E-mail: jakob.scholbach@math.uni-freiburg.de

Mathematics Subject Classification (2000) 14F42

Keywords motives, h-topology, sheaves with transfers Abstract The aim of this paper is to give a detailed proof of a comparison of Voevodsky’s categories of geometric motives with and without transfers, respectively. The latter category is defined by means of h-topology introduced by Voevodsky, a topology essentially given by Zariski coverings, finite coverings and blowups.

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Noname manuscript No. (will be inserted by the editor)

Geometric motives and the h-topology

Jakob Scholbach

the date of receipt and acceptance should be inserted later

Introduction

The category of geometric motives [Voe00b] is roughly defined to be the localisation of the category of smooth correspondences with respect to the affine line and the Zariski topology. Before, Voevodsky had defined a category of motives using the h-topology [Voe96]. The objective of this paper is to give a detailed proof of a theorem of Voevodsky stating the equivalence of these two concepts in the case of rational coefficients and a base field of characteristic zero.

We see two principal facets of this description: on the one hand it clarifies why the technique of proper descent, which is often applied by replacing some scheme by a proper hypercovering in order to resolve its singularities, fits into the motivic framework (e.g. in [GS96]). On the other hand, the theorem is a hint that correspondences, which make the handling of the several categories of motives technically intricate, should not be necessary in the case of rational coefficients.

We define a category of effective quasigeometric motives with rational coefficients as follows: Let Sch denote the category of schemes, by which we mean schemes of finite type over a field k. Let ℚSch be the ℚ-linear hull of Sch (i.e., morphisms are given by ℚ-linear combinations of morphisms of schemes) and let \overline{ℚSch} be its closure under countable direct sums. Finally let K^{−} (\overline{ℚSch}) denote the homotopy category of bounded above complexes in \overline{ℚSch}. We define the category

DM_{\overline{gm},h}^{eff}(ℚ) := K^{−} (\overline{ℚSch}) /\{𝔸^{1}-complexes, h-coverings\}

of effective quasigeometric motives to be the localization of K^{−} (\overline{ℚSch}) with respect to the following complexes:

– 𝔸^{1}-complexes: X × 𝔸^{1} → X, X smooth

– Čech nerves ... → U ×_{X} U → U → X of h-coverings U → X, for U smooth, X ∈ Sch.

The h-topology is essentially given by Zariski open coverings, finite surjective morphisms and blowups (Definition 1.5, Proposition 1.6). The subcategory DM_{gm,h}^{eff}(ℚ) of bounded complexes in ℚSch is called category of effective geometric motives.

Universität Freiburg, Mathematisches Institut, Eckerstr. 1, D-79104 Freiburg, Germany

On the other hand, the category of effective geometric motives with transfers is defined via the category ℚSmCor, consisting of smooth schemes over k and morphisms given by finite correspondences, i.e. ℚ-linear combinations of cycles W ⊂ X × Y finite over X. Again, we first define a quasigeometric category by

DMeff \overline{gm}(ℚ) := K^{−} (\overline{ℚSmCor}) /\{𝔸^{1}-complexes, Mayer-Vietoris\} ,

where \overline{ℚSmCor} is the closure of ℚSmCor under countable direct sums, 𝔸^{1}-complexes are as above, and Mayer-Vietoris complexes are given by U ∩V → U ⊔V → U ∪V (U, V smooth). Its subcategory DM_{gm}^{eff}(ℚ) of complexes in K^{b}(ℚSmCor) is Voevodsky’s category of effective geometric motives with transfers. Theorem 4.1. Let k be of characteristic zero. Then there is an equivalence between the categories of effective geometric motives with and without transfers, respectively (with rational coefficients):

DM_{gm}^{eff}(ℚ) ⟶^{≅} DM_{gm,h}^{eff}(ℚ).

This equivalence derives from an analogous equivalence of the quasigeometric variants. These equivalences extend to the non-effective motivic categories (see Definition 4.4 for their definition).

It should be pointed out that the above comparison is – using Neeman’s compacity notion [Nee01] – a formal corollary of the equivalence of the two categories of effective motivic complexes, which is the non-torsion part of [Voe00b, Theorem 4.1.12]. Our aim is to give a detailed proof. It closely follows the ideas developed by Friedlander, Suslin and Voevodsky [VSF00,Voe96,SV96], especially concerning sheaves with transfers and their cohomology comparison results. For example, the precursor for the embedding theorem (3.14) and its proof is [Voe00b, Theorem 3.2.6]. Overview of the proof and of the paper: A principal notion are presheaves with transfers. These are just contravariant functors SmCor → Ab. A presheaf is called homotopy invariant, if the canonical map F (X) → F (X × 𝔸^{1}) is an isomorphism.

The general idea is to refine the topology to avoid correspondences. The h-topology is well suited for this task: firstly, every h-sheaf has a unique transfer structure (Proposition 2.2). Secondly, for an homotopy invariant presheaf with transfers with rational coefficients, its sheafifications and cohomology groups with respect to the Zariski-, Nisnevich-, étale, cdh-, qfh- or h-topology agree (Theorem 2.11).

These facts give an equivalence between the categories of motivic complexes with rational coefficients, with and without transfers, respectively (Theorem 3.7):

DM_{Nis,tr}^{eff,−}(ℚ) ≅ DM_{h}^{eff,−}(ℚ),

where the former category is the full subcategory of the derived category of Nisnevich sheaves with transfers D^{−}(Shv_{Nis,tr} (Sm)) consisting of complexes with homotopy invariant cohomology-Nisnevich-sheaves and the latter is defined similarly using hsheaves instead.

Applying the cohomology comparison results, we obtain the embedding theorem (3.14):

DM_{gm,h}^{eff}(ℚ) ⊂ DM_{\overline{gm},h}^{eff}(ℚ) ⊂ DM_{h}^{eff,−}(ℚ).

The same statement is known for the situation with transfers (Voevodsky, see Theorem 3.15). More precisely, in either the situation with or without transfers, the geometric category is the subcategory of compact objects and the quasigeometric category is the one of ℵ_{1}-compact objects (Definition A.3). This completes the proof.

We restrict to characteristic zero in order to use resolution of singularities (see Proposition 1.1). We work with ℚ-coefficients in order to ensure the finiteness of the cohomological dimension and the equality of Nisnevich and étale cohomologies. Another technically important point is Lemma 2.6, which is used to close the gap – essentially consisting of finite surjective morphisms – between Nisnevich and qfh-topology. Analogous cohomology comparison results for sheaves with integral coefficients are not true (see Remark 2.12).

I would like to thank Annette Huber for her constant (in)valuable support while I was writing this paper.

1 Notations and Preliminary Results

We fix the following notations. The base field is denoted by k. By a scheme, we always mean a separated scheme of finite type over Spec k. The category of schemes is denoted Sch, its subcategory of smooth schemes is denoted Sm; ℤSch is the category whose objects are schemes, but morphisms are given by Hom_{ℤSch}(X, Y) := ⊕_{i}ℤ[Hom_{Sch}(X_{i}, Y)], where X_{i} are the connected components of X. By formally adding a zero object, we obtain an additive category.

Recall the definition of Voevodsky’s category of finite correspondences SmCor (see, for example [MVW, esp. Lecture 1]): its objects are smooth schemes and its morphisms are defined by

Hom_{SmCor}(X, Y) := ℤ[W ⊂ X × Y, W irreducible, surjective and finite over X],

if X is connected and Hom_{SmCor}(⊔X_{i}, Y) := ⊕Hom_{SmCor}(X_{i}, Y) otherwise. The composition of correspondences is defined in the usual way (see loc. cit.). We obtain an additive category. Assigning to a morphism in Sm its graph gives a faithful functor Sm → SmCor (see [Ful98, §16] for associativity of the composition of correspondences and functoriality).

Categories of presheaves are denoted by PSh(−), where − is either Sm or Sch. If not mentioned explicitly otherwise, presheaves will be presheaves of abelian groups. All presheaves will be understood to be additive, i.e. F (X ⊔ Y) ≅ F (X) ⊕ F (Y). If t is a topology, t-sheaves are denoted by Shv_{t} (−). The t-sheafification of a presheaf F is denoted by F_{t}. In general we add a subscript −_{tr} when there is some category or object “with transfers”: the categories of t-sheaves with transfers (i.e. contravariant functors from SmCor to abelian groups whose restriction to Sm is a t-sheaf) are denoted by Shv_{t,tr} (Sm) and similarly for presheaves with transfers: PSh_{tr}(Sm).

The representable presheaves on schemes are written ℤ(−) (and ℤ_{tr}(−) for correspondences, i.e. ℤ_{tr}(X)(V) = Hom_{SmCor}(V, X)).

The ℚ-rational variants of these constructions are denoted by ℚSch, ℚSmCor, ℚ_{tr}, PSh(Sch, ℚ), Shv_{Nis,tr} (Sm, ℚ) etc.

Let Com^{∗}(−) , ∗ ∈ {−, b} denote the category of bounded above and bounded complexes, respectively (with differentials having degree +1), K^{∗}(−) the corresponding homotopy category and D^{∗}(−) the derived category. The total complex of a double complex is denoted by Tot(−) (see [Wei94, 1.2.6] for choice of signs). If C, D are categories, C ⊂ D denotes a full embedding.

We say “k admits weak resolution of singularities”, if every scheme X admits a proper surjective morphism Y ⟶ X, where Y is smooth.

We say “k admits strong resolution of singularities”, if furthermore for any abstract blow-up p : Y ⟶ X with nowhere dense closed center Z ⊂ X of any scheme X (i.e. any proper surjective map, such that p^{−1}(X −Z) ≅ X −Z) there is a finite composition of blow-ups of smooth centers q : X̃ = X_{n} → ... → X_{0} = X factoring through p.

