lα_{π#}E) ≈ A/(π) → Hom(𝟙_{k}, E[1])

It follows that Hom(𝟙_{k}, E) is a torsion-free A-module (hence abelian group). Thus by transfer it follows that Hom(𝟙_{k}, E) → Hom(𝟙_{l}, g^{∗}E) ≈ A is injective. Let us denote the image by I ⊂ A. This is a free A-module (of rank zero or one). Since Hom(𝟙_{l}, g^{∗}(E)[1]) = 0 it follows by transfer that Hom(𝟙_{k}, E[1]) is [l : k]- torsion. Choosing π of sufficiently large characteristic, we find that A/(π) → Hom(𝟙_{k}, E[1]) is the zero map. Thus I = Hom(𝟙_{k}, E) ≠ 0, i.e. I ≈ A. It follows that Hom(𝟙_{k}, E) → Hom(𝟙_{A/(π)}, Lα_{π#}E) ≈ A/(π) is surjective for each π. Consider the commutative diagram

Hom(𝟙_{k}, E)  ⟶  Hom(𝟙_{l}, g^{∗}E) ≈ A
(∗)↓  ^{(∗∗)}↓
Hom(𝟙_{A/(π)}, Lα_{π#}E) ⟶^{≈} Hom(𝟙_{l,A/(π)}, Lα_{π#}g^{∗}E) ≈ A/(π)

The map (**) is the natural surjection and (*) is surjective as we just proved. It follows that I + (π) = A for each π and so I = A. Thus there exists t′ ∈ Hom(𝟙_{k}, E) with g^{∗}(t′) = t an isomorphism. Considering the diagram again one finds that Lα_{π#}(t′) is also an isomorphism. Thus t′ is an isomorphism (by the first part, again) and we are done.

We need two more auxiliary results. For the first, let f : Spec(l) → Spec(k) be a Galois extension with group G. If M ∈ DM^{gm}(k, A) then the Amodule Hom(𝟙, f^{∗}M) ≈ Hom(M(Spec(l)), M) has a natural action by G (coming from automorphisms of Spec(l)). We denote this action by κ_{M} : G → Aut(Hom(𝟙, f^{∗}M)).

Proposition 15. Let f : Spec(l) → Spec(k) be (finite) Galois and [l : k] invertible in A. Then the above construction yields an injective homomorphism

κ : Ker(f^{∗} : P ic(DM^{gm}(k, A)) → P ic(DM^{gm}(l, A))) → Hom(Gal(l/k), A^{×}).

Proof. Suppose that M ∈ P ic(DM^{gm}(k, A)), f^{∗}M ≃ 𝟙 and let us show that M ≃ 𝟙 if and only if the action is trivial. Necessity is clear, we show sufficiency. Independent of the assumptions on [l : k] and M I claim we have the following: if t : 𝟙_{L} → f^{∗}M is any morphism, then f^{∗}(tr(t)) : 𝟙_{L} → f^{∗}M is the sum of the conjugates under the G-action. Indeed the action on Hom_{L}(𝟙_{L}, f^{∗}M) ≈ Hom_{k}(𝟙_{L}, M) comes from premultiplication by elements of Hom_{k}(𝟙_{L}, 𝟙_{L}), whereas transfer comes from premultiplication with the adjunction morphism. Thus to prove the claim we may assume that M = 𝟙 and t = id, in which case the result follows from [16, Exercise 1.11]. Thus reinstating our assumptions, let t : 𝟙_{L} → f^{∗}M be an isomorphism and assume that the G-action is trivial. Then tr(t/[l : k]) : 𝟙 → M is an isomorphism since f^{∗}(tr(t/[l : k])) = t is, by Proposition 9. Finally we have to prove that κ is a homomorphism, i.e. that κ_{M⊗N} = κ_{M}κ_{N}. For this let us denote the adjunction isomorphism Hom_{DM(k,A)}(M(l), T) → Hom_{DM(l,A)}(𝟙, f^{∗}T) generically by ad. One checks that given f ∈ Hom(M(l), M), g ∈ Hom(M(l), N) then ad(f) ⊗ ad(g) = ad((f ⊗ g) ∘ α), where α : M(l) → M(l) ⊗ M(l) is the map corresponding to l ⊗ l → l, a ⊗ b → ab. Next observe that α is G-equivariant if G acts diagonally on M(l) ⊗ M(l). The result follows.

For the statement of the next result, we need DM(l, A) even if l is not perfect. It is explained in the next section what we mean by that. Under our assumptions on A, it is equivalent to DM(l^{p}, A), where l^{p} is the perfect closure of l.

Lemma 16. Let k be a perfect field, X/k a smooth variety, A a ring in which the exponential characteristic of k is invertible, and M ∈ DM(k, A). If for all n ∈ ℤ and all x ∈ X (not necessarily closed) we have that Hom_{DM(x,A)}(𝟙{n}, M_{x}) = 0, then also for all n ∈ ℤ we have Hom_{DM(k,A)}(MX{n}, M) = 0.

Proof. We will prove the result by induction on dim X. Thus in order to prove it for X we may assume that Hom_{DM(k,A)}(MX′{n′}, M) = 0 for every smooth, locally closed X′ ⊂ X with dim X′ < dim X, and every n′ ∈ ℤ (because the residue fields of X′ form a subset of those of X). If dim X = 0 then X is a disjoint union of spectra of fields, and the result is clear. To prove the general case, we may assume that X is connected. Let n ∈ ℤ and α ∈ Hom(MX{n}, M). It suffices to show that α = 0. By considering the generic point and using continuity [7, Example 2.6(2)] we conclude that there exists a non-empty open subvariety U ⊂ X such that α|_{U} = 0. Let Z = X \ U. If Z is empty there is nothing to do. Otherwise there exists a non-empty, smooth, connected open subvariety U_{1} ⊂ Z, since k is perfect. Let Z′ = Z \ U_{1}, U′ = U ∪ U_{1} = X \ Z′. Then U′ is smooth open in X and we have X \ U′ = Z′, which is strictly smaller than Z. We shall prove that α|_{U′} = 0. By repeating this argument with U replaced by U′ (i.e. Noetherian induction on Z) it will follow that α = 0. Note that U_{1} = U′ \ U is closed in U′, say of codimension c. Thus we get the distinguished Gysin triangle

MU{n} → MU′{n} → MU_{1}{n − c}.

Now Hom(MU_{1}{n − c}, M) = 0 by the induction on dimension. Thus Hom(MU′{n}, M) → Hom(MU{n}, M) is injective. But (α|_{U′})|_{U} = α|_{U} = 0 by assumption, so α|_{U′} = 0.

4 Weight Structures and the Geometric Fixed Points Functors

In this section, we will use Bondarko’s theory of weight structures to construct “generalised geometric fixed points functors” and prove that they have good properties. We shall fix a coefficient ring 𝔽 on which an integer e is invertible, and only work with fields of exponential characteristic e. We shall have to deal with DM(k, 𝔽) for k an imperfect field. There is now a fairly complete theory of DM(X, 𝔽) for Noetherian schemes over a field of exponential characteristic e (assumed invertible in 𝔽) [7]. It satisfies the six functors formalism, in particular continuity. We recall that if k is an imperfect field with perfect closure k^{p}, then the pullback DM(k, 𝔽) → DM(k^{p}, 𝔽) is an equivalence of categories [7, Proposition 8.1 (d)]. This means that essentially all properties known over perfect fields hold over imperfect fields as well. We also mention that all of the categories DM(X, 𝔽) afford DG-enhancements. (This is well known if k is a perfect field and hence holds for k any field by the previous remark, and this is all we need. But it is actually clear that the constructions in [7] all yield DG categories.) We shall work extensively in this section with weight structures [2], which we now review rapidly. Recall that given a category C and a full subcategory D ⊂ C, we call D Karoubi-closed in C if D is closed under retracts [2, p. 11]. In other words whenever X ∈ C and id_{X} factorises through an object of D, then X ∈ D. For example, if C is Karoubi-closed itself, then D is Karoubi-closed in C if and only if D is a Karoubi-closed category, and strictly full in C. Similarly, given a category C and a full subcategory D ⊂ C, by the Karoubiclosure of D in C we mean the full subcategory of C spanned by all the objects which are retracts of objects of D. For example, if D ⊂ C is strictly full and D is a Karoubi-closed category, then D is Karoubi-closed in C.

