Framed transfers and motivic fundamental classes

We relate the recognition principle for infinite P1 ‐loop spaces to the theory of motivic fundamental classes of Déglise, Jin and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with An/(An−0) , and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite R ‐correspondences for R a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calmès and Fasel's category of finite Milnor–Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of R ‐module spectra factors through the category of finite R ‐correspondences.


Introduction
This paper connects two recent developments in our understanding of certain cohomology theories for schemes, namely those that are represented in the Morel-Voevodsky category of motivic spectra [38]. On the one hand, the work of Levine [36] and Déglise, Jin and Khan [17] develops the theory of fundamental classes in the setting of motivic homotopy theory. This results in a vast generalization of Fulton's operations in Chow groups [24] to these cohomology theories. On the other hand, the work of Garkusha, Panin, Ananyevskiy and Neshitov [2,[25][26][27], building on some insights of Voevodsky [49], develops a theory of framed motives. One achievement of their work is to give explicit models for motivic suspension spectra of smooth schemes. Recall that if E ∈ SH(S) is a motivic spectrum over a scheme S, there is an associated bigraded cohomology theory on smooth S-schemes: Both the theory of fundamental classes and that of framed motives imply the existence of certain transfers, called framed transfers, in such a cohomology theory. These transfers can be encoded by an extension of E * , * (−) to the category hCorr fr (Sm S ) of framed correspondences: In the first part of this paper, we show that the framed transfers produced by both theories agree. This is nontrivial as their respective constructions are based on different geometric ideas. In the second part of this paper, we introduce the category hCorr R (Sm S ) of finite R-correspondences for R a motivic ring spectrum, and we construct a further interesting extension when E is a module over R. The category hCorr R (Sm S ) recovers Voevodsky's category of finite correspondences when R is the motivic Eilenberg-Mac Lane spectrum HZ, and it recovers Calmès and Fasel's category of finite Milnor-Witt correspondences when R = HZ. Thus, our construction unifies those of Voevodsky and of Calmès-Fasel, as well as their relationship with the category of framed correspondences.
One of the key results in [20] is that the presheaves Corr efr S (−, Y ) and Corr fr S (−, Y ) are motivically equivalent. This implies that Voevodsky's map factors through the ∞-groupoid Corr fr S (X, Y ). In particular, α induces a map Σ ∞ T X + → Σ ∞ T Y + in SH(S), whence a map α * : E(Y ) → E(X) in cohomology.
(3) Via framed motivic spectra. In [20], we constructed the ∞-category SH fr (S) of framed motivic spectra over S, in which the functoriality with respect to framed correspondences is hard-coded. In particular, α induces a morphism Σ ∞ T,fr (α): Σ ∞ T,fr X → Σ ∞ T,fr Y in SH fr (S). The reconstruction theorem of [20] (generalized to arbitrary schemes in [31]) gives an equivalence of ∞-categories SH fr (S) SH(S), under which Σ ∞ T,fr (α) corresponds to a morphism Σ ∞ T X + → Σ ∞ T Y + as in (2). In § 3, we show that these three constructions agree.
We note that each construction has its own useful features. Construction (1) connects framed correspondences with the powerful formalism of six operations. As we explained in the introduction to [20], it was this hypothetical connection that led us to the correct formulation of the recognition principle for infinite P 1 -loop spaces. Construction (2) is helpful to perform explicit computations. For example, Bachmann and Yakerson employ the Voevodsky transfer to show that for a strictly homotopy invariant Nisnevich sheaf of abelian groups M on Sm k the double contraction M −2 has an infinite G m -delooping (at least when char k = 0) [7]. Construction (3) has the advantage that it is coherently compatible with the composition of framed correspondences, that is, it gives a functor Corr fr (Sm S ) → SH(S).
Our comparison theorems can therefore also be viewed as coherence theorems for the first two types of transfers.