Proposition 1.1 Any field admits weak resolution of singularities. Any field of characteristic 0 admits strong resolution of singularities.

Proof For the weak resolution, see [Hir64, Main Theorem I, S. 132] (char k = 0) and [dJ96, Theorem 4.1; §1] (char k > 0).

For the strong resolution over a field of characteristic zero one applies the resolution of indeterminacies to p^{−1}, see [Hir75] in the analytical setting and [AKMW02, §1.2.4], [RG71, Corollaire 5.7.12], [Bon02, §1.2] for the algebraic setting.

Definition 1.2 For a simplicial scheme X we denote by ℤ(X) the Moore complex of representable presheaves

... ⟶ ℤ(X_{2}) ⟶^{δ_{2}} ℤ(X_{1}) ⟶^{δ_{1}} ℤ(X_{0}),

where δ_{n} = ∑_{i=0}^{n+1}(−1)^{i}∂_{n}^{i} and ∂_{n}^{i} : X_{n} → X_{n−1} is the i-th boundary map of X.

For a morphism U → X set U_{X}^{n} := U ×_{X} ... ×_{X} U (n factors). We denote by U_{X} the simplicial scheme ... → U_{X}^{3} → U_{X}^{2} → U with the obvious boundary and degeneracy maps. If U → X is a covering with respect to a (pre-)topology t, U_{X} is a t-hypercovering of X [SGA72, Exp. V, Def. 7.3.1.1.].

Lemma 1.3 Let X = (X_{n})_{n∈ℕ} be a hypercovering in (Sch/X)_{t}, t an arbitrary topology. The augmented Moore complex of the corresponding representable t-sheaves ℤ_{t}(X) is exact:

... → ℤ_{t}(X_{2}) → ℤ_{t}(X_{1}) → ℤ_{t}(X_{0}) → ℤ_{t}(X) → 0.

Proof A theorem of Verdier [SGA72, Exposé V, Théorème 7.3.2(3)] asserts that for any hypercovering X in (Sch/X)_{t} the sheafification of the following complex is exact:

... → ℤ_{X}(X_{2}) → ℤ_{X}(X_{1}) → ℤ_{X}(X_{0}) → ℤ_{X}(X) = ℤ → 0,

where ℤ_{X}(U)(V) := ℤ[Hom_{Sch/X}(V, U)]. The proof of loc. cit. is valid in our situation, if we replace ℤ_{X,t}(−) by ℤ_{Spec k,t}(−) = ℤ_{t}(−).

Lemma 1.4 Let f : U → X be an arbitrary morphism. Then the complex

ℤ(U_{X}) → ℤ(X) → coker ℤ(f) → 0

is an exact complex (of presheaves).

Proof This is [Voe96, Proposition 2.1.4]. We give a proof for the convenience of the reader. Let A be an arbitrary connected scheme. For brevity, (A, −) and [A, −] denote Hom_{Sch}(A, −) and ℤ(−)(A) = Hom_{ℤSch}(A, −), respectively. We have to show that the following complex is exact:

... → [A, U_{X}^{3}] → [A, U_{X}^{2}] → [A, U] → [A, X] → coker ℤ(f)(A) → 0.

We may suppose (A, U) ≠ ∅. Fix an arbitrary element ∗ ∈ (A, U). Consider the simplicial set

(A, U_{X}) : ... → (A, U_{X}^{3}) → (A, U_{X}^{2}) → (A, U).

The very definition of homotopy groups [GJ99, I.7.] p_{n} := π_{n} ((A, U_{X}), ∗) and the universal property of the fiber product yields p_{n} = 0 for n ≥ 1. The homotopy set p_{0} is a quotient of (A, U): morphisms f, g : A → U are identified iff there is a morphism h : A → U ×_{X} U such that f = pr_{1} ∘ h, g = pr_{2} ∘ h. For the simplicially constant complex Q = (p_{0})_{n≥0} we obviously have π_{n}(Q) = 0 for n ≥ 1, π_{0}(Q) = p_{0}. Thus, the canonical morphism of simplical sets (A, U_{X}) → Q is a weak equivalence in the sense of [GJ99, I.7., page 32]. Weak equivalences are preserved by ℤ(−), thus the associated morphism of Moore complexes [A, U_{X}] → ℤ(Q) is a quasi-isomorphism [GJ99, Proposition III.2.16., Corollary III.2.7.]. So, the complex is exact at [A, U_{X}^{n}], n > 1.

Exactness at [A, U] follows from the definition of the fiber product: Let s = ∑_{i∈I} n_{i}· s_{i} be mapped to 0 under f. One inductively reduces to the case #I = 2. Then it is obvious that there is g ∈ [A, U_{X}^{2}], such that g ↦ s.

1.1 H-topologies

Definition 1.5 [Voe96] A morphism of schemes f : X → Y is called a topological epimorphism, if the topological space associated to Y is a quotient topological space of X, i.e. if f is surjective and A ⊂ X is open iff f(A) ⊂ Y is open. f is called a universal topological epimorphism, if this holds for any base change.

Coverings of the h-topology on Sch are by definition finite families {f_{i} : U_{i} → X} of morphisms of finite type, such that ⊔f_{i} : ⊔U_{i} → X is a universal topological epimorphism. If the f_{i} are furthermore quasi-finite, this family is defined to be a covering of the qfh-topology.

The cdh-topology is the minimal Grothendieck topology containing Nisnevich coverings and abstract blow-ups (see above).

Normal qfh-coverings {f_{i} : U_{i} → X} are qfh-coverings having a factorization f_{i} = f ∘ j_{i}, where {j_{i} : U_{i} → U} is a Zariski covering by open subsets and f : U → X is a finite surjective morphism.

A family of morphisms {f_{i} : U_{i} → X} is called normal h-covering, if it has a factorization f_{i} = b ∘ f ∘ j_{i}, where {j_{i} : U_{i} → U} is an open covering, f : U → X_{Z} a finite surjective map and b : X_{Z} → X the blow-up of X in a closed, nowhere dense center Z.

Any scheme is h-locally smooth by Proposition 1.1. We have the following inclusions of topologies on Sch: Zar ⊆ Nis ⊆ ét ⊆ qfh ⊆ h and Nis ⊆ cdh ⊆ h.

Proposition 1.6 [Voe96, Theorem 3.1.9], [SV96, Lemma 10.4]. Every h-covering of a scheme admits a normal refinement. The same is true for qfh-coverings of normal schemes.

Lemma 1.7 Let A be an abelian category closed under countable direct sums. Let B ⊂ D := D^{−}(A) be a full triangulated subcategory closed under existing countable direct sums (i.e. ℵ_{1}-localising in the sense of Definition A.3). If all components F^{n} of some complex F ∈ D are in B, then F is in B.

Proof Compare [MVW, Lemma 9.4.].

We write D′ := D^{−}(PSh(Sch)) and D := D^{−}(Shv_{h} (Sch)) in the following lemmas. The localising subcategory generated by a class of objects T, i.e. the minimal triangulated subcategory closed under existing direct sums containing T will be denoted by ⟨T ⟩ (see Definition A.2).

Lemma 1.8 We have ⟨ℤ_{h}(X), X ∈ Sm⟩ = D and ⟨ℤ(X), X ∈ Sch⟩ = D′.

Proof For any h-sheaf F, the morphism ⨁_{X∈Sm,a∈F (X)} ℤ_{h}(X) ⟶^{⊕a} F is surjective in Shv_{h} (Sch) because of weak resolution of singularities. This easily yields the assertion (Lemma 1.7, applied to B = ⟨ℤ_{h}(X), X ∈ Sm⟩). The statement for presheaves is proven similarly.

Lemma 1.9 Let T_{h} ⊂ D′ denote the class of augmented Moore complexes

... → U ×_{X} U → U → X

of h-coverings U → X, where U ∈ Sm, X ∈ Sch (Definition 1.2). The sheafification functor D′ → D factorizes over D′/ ⟨T_{h}⟩ and induces an equivalence of categories D′/ ⟨T_{h}⟩ ≅ D.

Proof Let C_{h} denote the localising subcategory of D′ generated by presheaves F with F_{h} = 0. With Lemma 1.7, it is quite obvious to see that we have D ≅ D′/C_{h}, so it suffices to show T_{h} = C_{h}. We have ⟨T_{h}⟩ ⊆ C_{h} (Lemma 1.3).

For the converse, choose a complex F ∈ C_{h}. We may assume that it is concentrated in degree 0 (Lemma 1.7). Therefore, there is an epimorphism of presheaves ⊕_{i}coker ℤ(f_{i}) → F, where f_{i} : U_{i} → X_{i} are h-coverings. Using weak resolution of singularities we may assume that the U_{i} are smooth. As (coker ℤ(f_{i}))_{h} = 0, we even get a resolution of F by sheaves of this form. We may thus assume F = coker f for an h-covering f : U → X (Lemma 1.7). The sequence

... → ℤ(U ×_{X} U) → ℤ(U) ⟶ ℤ(X) ⟶ coker f ⟶ 0

is exact (Lemma 1.4). The augmented Moore complex lies in ⟨T_{h}⟩ and so does F = coker f.

Remark 1.10 The proof shows that the proposition still holds if we enlarge T_{h} by adding complexes becoming exact when they are h-sheafified. In particular one may add arbitrary h-hypercoverings (Lemma 1.3). On the other hand, it suffices to localise with respect to h-coverings ℤ(U_{X}) → ℤ(X), where U → X is of normal form (Proposition 1.6).