Definition. Let C be a triangulated category and C^{w≥0}, C^{w≤0} ⊂ C two classes of objects. We call this a weight structure if the following hold:

(i) C^{w≥0}, C^{w≤0} are additive and Karoubi-closed in C.

(ii) C^{w≥0} ⊂ C^{w≥0}[1], C^{w≤0}[1] ⊂ C^{w≤0}

(iii) For X ∈ C^{w≥0}, Y ∈ C^{w≤0} we have Hom(X, Y [1]) = 0.

(iv) For each X ∈ C there is a distinguished triangle

B[−1] → X → A

with B ∈ C^{w≥0} and A ∈ C^{w≤0}.

These axioms look quite similar to those of a t-structure, but in practice weight structures behave rather differently. We call a decomposition as in (iv) a weight decomposition. It is usually far from being unique. We put C^{w≥n} = C^{w≥0}[−n] and C^{w≤n} = C^{w≤0}[−n]. We also write C^{w>n} = C^{w≥n+1} etc. The intersection C^{w=0} := C^{w≥0} ∩ C^{w≤0} is called the heart of the weight structure.

A weight structure is called non-degenerate if ∩_{n}C^{w≥n} = 0 = ∩_{n}C^{w≤n}. It is called bounded if ∪_{n}C^{w≥n} = C = ∪_{n}C^{w≤n} A functor F : C → D between categories with weight structures is called wexact if F (C^{w≤0}) ⊂ D^{w≤0} and F (C^{w≥0}) ⊂ D^{w≥0}. It is called w-conservative if given X ∈ C with F (X) ∈ D^{w≤0} we have X ∈ C^{w≤0}, and similarly for w ≥ 0. Note that a w-conservative functor on a non-degenerate weight structure is conservative. In the following proposition we summarise properties of weight structures we use.

Proposition 17. (1) C^{w≤0} and C^{w≥0} are extension-stable: if A → B → C is a distinguished triangle and A, C ∈ C^{w≤0} (respectively A, C ∈ C^{w≥0}) then B ∈ C^{w≤0} (respectively B ∈ C^{w≥0}).

Moreover X ∈ C^{w≥0} if and only if Hom(X, Y) = 0 for all Y ∈ C^{w<0}, and similarly X ∈ C^{w≤0} if and only if Hom(Y, X) = 0 for all Y ∈ C^{w>0}.

(2) Bounded weight structures are non-degenerate.

(3) If C admits a DG-enhancement and the weight structure is bounded, then there exists a w-exact, w-conservative triangulated functor

t : C → K^{b}(C^{w=0})

called the weight complex. Its restriction to C^{w=0} is the natural inclusion.

(4) If the weight structure is bounded and C^{w=0} is Karoubi-closed then so is C.

(5) If H ⊂ C is a negative subcategory of a triangulated category (i.e. for X, Y ∈ H we have Hom(X, Y [n]) = 0 for n > 0) generating it as a thick subcategory, then there exists a unique weight structure on C with H ⊂ C^{w=0}. Moreover C^{w≤0} is the smallest extension-stable Karoubi-closed subcategory of C containing ∪_{n≥0}H[n], and similarly for C^{w≥0}. The weight structure is bounded and C^{w=0} is the Karoubi-closure of H in C.

(6) If D ⊂ C is a triangulated subcategory such that D^{w≤0} := D ∩ C^{w≤0} and D^{w≥0} := D ∩ C^{w≥0} define a weight structure on D (we say the weight structure restricts to D) then the Verdier quotient C/D affords a weight structure with (C/D)^{w≤0} the Karoubi-closure of the image of C^{w≤0} in C/D, and similarly for (C/D)^{w≥0}, (C/D)^{w=0}.

The natural “quotient” functor Q : C → C/D is w-exact. If X, Y ∈ C^{w=0} then

Hom(QX, QY) = Hom(X, Y)/Σ_{Z∈D^{w=0}} Hom(Z, Y) ∘ Hom(X, Z).

The weight structure on C/D is bounded if the one on C is.

Proof. (1) [2, Proposition 1.3.3 (1-3)]. (2) [2, Proposition 1.3.6 (3) and comment after proof]. (3) [2, Proposition 3.3.1 (I), (IV) and Section 6.3]. (4) [2, Lemma 5.2.1]. (5) [2, Theorem 4.3.2 (II) and its proof], [4, Remark 2.1.2].

(6) [2, Proposition 8.1.1]. Weight exactness holds by definition of the weight structure on C/D.

We shall call a triangulated category with a fixed weight structure a w-category.

Lemma 18. Let C be a w-category with heart H, and H′ ⊂ H an additive subcategory. Let C′ be the thick triangulated subcategory generated by H′ inside C. Then the weight structure of C restricts to C′. In particular, if X ∈ C′ then we may choose a weight decomposition A → X → X′ (i.e. A ∈ C^{w≥0} and X′ ∈ C^{w<0}) with A, X′ ∈ C′.

Proof. This is just Proposition 17 (5) which says that C′, being negatively generated by H′, carries a natural unique weight structure. By the description provided we find C^{′w≤0} ⊂ C^{w≤0}, C^{′w≥0} ⊂ C^{w≥0}. Hence a weight decomposition in C′ is also a weight decomposition in C. The rest follows from the definitions. (It follows from the orthogonality characterisation that C^{′w≤0} = C^{w≤0} ∩ C′, but we do not need this.)

Lemma 19. Let F : C → D be a triangulated functor of w-categories, and assume that the weight structure on C is bounded. Then F is w-exact if and only if F (C^{w=0}) ⊂ D^{w=0}.

Proof. Necessity is clear, we show sufficiency. We find by induction that the thick subcategory of C generated by C^{w=0} contains C^{w≤n} ∩C^{w≥−n} for all n, and hence all of C by boundedness. It follows that the weight structure on C is the one described in Proposition 17 (5), i.e. C^{w≥0}, C^{w≤0} are obtained as extension closures of ⋃_{n≥0} C^{w=n}, ⋃_{n≤0} C^{w=n}. The result follows since D^{w≥0}, D^{w≤0} are extension-stable.

Lemma 20. Let C be a w-category which is also a tensor category. Assume that 𝟙_{C} ∈ C^{w=0} and that tensoring is weight-bi-exact, i.e. that C^{w≤0} ⊗C^{w≤0} ⊂ C^{w≤0} and similarly for C^{w≥0}. Then the weight complex functor is tensor whenever C affords a tensor DGenhancement and Pic-injective whenever additionally the weight structure is bounded. If moreover C is rigid then the dualisation D : C^{op} → C is w-exact (i.e. D(C^{w≥0}) ⊂ C^{w≤0} and vice versa).

Proof. If D is a negative DG tensor category, then H^{0}(D) is tensor in a natural way and the weight complex functor t manifestly respects the tensor structure. If C is a tensor DG category with the property that H^{n}(Hom(X, Y)) = 0 for all X, Y ∈ D and n > 0 then the good truncation τ_{≤0}D is tensor in a natural way, and the quasi-equivalence τ_{≤0}D → D is a tensor equivalence.

Hence the weight complex functor is tensor as soon as there is any tensor DG enhancement of C^{w=0}. Moreover by Proposition 17 (3) if the weight structure is bounded then t is w-conservative. Since it induces an isomorphism on hearts it is a fortiori Pic-injective. This proves the first part. For the second part, let X ∈ C. The category C being rigid means that there exists an object DX such that ⊗DX is both right and left adjoint to ⊗X. If X ∈ C^{w≥0} and Y ∈ C^{w>0} then Hom(Y, DX) = Hom(Y ⊗ X, 𝟙) = 0 because Y ⊗ X ∈ C^{w>0} whereas 𝟙 ∈ C^{w=0}. It follows that DX ∈ C^{w≤0} by Proposition 17 (1). The case of X ∈ C^{w≤0} is similar.