Finite correspondences for motivic ring spectra
In § 4, we introduce categories of finite correspondences that encode the functoriality of R-cohomology for a given motivic ring spectrum R ∈ SH(S). We define for X, Y ∈ Sm S an ∞-groupoid Corr R S (X, Y ) of finite R-correspondences such that Corr R S (X, S) R(X). We expect that these are the mapping spaces of an ∞-category Corr R (Sm S ) with the following properties.
(1) There is a functor M R : Corr R (Sm S ) → Mod R (SH(S)) sending X to R ⊗ Σ ∞ T X + . (2) There is a functor Φ R : Corr fr (Sm S ) → Corr R (Sm S ) sending X to X. (3) The following square of ∞-categories commutes: In this paper we restrict ourselves to constructing the homotopy category hCorr R (Sm S ), and we establish the above properties at the level of homotopy categories (enriched in the homotopy category of spaces). The functor M R exists essentially by design, and the functor Φ R is defined using the formalism of fundamental classes. Property (3) is an application of the main result of § 3.
We then consider the cases R = HZ and R = HZ for S essentially smooth over a Dedekind domain and a field, respectively. In these two cases the mapping spaces Corr R S (X, Y ) are discrete, so the ∞-category Corr R (Sm S ) is defined and is a 1-category. Moreover, we prove the following comparison results.
(5) Corr HZ (Sm S ) is equivalent to the category of finite Milnor-Witt correspondences, constructed by Calmès and Fasel [11], and the functor Φ HZ refines the one defined by Déglise and Fasel in [16].
If k is a field and R ∈ SH(k) is an MSL-algebra, the category hCorr R (Sm k ) is equivalent to that constructed by Druzhinin and Kolderup in [19]. For R = KGL (respectively, R = KO if char k = 2), it receives a functor from Walker's category of finite K 0 -correspondences [47] (respectively, from Druzhinin's category of finite GW-correspondences [18]). However, a novel feature of our category is that it is enriched in the homotopy category of spaces, hinting that it is the homotopy category of a more fundamental ∞-category. Its mapping spaces are discrete if and only if R is 0-truncated in the effective homotopy t-structure, a condition which implies that R is an HZ-algebra. The Calmès-Fasel category Corr HZ (Sm k ) is thus in a precise sense the most general 1-category of finite correspondences.
Assuming that the ∞-category Corr R (Sm S ) has been constructed, one can consider the ∞category DM R (S) of T-spectra in A 1 -invariant Nisnevich-local presheaves on Corr R (Sm S ). When S is the spectrum of a field of characteristic zero and R = HZ or R = HZ, it is well known that DM R (S) Mod R (SH(S)) [21,43] and that the 'cancellation theorem' holds for Corr R (Sm S ) [23,51]. We will not attempt here to generalize these results. However, we note that the conjectural properties listed above imply that SH(S) is always a retract of DM 1 (S).

Conventions and notation
Our terminology and notation follows [20]. In particular: • Spc is the ∞-category of spaces/∞-groupoids, Spt that of spectra; • Maps(X, Y ) is the space of maps from X to Y in an ∞-category; • if C is an ∞-category, we denote by hC its homotopy category; • Perf (X) is the ∞-category of perfect complexes over X; • SH(S) is the stable motivic homotopy ∞-category over S; • DM(S) is Voevodsky's ∞-category of motives over S; • G, T and P denote the pointed presheaves (G m , 1), A 1 /G m and (P 1 , ∞); Σ G , Σ T and Σ P are the corresponding suspension functors and Ω G , Ω T and Ω P their right adjoints.
• If C is an ∞-category and M, N are two collections of morphisms in C that are stable under composition and pullback along one another, we write Corr(C, M, N ) for the ∞-category of spans with backward maps in M and forward maps in N ; see [9, § 5] for details on the construction of this ∞-category.

Preliminaries
In this section, we review some aspects of the formalism of six functors in stable motivic homotopy theory [3,13].
In § 2.1, we discuss various (co)homology theories associated with a motivic spectrum and their basic properties. In § 2.2, we review the formalism of fundamental classes for local complete intersection morphisms.
If f is smooth, there is a further left adjoint If f is locally of finite type † , we also have an adjunction The basic properties of these functors are summarized by the existence of a functor where 'lft' is the class of morphisms locally of finite type (see [30, § 6.2; 32, Chapter 2, § 5.2]). We will often use this implicitly when discussing the functoriality of certain constructions.