2 Sheaves with transfers

We are going to study the relationship between sheaves with and without transfers. It turns out that every h-sheaf has a unique transfer structure. Note that the sheafification of a presheaf with transfers is in general (i.e. for an arbitrary topology) not necessarily a presheaf with transfers.

Lemma 2.1 The canonical (mono-)morphisms ℤ(X) → ℤ_{tr}(X), X ∈ Sm induce isomorphisms

ℤ_{h}(X) ≅ ℤ_{tr,h}(X), ℚ_{h}(X) ≅ ℚ_{tr,h}(X).

They induce a transfer structure on any h-sheaf. We call it the canonical transfer structure.

Proof The second part of the lemma is stated by Voevodksy [Voe00a, §3.4.]. The lemma was also proven independently by Ivorra [Ivo05, Corollaire 3.2.19].

Fix a smooth scheme V. By definition, c(X/k, 0)(V) = c(X ×_{k} V/V, 0) is the group of universally integral cycles of X ×_{k} V, whose support is proper of relative dimension 0 over V [SV00, §2.3., 3.1.1-3.1.3, definition after Lemma 3.3.9]. This presheaf is obviously additive. Since all presheaves we are dealing with are additive, we may suppose that V is connected. To be proper of relative dimension 0 is the same as to be finite over V. As V is regular, any relative cycle is universally integral [SV00, Proposition 3.3.15]. V is normal, thus geometrically “unibranche” [Gro67, 6.15.1, 0.23.2.1], therefore any subset Z ⊆ X which is equidimensional over V defines a relative cycle [SV00, Theorem 3.4.2].

c(X/k, 0)(V) ⊃ ℤ [ \begin{matrix}Z ⊂ X × V, finite over \\ V, equidimensional over V \end{matrix} ]

Any effective cycle is equidimensional [SV00, Proposition 3.1.7]. Thus we obtain for any smooth connected V:

c^{eff}(X/k, 0)(V) = ℕ[Z ⊆ X × V, Z finite over V, Z irred.] =: ℕ_{tr}(X)(V),

i.e. c^{eff}(X/k, 0)(V) = Hom_{ℕSmCor}(V, X) (the free abelian monoid generated by elementary correspondences from V to X). The functoriality of the two presheaves agrees [SV00, Corollary 3.3.11]. We obtain an equality of presheaves on Sm: c^{eff}(X/k, 0)|_{Sm} = ℕ_{tr}(X), and a fortiori c_{h}^{eff}(X/k, 0) = ℕ_{tr,h}(X). Moreover, the map ℕ(X) → c^{eff}(X/k, 0) mapping a morphism of schemes to its graph induces an isomorphism ℕ_{h}(X) ⟶^{≅} c_{h}^{eff}(X/k, 0) [SV00, Theorem 4.2.12(2)].

Assembling these isomorphisms together we conclude that the canonical monomorphism ℕ(X) → ℕ_{tr}(X) becomes an isomorphism when we apply h-sheafification. This yields the assertions for ℤ_{tr} and ℚ_{tr} as well.

The canonical transfer structure is given as follows: let F be an h-sheaf (without transfers), X ∈ Sm. Then we have

F (X) = Hom_{PSh(Sm)}(ℤ(X), F) = Hom_{Shv_{h}(Sm)}(ℤ_{h}(X), F)
≅ Hom_{Shv_{h}(Sm)}(ℤ_{tr,h}(X), F) = Hom_{PSh(Sm)}(ℤ_{tr}(X), F).

Let Z ∈ Hom_{SmCor}(Y, X) = Hom_{PSh_{tr}(Sm)}(ℤ_{tr}(Y), ℤ_{tr}(X)). Now, F (Z) : F (X) → F (Y) is defined using the above isomorphism by a ↦ a ∘ ℤ_{tr}(Z). This clearly defines a transfer structure on F whose restriction to Sm coincides with the original h-sheaf on Sm.

Proposition 2.2 There is an equivalence of categories of h-sheaves with and without transfers:

Shv_{h,tr} (Sm) ≅ Shv_{h} (Sch) .

Proof Note that h-sheaf on SmCor (i.e. on Sm) is meant in the sense of the induced topology under the embedding Sm ⊂ Sch [SGA, Exposé II, 1.1., 3.1.]. As any scheme is h-locally smooth, we have an equivalence of h-sheaves on Sch and h-sheaves on Sm [SGA, Exposé III, Théorème 4.1].

We make the following obvious remark: for any presheaf with transfers G the following functors on Sm are naturally isomorphic:

G(−) ≅ Hom_{PSh_{tr}(Sm)}(ℤ_{tr}(−), G) ≅ Hom_{PSh(Sm)}(ℤ_{tr}(−), G).

The first bijection is the Yoneda lemma. The second one is shown similarly using the transfer structure of G.

First, we show that for any presheaf with transfers F the canonical morphism F → F_{h} is a morphism of presheaves with transfers. F_{h} has its canonical transfer structure (Lemma 2.1), in particular we have F_{h}(X) = Hom_{PSh(Sm)}(ℤ_{tr}(X), F_{h}). For a correspondence Z ∈ Hom_{SmCor}(Y, X) the morphisms F (X) → F (Y) and F_{h}(X) → F_{h}(Y) are induced by the map ℤ_{tr}(Y) → ℤ_{tr}(X), by the Yoneda lemma and by construction of the canonical transfer structure, respectively. Consider the following diagram:

\xymatrix{F (X) = Hom_{PSh(Sm)} (ℤ_{tr}(X), F) \ar[r] \ar[d] & Hom_{PSh(Sm)} (ℤ_{tr}(X), F_{h}) = F_{h}(X) \ar[d] \\ F (Y) = Hom_{PSh(Sm)} (ℤ_{tr}(Y), F) \ar[r] & Hom_{PSh(Sm)} (ℤ_{tr}(Y), F_{h}) = F_{h}(Y)}

The horizontal maps are induced by morphisms of presheaves (without transfers) F → F_{h}. The commutativity of the diagram yields the first claim.

Secondly, one analogously proves that morphisms of h-sheaves are morphisms of h-sheaves with transfers.

Thirdly, we show that any h-sheaf admits only one transfer structure. Considering any h-sheaf F, the canonical morphism F → F_{h} is an isomorphism of presheaves without transfer. According to the first claim, it is also a morphism of sheaves with transfers, where F_{h} carries the canonical transfer structure. This implies that F carries the canonical transfer structure, as well.

2.1 Cohomology comparison results

Recall that a presheaf F on Sch (or Sm) is called homotopy invariant, if the canonical projection p : X × 𝔸^{1} → X induces an isomorphism p^{∗} : F (X) → F (X × 𝔸^{1}) for any X in Sch (or Sm, respectively). One easily sees that for any complex F^{∗} of homotopy invariant presheaves the cohomology presheaves H^{n}(F^{∗}) are homotopy invariant, as well.

First of all, we gather several results of Friedlander, Suslin and Voevodsky in the following theorem. These facts constitute the nontrivial input to the comparison result:

Theorem 2.3 Let k be a field admitting strong resolutions of singularities. Let F be a homotopy invariant presheaf of ℚ-vector spaces with transfers. Let t be either the Zariski, Nisnevich, étale or cdh-topology. Then the cohomology presheaves on Sm

X ↦ H_{t}^{n}(X, F_{t}), n ≥ 0

are homotopy invariant presheaves with transfers and are (up to canonical isomorphisms) independent of t.

Proof Presheaves with transfers are pretheories [Voe00b, Proposition 3.1.11]. Hence [Voe00a, Theorem 4.27, Proposition 5.28] (Zariski-topology), [Voe00a, Theorem 5.7.] (Nisnevich), [Voe00a, Propositions 5.24, 5.27] (étale topology, this requires rational coefficients) and finally the proof of [FV00, Theorem 5.5] (cdh-topology, this requires strong resolution of singularities) apply.

We want to generalize this theorem to the qfh- and h-topology. We first observe that the h-topology is the “union” of the cdh- and the qfh-topology, in the sense made precise by Lemma 2.4 below. Looking at the definitions of these topologies and Proposition 1.6, this statement is not surprising. We then use the usual Galois descent method in order to overcome the gap between Nisnevich and qfh-topology (Lemma 2.6). The cohomology comparison result between étale and qfh-topology (for a qfh-sheaf, Theorem 2.7) and the adaptation of the cohomology comparison result of Nisnevich versus cdh-topology to the case qfh- versus h-topology then give the result.

Lemma 2.4 Let F be a presheaf on Sch. We assume that F is a qfh-sheaf and a cdhsheaf. Then F is an h-sheaf.

Proof Consider the exact sequence of presheaves

0 ⟶ ker ⟶ F ⟶ F_{h} ⟶ coker ⟶ 0.

Sheafifying it with respect to either the cdh- or the qfh-topology does not affect the middle terms and shows that both ker and coker are cdh- and qfh-sheaves. Thus, we are reduced to show F = 0 for any F which is both a qfh- and cdh-sheaf satisfying F_{h} = 0.

Let X ∈ Sch, without loss of generality X is reduced (as F is a qfh-sheaf), a ∈ F (X). There is an h-covering {g_{i} : U_{i} → X}, such that g_{i}^{∗}(a) = 0. We may assume that the covering is of normal form (Proposition 1.6):

g_{i} : U_{i} ⟶^{f_{i}} U ⟶^{f} X_{X_{1}} ⟶^{p} X.

Clearly {U_{i} ⟶^{f∘f_{i}} X_{X_{1}}} is a qfh-covering. Hence p^{∗}(a) = 0, for F is a qfh-sheaf. Since F is a cdh-sheaf and X_{X_{1}} ⊔ X_{1} → X is a cdh-covering, it suffices to show F (X_{1}) = 0.