We point out that for any field k, the category DM^{gm}(k, 𝔽) carries a canonical weight structure [3]. (Note that the perfectness assumption in that article can be dispensed with by passing to the equivalent category DM^{gm}(k^{p}, 𝔽).) It is bounded and DM^{gm}(k, 𝔽) is also a rigid tensor category with the tensor structure satisfying the assumptions of Lemma 20. The base change functors f^{∗} : DM^{gm}(l, 𝔽) → DM^{gm}(l′, 𝔽) for f : Spec(l′) → Spec(l) are w-exact by Lemma 19. The heart of the weight structure is Chow(k^{p}, 𝔽) which contains Chow(k, 𝔽) as a full subcategory by Lemma 1. In the remainder of this section we will be dealing with the following situation. The coefficient ring 𝔽 is a finite field of characteristic p (necessarily p ≠ e, where e is the exponential characteristic of the ground field k). For every extension l/k we are given a set S_{l} ⊂ SmP roj(l) such that for all closed points x ∈ X ∈ S_{l} we have p|deg(x). Recall the categories ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} of Section 2. We will assume that they are stable by base change, i.e. that for X ∈ S_{l} and l′/l another extension we have MX_{l′} ∈ ⟨S_{l′}⟩_{Chow(l′,𝔽)}^{⊗,T}. We write D⟨S⟩TM(l, 𝔽) for the thick triangulated subcategory of DM(l, 𝔽) generated by ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} ⊂ Chow(l, 𝔽) ⊂ DM(l, 𝔽)^{w=0}. It is tensor. The categories D⟨S⟩TM(l, 𝔽) are also stable by base change in the sense that if f : Spec(l′) → Spec(l) is a field extension then f^{∗}(D⟨S⟩TM(l, 𝔽)) ⊂ D⟨S⟩TM(l′, 𝔽). By Proposition 17 (5) the weight structure on DM^{gm}(l, 𝔽) restricts to D⟨S⟩TM(l, 𝔽), and the heart is ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T}. We write ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗} for the Karoubi-closed tensor subcategory of Chow(l, 𝔽)

generated by Tate twists of motives of varieties in S_{l} (i.e. this is ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} “without the Tate motives”). By Proposition 7 this subcategory consists of Tate-free objects. Let ⟨S_{l}⟩^{tri} ⊂ D⟨S⟩TM(l, 𝔽) be the thick triangulated subcategory generated by ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗}. As before, the weight structure restricts to ⟨S_{l}⟩^{tri}. We write φ_{0}^{l} : D⟨S⟩TM(l, 𝔽) → D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri} for the Verdier quotient.

Proposition 21. The category D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri} carries natural weight and tensor structures, and φ_{0}^{l} is a w-exact tensor functor. The composite

T ate(𝔽) → D⟨S⟩TM(l, 𝔽) → D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri}

is a full embedding with essential image (D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri})^{w=0}.

Proof. The existence of the weight structure and weight exactness is Proposition 17 (6). This also says that (D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri})^{w=0} is generated as a Karoubi-closed subcategory by φ_{0}^{l} (D⟨S⟩TM(l, 𝔽)^{w=0}). If M ∈ D⟨S⟩TM(l, 𝔽)^{w=0} = ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} then we may write M ≈ M′ ⊕ T with T a Tate and M′ ∈ ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗}, by Proposition 7. Thus φ_{0}^{l}(M) ≈ φ_{0}^{l}(T)

and so φ_{0}^{l} : T ate(𝔽) → (D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri})^{w=0} is essentially surjective up to (relative) Karoubi-closing. We shall show it is fully faithful whence its essential image is (absolutely) Karoubi-closed and so φ_{0}^{l} : T ate(𝔽) → (D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri})^{w=0} will be an equivalence. But by the description in Proposition 17 (6) it suffices to prove that any morphism between Tate objects factoring through ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗} is zero. This follows from Lemma 4. For the existence of the tensor structure we need ⟨S⟩^{tri} ⊗ D⟨S⟩TM(l, 𝔽) ⊂ ⟨S⟩^{tri}; then φ_{0}^{l} is automatically tensor. Considering generators, it suffices to show that ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗} ⊗ ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} ⊂ ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗}. This follows from Proposition 7.

Let l/k be any extension. We write Φ^{l} : D⟨S⟩TM(k, 𝔽) → K^{b}(T ate(𝔽)) for the composite

Φ^{l} : D⟨S⟩TM(k, 𝔽) → D⟨S⟩TM(l, 𝔽) → D⟨S⟩TM(l, 𝔽)/⟨S_{l}⟩^{tri}

⟶^{t} K^{b} ((D⟨S⟩TM(l, 𝔽)/⟨S⟩^{tri})^{w=0}) ≈ K^{b}(T ate(𝔽))

of base change, the Verdier quotient functor φ_{0}^{l}, and the weight complex t. It is a w-exact triangulated tensor functor. We can now state the main theorem of this section.

Theorem 22. Let k be a ground field of exponential characteristic e, 𝔽 a finite field of characteristic p ≠ e. Suppose given for each field extension l/k a set S_{l} ⊂ SmP roj(l) and a function ex = ex_{l} : S_{l} → ℕ. Assume that the following hold (for all fields l/k):

(1) For x ∈ X ∈ S_{l} closed, p|deg(x).

(2) If l′/l is a field extension and X ∈ S_{l} has no rational point over l′, then X_{l′} is isomorphic to an object of S_{l′} and ex(X_{l′}) ≤ ex(X).

(3) If l′/l is a field extension and X ∈ S_{l} has a rational point over l′, then MX_{l′} is a summand of a motive of the form

T ⊕ M(X_{1}^{(1)} ⊗ ⋯ ⊗ X_{n_{1}}^{(1)}){i_{1}} ⊕ ⋯ ⊕ M(X_{1}^{(m)} ⊗ ⋯ ⊗ X_{n_{m}}^{(m)}){i_{m}},

with T ∈ T ate(𝔽), X_{i}^{(j)} ∈ S_{l′} and ex(X_{i}^{(j)}) < ex(X) for all i, j.

Then the family {Φ^{l}}_{l}, as l runs through finitely generated extensions of k, is w-conservative (so in particular conservative) and Pic-injective.

We note that (2) and (3) imply that ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} are stable by base change, i.e. we are in the situation we have been discussing. Also (1) implies that none of the X ∈ S_{l} have rational points over l. The somewhat obscure functions ex_{l} are necessary to make an induction step in the proof work. We will always use ex = dim in applications. Before proving the result we explain how to compute Φ^{l} in the case that k is perfect (but l need not be).

Proposition 23. Assume in addition that k is perfect. Let l/k be a field extension. There exists an essentially unique additive functor Φ_{0}^{l} : ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T} → T ate(l, 𝔽) such that Φ_{0}^{l}|_{T ate(l,𝔽)} = id and Φ_{0}^{l}(M) = 0 if M is Tate-free. It is tensor and the following diagram commutes (up to natural isomorphism; the lower horizontal arrow is base change of Chow motives):

D⟨S⟩TM(k, 𝔽) ⟶^{Φ^{l}} K^{b}(T ate(𝔽))
t↓  ^{Φ_{0}^{l}}↑
K^{b} (⟨S_{k}⟩_{Chow(k,𝔽)}^{⊗,T}) ⟶ K^{b} (⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T})

Proof. Certainly Φ_{0}^{l} is essentially unique, using e.g. Proposition 7. The functor t∘φ_{0}^{l} satisfies the required properties, so Φ_{0}^{l} exists. It is tensor by construction. To establish the commutativity claim, consider the diagram

D⟨S⟩TM(k, 𝔽) ⟶^{t} K^{b} (⟨S_{k}⟩_{Chow(k,𝔽)}^{⊗,T})
↓  ↓
D⟨S⟩TM(l, 𝔽)  ⟶^{t} K^{b} (⟨S_{l}⟩_{Chow(k,𝔽)}^{⊗,T})
φ_{0}^{l}↓  ^{Φ_{0}^{l}}↓
D⟨S⟩TM(l, 𝔽)/⟨S_{l}⟩^{tri} ⟶^{t} K^{b}(T ate(𝔽)).

It suffices to prove that the two squares commute (up to natural isomorphism). This is most readily seen using DG-enhancements: let D(r) be a functorial negative DG-enhancement of ⟨S_{r}⟩_{Chow(r,𝔽)}^{⊗,T} ⊂ D⟨S⟩TM(r, 𝔽), for fields r/k.