Thom transformations.
Let S be a scheme, E a locally free O S -module of finite rank, and V = Spec(Sym(E)) the associated vector bundle. If p : V → S is the structure morphism and s : S → V the zero section, then the adjoint functors are SH(S)-linear equivalences of ∞-categories called Thom transformations. In particular, Σ E Σ E 1 S ⊗ (−), and the object Σ E 1 S ∈ SH(S) is invertible with inverse Σ −E 1 S . The Thom transformations Σ E are defined more generally for E a perfect complex of O S -modules, and they assemble into a morphism of grouplike E ∞ -spaces natural in S, called the motivic J-homomorphism (see [6, § 16.2]). In particular, for every cofiber sequence Purity equivalences. For f : X → S a smooth morphism with sheaf of relative differentials Ω f , we have canonical equivalences A motivic spectrum E ∈ SH(S) gives rise to various (co)homology theories for S-schemes, which can be twisted by K-theory classes. Let p : X → S be a morphism † and let ξ ∈ K(X). We will consider the following mapping spaces.
(1) The ξ-twisted cohomology of X with coefficients in E is (2) The ξ-twisted Borel-Moore homology of X with coefficients in E is We omit the second parameter when ξ = 0. Moreover, it is understood that an element ξ ∈ K(X) is allowed to twist the cohomology of any X-scheme: if f : X → X is a morphism, we will often write E(X , ξ) instead of E(X , f * ξ), and similarly for Borel-Moore homology. There are also twisted versions of compactly supported cohomology and of homology (see [17,Definition 2.2.1]), but we shall not use these theories in this paper.
Remark 2.1. 9. In what follows, we often fix a motivic spectrum E ∈ SH(S) and talk about E-cohomology spaces in the interest of readability. However, E-cohomology spaces can generally be replaced by the corresponding endofunctors of SH(S). In particular, the naturality in E of all constructions and statements will be implicit.
2.1.5. Twisted motives. One can also define various twisted motives in SH(S): if p : X → S is a morphism and ξ ∈ K(X), we let S (X, ξ), E) by adjunction. The relationship between M S (X, ξ) and cohomology is more subtle. There is a canonical map p * E → Hom(p ! 1 S , p ! E) adjoint to the composite Applying Maps(Σ ξ 1 X , −), we obtain a canonical map it is an equivalence when X is smooth over S by purity (2.3), whence when X is cdh-locally smooth since motivic spectra satisfy cdh descent [12,Proposition 3.7]. However, it is not known to be an equivalence in general.
2.1.6. Functoriality. The cohomology space E(X, ξ) is contravariant in the pair (X, ξ). More precisely, if (Sch S ) /K → Sch S denotes the Cartesian fibration classified by K : Sch op S → Spc, then (X, ξ) → E(X, ξ) is a contravariant functor on (Sch S ) /K . In particular, for every S-morphism f : Y → X, there is a pullback map induced by the unit transformation id → f * f * . † Whenever the functors p ! or p ! are used, it is implicitly assumed that p is locally of finite type.
On the other hand, Borel-Moore homology E BM (X/S, ξ) is covariant in (X, ξ) for proper maps and contravariant in (X, ξ) forétale maps. This bivariance can be expressed coherently using the ∞-category of correspondences Corr((Sch S ) /K , prop,ét). In addition, Borel-Moore homology is contravariantly functorial in the base S. In particular, for a morphism f : S → S, there is a base change map f * : E BM (X/S, ξ) → E BM (X × S S /S , π * 1 ξ) induced by the exchange transformations Ex * * and Ex * ! .
2.1.7. Cohomology with support. Let X be an S-scheme, i : Y → X an immersion, and Given a Cartesian square the unit transformation id → g * g * and the exchange transformations Ex * ! : which induces a pullback in cohomology with support (2.5) If k : V → Y is another immersion, we also have a 'forgetful' map If Y is closed in X and both are smooth over a common base, we have a purity equivalence by (2.4).