We proceed by noetherian induction with respect to the strictly descending chain of closed subsets X =: X_{0} ⊋ X_{1} ⊋ .... The induction stops when the chain becomes stationary or if the normal refinement of the covering of X_{i} does not contain any blow-up. The former is impossible, thus at some point of the noetherian induction the h-covering of X_{i} must even be a qfh-covering.

Theorem 2.5 (Comparison of Nisnevich and (qf)h-sheafification) We assume that k admits strong resolution of singularities. Let F be a homotopy invariant presheaf of ℚ-vector spaces with transfers (on Sm). Then the Nisnevich, qfhand h-sheafifications F_{Nis}, F_{qfh}|_{Sm} and F_{h}|_{Sm} are isomorphic.

Consider the presheaves ℚ_{tr}′(X) [SV96, §5 and §6], where X ∈ Sch:

ℚ_{tr}′(X)(Y) := ℚ [ \begin{matrix}W ⊂ X × Y, closed, integral, \\ finite and surjective over Y \end{matrix} ] ,

where Y is an arbitrary normal connected scheme. Analogously we have a presheaf ℤ[1/p]_{tr}′(X) , where p is the exponential characteristic of k. Functoriality is given as follows: Let f : Z → Y be a morphism of normal connected schemes, W ⊂ X × Y an integral cycle, finite and surjective over Y. Set f^{∗}([W]) := ∑ n_{i}[(V_{i})_{red}], where V_{i} are the irreducible components of W ×_{Y} Z and (V_{i})_{red} are the associated reduced (thus integral) schemes. The multiplicities n_{i} are defined as described in [SV96, definition after Lemma 5.6]. They are always positive, see loc. cit.

Lemma 2.6 Let f : Y ⟶ X be the normalization of a smooth (thus excellent) connected affine scheme X in a normal finite field extension L of k(X). Then there exists a correspondence

f_{∗} ∈ ℤ [ \frac{1}{p} ]_{tr_{′}} (Y)(X)

(finite integral correspondence from X to Y with coefficients in ℤ[1/p]), such that f ∘ f_{∗} = d · Id_{X}, d ∈ ℕ^{>0}.

Proof (compare [Voe96, Theorem 3.3.8] for ℤ_{qfh}).

The automorphism group G := Aut(Y/X) = Gal(k(Y)/k(X)) is a finite group. Considering g ∈ G as the graph of the corresponding automorphism, the sum a = ∑_{g∈G} g is a G-invariant element of ℤ_{tr}′(Y)(Y) (a is an integral cycle as Y is integral). According to [SV96, Corollary 6.2], a has a preimage f_{∗} ∈ ℤ[1/p]_{tr}′(Y)(X):

\xymatrix{Y \ar[d]^{f} \ar[dr]^{a} \\ X \ar[r]^{f_{∗}} & Y \ar[r]^{f} & X}

Let d denote the cardinality of G. Then f ∘ f_{∗} ∘ f = f ∘ a = d · f.

We show that for any surjective morphism f : Y → X of schemes, where Y is assumed to be integral, and X to be smooth and every effective correspondence c ∈ ℕ_{tr}′(X)(X) with c ∘ f = d · f the equality c = d · Id_{X} holds. This provides the assertion of the lemma, for f is surjective in our situation and – by the proof of loc. cit. – f_{∗}, being the preimage of an effective correspondence, is effective as well.

Let c = ∑_{j} n_{j} · c_{j}, n_{j} ∈ ℕ, c_{j} ⊂ X × X integral and finite over X. By definition

c ∘ f = ∑_{i,j} n_{j} · m_{ij} · W_{ij} =^{!} d · Γ_{f} ,

where W_{ij} ⊂ T_{j} are the associated reduced schemes of the irreducible components of T_{j} := c_{j} ×_{X} Y, the morphism c_{j} → X is the projection to the first component. The multiplicities of the components m_{ij} are positive (see above). As all coefficients are positive, it follows W_{ij} = Γ_{f} for all i, j and ∑_{i,j} n_{j}m_{ij} = d.

Using the surjectivity of f it is easy to see (using Hilbert’s Nullstellensatz) that the k̄-valued points of Δ_{X} and c_{j} agree. Thus Δ_{X} = c_{j} on the level of topological spaces (Hilbert’s Nullstellensatz). As X, and thus Δ_{X} is reduced, they agree as schemes, too. Hence we have #{j} = 1 and #{i} = 1 as T = Γ_{f} ≅ Y is irreducible. Moreover m = 1, for Y is integral by assumption.

Proof (Theorem 2.5) In view of Theorem 2.3 we may assume that F is an homotopy invariant cdh-sheaf with transfers, which implies that F_{qfh} is an h-sheaf (Lemma 2.4), i.e. F_{qfh} = F_{h}. Consider the exact sequence of Nisnevich sheaves with transfers:

0 ⟶ F′ ⟶ F ⟶ F_{qfh} ⟶ F′′ ⟶ 0.

(Recall that Shv_{Nis,tr} (Sm) is an abelian category [Voe00b, Theorem 3.1.4] and that F → F_{qfh} = F_{h} is a morphism of sheaves with transfers, see Proposition 2.2).

As in the proof of Lemma 2.4 it suffices to show the following: Let F be a (not necessarily homotopy invariant) cdh-sheaf on Sch, whose restriction to Sm is a sheaf with transfers, such that F_{qfh} = 0. Then we have F = 0.

Let X ∈ Sch, without loss of generality affine and smooth (as F is a cdh-sheaf), and a ∈ F (X) an arbitrary section. We show a = 0. There is a qfh-covering of normal form {f_{i} : U_{i} → U → X}, such that f_{i}^{∗}(a) = 0 (Proposition 1.6). As F is a Zariski sheaf it suffices to consider U → X. We replace the excellent scheme U by its normalization in a normal finite extension of k(X). Choose a cdh-covering Ũ → U with Ũ ∈ Sm (strong resolution of singularities).

As F is a sheaf with transfers,

F (V) = Hom_{PSh_{tr}(Sm)}(ℚ_{tr}(V), F)
= Hom_{PSh(Sm)}(ℚ_{tr}(V), F)
= Hom_{PSh(Sm)}(ℚ_{tr}′(V), F)

for any smooth V (see the proof of Proposition 2.2). As F is a cdh-sheaf, we may cdh-sheafify the first presheaf in all of these Hom-sets without changing the Hom-sets. Hence, the morphism F (X) → F (Ũ) is induced by

ℚ_{tr′,cdh}(Ũ) ⟶^{α} ℚ_{tr′,cdh}(U) ⟶^{β} ℚ_{tr′,cdh}(X).

We know ℚ_{tr}′(W)|_{Sm} = ℚ_{qfh}(W)|_{Sm} [SV96, Theorem 6.3] for any scheme W. Thus, a fortiori, ℚ_{tr}′(W)_{cdh} = ℚ_{qfh,cdh}(W) = ℚ_{h}(W). As Ũ → U is an h-covering, α is an epimorphism [Voe96, Proposition 3.2.5.2]. According to Lemma 2.6, β has a section, so it is an epimorphism. Hence F (X) → F (Ũ) is a monomorphism, i.e. a = 0.

Theorem 2.7 (Comparison of etale and qfh-cohomology) We assume that k admits strong resolution of singularities. Let F be a homotopy invariant étale sheaf with transfers with values in ℚ-vector spaces and let X be a normal scheme. Then H_{ét}^{n}(X, F) ≅ H_{qfh}^{n}(X, F_{qfh}), n ∈ ℕ.

Proof The case n = 0 is Theorem 2.5, so we may assume that F is a qfh-sheaf. Then [Voe96, Theorem 3.4.1] is what we assert.

Theorem 2.8 (Comparison of h- and qfh-cohomology) Let k be a field admitting strong resolution of singularities and let F be a homotopy invariant presheaf with transfers of ℚ-vector spaces. Then for all X ∈ Sm, n ≥ 0 the following holds:

H_{h}^{n}(X, F_{h}) = H_{qfh}^{n}(X, F_{qfh}).

Lemma 2.9 Let X ∈ Sm, Z ⊂ X a smooth closed subscheme. Let p_{Z} : X_{Z} → X denote the blow-up of Z in X. Further, let F be a homotopy invariant presheaf with transfers of ℚ-vector spaces. Then it holds for i ≥ 0

Ext_{Shv_{qfh}(Sm)}^{i}(coker ℤ_{qfh}(p_{Z})), F_{qfh}) = 0.

Proof An analogous statement for the Nisnevich topology is proven in [Voe00a, Proposition 5.21] exploiting the fact that Nisnevich cohomology groups of F_{Nis} are homotopy invariant. The proof of loc. cit. remains valid for any finer topology t, which fulfills H_{t}^{i}(X, F_{t}) ≅ H_{t}^{i}(X × 𝔸^{1}, F_{t}). In particular it holds (for sheaves of ℚ-vector spaces) for the qfh-topology (Theorem 2.3, Theorem 2.7).

Lemma 2.10 Let k be a field admitting strong resolution of singularities, F a qfh-sheaf, such that F_{h} = 0, G a homotopy invariant presheaf with transfers of ℚ-vector spaces. Then we have

Ext_{Shv_{qfh}(Sm)}^{n}(F, G_{qfh}) = 0, n ≥ 0.

Proof The proof of [FV00, Lemma 5.4] goes through (using Lemma 2.9 instead of [FV00, Lemma 5.3]), if we replace Nisnevich and cdh-topology by qfh- and h-topology respectively.

Proof (Theorem 2.8) The proof of [FV00, Theorem 5.5] goes through verbatim using Lemma 2.10 instead of [FV00, Lemma 5.4].