(In other words D(r) is a DG-category with the same objects as ⟨S_{r}⟩_{Chow(r,𝔽)}^{⊗,T} and mapping complexes concentrated in non-positive degrees, such that these mapping complexes compute morphisms in D⟨S⟩TM(r, 𝔽). Finally we ask that when varying r, the assignment r ↦ D(r) is a (pseudo-)functor and H^{∗}(D(r)) → D⟨S⟩TM(r, 𝔽) is a (pseudo-)natural transformation.) Then it suffices to establish strict commutativity of the diagram

D(k) ⟶  D(k)_{0}
↓  ↓
D(l)  ⟶ D(l)_{0} ≈ ⟨S_{l}⟩_{Chow(l,𝔽)}^{⊗,T}
↓  ^{Φ_{0}^{l}}↓
D(l)/⟨S_{l}⟩^{tri} ⟶ (D(l)/⟨S_{l}⟩tri)_{0} ≈ T ate(𝔽),

where D_{0} for a negative DG-category means zero-truncation. (Indeed the previous diagram is obtained by passing to Ho(Pre-Tr(•)).) The upper square commutes by definition and the lower square commutes if and only if it commutes on degree zero morphisms, which is true essentially by definition of Φ_{0}^{l}.

We establish Theorem 22 through a series of lemmas.

Lemma 24. Let C be a w-category, X ∈ C^{w≤0}. Suppose given weight decompositions A → X → X′ and B[1] → X′ → X′′ (i.e. A, B ∈ C^{w≥0}, X′ ∈ C^{w<0} and X′′ ∈ C^{w<−1}). Then A, B ∈ C^{w=0} and for T ∈ C^{w=0} there is a natural exact sequence

Hom(T, B) → Hom(T, A) → Hom(T, X) → 0.

Proof. We have A, B ∈ C^{w=0} by (the dual of) [2, Proposition 1.3.3 (6)]. There is an exact sequence

Hom(T, X′[−1]) → Hom(T, A) → Hom(T, X) → Hom(T, X′) = 0

where the last term is zero because T ∈ C^{w≥0}, X′ ∈ C^{w<0}. In particular Hom(T, A) → Hom(T, X) is surjective. Applying the same reasoning to Hom(T, X′[−1]) we find that Hom(T, B) → Hom(T, X′[−1]) is surjective and hence

Hom(T, B) → Hom(T, A) → Hom(T, X) → 0

is exact. This concludes the proof.

Corollary 25. Let X ∈ D⟨S⟩TM(k, 𝔽)^{w≤0} have a weight decomposition T → X → X′ with T ∈ T ate(𝔽) (and X′ ∈ D⟨S⟩TM(k, 𝔽)^{w<0}). Suppose that φ^{k}(X) ∈ (D⟨S⟩TM(k, 𝔽)/⟨S_{k}⟩^{tri})^{w<0}. Then for T′ ∈ T ate(𝔽) we have Hom(T′, X) = 0.

Proof. Let B[1] → X′ → X′′ be a further weight decomposition. Naturality in the above lemma yields the following commutative diagram with exact rows

Hom(T′, B)  ⟶^{γ}  Hom(T′, T)  ⟶  Hom(T′, X)  ⟶ 0
α↓  ^{β}↓  ↓
Hom(φ^{k}(T′), φ^{k}(B)) ⟶^{δ} Hom(φ^{k}(T′), φ^{k}(T)) ⟶ Hom(φ^{k}(T′), φ^{k}(X)) ⟶ 0.

Since φ^{k} is weight exact we have Hom(φ^{k}(T′), φ^{k}(X)) = 0 and so δ is surjective. The construction of φ^{k} (in particular Proposition 21) implies that α is surjective and β is an isomorphism. It follows that γ is surjective, whence Hom(T′, X) = 0. This concludes the proof.

The main work in proving our theorem is the following lemma. We let φ^{l} : D⟨S⟩TM(k, 𝔽) → D⟨S⟩TM(l, 𝔽)/⟨S_{l}⟩^{tri} be the composite of φ_{0}^{l} and base change.

Lemma 26. Let X ∈ D⟨S⟩TM(k, 𝔽)^{w≤0} and suppose that for all l/k finitely generated, φ^{l}(X) ∈ (D⟨S⟩TM(l, 𝔽)/⟨S_{l}⟩^{tri})^{w<0}. Then X ∈ D⟨S⟩TM(k, 𝔽)^{w<0}.

Proof. We begin by pointing out that Lemma 16 also applies if k is not perfect. Indeed if k^{p}/k is the perfect closure then X_{k^{p}} is homeomorphic to X, so has the same set of points, and the residue field extensions of X_{k^{p}} → X are purely inseparable, so induce equivalences on DM(?, 𝔽). Thus the Lemma holds over k if and only if it holds over k^{p}. Let R be the set of finite multi-subsets of ℕ (i.e. the set of finite nonincreasing sequences in ℕ). It is well-ordered lexicographically and so can be used for induction. We extend ex to a function ex_{l} : D⟨S⟩TM(l, 𝔽) → R. First, for X_{1}, …, X_{n} ∈ S_{l} put ex(X_{1}, …, X_{n}) = {{ex(X_{1}), …, ex(X_{n})}}. Next, if Y ∈ D⟨S⟩TM(l, 𝔽) then there exist X_{1}, …, X_{n} ∈ S_{l} such that Y ∈ ⟨T ate(𝔽), X_{1}, …, X_{n}⟩^{tri}, i.e. Y is in the thick tensor triangulated subcategory generated by the MX_{i} and the Tate motives. We let ex(Y) be the minimum of ex(X_{1}, …, X_{n}) such that this holds. We shall abuse notation and write ex(Y) = ex(X_{1}, …, X_{n}) to additionally mean that Y ∈ ⟨T ate(𝔽), X_{1}, …, X_{n}⟩^{tri}. Let us observe that if ex(Y) = ex(X_{1}, …, X_{n}) and l′/l is an extension in which one of the X_{i} acquires a rational point, then ex(Y_{l′}) < ex(Y_{l}), using assumptions (2) and (3). We shall prove the result by induction on ex(X). Note that is suffices to prove that there is a weight decomposition A ⟶^{α} X → X′ (i.e. A ∈ D⟨S⟩TM(k, 𝔽)^{w=0} and X′ ∈ D⟨S⟩TM(k, 𝔽)^{w<0}) with α = 0 (because then X′ ≈ X ⊕ A[1] and so X ∈ D⟨S⟩TM(k, 𝔽)^{w<0}, the latter being Karoubi-closed by definition). If ex(X) = ∅ then X must must be Tate. By Lemma 18 we may choose a weight decomposition T ⟶^{α} X → X′ with T ∈ T ate(𝔽). By the corollary above (applied to T′ = T) we find that α = 0. This finishes the base case of our induction. Suppose now ex(X) = ex(X_{1}, …, X_{n}) > ∅. If l/k is any extension such that one of the X_{1}, …, X_{n} acquires a rational point over l, then we may assume the lemma proved over l by induction, so X_{l} ∈ D⟨S⟩TM(l, 𝔽)^{w<0}. Let A ⟶^{α} X → X′ be a weight decomposition; as before way may choose A ∈ ⟨{X_{1}, …, X_{n}}⟩_{Chow(k,𝔽)}^{⊗,T}. Write A ≈ T ⊕ A′ as in Proposition 7. I claim that α|_{A′} = 0. It is enough to show that if Y is a product of the X_{i} then Hom(MY {n}, X) = 0 for all n. By Lemma 16, it is enough to show that for all n ∈ ℤ and p ∈ Y we have that Hom_{DM(p,𝔽)}(𝟙{n}, X_{p}) = 0. But every variety has a rational point after base change to any one of its points, so X_{p} ∈ D⟨S⟩TM(p, 𝔽)^{w<0} by induction. This proves the claim.

We thus have a weight decomposition T ⊕ A′ ⟶^{(α,0)^{T}} X → X′. Let Y be a cone on α : T → X. We find that X′ ≈ Y ⊕A′[1] and hence Y ∈ D⟨S⟩TM(k, 𝔽)^{w<0}. Thus T ⟶^{α} X → Y is a weight decomposition. Using the corollary again we get Hom(T, X) = 0 and so α = 0. This finishes the induction step.