Localization. Suppose that we have a diagram in Sch
where i is a closed immersion and j is the complementary open immersion, and let ξ ∈ K(X). Then the localization sequence gives the fiber sequence Dually, the localization sequence gives the fiber sequence In Borel-Moore homology, we similarly obtain the fiber sequence 2.1.9. Descent properties. Recall that the functor Sch op S → Fun(SH(S), SH(S)), (p : X → S) → p * p * , is an A 1 -invariant cdh-sheaf on Sch S [12,Proposition 3.7]. Consequently, cohomology is an A 1invariant cdh-sheaf and Borel-Moore homology is a Nisnevich sheaf. In particular, if f : X → Y is an A 1 -cdh-equivalence (that is, f induces an equivalence between the associated A 1 -invariant cdh sheaves) and ξ ∈ K(Y ), then the induced map is an equivalence.
In fact, we have the following more precise excision properties. Let Y ⊂ X be a subscheme and f : X → X a morphism such that f −1 (Y ) Y . For any ξ ∈ K(Y ), the pullback is an equivalence under either of the following conditions: (1) For any S-scheme X, subschemes Z, Z ⊂ X and K-theory classes ξ ∈ K(Z) and ξ ∈ K(Z ), we have the usual cup product (2) For any S-scheme X, subschemes T ⊂ Z ⊂ X and K-theory classes ξ ∈ K(Z) and ζ ∈ K(T ), we have the refined cup product We refer to [15, 1.2.8] for the definition. This refines the cup product from (1) as follows: there is a commutative square where i : Z → X and i : Z → X are the inclusions.
(3) Suppose Z → Y → X are S-morphisms locally of finite type and let ξ ∈ K(Z) and ζ ∈ K(Y ). Then we have the composition product We refer to [14, 1.2.8] for the definition.
Of course, the cup product and the composition product are associative or unital (up to homotopy) if the multiplication on E is. 2.1.11. Borel-Moore homology as cohomology with support. Let f : Z → S be a morphism locally of finite type. We say that f is smoothable if there exists a factorization where i is a closed immersion and p is smooth. For example, if S has the resolution property (that is, every finitely generated quasi-coherent sheaf is a quotient of a locally free sheaf of finite rank), then every quasi-projective morphism f : Z → S is smoothable.
In the above situation, if E ∈ SH(S) and ξ ∈ K(Z), the purity equivalence (2.3) induces a canonical equivalence (2.10) We record the following compatibility properties of the equivalence (2.10), which follow easily from the definitions. We state them without twists for simplicity.
(2) Pushforwards. Consider a commutative diagram where f and g are smooth, h is proper and i and k are closed immersions. Then the following diagram commutes: Here, h * is the proper pushforward and g ! is the Gysin map induced by the purity equivalence for g (see 2.2.3 for the definition of g ! in a more general context). As a special case, if t : W → Z is a closed immersion, then the proper pushforward t * : Suppose that E is equipped with a multiplication μ : E ⊗ E → E, and consider a commutative diagram where the vertical maps are smooth, the horizontal maps are closed immersions, and the square is Cartesian. Then the following diagram commutes:

Fundamental classes
2.2.1. We briefly recall the formalism of fundamental classes from [17]. Let f : X → Y be a smoothable lci morphism. The fundamental class of f is a canonical element The associated purity transformation where the last morphism is the canonical one (see, for example, [17, 2.1.10]). The following proposition summarizes the key properties of fundamental classes: and hence f are smoothable. Then the following diagram commutes: Here, the left vertical arrow uses the equivalence (ii) Given a tor-independent Cartesian square where f is lci and smoothable, the following diagrams commute: Here, the left vertical arrows use the coincides with the purity equivalence (2.4).

Gysin maps in cohomology. Consider a commutative square of S-schemes
where f is smoothable and lci, i and k are closed immersions and g is proper † . For every ξ ∈ K(T ), we have a pushforward morphism or Gysin map defined by the composition Let us emphasize two special cases.
(1) If i : Z → X is a regular closed immersion and ξ ∈ K(Z), we have the Gysin map (2) If f : X → Y is smoothable, lci and proper, and if ξ ∈ K(Y ), we have the Gysin map Properties (i) and (ii) of Proposition 2.2.2 imply obvious compatibilities of these Gysin maps with composition and pullback.