The following theorem gathers the above results. It points out the quite special nature of homotopy invariant Zariski sheaves with transfers of ℚ-vector spaces: they are already sheaves with respect to the (much finer) h-topology.

Theorem 2.11 Let t ∈ {Zar, Nis, ét, cdh, qfh, h}. Let F^{∗} be a bounded above complex of presheaves of ℚ-vector spaces with transfers, whose cohomology-t-sheaves are homotopy invariant. We assume that k admits strong resolution of singularities. Then the hypercohomology presheaves X ↦ ℍ_{t}^{n}(X, F_{t}^{∗}), X ∈ Sm are homotopy invariant presheaves with transfers and are independent of the choice of the topology t.

The assumption on F^{∗} is satisfied, if the cohomology presheaves of F^{∗} are homotopy invariant.

Proof If F^{∗} is concentrated in degree 0, we are done because of Theorem 2.3, Theorem 2.7, Theorem 2.5, Theorem 2.8, as hypercohomology is just cohomology in this case. This also shows the last claim.

In general, let denote (H^{q}(F^{∗}))_{t} the t-sheafification of the q-th cohomology presheaf of the complex F^{∗}, which is an homotopy invariant presheaf with transfers. The presheaves X ↦ ℍ_{Nis}^{n}(X, F_{Nis}^{∗}) have transfers [MVW, Exercise 13.5.]. So, the other hypercohomology groups, which are canonically isomorphic to the Nisnevich hypercohomology, have transfers as well. The hypercohomology spectral sequence

E_{2}^{p,q} = H_{t}^{p}(X, (H^{q}(F^{∗}))_{t}) = Ext^{p}(ℚ_{t}(X), (H^{q}(F^{∗}))_{t}) ⟹ ℍ_{t}^{p+q}(X, F_{t}^{∗})

converges because of finite cohomological dimension for sheaves of ℚ-vector spaces (see [Voe96, Theorem 3.4.6] for the h-topology). Comparing the limit terms for X and X × 𝔸^{1} yields the assertion.

Remark 2.12 We restrict to ℚ-linear categories as the comparison of Nisnevich and qfh-sheaves is valid only for sheaves of ℚ-vector spaces. An analogous statement for sheaves with torsion does not hold: Consider the Nisnevich sheaf with transfers O^{∗}. If R is a ring, then O^{∗}(R[t]) = R^{∗} × {1 + ∑_{i=1}^{n} f_{i}t^{i}, (f_{i})^{N} = 0, N ≫ 0}. In particular, O^{∗} is homotopy invariant on Sm, but not on Sch. See [Voe00a, Remark 5.8] for a counterexample for the Nisnevich-flat topology.

The restriction to characteristic zero is only needed for the cohomology comparison result of Nis- and cdh-topology (and the analogue qfh- vs. h-topology). It seems reasonable to aim at a generalization to positive characteristic at least for ℚ-coefficients using Noether normalization and de Jong’s resolution of singularities, but we did not succeed to prove it.

3 Motivic complexes and geometric motives

3.1 Motivic complexes

As we have outlined in the introduction, we will embed the several categories of geometric motives – which we are mostly interested in – into the more flexible categories of motivic complexes. Voevodsky defines a category of motivic complexes using Nisnevichsheaves with transfers. Parallely, we also use a different version using h-sheaves without transfers [Voe96]. In the case of ℚ-vector space sheaves both categories are equivalent (Theorem 3.7). This theorem relies on the cohomology comparison results of Chapter 2.

Definition 3.1 [Voe00b, §3.1.] The category of effective motivic complexes with transfers is defined to be the full subcategory DM_{Nis,tr}^{eff,−} of D^{−}(Shv_{Nis,tr} (Sm)) consisting of complexes whose Nisnevich cohomology sheaves are homotopy invariant. (This category is denoted by DM_{Nis}^{eff,−}(k) in loc. cit.).

Similarly, DM_{h}^{eff,−} ⊂ D^{−}(Shv_{h} (Sch)) denotes the full subcategory, whose objects have homotopy invariant cohomology h-sheaves. This category is called category of effective motivic complexes without transfers.

We write DM_{h}^{eff,−}(ℚ) and DM_{Nis,tr}^{eff,−}(ℚ) for the corresponding ℚ-linear categories.

Definition 3.2 [MVW, §2] Let Δ^{∗} denote the standard cosimplicial object in Sch, i.e. Δ^{n} = {∑ x_{i} = 1} ⊂ 𝔸^{n+1}. For any presheaf F, let C_{∗}(F) denote the Suslin complex F (Δ^{∗} × −), i.e. C_{n}(F)(X) = F (Δ^{n} × X); its differentials are alternating sums of δ_{j}. We get an exact functor C_{∗} : PSh(Sch) → Com(PSh(Sch)). If F is a sheaf with respect to an arbitrary (pre)topology, then C_{∗}(F) is a complex of sheaves as well as well.

Recall from Definition A.2 that ⟨−⟩ denotes the minimal localising subcategory containing a given class of objects. For the rest of this subsection, we will use the following notation: D_{h} := D^{−}(Shv_{h} (Sch, ℚ)), A_{h} := ⟨{ℚ_{h}(X × 𝔸^{1}) → ℚ_{h}(X), X ∈ Sm}⟩ ⊂ D_{h}.

Remark 3.3 Morphisms whose cone is in A_{h} are called weak 𝔸^{1}-equivalences. Objects L ∈ D_{h} are called 𝔸^{1}-local, if for all weak 𝔸^{1}-equivalences K → K′ the induced map Hom(K′, L) → Hom(K, L) is a bijection.

Theorem 2.11 implies that DM_{h}^{eff,−}(ℚ) is a triangulated subcategory of D_{h}: For a distinguished triangle A → B → C → A[1] in D_{h} with A, B ∈ DM_{h}^{eff,−} we have that Hom_{D_{h}}(ℚ_{h}(−), A[n]) = ℍ_{h}^{n}(−, A) is homotopy invariant, analogously for B. The long exact Hom_{D_{h}}-sequence coming from this triangle shows that C is 𝔸^{1}-local as well, compare [MVW, Lemma 9.17.]. Hence, C has homotopy invariant cohomology sheaves. The latter can be seen as in [MVW, Proposition 14.5] (using Theorem 2.11). As in loc. cit. one sees that bounded above complexes (of sheaves of ℚ-vector spaces) with homotopy invariant cohomology sheaves are 𝔸^{1}-local.

Proposition 3.4 With these notations there is an equivalence of categories D_{h}/A_{h} ≅ DM_{h}^{eff,−}(ℚ) induced by a triangulated functor RC : D_{h} ⟶ DM_{h}^{eff,−}(ℚ), whose restriction on Shv_{h} (Sch) is given by C_{∗}.

Proof (compare [Voe00b, Proposition 3.2.3]). For brevity, we omit the specification of ℚ-vector space sheaves in the notations. We show that the restriction of the projection map Φ : DM_{h}^{eff,−} ⊂ D_{h} ⟶^{Proj.} D_{h}/A_{h} is an equivalence of categories and that the restriction to h-sheaves of the inverse equivalence Φ^{−1} is just C_{∗}:

\xymatrix{Shv_{h} (Sch) \ar[dr]_{C_{∗}} \ar[r] & D_{h}/A_{h} \ar[d]^{Φ^{−1}} \\  & _{DM{eff,− \atop h}}}

Then, we set RC := Φ^{−1} ∘ Proj. According to Remark 3.3 and [Nee01, Theorem 2.1.8], Φ is triangulated and so is RC.

Firstly, we show that Φ is fully faithful. Secondly, we show the essential surjectivity. Both parts will use the cohomology comparison results for homotopy invariant sheaves with transfers.

Φ is fully faithful: Let T ∈ DM_{h}^{eff,−} be arbitrary. According to Proposition 2.2, T |_{Sm} is a complex of sheaves with transfers, which implies (Theorem 2.11) for all X ∈ Sm, n ∈ ℤ

Hom_{D_{h}}(ℚ_{h}(X), T [n]) ≅ Hom_{D_{h}}(ℚ_{h}(X × 𝔸^{1}), T [n]).

Hence (compare [MVW, Lemma 9.17]) T is an 𝔸^{1}-local object in D_{h}. Then we have for all S ∈ D_{h}: Hom_{D_{h}/A_{h}}(S, T) ≅ Hom_{D_{h}}(S, T) [Nee01, Lemma 9.1.5].

Φ is essentially surjective: we know K ≅ Tot(C_{∗}K) in D_{h}/A_{h} (see [MVW, Lemma 9.14], the proof without transfers is strictly parallel, as the homotopy providing the quasi-isomorphism is induced by the multiplication map μ : 𝔸^{2} → 𝔸^{1}). According to Proposition 2.2, K and Tot(C_{∗}(K)) are complexes of h-sheaves with transfers (of ℚ-vector spaces). The cohomology presheaves of Tot(C_{∗}(K)) are homotopy invariant [MVW, Lemma 2.17], thus the associated h-sheaves of the cohomology presheaves restricted to Sm are homotopy invariant, as well (Theorem 2.11). Using the equivalence of h-sheaves on Sm and on Sch (see the proof of Proposition 2.2), this easily implies that the same holds without restricting to Sm. Hence Tot(C_{∗}(K)) ∈ DM_{h}^{eff,−} and we are done.

Remark 3.5 If one attempts to define explicitly Φ^{−1} : K ↦ Tot(C_{∗}K), one has to deal with the question, whether C_{∗} is an exact functor. It suffices to show the rightexactness of C_{∗}. The presheaf cokernel F of an epimorphism of h-sheaves has transfers (Proposition 2.2) and satisfies F_{h} = 0. The cohomology comparison result for the htopology (Theorem 2.11) instead of the Nisnevich analogue [MVW, Theorem 13.11.] yields that the h-sheafification of C_{∗}(F) is exact. The h-analogue of [MVW, Corollary 13.13.] shows that C_{∗} is well-defined on D_{h}.