The rest of Theorem 22 is relatively easy to establish now. We begin with the following.

Lemma 27. Let C, D be w-categories with bi-w-exact tensor structures. Suppose that C is rigid and its weight structure is bounded. Let ϕ : C → D be a w-exact tensor functor such that whenever X ∈ C^{w≤0} and ϕ(X) ∈ D^{w<0} then X ∈ C^{w<0}. Then ϕ is w-conservative.

Proof. Let X ∈ C. If ϕ(X) ∈ D^{w≤0} then also X ∈ C^{w≤0}. Indeed since the weight structure on C is bounded we have X ∈ C^{w≤N} for some N. If N > 0 then the assumptions imply that X ∈ C^{w≤N−1}, and so on. Suppose now instead that ϕ(X) ∈ D^{w≥0}. We need to show that X ∈ C^{w≥0}. But X ∈ C^{w≥0} if and only if DX ∈ C^{w≤0} by Lemma 20 (use that X ≈ D(DX)), and ϕ commutes with taking duals (since C is rigid). Thus ϕ(DX) = Dϕ(X) ∈ D^{w≤0}, so DX ∈ C^{w≤0} and we are done.

It follows from Lemmas 26 and 27 that {φ^{l}}_{l} is a w-conservative family. But all our weight structures are bounded so the weight complex functors are wconservative, and thus {Φ^{l}}_{l} is also a w-conservative family. Finally for P ic-injectivity, let X ∈ D⟨S⟩TM(k, 𝔽) be invertible with Φ^{l}(X) ≈ 𝟙 for all l. Since 𝟙 ∈ K^{b}(T ate(𝔽))^{w=0}, w-conservativity implies that X ∈ D⟨S⟩TM(k, 𝔽)^{w=0} = ⟨S_{k}⟩_{Chow(k,𝔽)}^{⊗,T}. Write X ≈ T ⊕ X′, with T Tate and X′ Tate-free. Then 𝟙 ≈ Φ^{k}(X) = T and so T ≈ 𝟙. It follows that Φ^{l}(X) = 𝟙 ⊕ Φ^{l}(X′) ∈ T ate(𝔽). For this to be invertible we need Φ^{l}(X′) = 0. Since this is true for all l, conservativity implies that X′ = 0. This finishes the proof of Theorem 22.

5 Application 1: Invertibililty of Affine Quadrics

We now begin to reap in the benefits of the work of the previous sections. First we construct the conservative and Pic-injective collection of functors we shall use in the remainder of this work. After that we study invertibility of affine quadrics. We will be dealing with quadratic forms. If l is a field and ϕ is a non-degenerate quadratic form over l, we write Y_{ϕ} = P roj(ϕ = 0) for the projective quadric. This does not really make sense if dim ϕ = 1 in which case we put Y_{ϕ} = ∅ by convention. Given a ∈ l^{×} we put Y_{ϕ}^{a} = P roj(ϕ = aZ^{2}) and X_{ϕ}^{a} = Spec(ϕ = a). All of these varieties are smooth. Fix a perfect field k of exponential characteristic e ≠ 2 and coefficient ring A containing 1/e. We denote by QM(k, A) the Karoubi-closed tensor subcategory of Chow(k, A) generated by the (motives of) smooth projective quadrics over k, and the Tate motives. By [16, Property (14.5.6)] the category Chow(k, A) embeds into DM^{gm}(k, A). We write DQM^{gm}(k, A) for the thick triangulated subcategory of DM^{gm}(k, A) generated by QM(k, A). This is a tensor category. We write QM(k) = QM(k, ℤ[1/e]) and DQM^{gm}(k) = DQM^{gm}(k, ℤ[1/e]). As promised, these categories contain the motives of all (smooth) affine quadrics.

Lemma 28. If ϕ is a non-degenerate quadratic form over the perfect field k of characteristic not two, and a ∈ k^{×}, then the affine quadric X_{ϕ}^{a} satisfies M(X_{ϕ}^{a}) ∈ DQM^{gm}(k, A).

Proof. We have X_{ϕ}^{a} = Y_{ϕ}^{a} \ Y_{ϕ} and M(Y_{ϕ}^{a}), M(Y_{ϕ}), 𝟙{1} ∈ DQM^{gm}(k, A), so the result follows from the Gysin triangle.

We recall the following result.

Lemma 29 ((Rost)). Let ϕ be an isotropic non-degenerate quadratic form. Then there exists a non-degenerate form ψ such that

M(Y_{ϕ}) ≈ 𝟙 ⊕ M(Y_{ψ}){1} ⊕ 𝟙{dim Y_{ϕ}}.

Moreover for a ∈ k^{×} the natural “inclusion” M(Y_{ϕ}) → M(Y_{ϕ}^{a}) is given by

(\begin{matrix}id & 0 & 0 \\ 0 & 0 & 0 \\ 0 & s\{1\} & i\{1\} \end{matrix}) : 𝟙 ⊕ 𝟙\{dim Y_{ϕ}\} ⊕ M(Y_{ψ})\{1\} → 𝟙 ⊕ 𝟙\{dim Y_{ϕ} + 1\} ⊕ M(Y_{ψ}^{a})\{1\},

where i : M(Y_{ψ}) → M(Y_{ψ}^{a}) is the natural “inclusion” and s : 𝟙{dim Y_{ψ}^{a}} → M(Y_{ψ}^{a}) is the fundamental class (dual of the structure map).

Proof. This is a result about Chow motives. It is basically [19, Proposition 2]. Rost starts with ϕ = ℍ ⊥ ψ, but this is equivalent to ϕ having a rational point.

For the explicit form of the “inclusion”, note first that all matrix entries shown as zero have to be so for dimensional reasons. The entries “id” and “i{1}” follow from naturality of Rost’s construction. For the final entry, we can argue as follows. Note that ℤ = CH^{0}(Y_{ψ}^{a}) = Hom(𝟙{dim Y_{ψ}^{a} + 1}, MY_{ψ}^{a}{1}) ≈ Hom(𝟙{dim Y_{ψ}^{a} + 1}, MY_{ϕ}) = CH^{1}(Y_{ϕ}). The induced map we are interested in corresponds under this identification to the cycle class of the closed subvariety Y_{ϕ} ⊂ Y_{ϕ}^{a}. So up to verifying a sign (which is irrelevant for all our applications), it is enough to show that this class is a generator, which one sees for example by considering the embedding into ambient projective space.

Lemma 30. For a field extension l/k let S_{l} be the set of anisotropic projective smooth quadrics over l, and let ex_{l} : S_{l} → ℕ be the dimension function ex(X) = dim X. Then Theorem 22 applies, with 𝔽 = 𝔽_{2}.

We note that D⟨S⟩TM(k, 𝔽_{2}) = DQM^{gm}(k, 𝔽_{2}), in the notation of the Theorem.

Proof. Points on an anisotropic quadric have degree divisible by two by Springer’s theorem [14, Chapter 7, Theorem 2.3], hence condition (1) holds. Condition (2) is satisfied essentially by definition. Finally condition (3) follows from Lemma 29.

It follows from Lemma 29 that motives of quadrics are geometrically Tate. Let f : Spec(k^{s}) → Spec(k) be a separable closure. It follows that the weight complex functor t : DQM^{gm}(k^{s}) → K^{b}(Chow(k^{s}, ℤ[1/e])) takes values in K^{b}(T ate(ℤ[1/e])). We write Ψ for the composite

Ψ : DQM^{gm}(k) ⟶^{f^{∗}} DQM^{gm}(k^{s}) ⟶^{t} K^{b}(T ate(ℤ[1/e])).

Let g : Spec(l) → Spec(k) be any field extension and α : ℤ[1/e] → 𝔽_{2} be the natural surjection. Via Lemma 30 and Theorem 22 we obtain functors Φ^{l} : DQM^{gm}(k, 𝔽_{2}) → K^{b}(T ate(𝔽_{2})). We abuse notation and denote the

composite with change of coefficients DQM^{gm}(k) ⟶^{Lα_{#}} DQM^{gm}(k, 𝔽_{2}) → K^{b}(T ate(𝔽_{2})) also by Φ^{l}.