Gysin maps in Borel
Properties (i) and (ii) of Proposition 2.2.2 imply obvious compatibilities of these Gysin maps with composition, proper pushforward and tor-independent base change.
Remark 2.2.6. We use the notation f ! and f ! for Gysin maps, rather than f * and f * , as a visual reminder that these maps use the purity transformation p f . It does not indicate a particular relation to the functors f ! and f ! .

Functoriality.
If M is a collection of morphisms of schemes that is closed under tor-independent base change, we let Fun cart, M (Δ 1 , Sch) ⊂ Fun(Δ 1 , Sch) † More generally, it suffices to assume that i and k are immersions and that the scheme-theoretic image of Z in X is proper over Y , so that f * i ! f ! i ! . This makes (2.6) a special case of (2.11).
be the subcategory whose objects are the morphisms in M and whose morphisms are the torindependent Cartesian squares. By Proposition 2.2.2(ii), the assignment f → η f is a section of the Cartesian fibration classified by the functor where 'slci' is the collection of smoothable lci morphisms.
We expect that the construction f → η f can be refined to a section of but this is a nontrivial task because the construction of η f depends on a choice of factorization of f . For our purposes, it will suffice to know that we do have such refinements on the subcategory of regular immersions or that of smooth morphisms. In the case of regular closed immersions, the construction of the fundamental class in [17, 2.2.5. We now discuss the functoriality of the commutative square of Proposition 2.2.2(i). Let Fun cart, M0,M1,M2 (Δ 2 , Sch) ⊂ Fun(Δ 2 , Sch) be the subcategory whose objects are triangles with f i ∈ M i and whose morphisms are natural transformations composed of tor-independent Cartesian squares. By Proposition 2.2.2(i), if f 0 and f 1 are lci and smoothable, the classes η f1 and η f2 · η f0 in π 0 1 BM (Z/X, L f1 ) are equal, where η f2 · η f0 is the composite ), This can be refined to a section of the functor , at least if each M i is either the class of regular immersions or that of smooth morphisms. One can reduce as in 2.2.7 to the case of regular closed immersions, where an explicit functorial homotopy η f1 η f2 · η f0 is given by a double deformation to the normal cone [17, 3.2.19].

Comparison of transfers
In this section we show that the framed transfers in cohomology provided by the motivic recognition principle are given by Gysin maps. In § 3.1, we define the fundamental transfer associated with a tangentially framed correspondence using Gysin maps. We then introduce in § 3.2 the Voevodsky transfer associated with an equationally framed correspondence, and we show that the Voevodsky transfer computes the fundamental transfer. Finally, in § 3.3, we show that the transfers obtained from the recognition principle agree with the Voevodsky transfer. Throughout this section, we fix a base scheme S and a motivic spectrum E ∈ SH(S). As explained in Remark 2.1.9, the spectrum E is only used for readability purposes.

The fundamental transfer
3.1.1. Recall that a tangentially framed correspondence between S-schemes X and Y is the data of a span over S, where f is finite syntomic, together with an equivalence τ : 0 L f in the ∞-groupoid K(Z). We denote by Corr fr S (X, Y ) the ∞-groupoid of tangentially framed correspondences from X to Y , defined as where the colimit is taken over the groupoid of spans with f finite syntomic.
3.1.2. Note that a finite syntomic morphism f : Z → X admits a canonical factorization which we use to define the fundamental class η f ∈ 1 BM (Z/X, L f ).
Using the functoriality of f → η f described in 2.2.7, we obtain a map . If X and Y are smooth over S, then by the Yoneda lemma we obtain a map , which we sometimes also denote by tr η .
Remark 3.1.4. Dually, a framed correspondence α as above also induces a map in SH(S) (see 2.1.10 for the notation M S (X, ξ)). By unpacking the definitions, it is easy to show that the natural transformation E(−) → Maps(M S (−), E) on S-schemes is also natural with respect to Gysin maps. In particular, if X and Y (but not necessarily Z) are cdh-locally smooth over S, applying Maps(−, E) to (3.1) yields the fundamental transfer tr η (α): 3.1.3. Example: the action of K-theory. The ∞-groupoid Corr fr S (S, S) contains ΩK(S) as a full subgroupoid. By construction, the composite is the action of ΩK(S) on 1 S induced by the motivic J-homomorphism K(S) → SH(S).
If S is a regular semilocal scheme over a field of characteristic not 2, π 0 1 S (S) is isomorphic to the Grothendieck-Witt group GW(S) of nondegenerate symmetric bilinear forms over S [6,Lemma 10.12]. This isomorphism is such that the J-homomorphism sends a unit a to the class a of the bilinear form (x, y) → axy.