Lemma 3.6 The categories D^{−}(Shv_{h} (Sch, ℚ)) and DM_{h}^{eff,−}(ℚ) are tensor triangulated categories.

Proof Starting with X ⊗ Y := X ×_{k} Y, X, Y ∈ Sch, one obtains a tensor structure on D^{−}(Shv_{h} (Sch)) in the same manner as in [MVW, Corollary 8.17].

The tensor structure on D_{h}/A_{h} is determined by the requirement that the canonical functor D_{h} → D_{h}/A_{h} is a tensor functor, compare [MVW, Corollary 9.4.]. Proposition 3.4, i.e. D_{h}/A_{h} ≅ DM_{h}^{eff,−}(ℚ) yields the second assertion.

Theorem 3.7 Let k be a field admitting strong resolution of singularities. Using the notations of Definition 3.1 there are natural equivalences of tensor triangulated categories

DM_{Nis,tr}^{eff,−}(ℚ) ≅ DM_{h}^{eff,−}(ℚ) ≅ DM_{h}(ℚ),

where DM_{h}(ℚ) denotes the homological category of h-sheaves (with rational coefficients) [Voe96, §4.1.].

As mentioned in the introduction, this is part of [Voe00b, Theorem 4.1.12]. We wish to give a detailed proof of it. It is essentially a corollary of the cohomology comparison result of Nisnevich and h-topology (Theorem 2.11) and the fact that h-sheafs have a unique transfer structure (Proposition 2.2).

Proof We permit ourselves to omit “ℚ” in the notations.

Let us write A_{Nis,tr} ⊂ D_{Nis,tr} := D^{−}(Shv_{Nis,tr} (Sm, ℚ)) for the localising subcategory generated by ℚ_{tr}(X × 𝔸^{1}) → ℚ_{tr}(X), X ∈ Sm. We have an equivalence of categories DM_{Nis,tr}^{eff,−} ≅ D_{Nis,tr}/A_{Nis,tr} [Voe00b, Proposition 3.2.3]. Consider the composition (−)_{h} : Shv_{Nis,tr} (Sm) → Shv_{Nis} (Sm) → Shv_{h} (Sch) of the forgetful and sheafification functor, where the latter uses the extension of h-sheaves on Sm to ones on Sch [SGA, Exposé II, Théorème 4.1]. It clearly extends to the corresponding derived categories, maps A_{Nis,tr} to 0 ∈ DM_{h}. It suffices to show that the complexes generating A_{Nis,tr} are mapped to 0, which is equivalent to K_{X} := Ker(ℚ_{tr}(X × 𝔸^{1}) → ℚ_{tr}(X)) ↦ 0, i.e. that (K_{X})_{h} is contractible in the sense of [Voe96, Definition 2.2.1]. (According to Lemma 2.1, ℚ_{tr}(X), X ∈ Sm maps to ℚ_{h}(X)). The latter holds obviously, as K_{X} = ℚ_{tr}(X)⊗K_{Spec k} and (K_{Spec k})_{h} is contractible [Voe96, Lemma 2.2.3]. We obtain a surjective functor Φ : D_{Nis,tr}/A_{Nis,tr} → DM_{h} (Proposition 2.2). (nota bene: We have not yet used rational coefficients).

\xymatrix{D_{Nis,tr} \ar@{->>}[r]^{(−)_{h}} \ar@{->>}[d] & D_{h} \ar@{->>}[d] \\ D_{Nis,tr}/A_{Nis,tr} \ar@{->>}[r]^{Φ} & _{DM_{h}}}

For the full faithfullness of Φ it suffices to show

Hom_{D_{Nis,tr}}(ℚ_{tr}(X), F [n]) = Hom_{DM_{h}}(ℚ_{h}(X), F_{h}[n])

for every F ∈ DM_{Nis,tr}^{eff,−}.

For brevity, we write Spec k for ℚ_{h}(Spec k), X for ℚ_{h}(X) etc. The triangle I^{1} → 𝔸^{1} → Spec k → I^{1}[1] is distinguished in D_{h}. As the category is tensor triangulated (Lemma 3.6), we obtain a long exact sequence

... ← Hom(X ⊗ I^{1}, F_{h}[n]) ← Hom(X ⊗ 𝔸^{1}, F_{h}[n]) ← Hom(X, F_{h}[n]) ← ....

The two terms to the right are the n-th hypercohomologies of F in X and X × 𝔸^{1}, respectively. They are homotopy invariant (Theorem 2.11). Hence we have Hom(X ⊗ I^{1}, F_{h}[n]) = 0, i.e. F_{h} is strictly homotopy invariant in the sense of [Voe96, definition 2.2.8], which implies Hom_{DM_{h}}(X, F_{h}[n]) = Hom_{D_{h}}(X, F_{h}[n]) = ℍ_{h}^{n}(X, F_{h}) [Voe96, Proposition 2.2.9].

The equivalence DM_{h}^{eff,−} ≅ DM_{h} is trivial (and does not need rational coefficients).

3.2 Geometric motives

We now define the categories of (quasi-)geometric motives with and without transfers. The prefix “quasi” indicates that we also consider countable direct sums (i.e. disjoint unions) of schemes. Moreover, the “quasi”-categories consist of bounded above complexes.

Recall from Definition A.3 that for a class T of objects of a triangulated category the notations ⟨T ⟩^{ℵ_{0}} (and ⟨T ⟩^{ℵ_{1}}, respectively) mean the minimal thick subcategory containing T (closed under countable existing direct sums, respectively).

Definition 3.8 The categories \overline{ℤSch} and \overline{SmCor} are defined as the closure under countable direct sums of ℤ : ℤSch → PSh(Sch) (the Yoneda embedding), and similarly for \overline{SmCor}. For brevity, we may write ⊕X_{i} := ⊕ℤ(X_{i}) for a family of schemes X_{i}.

Definition 3.9 [Voe00b, §2.1.] Consider the class T consisting of the following complexes

– X × 𝔸^{1} → X, X ∈ Sm.

– U ∩ V ⟶^{ι_{U} ⊕ι_{V}} U ⊕ V ⟶^{ι^{U} ⊕(−ι^{V} )} X for any open covering X = U ∪ V of a smooth scheme X.

The category of effective quasigeometric motives with transfers DMeff \overline{gm} is defined as follows: DMeff \overline{gm} := K^{−} (\overline{SmCor}) / ⟨T ⟩^{ℵ_{1}}, i.e. the localisation of K^{−} (\overline{SmCor}) with respect to the minimal triangulated subcategory closed under countable sums containing T. The subcategory DM_{gm}^{eff} consisting of K^{b}(SmCor) is called category of effective geometric motives with transfers. (This coincides with Voevodsky’s definition K^{b} (SmCor) / ⟨T ⟩^{ℵ_{0}}, see Theorem 3.15).

Definition 3.10 We define the category DM_{\overline{gm},h}^{eff} of effective quasigeometric motives without transfers over a field k: Let T denote the union of the following two classes of complexes in K^{−} (\overline{ℤSch}):

– T_{𝔸^{1}} = {X × 𝔸^{1} ⟶^{pr_{1}} X, X ∈ Sch}.

– T_{h} = augmented Moore-complexes of h-coverings U → X, U ∈ Sm, X ∈ Sch (Definition 1.2). We define DM_{\overline{gm},h}^{eff} to be the localisation of K^{−} (\overline{ℤSch}) with respect to the ℵ_{1}-

localising subcategory generated by T:

DM_{\overline{gm},h}^{eff} := K^{−} (\overline{ℤSch}) / ⟨T ⟩^{ℵ_{1}} .

The subcategory DM_{gm,h}^{eff} of bounded complexes of schemes is called category of effective geometric motives without transfers. The canonical functor M_{gm} : Sch → DM_{gm,h}^{eff} maps a scheme to its geometric motive.

If we consider rational coefficients instead of integral ones, we write DM_{gm,h}^{eff}(ℚ) etc.

Remark 3.11 Note that the quasigeometric categories of motives are pseudo-abelian, as they are closed under direct sums of the form ⊕_{i∈ℕ}A, where A ∈ DMeff \overline{gm} [Lev98, Lemma II.2.2.4.8.1].

Lemma 3.12 The category DM_{\overline{gm},h}^{eff} is a triangulated tensor category, whose tensor structure is induced by M_{gm}(X)⊗M_{gm}(Y) = M_{gm}(X ×Y). The subcategory DM_{gm,h}^{eff} is closed under this tensor product.

Proof The tensor structure on K^{−} (\overline{ℤSch}) is determined via the one on Sch (X ⊗Y := X × Y) and (⊕X_{i}) ⊗ (⊕Y_{j}) := ⊕_{i,j}X_{i} ⊗ Y_{j} and K^{∗} ⊗ L^{∗} := Tot ((K^{n} ⊗ L^{m})^{n,m}). We show that for all A ∈ T (Definition 3.10) and B ∈ K^{−} (\overline{ℚSch}) the tensor product A⊗B lies in ⟨T ⟩^{ℵ_{1}}. Then [Lev98, II.2.3.4] yields the claim. The subcategory B ⊂ K^{−} (\overline{ℚSch}) containing the complexes B with A ⊗ B ∈ ⟨T ⟩^{ℵ_{1}} is ℵ_{1}-localising. Using Lemma 1.7 we reduce to the case B ∈ Sch, which is obvious. The second claim is clear from the definitions.