Theorem 31. The functors Ψ, Φ^{l} are tensor triangulated. Together (as l ranges over all finitely generated extensions of k) they are conservative and Pic-injective.

Proof. The functors are composites of tensor triangulated functors, so are tensor triangulated. By Theorem 14 the collection f^{∗}, {Lα_{p#}}_{p} (where p ranges over all primes) is conservative and Pic-injective. Since all weight complex functors are conservative and Pic-injective by Lemma 20, the collection tf^{∗}, {tLα_{p#}}_{p} is conservative and Pic-injective. We have tf^{∗} = Ψ. By Theorem 22 we may replace tLα_{2#} in our collection by {Φ^{l}}_{l}.

It remains to deal with Lα_{p#} at odd p. Let M ∈ DQM^{gm}(k, ℤ[1/e]). By repeated application of Lemma 29 we can find an extension L/k (which we may assume Galois) of degree a power of 2, such that M_{L} is in the triangulated subcategory generated by the Tate motives. In particular t(Lα_{p#}M_{L}) ≈ Lα_{p#}Ψ(M) (as complexes of Tate motives). Since [L : k] is a power of two, base change along L/k is conservative in odd characteristic by Proposition 9. Thus if Ψ(M) ≃ 0 then also Lα_{p#}M ≃ 0 and our collection is conservative. We need to work a bit harder for Pic-injectivity. Let M ∈ DQM^{gm}(k, ℤ[1/e]) be invertible with Φ^{l}(M) ≃ 𝟙[0] for all l/k and Ψ(M) ≃ 𝟙[0]. Then we know that Lα_{2#}(M) ≃ 𝟙 by Theorem 22. We also have t(M_{L}) = Ψ(M) ≃ 𝟙, so M_{L} ≃ 𝟙 by Lemma 20. Consider the mod 2 Bockstein sequence

Hom(𝟙, Lα_{2#}M[−1]) = 0 → Hom(𝟙, M) ⟶^{2} Hom(𝟙, M) →
→ Hom(𝟙, Lα_{2#}M) → Hom(𝟙, M[1]) ⟶^{2} Hom(𝟙, M[1]) →
→ Hom(𝟙, Lα_{2#}M[1]) = 0.

The extremal terms are zero because Lα_{2#}M ≃ 𝟙, and for the same reason we have that Hom(𝟙, Lα_{2#}M) ≈ 𝔽_{2}. Thus Hom(𝟙, M) has no 2-torsion, whereas Hom(𝟙, M[1]) has no 2-cotorsion. The composite M → M_{L} → M of base change and transfer is multiplication by [L : k] = 2^{N}. We conclude that Hom(𝟙, M) injects into Hom_{L}(𝟙_{L}, M_{L}) ≈ ℤ[1/e] and that the kernel of Hom(𝟙, M[1]) → Hom_{L}(𝟙_{L}, M_{L}[1]) = 0 (i.e. the whole group) is contained in the 2^{N}-torsion. But multiplication by 2 is surjective on Hom(𝟙, M[1]), whence so is multiplication by 2^{N}, and we conclude that Hom(𝟙, M[1]) = 0. Consequently we have Hom(𝟙, M) ≈ ℤ[1/e] (since it is an ideal of ℤ[1/e] with a non-vanishing quotient, i.e. 𝔽_{2}). We shall now apply Proposition 15. As we have seen M_{L} ≃ 𝟙, so we obtain a G = Gal(L/k)-action on Hom(𝟙, M_{L}) ≈ ℤ[1/e], i.e. a group homomorphism κ_{M} : G → ℤ[1/e]^{×}. Since e is prime we have ℤ[1/e]^{×} = {±1} × {e^{k}|k ∈ ℤ} and since G is finite the image of κ_{M} must be contained in {±1}. Note that if κ_{M} = 1 then M ≃ 𝟙 and we are done. Indeed it suffices by Theorem 14 to show that Lα_{p#}M ≃ 𝟙 for odd p. Since (Lα_{p#}M)_{L} ≃ 𝟙, by Proposition 15 this happens if and only if an appropriate Galois action is trivial, but this action is just the reduction G ⟶^{κ_{M}} ℤ[1/e]^{×} → (ℤ/p)^{×}. So assume now that κ_{M} is non-trivial. Let β : ℤ[1/e] → ℤ[1/(2e)] be the natural map. Note that κ_{M} : G → {±1} has a kernel index 2, i.e. corresponds to a quadratic subextension k ⊂ k_{2} ⊂ L. I claim that Lβ_{#}M ≃ Lβ_{#}M̃Spec(k_{2}). Indeed this follows from Proposition 15 applied to A = ℤ[1/(2e)], where f^{∗} becomes conservative, and the observation that κ_{M̃Spec(k_{2})} = κ_{M}. In particular we must have Hom(𝟙, Lβ_{#}M̃Spec(k_{2})) ≈ Hom(𝟙, M) ⊗_{ℤ[1/e]} ℤ[1/(2e)] = ℤ[1/(2e)], by Proposition 12 and our previous computation. But one may compute easily that Hom(𝟙, Lβ_{#}M̃Spec(k_{2})) = 0. This contradiction concludes the proof.

Note that if A is a PID, then P ic(K^{b}(T ate(A))) = ℤ ⊕ ℤ. Consequently we have the following corollary.

Corollary 32. The abelian group P ic(DQM^{gm}(k, ℤ[1/e])) is torsion-free (where k is a perfect field of exponential characteristic e ≠ 2).

As the proof of the theorem shows, this is completely false with ℤ[1/(2e)] coefficients (or in the étale topology), where we have P ic = ℤ ⊕ ℤ ⊕ F where F is an 𝔽_{2}-vector space. We can now prove that affine quadrics are invertible.

Theorem 33. Let k be a perfect field of characteristic not two, ϕ a nondegenerate quadratic form over k and a ∈ k^{×}. Then M̃(X_{ϕ}^{a}) is invertible in DM^{gm}(k, ℤ[1/e]).

Proof. We have M̃X_{ϕ}^{a} ∈ DQM^{gm}(k) by Lemma 28 and so we can use Theorem 31. Since the category DQM^{gm}(k) is generated by rigid objects (Chow motives) it is rigid and so conservative tensor functors detect invertibility, by standard arguments. We thus need to show that Ψ(M̃X_{ϕ}^{a}) is invertible and that for each l/k, Φ^{l}(M̃X_{ϕ}^{a}) is invertible. Let d+2 = dim ϕ. Let us put V_{ϕ}^{a} = D(MX_{ϕ}^{a}){d+1} and Ṽ_{ϕ}^{a} = D(M̃X_{ϕ}^{a}){d+1}. Then M̃X_{ϕ}^{a} is invertible if and only if Ṽ_{ϕ}^{a} is. From the closed inclusion i : Y_{ϕ} → Y_{ϕ}^{a} with complement X_{ϕ}^{a} we get the dual Gysin triangle

MY_{ϕ} ⟶^{i} MY_{ϕ}^{a} → V_{ϕ}^{a}.

It follows that t(V_{ϕ}^{a}) = [MY_{ϕ} ⟶^{i} ṀY_{ϕ}^{a}]. Here the dot is used to indicate the term of degree zero in the chain complex. Dualising the defining triangle of M̃X_{ϕ}^{a} we obtain

𝟙{d + 1} ⟶^{s} V_{ϕ}^{a} → Ṽ_{ϕ}^{a},

where s is the fundamental class (dual of the structure map). Hence we finally obtain

t(Ṽ_{ϕ}^{a}) = [MY_{ϕ} ⊕ 𝟙{d + 1} ⟶^{(i,s)} ṀY_{ϕ}^{a}] =: C(ϕ).