Example: finiteétale transfers. There is a canonical map Corr
in SH(S); such a diagram exists if E is an MGL-module in the homotopy category hSH(S), and it is given if E is an MGL-module in SH(S). Then the fundamental transfers in E-cohomology are independent of the tangential framings. More precisely, given X, Y ∈ Sch S , there is a canonical factorization This follows at once from the fact that the MGL-linearized J-homomorphism

The Voevodsky transfer
3.2.1. Let X and Y be S-schemes and let α ∈ Corr efr,n S (X, Y ) be an equationally framed correspondence of level n from X to Y [20, Definition 2.1.2]. We display α as the diagram (3.2) where f is finite, u is anétale neighborhood of Z in A n X , 0 is the zero section and the right-hand square is Cartesian.
We will denote by P ×n the n-fold product (P 1 ) ×n , regarded as a compactification of A n , and by ∂P ×n ⊂ P ×n the complementary reduced closed subscheme that is the union of the 'faces' P ×i−1 × {∞} × P ×n−i : Here, the maps E Z (P ×n (2.6). To see that the latter is an equivalence, first note that it fits in the diagram where the rows are the fiber sequences (2.8) and (2.9). By 2.1.19, the claim then follows from the following lemma: Proof. We consider the commutative square of inclusions in PSh(Sch): The upper horizontal map is a covering sieve in the closed topology, the lower horizontal map is a covering sieve in the open topology, and the left vertical map is the colimit of an n-dimensional cube of A 1 -homotopy equivalences. In particular, these three maps are A 1 -cdh-equivalences, hence so is the right vertical map.
Remark 3.2.5. In general, U is an algebraic space and not a scheme, but this does not matter. Indeed, the inclusion of schemes into (Zariski-locally quasi-separated) algebraic spaces induces an equivalence between the ∞-categories of Nisnevich sheaves, by [28, Proposition 5.7.6]. As a result, we may tacitly extend any Nisnevich sheaf, such as E(−) or SH(−), to algebraic spaces. However, we can assume that U is a scheme in many cases [20, Lemma A.1.2(iv)]. [20,Corollary A.1.7], the equationally framed correspondence α is equivalently a morphism of pointed presheaves Σ n P X + → L nis Σ n T Y + . Explicitly, it is given by the following zig-zag in PSh(Sch S ) * :

By Voevodsky's lemma
Here,∂P ×n ⊂ P ×n is the subpresheaf defined as the unioñ  Proof. Let α be an equationally framed correspondence as in (3.2), and let τ : 0 L f be the induced trivialization in K(Z). We must show that the following diagram commutes: To do so, we subdivide this diagram as follows: The rectangle (1) commutes by the base change property of Gysin maps (Proposition 2.2.2(ii)) applied to the Cartesian square which is tor-independent since i is a regular immersion of codimension n. Thus, the unnamed equivalence in (1) is induced by the isomorphism N i h * (N 0 ) O n . This isomorphism also induces the trivialization τ , whence the commutativity of the square (2). The triangles (3), (4) and (5) all commute by the compatibility of Gysin maps with composition (Proposition 2.2.2(i)), where the commutativity of (5) means that going around starting from the lower left corner gives the identity.
To conclude the proof, we must show that the diagram (3.5) can be promoted to a functor of the triple (X, Y, α). This follows from the functoriality properties of Gysin maps discussed in 2.2.7 and 2.2.8. For the triangle (4), we must recall that the fundamental class η f was defined using the canonical factorization Z → V(f * O Z ) → X. The commutativity of (4) can be made functorial using the triangles in which the upper three maps are regular closed immersions and the other five are smooth. This concludes the proof of the theorem.