Remark 3.13 A similar statement holds for the variants with transfers DMeff \overline{gm} (proof as above) and DM_{gm}^{eff} [Voe00b, Proposition 2.1.3].

3.3 Embedding theorems

Theorem 3.14 Let k admit the strong resolution of singularities. Then we have a commutative diagram of tensor-triangulated functors:

\xymatrix{K^{−} (\overline{ℚSch}) \ar[r]^{ℚ_{h}} \ar[d] & D^{−}(Shv_{h} (Sch, ℚ)) \ar[d]^{RC} \\ _{DM{eff \atop \overline{gm},h}} \ar[r]^{i}_{⊂} & DM_{h}^{eff,−}(ℚ)}

where i is a full embedding of quasigeometric effective motives (Definition 3.10) into the category of effective motivic complexes (Definition 3.1). The image of i is precisely the subcategory of ℵ_{1}-compact objects of DM_{h}^{eff,−}(ℚ). The idempotent completion of the image of K^{b}(ℚSch) is the subcategory of compact objects in DM_{h}^{eff,−}(ℚ).

Proof (compare [Voe00b, Theorem 3.2.6]). In the proof, we write A_{h} := ⟨T_{𝔸^{1}}⟩ (see Definition 3.10) and D_{h} := D^{−}(Shv_{h} (Sch, ℚ)).

Existence of i: By its very definition it is clear that the functor RC commutes with direct sums. We show that RC(ℚ_{h}(T_{𝔸^{1}})) = 0 and RC(ℚ_{h}(T_{h})) = 0, which yields the existence of i. The first claim directly follows from the construction of RC (see

Proposition 3.4, A_{h} = ℚ_{h}(T_{𝔸^{1}})). RC : D ⟶ D/A_{h} ⟶^{Φ^{−1}} DM_{h}^{eff,−}. For the second claim, we note RC(ℚ_{h}(Y)) = C_{∗}(ℚ_{h}(Y)) =: C_{∗}(Y). The inverse equivalence Φ^{−1} of Proposition 3.4 maps a complex K of sheaves to the total complex Tot(C_{∗}(K)). We have to show that the total complex of the double complex

C_{∗}(U_{X} → X → 0) : ... ⟶ C_{∗}(U ×_{X} U) ⟶ C_{∗}(U) ⟶ C_{∗}(X) ⟶ 0

is exact for any h-covering U → X, X ∈ Sch. But C_{∗}(U_{X} → X → 0) is exact in view of the exactness of C_{∗} (see the proof of Proposition 3.4) and Lemma 1.3. As Tot preserves quasi-isomorphisms, we are done.

Full faithfulness of i: Let us denote D′ := D^{−}(PSh(Sch, ℚ)). Consider the following diagram

\xymatrix{K^{−} (\overline{ℚSch}) \ar[r]^{ℚ(−)} \ar[dr]^{ℚ_{h}(−)} \ar[dd] & D′ \ar[d] \\  & D ≅^{1.9} D′/ ⟨ℚ(T_{h})⟩ \ar[d]^{RC} \\ K^{−} (\overline{ℚSch}) / ⟨T_{𝔸^{1}}, T_{h}⟩^{ℵ_{1}} \ar[r] & D/ ⟨ℚ(T_{𝔸^{1}})⟩ ≅ DM_{h}^{eff,−}}

The functor ℚ : K^{−} (\overline{ℚSch}) ⟶ D′ is a full embedding [Wei94, Corollary 10.4.7] as the presheaves ⊕_{i}ℚ(X_{i}) are direct sums of projective objects, thus projective. Proposition A.5 precisely grants the extension of the embedding K^{−} (\overline{ℚSch}) ⊂ D′ to the categories in question and the asserted compacity statements.

i is tensor-triangulated: The functors RC and ℚ_{h} are tensor functors by definition. It is clear that ℚ_{h} is tensor-triangulated. In order to see that RC is tensor-triangulated, it suffices to remark that the projection D → D/ ⟨ℚ_{h}(T_{𝔸^{1}})⟩ is tensor-triangulated. Hence, RC ∘ ℚ_{h} is tensor-triangulated, which yields the claim.

Theorem 3.15 There is a full embedding

i_{tr} : DMeff \overline{gm} ⊂ DM_{Nis,tr}^{eff,−}.

Similarly to the theorem above, its essential image is exactly the subcategory of ℵ_{1}- compact objects and the idempotent completion of i_{tr}(K^{b}(SmCor)) are the compact objects of DM_{Nis,tr}^{eff,−}.

Proof The proof of [Voe00b, Theorem 3.2.6] generalizes to the quasi-geometric situation (using the ℵ_{1}-compacity arguments as in Theorem 3.14, Remark A.6).

4 Reinterpretation of geometric motives

We are now ready to gather the results of the preceding sections in order to achieve the main result of the present paper:

Theorem 4.1 Let k be a field admitting strong resolution of singularities. Then there is an equivalence of tensor triangulated categories between the categories of effective geometric motives (with rational coefficients) with and without transfers, respectively (Definitions 3.9, 3.10):

DM_{gm,h}^{eff}(ℚ) ≅ DM_{gm}^{eff}(ℚ).

The same is true for the quasigeometric categories:

DM_{\overline{gm},h}^{eff}(ℚ) ≅ DMeff \overline{gm}(ℚ).

The theorem simplifies the handling of geometric motives insofar as one does not have to work with correspondences. Moreover, singular schemes are contained in the category in a natural manner. Last but not least, the localization with respect to the h-topology permits greater flexibility in the manipulation of objects. The price for this flexibility is the loss of torsion information.

Proof We embed the categories of effective (quasi)-geometric motives into the categories of motivic complexes, see Theorems 3.14, 3.15. Either in the case with and without transfers, these are exactly the (ℵ_{1})-compact objects (Definition A.3, Proposition A.5). The equivalence DM_{h}^{eff,−}(ℚ) ≅ DM_{Nis,tr}^{eff,−}(ℚ) (Theorem 3.7) yields the assertion.

Remark 4.2 An attempt to prove an analogous statement for a category of geometric motives defined by inverting only smooth h-hypercoverings fails, because the Moore complex of a hypercovering is in general not exact at the level of presheaves.

At first glance, the equivalence of the geometric categories, i.e. bounded complexes might be surprising, as a non-smooth scheme is ad hoc only replaced by an unbounded smooth h-hypercovering. We describe a bounded complexes of smooth schemes isomorphic to a scheme X: there is a proper surjective morphism p : X̃ → X, which is an isomorphism outside some closed subset Z ⊊ X (strong resolution of singularities). We obtain a distinguished triangle in DMeff \overline{gm} of the form

M_{gm}(p^{−1}(Z)) → M_{gm}(Z) ⊕ M_{gm}(X̃) → M_{gm}(X) → M_{gm}(p^{−1}(Z))[1]

[Voe96, Proposition 4.1.5]. Inductively applying this resolution technique to Z and p^{−1}(Z) one constructs a so-called hypercube of schemes, one of whose vertices is X, the other vertices are smooth. According to the above exact triangle, the motive of X is isomorphic to the total complex of the rest of the hypercube. The induction stops after at most dimX steps. See [GNAPGP88, especially Theorem 2.15] for the construction of cubical hyperresolutions.

In the sequel, we will draw some corollaries of the theorem. We will always assume that k admits strong resolution of singularities and we will work with rational coefficients.

Corollary 4.3 If A, B are closed subschemes of a scheme X, such that A∪B = X, then there is a distinguished triangle in DM_{\overline{gm},h}^{eff} of the following form:

M_{gm}(A ×_{X} B) → M_{gm}(A) ⊕ M_{gm}(B) → M_{gm}(X) → M_{gm}(A ×_{X} B)[1].

In particular, as a motive, a scheme is isomorphic to the associated reduced scheme: M_{gm}(X_{red}) ≅ M_{gm}(X).

Proof This follows from the fact that A ⊔ B → X is a qfh-covering and Lemma 1.4.

Definition 4.4 Inverting the Tate-Object ℚ(1) in the category of effective geometric motives (with transfers), Voevodsky defines the category of geometric motives, which we denote by DM_{gm} [Voe00b, §2.1, page 192]. Strictly parallely, we define the category of geometric motives (without transfers) DM_{gm,h} and their quasi-geometric analogs DM_{\overline{gm},h} and DM\overline{gm}.

Theorem 4.5 The above equivalence of effective geometric motives induces an equivalence of geometric motives with and without transfers (with rational coefficients):

DM_{gm,h}(ℚ) ≅ DM_{gm}(ℚ).

Moreover, the canonical functor DM_{gm,h}^{eff}(ℚ) → DM_{gm,h}(ℚ) is a full embedding. These statements are also true for the quasigeomtrical categories.

Proof Under the equivalence of tensor categories DM_{gm,h}^{eff}(ℚ) ≅ DM_{gm}^{eff}(ℚ), ℚ(n) ∈ DM_{gm,h}^{eff}(ℚ) maps to ℚ(n) ∈ DM_{gm}^{eff}(ℚ) (note that the equivalence restricted to complexes of smooth schemes is the identity). So, the first statement in either the geometric or the quasi-geometric case is clear from Theorem 4.1. The second statement is then a reformulation of [Voe00b, Theorem 4.3.1] in the geometric case. The quasi-geometric case follows from the proof of loc. cit. and the fact that ⊗ commutes with ⊕.

Another consequence is a conceptual understanding of realizations of motives, similar to results of Huber [Hub00, Theorem 2.1.6.]:

Corollary 4.6 Let A be any ℚ-linear abelian category. Let R : ℚSch → D(A) be a functor mapping Čech nerves of h-coverings and 𝔸^{1}-complexes (see Definition 3.10) to complexes quasi-isomorphic to zero. Then R extends to a functor DM_{\overline{gm},h}^{eff} → D(A). If furthermore A is a tensor category and tensoring in A with R(ℚ(1)) is an equivalence of categories, R extends to a functor DM_{\overline{gm},h} → D(A). For example, singular realization of varieties factors over geometric motives.