The functor Ψ is computed by first applying geometric base change, so ϕ becomes completely split. In particular it has to be isotropic. An induction on dimension using Lemma 34 below shows that we may reduce to dim ϕ = 1 or 2, i.e. {x^{2} = 1} or {xy = 1} (recall that completely split quadrics are characterised by their dimension, so we can choose any non-degenerate model quadric of the correct dimension). But M̃({x^{2} = 1}) = 𝟙 and M̃({xy = 1}) = M̃(𝔾_{m}) are both invertible. Dealing with Φ^{l} is a bit harder. The expression C(ϕ) ∈ K^{b}(QM(k)) makes sense even if k is not perfect. Using Proposition 23 it suffices to prove: if l/k is any field extension, then Φ_{0}^{l}C(ϕ_{l}) is invertible. We drop the subscript zero from now on. We may as well prove: if k is any field and ϕ is any non-degenerate quadratic form over k, then Φ^{k}(C(ϕ)) is invertible. By Lemma 34 below, if ϕ ≈ ψ ⊥ ℍ then C(ϕ) ≃ C(ψ){1}. We may thus assume that either ϕ is anisotropic, or ϕ = ℍ, or ϕ is of dimension one. If ϕ = ℍ then Y_{ϕ} ≈ Spec(k × k), Y_{ϕ}^{a} ≈ ℙ^{1} and the result follows easily. If ϕ is of dimension one then MY_{ϕ} = 0 and either MY_{ϕ}^{a} = 𝟙 ⊕ 𝟙 or MY_{ϕ}^{a} = M(k′), where k′/k is a quadratic extension. Again the result follows easily. So we may assume that ϕ is anisotropic. There are three cases. If ϕ ⊥ ⟨−a⟩ is also anisotropic, then none of MY_{ϕ}, MY_{ϕ}^{a} afford Tate summands, by Proposition 7. Thus Φ^{k}(C(ϕ)) = 𝟙{d + 1}[1] is invertible. If ϕ ⊥ ⟨−a⟩ is isotropic, then ϕ ⊥ ⟨−a⟩ = ψ ⊥ ℍ. Suppose that ψ has dimension greater than one. Then by (the contrapositive of) Lemma 35 below, ψ is anisotropic. It follows that MY_{ϕ}^{a} ≈ 𝟙 ⊕ 𝟙{d + 1} ⊕ MY_{ψ}^{a} and Φ^{k}(C(ϕ)) = [𝟙{d + 1} → ˙𝟙 ⊕ 𝟙{d + 1}]. The component 𝟙{d + 1} → 𝟙{d + 1}] comes from the fundamental class of M_{ψ}^{a} and so is an isomorphism. Thus Φ^{k}(C(ϕ)) ≃ 𝟙 is invertible. Finally it might be that ψ has dimension one. Then Y_{ϕ}^{a} ≈ ℙ^{1} whereas MY_{ϕ} affords no Tate summands, and the result follows as in the case of dimension greater than one. This concludes the proof.

Lemma 34. Notation as in the theorem. If ϕ = ψ ⊥ ℍ then C(ϕ) ≃ C(ψ){1}.

Proof. Using the explicit form for the inclusion MY_{ϕ} → MY_{ϕ}^{a} from Lemma 29 we find that

C(ϕ) = [(𝟙 ⊕ 𝟙{d} ⊕ MY_{ψ}{1}) ⊕ 𝟙{d + 1} ⟶^{α} ˙𝟙 ⊕ 𝟙{d + 1} ⊕ MY_{ψ}^{a}{1}],

where α is given by the matrix

(\begin{matrix}id & 0 & 0 & 0 \\ 0 & 0 & 0 & f \\ 0 & s\{1\} & i\{1\} & 0 \end{matrix}) .

Here f comes from the fundamental class and so is an isomorphism. It follows that C(ϕ) ≈ C(ψ){1} ⊕ cone(id_{𝟙})[−1] ⊕ cone(id_{𝟙{d+1}})[−1] ≃ C(ψ){1}. This is the desired result.

Lemma 35. If ϕ ⊥ ⟨a⟩ ≈ ψ ⊥ ℍ ⊥ ℍ, then ϕ is isotropic.

Proof. Let X = Y_{⟨a⟩⊥ϕ}. Then Y_{ϕ} = X ∩ {X_{0} = 0}. Since ⟨a⟩ ⊥ ϕ ≈ ψ ⊥ ℍ ⊥ ℍ, we find that Y_{ℍ⊥ℍ} ⊂ X. Then Y_{ϕ} ∩ Y_{ℍ⊥ℍ} = Y_{ℍ⊥ℍ} ∩ {X_{0} = 0} (intersecting inside X). Now we know that after a linear change of coordinates (X_{0} : ⋯: X_{r}) ↦ (T_{0} : ⋯: T_{r}) the subvariety Y_{ℍ⊥ℍ} of X is given by the equations T_{0}T_{1} + T_{2}T_{3} = 0, T_{i} = 0 for i > 3. Thus Y_{ϕ} ∩ Y_{ℍ⊥ℍ} is obtained by adding a further linear constraint in the T_{0}, T_{1}, T_{2}, T_{3}. It is easy to see that there must be a rational, non-zero solution, so Y_{ϕ} has a rational point. This was to be shown.

6 Application 2: Po Hu’s Conjectures for Motives

In this final section we prove a version for motives of Po Hu’s conjectures [12, Conjecture 1.4]. We retain notation from the previous section. For \underline{a} = (a_{1}, …, a_{n}) ∈ (k^{×})^{n}, b ∈ k^{×} let us put

U b ; \underline{a} = X_{⟨⟨a_{1},...,a_{n}⟩⟩}^{b},

where ⟨⟨a_{1}, …, a_{n}⟩⟩ is the n-fold Pfister quadric associated with the symbol \underline{a}. We use notation such as \underline{a}, a′ = (a_{1}, …, a_{n}, a′) ∈ (k^{×})^{n+1} for concatenation of tuples.

Theorem 36. Let k be a perfect field of characteristic not two, and \underline{a} ∈ (k^{×})^{n}, b ∈ k^{×}. In DM^{gm}(k) there is an isomorphism

M̃(U_{\underline{a},b}^{1}) ⊗ M̃(U b ; \underline{a})[1] ≈ M̃(U 1 ; \underline{a})\{2^{n}\}.  (6.1)

To prove this, we have to recall some facts about Rost motives. If \underline{a} ∈ (k^{×})^{n}, then there is the associated Rost motive R\underline{a} ∈ QM(k). Recall that one has H_{et}^{1}(k, 𝔽_{2}) = k^{×}/2, and hence cup product yields a natural map ∂ = ∂^{k} : (k^{×})^{n} → H_{et}^{n}(k, ℤ/2). The Rost motives have the remarkable property that R\underline{a} is irreducible if and only if ∂(\underline{a}) ≠ 0. In fact there are canonical maps

𝟙\{2^{n−1} − 1\} → R\underline{a} → 𝟙  (6.2)

(which we call structure maps) and if ∂(\underline{a}) = 0 then this is a splitting distinguished triangle. The same statements hold true with 𝔽_{2} coefficients. These results follow from the work of a number of people, see [18] for an overview. The relationship between Rost motives and U b ; \underline{a} is encapsulated in the following proposition.

Proposition 37. For \underline{a} ∈ (k^{×})^{n}, b ∈ k^{×} there is a distinguished triangle

M̃(U b ; \underline{a}) → R_{\underline{a},b} → R\underline{a}\{2^{n−1}\} ⊕ 𝟙.

Here R_{\underline{a},b} → 𝟙 is the structure map, and the composite

𝟙\{2^{n} − 1\} → R_{\underline{a},b} → R\underline{a}\{2^{n−1}\}

is the \{2^{n−1}\} twist of the structure map 𝟙\{2^{n−1} − 1\} → R\underline{a}.