The transfer from the recognition principle
3.3.1. Recall that there is an ∞-category Corr fr (Sm S ) whose objects are smooth S-schemes and whose mapping spaces are the ∞-groupoids Corr fr S (X, Y ), which gives rise to the ∞-category SH fr (S) of framed motivic spectra [20, § 3]. The 'graph' functor γ : Sm S+ → Corr fr (Sm S ), (f : induces an adjunction γ * : SH(S) SH fr (S) : γ * such that the following square commutes: By the reconstruction theorem [31,Theorem 16], the functor γ * : SH(S) → SH fr (S) is an equivalence of ∞-categories. It follows that E-cohomology of smooth S-schemes acquires canonical framed transfers: The goal of this section is to show that these transfers coincide with the Voevodsky transfers, hence with the fundamental transfers.
3.3.2. We need a technical preliminary result, which we formulate in a more general context. Let C be a presentably symmetric monoidal ∞-category and let T ∈ C be an object. For any presentable C-module M , we have an adjunction where Σ T = T ⊗ (−). We can then form the diagram where the transition maps use the unit transformation id → Ω T Σ T and (in one direction) the cyclic permutations Σ T Σ m T Σ m T Σ T and Ω m T Ω T Ω T Ω m T . We denote by Spt T (M ) the ∞-category of T-spectra in M , defined as the limit We have an adjunction where Ω ∞ T is the projection to the last copy of M . If Ω T preserves sequential colimits, then Lemma 3.3.3. With the above notation, suppose that Ω T : M → M preserves sequential colimits and that the cyclic permutation of T ⊗n is homotopic to the identity for some n 2.

Then the natural transformations
between endofunctors of M are homotopic equivalences.
Proof. We have a commutative diagram where F 0 = Σ ∞ T and F 1 = Spt T (Σ ∞ T ). Let G i be the right adjoint to F i and u i : id → G i F i the unit transformation. Then the given natural transformations are Ω ∞ T u i Σ ∞ T for i = 0, 1. By the assumption on T, the functor Spt T (−) is a left localization of the ∞-category of presentable C-modules [42,Corollary 2.22]. This implies that F 0 and F 1 are equivalences and moreover that there is a natural equivalence α : F 0 F 1 such that αΣ ∞ T is the identity. In particular, the unit transformations u i are equivalences, and α and its mate G 0 G 1 give the desired homotopy.
3.3.3. We now prove that the Voevodsky transfer coincides with the transfer coming from the reconstruction theorem. For X, Y ∈ Sm S , we can regard the Voevodsky transfer as a map → Ω n P L nis Σ n T : PSh Σ (Sm S ) * → PSh Σ (Sm S ) * , extending an equivalence on representables. Composing with L nis → L mot and taking the colimit over n, we obtain a transformation h efr S → Ω ∞ T Σ ∞ T L mot , which extends (3.6) to pointed presheaves. We will prove more generally that the following diagram of endofunctors of PSh Σ (Sm S ) * commutes: which is an equivalence since γ * : SH(S) → SH fr (S) is fully faithful. Hence, the bottom horizontal map is an equivalence, as desired.
Note that we already have such an equivalence by [20,Corollary 3.5.16]. The point of this corollary is that this equivalence is induced by Voevodsky's lemma, as one would expect.
Proof. If we plug F in the diagram (3.7), we obtain a commutative square of Proof. This follows immediately from Theorem 3.3.10.

Finite correspondences for motivic ring spectra
In this section we introduce finite R-correspondences for a motivic ring spectrum R, generalizing the finite correspondences of Voevodsky and the finite Milnor-Witt correspondences of Calmès and Fasel. In § 4.1, we construct the homotopy category hCorr R (Sm S ) of finite R-correspondences between smooth S-schemes, together with a functor to the homotopy category of R-modules. In § 4.2, we construct a functor from the category of (tangentially) framed correspondences to that of finite R-correspondences and compare it with the free Rmodule functor. Finally, in § 4.3, we compare our constructions with those of Voevodsky and of Calmès-Fasel. Throughout this section, S is a fixed base scheme. All S-schemes are assumed to be separated.