A Compact objects in categories of bounded above complexes

In order to motivate the somewhat lengthy definition of ℵ_{1}-compact objects, we first outline the application of these notions. It turns out that the abstract characterization of complexes of schemes in terms of compacity notions makes the proof of Theorem 4.1 very simple and formal. Consider the following special case of a theorem of Neeman [Nee01, Theorem 4.4.9], which is (tacitly) used in the proof of the embedding theorem of Voevodsky [Voe00b, Theorem 3.2.6]:

Proposition A.1 The full embedding K^{b} (ℤSch) ⊂ D(PSh(Sch)) identifies the former category with the subcategory of compact objects of the latter category. Localizing with respect to some compact objects, for example Mayer-Vietoris complexes U ∩ V → U ⊕ V → U ∪ V the above embedding extends to an embedding

K^{b} (ℤSch)/M.-V. ⊂ D(PSh(Sch))/M.-V.

Note that the right hand side is just D(Shv_{Zar} (Sch)). Instead of the Zariski topology, we want to invert the h-topology. In this setting, we cannot replace the unbounded Moore complexes of h-coverings by some bounded complexes. Hence we are looking for a similar statement for K^{−} (ℤSch) instead of K^{b} (ℤSch). In order to apply the aforementioned theorem of Neeman, we enlarge K^{−} (ℤSch) insofar as we add countable existing direct sums; we arrive at K^{−} (\overline{ℤSch}), which is precisely the subcategory of so-called ℵ_{1}-compact objects (see below). Then a similar statement as above holds (see Proposition A.5 for the precise statement).

Neeman theorem handles the case that the larger category is closed under arbitrary direct sums. So, we will have to make some minor modifications to Neeman’s definitions in order to adapt them to bounded above categories like D^{−}(PSh(Sch)). The idea is to add boundedness conditions in all definitions and to check that the proof of [Nee01, Theorem 4.4.9] remains valid.

In the following definitions, let β be an infinite cardinal, let A be an abelian category having all direct sums, e.g. A = PSh(Sch), and D := D^{−}(A).

Definition A.2 A triangulated subcategory S ⊂ D is called β-localising, if it is thick and closed under direct sums of cardinality < β that exist in D. The smallest β-localising subcategory containing a class of objects S will be denoted ⟨S⟩ (⟨S⟩^{β}, respectively). Hence ⟨S⟩^{ℵ_{0}} is the minimal thick subcategory containing S and ⟨S⟩^{ℵ_{1}} is the minimal subcategory closed under countable existing direct sums. We only need these two cases.

To any complex C^{∗} ∈ D, we associate an integer |C| := max{n, C^{n} ≠ 0}. (Recall that our complexes have differentials of degree +1). Let {X_{λ}, λ ∈ Λ} be a set of objects in D. It is called bounded if sup{|X_{λ}|, λ ∈ Λ} is finite, which implies that the direct sum ⊕X_{λ} exists (in D). We make the following definitions (note that working in the category D(A) and dropping the boundedness conditions, we exactly arrive at Neeman’s definitions [Nee01, §§3,4]).

Definition A.3 – A class S of objects of D is called β-perfect if it contains the zero object and the following conditions hold:

1. Let {X_{λ}, λ ∈ Λ} be a bounded collection of cardinality < β of objects in D. Any morphism k → ⊕X_{λ} with k ∈ S factors as k → ⊕k_{λ} ⟶^{⊕f_{λ}} ⊕X_{λ}, where k_{λ} ∈ S. Furthermore, if Λ is infinite, we also require sup |k_{λ}| ≤ sup |X_{λ}| (which ensures that ⊕k_{λ} exists).

2. If any such morphism k → ⊕k_{λ} ⟶^{⊕f_{λ}} ⊕X_{λ} vanishes, then there are objects l_{λ} ∈ S such that f_{λ} is factorized by k_{λ} → l_{λ} → X_{λ} and k → ⊕k_{λ} → ⊕l_{λ} vanishes. Again, if Λ is an infinite set, we require sup |l_{λ}| ≤ sup |X_{λ}|.

– There is a maximal β-perfect class of objects, contained in S [Nee01, Corollary 3.3.10]. It will be denoted S_{β}.

– An object k ∈ D is called β-small, if any morphism k → ⊕_{λ∈Λ}X_{λ}, where the family X_{λ} is supposed to be bounded, factors as k → ⊕_{λ∈Λ′}X_{λ} → ⊕_{λ∈Λ}X_{λ}, where Λ′ ⊂ Λ is a subset of cardinality < β. The full subcategory of D consisting of β-small objects is denoted D^{(β)}.

– The subcategory of β-compact objects in D is defined by D^{β} := {D^{(β)}}_{β}. If β = ℵ_{0}, we will usually speak of compact objects.

Lemma A.4 The compact objects of D′ := D^{−}(PSh(Sch)) are given by the subcategory (under the Yoneda embedding) K^{b} (ℤSch) of bounded complexes of schemes. The ℵ_{1}-compact objects of D′ are precisely given by bounded above complexes of countable direct products of schemes, i.e. K^{−} (\overline{ℤSch}). The same is true for D := D^{−}(Shv_{h} (Sch)).

Proof A scheme, considered as an object of D′ under the Yoneda embedding, is cleary ℵ_{0}-small (i.e. Hom_{D′}(ℤ(X), −)) commutes with direct sums). The same is true in the category D. In fact, this holds for any topology t whose coverings are open morphisms: by convention our schemes are quasi-compact, so every covering has a finite subcovering, which implies that the direct sum of a set of t-sheaves F_{λ} is given by X ↦ ⊕F_{λ}(X).

Via induction on the length of a complex one sees that a bounded complex of schemes is ℵ_{0}-small. All ℵ_{0}-small objects are (ℵ_{0}-)compact [Nee01, Example 3.3.16]. As the subcategory of β-compact objects is β-localising for a regular cardinal β (such as ℵ_{0} or ℵ_{1}) [Nee01, Lemma 4.2.5] countable direct sums of schemes are ℵ_{1}-compact. Bounded above complexes of such objects are ℵ_{1}-compact as well (Lemma 1.7). We saw in Lemma 1.8 that ⟨K^{b}(ℤSch)⟩ = D′, so [Nee01, Lemma 4.4.5] shows K^{b}(ℤSch) = D^{′ℵ_{0}} and

K^{−} (\overline{ℤSch}) = D^{′ℵ_{1}} and similarly for D.

Proposition A.5 We write T_{h} := {ℚ(U_{X}) → ℚ(X), U → Xan h-covering} (see Definition 1.2) and T_{𝔸^{1}} = {ℚ(X × 𝔸^{1}) → ℚ(X), X ∈ Sm}.

The natural functor K^{−} (\overline{ℚSch}) / ⟨T_{h}⟩^{ℵ_{1}} → D := D^{−}(Shv_{h} (Sch, ℚ)) is a full embedding whose image consists of the ℵ_{1}-compact objects of D.

The natural functor

K^{−} (\overline{ℚSch}) / ⟨T_{h}, T_{𝔸^{1}}⟩^{ℵ_{1}} → D/ ⟨T_{𝔸^{1}}⟩

is a full embedding. Its image consists of the ℵ_{1}-compact objects of D. Moreover, the idempotent completion of the image of bounded complexes of schemes are precisely the compact objects of D.

Proof For the convenience of the reader, we indicate the necessary changes to the proof of [Nee01, Theorem 4.4.9]: The second part of the proof [Nee01, Lemma 3.3.3] does not work, but this lemma is only applied to triangulated subcategories, so the second part is not needed in view of [Nee01, Lemma 3.3.5]. [Nee01, Lemma 3.3.7] only holds for α ≤ β or β > ℵ_{0} in the notations of loc. cit. This lemma is only used in [Nee01, Theorem 3.3.9], which is only needed with α = β. [Nee01, Theorem 4.3.3] has to be stated as follows: Let β be a regular cardinal, S a class of objects in D^{β}. Let f : x → z be a morphism with x ∈ D^{β} and z ∈ ⟨S⟩. Then there is an object y ∈ ⟨S⟩^{β}, such that |y| ≤ |z| and f factors as x → y → z. Its proof is the same as in loc. cit. This sharper statement is necessary to prove [Nee01, Lemma 4.4.8] in our situation. The proof of the fact that localisation preserves direct sums [Nee01, Lemma 3.2.10] does a priori not work in general, but when we localise with respect to h-coverings, the claim holds, as schemes are quasi-compact (cf. the proof of Lemma A.4). When we localise with respect to 𝔸^{1}-complexes, the assertion is true because of Proposition 3.4.

The first claim of the proposition is now precisely the statement of [Nee01, Theorem 4.4.9]: K^{−} (\overline{ℤSch}) are the ℵ_{1}-compact objects in D^{−}(PSh(Sch)), which is generated by these complexes and D is the localisation of the D′ with respect to ⟨T_{h}⟩ (Lemma 1.9). The second claim follows from the first one (using [Nee01, Theorem 4.4.9] again).

Remark A.6 The above proposition also holds for Nisnevich sheaves with transfers. Indeed, by [Voe00b, Proposition 3.1.9] Hom_{D^{−}(Shv_{Nis,tr}(Sm))}(X, K[i]) = ℍ_{Nis}^{i}(X, K), Nisnevich hypercohomology commutes with direct sums and Proposition 3.4 is replaced by its predecessor [Voe00b, Proposition 3.2.3].

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