Proof. This is essentially [13, proof of Proposition 5.5]. We know that U := U b ; \underline{a} is the complement of X := Y_{⟨⟨\underline{a}⟩⟩} in Y := Y_{⟨⟨\underline{a}⟩⟩}^{b}. By the work of Rost [19, Theorem 17 and Proposition 19], if we put R_{n} := R_{\underline{a},b} and R_{n−1} = R\underline{a}, then

M(Y) = R_{n} ⊕ ⨁_{k=1}^{2^{n−1}−1} R_{n−1}{k} := R_{n} ⊕ R′,

M(X) = ⨁_{k=0}^{2^{n−1}−1} R_{n−1}{k} := R_{n−1} ⊕ R′

and the natural map M(X) → M(Y) is the identity on R′. The localisation triangle M^{c}(X) = M(X) → M^{c}(Y) = M(Y) → M^{c}(U) fits into the following commutative diagram of (distinguished) triangles:

R′  R′
↓  ↓
M(X) ⟶ M(Y) ⟶ M^{c}(U)
↓  ↓
R_{n−1}  R_{n}

An application of the octahedral axiom yields a distinguished triangle R_{n−1} → R_{n} → M^{c}(U). Noting that DM^{c}(U) = M(U){−(2^{n} − 1)}, DR_{n} = R_{n}{−(2^{n} − 1)} and DR_{n−1} = R_{n−1}{−(2^{n−1} − 1}, by dualising and twisting the triangle, we find a distinguished triangle M(U) → R_{n} → R_{n−1}{2^{n−1}}. Adding in the copy of 𝟙 implied in M̃(U), we get the claimed triangle with the correct map R_{n} → 𝟙. To see the second claim about the differential, the important point is that in the triangle R_{n−1} → R_{n} → M^{c}(U) the map R_{n−1} → R_{n} is induced from the inclusion M(X) → M(Y) by passing to the appropriate summands. It follows that R_{n−1} → R_{n} → 𝟙 is the structure map of R_{n−1} → 𝟙. The desired result now follows by dualising.

Proof of Theorem 36. By Lemma 28, we have M̃(U b ; \underline{a}) ∈ DQM^{gm}(k), etc. We also know by Theorem 33 that both sides of equation (6.1) are invertible. Hence if F : DQM^{gm}(k) → C is a Pic-injective functor, it suffices to prove that F (LHS) ≈ F (RHS). Of course we use the Pic-injective collection from Theorem 31. ¿From Proposition 37 we know that

t(M̃(U b ; \underline{a}) = [Ṙ_{\underline{a},b} → R\underline{a}\{2^{n}\} ⊕ 𝟙],

and we also know certain things about the differential. To compute Ψ, we have to consider geometric base change, where the triangle (6.2) is splitting distinguished. One obtains

Ψ(M̃(U b ; \underline{a})) = [ ˙𝟙 ⊕ 𝟙\{2^{n} − 1\} → 𝟙\{2^{n−1}\} ⊕ 𝟙\{2^{n} − 1\} ⊕ 𝟙]

and from the information about the differential given in proposition 37 we deduce that Ψ(M̃(U b ; \underline{a}) ≃ 𝟙\{2^{n−1}\}[−1]. Thus Ψ(LHS) ≈ Ψ(RHS) reads

𝟙{2^{n}}[−1] ⊗ 𝟙{2^{n−1}}[−1][1] ≈ 𝟙{2^{n−1}}[−1]{2^{n}},

which is certainly true. Now let l/k be an arbitrary field extension. We need to prove Φ^{l}(LHS) ≈ Φ^{l}(RHS). This involves R\underline{a}, R_{\underline{a},b}, R_{\underline{a},1} and R_{\underline{a},b,1}. Depending on l these may Table 1: Terms needed to compute Φ^{l}.

∂^{l}(\underline{a}, b) ≠ 0  ∂^{l}(\underline{a}, b) = 0 but ∂^{l}(\underline{a}) ≠ 0
Φ^{l}(U_{\underline{a},b}^{1}) [ ˙𝟙 ⊕ 𝟙\{2^{n+1} − 1\} → 𝟙]  [ ˙𝟙 ⊕ 𝟙\{2^{n+1} − 1\} → …

⋯ → 𝟙{2^{n}} ⊕ 𝟙{2^{n+1} − 1} ⊕ 𝟙]

Φ^{l}(U b ; \underline{a}) [˙0 → 𝟙] [ ˙𝟙 ⊕ 𝟙\{2^{n} − 1\} → 𝟙]
Φ^{l}(U 1 ; \underline{a}) [ ˙𝟙 ⊕ 𝟙\{2^{n} − 1\} → 𝟙]  [ ˙𝟙 ⊕ 𝟙\{2^{n} − 1\} → 𝟙]

Table 2: Terms needed to compute Φ^{l}, simplified form.

∂^{l}(\underline{a}, b) ≠ 0 ∂^{l}(\underline{a}, b) = 0 but ∂^{l}(\underline{a}) ≠ 0
Φ^{l}(U_{\underline{a},b}^{1}) 𝟙\{2^{n+1} − 1\}  𝟙\{2^{n}\}[−1]
Φ^{l}(U b ; \underline{a}) 𝟙[−1] 𝟙\{2^{n} − 1\}
Φ^{l}(U 1 ; \underline{a}) 𝟙\{2^{n} − 1\}  𝟙\{2^{n} − 1\}

or may not split into Tate motives, so may or may not survive Φ. We see that R_{\underline{a},1} and R_{\underline{a},b,1} always split (because ∂^{l}(1) = 0), and that R_{\underline{a},b} splits whenever R\underline{a} splits (because ∂(\underline{a}, b) = ∂(\underline{a}) ∪ ∂(b)). If R\underline{a} splits then everything is split and Φ^{l} is just mod two reduction of Ψ, so we know the equation is satisfied. Thus there are just two cases and three things in each to compute, which we gather in Table 1. The differentials can again be figured out using Proposition 37. Using these one can simplify the expressions. We have gathered the results in Table 2. To complete the proof, we check that Φ^{l}(LHS) ≈ Φ^{l}(RHS) in both cases. This is easy.

References

[1] Tom Bachmann, On some conservative functors in motivic homotopy theory, submitted (2016), arXiv:1506.07375.

[2] Mikhail V. Bondarko, Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology 6 (2010), no. 03, 387–504.

[3] , Z[1/p]-motivic resolution of singularities, Compositio Mathematica 147 (2011), no. 05, 1434–1446.

[4] Mikhail V. Bondarko and Vladimir A. Sosnilo, On constructing weight structures and extending them onto idempotent extensions, arXiv preprint arXiv:1605.08372 (2016).

[5] Denis-Charles Cisinski and Frédéric Déglise, Local and stable homological algebra in grothendieck abelian categories, Homology, Homotopy and Applications 11 (2009), no. 1, 219–260.

[6] , Triangulated categories of mixed motives, arXiv preprint arXiv:0912.2110 (2009).

[7] , Integral mixed motives in equal characteristic, Documenta Mathematica Extra Volume Merkurjev (2015), 145–194.

[8] P. Deligne and J.S. Milne, Tannakian Categories, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer Berlin Heidelberg, 1981, pp. 101–228 (English).

[9] H. Fausk, Picard groups of derived categories, Journal of Pure and Applied Algebra 180 (2003), no. 3, 251 – 261.

[10] H. Fausk, L.G. Lewis, and J.P. May, The Picard group of equivariant stable homotopy theory, Advances in Mathematics 163 (2001), no. 1, 17–33.

[11] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1984.

[12] Po Hu, On the Picard group of the stable 𝔸^{1}-homotopy category, Topology 44 (2005), no. 3, 609–640.

[13] Po Hu and Igor Kriz, Some remarks on Real and algebraic cobordism, K-theory 22 (2001), no. 4, 335–366.

[14] Tsit-Yuen Lam, The algebraic theory of quadratic forms, Lecture Notes Series in Mathematics, Benjamin/Addison-Wesley, 1973.

[15] L.G. Lewis, J.P. May, and M. Steinberger, Equivariant Stable Homotopy Theory, Lecture notes in mathematics 1213 (1986).

[16] Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, American Mathematical Soc., 2006.

[17] J.S. Milne, Etale Cohomology (PMS-33), Princeton mathematical series, Princeton University Press, 1980.

[18] F. Morel, Voevodsky’s proof of Milnor’s conjecture, Bull. Amer. Math. Soc. (1998).

[19] Markus Rost, The motive of a pfister form, preprint (1998).

[20] Vladimir Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories 143 (2000), 188–238.

[21] , Cancellation theorem, Documenta Mathematica Extra Volume: A. Suslin’s Sixtieth Birthday (2010), 671–685.

Tom Bachmann

Mathematisches Institut Ludwig-Maximilians-Universität München Theresienstr. 39 80333 München Germany tom.bachmann@zoho.com 396

Documenta Mathematica 22 (2017)