The category of finite R-correspondences
Given an associative ring spectrum R ∈ SH(S), we will construct an hSpc-enriched category hCorr R (Sm S ) of finite R-correspondences between smooth S-schemes.
To motivate our construction, recall that a morphism from X to Y in Voevodsky's category of finite correspondences over a regular scheme S is an element of the free abelian group generated by integral closed subschemes Z ⊂ X × S Y that are finite and surjective over a component of X. Alternatively, we can think of a morphism in this category as a reduced closed subscheme Z ⊂ X × S Y , each of whose irreducible components is finite and surjective over a component of X and labeled by an integer. The category hCorr R (Sm S ) will admit a similar description, but with integers replaced by Borel-Moore R-homology classes of Z over X.

Let S be a scheme and R ∈ SH(S).
For separated S-schemes X, Y ∈ Sch S , define where the colimit is taken over the filtered poset of reduced subschemes Z ⊂ X × S Y that are finite and universally open † over X. To form this colimit, we use the covariant functoriality of Borel-Moore homology with respect to proper morphisms (see 2.1.11).

Suppose now that R ∈ SH(S)
is a homotopy associative ring spectrum, that is, R is an associative algebra in the homotopy category hSH(S). There is a map as the composition Here, μ BM is the composition product (which uses the ring structure on R, see 2.1.20(3)), and p * is the proper pushforward in Borel-Moore homology. More succinctly, The map (4.1) is then the filtered colimit over Z and Z of the maps θ BM .
Lemma 4.1.5. The composition law (4.1) is unital and associative up to homotopy, with identity Γ R (id S ) ∈ Corr R S (X, X).
Proof. Let X 1 , X 2 , X 3 and X 4 be smooth S-schemes, and let Z i,i+1 ⊂ X i × S X i+1 be reduced subschemes, finite and universally open over where the squares are Cartesian. It suffices to note that the two possible ways of composing the elements x i are both equal to where g : Z 12 → X 2 , h : Z 123 → X 3 and p : Z 1234 → Z 34 • Z 23 • Z 12 . This follows directly from the properties of the composition product listed in [14, 1.2.8]. The fact that Γ R (id X ) is the identity is trivial.

The category of finite R-correspondences.
In view of Lemma 4.1.5, we can define a category hCorr R (Sch S ) as follows.
• The objects of hCorr R (Sch S ) are separated S-schemes.
• The set of morphisms from X to Y is π 0 Corr R S (X, Y ). • The identity morphism at X is [Γ R (id X )] ∈ π 0 Corr R S (X, X). • The composition law is given by π 0 of the composition law (4.1).
It is moreover easy to show that the morphisms Γ R defined in 4.1.2 assemble into a functor For any full subcategory C ⊂ Sch S , we denote by hCorr R (C) the corresponding full subcategory of hCorr R (Sch S ).
Remark 4.1.7. By construction, hCorr R (C) is enriched in the homotopy category hSpc. If R is an A ∞ -ring spectrum, we expect that with more effort one can construct an ∞-category Corr R (C) with mapping spaces Corr R S (X, Y ), whose homotopy category is hCorr R (C); this explains our notation for the latter category. In our two main examples, when C is the category of smooth schemes over a field and R = HZ or R = HZ, we will see that the spaces Corr R S (X, Y ) are always discrete, so that Corr R (C) = hCorr R (C). 4.1.5. We note that the category hCorr R (Sch S ) is semiadditive, with the sum given by the disjoint union of schemes. In fact, we have canonical equivalences of spaces We shall use the fact that the four functors f * , f * , f ! , and f ! preserve R-modules, in the sense that they lift canonically from hSH(−) to Mod R (hSH(−)).
Let Z ⊂ X × S Y and α ∈ R BM (Z/X) define a finite R-correspondence from X to Y . Consider the diagram As in 2.2.1, α gives rise to a natural transformation The functor M R then sends (Z, α) to the composition

Compatibility with composition is a straightforward verification.
Remark 4.1.11. If R is an A ∞ -ring spectrum, then the four functors f * , f * , f ! and f ! lift canonically from SH(−) to Mod R (SH(−)), and the above construction actually defines an hSpc-enriched functor Following Remark 4.1.7, we expect that it can be refined to a functor of ∞-categories 4.1.7. The symmetric monoidal structure. Let R ∈ SH(S) be a homotopy commutative ring spectrum. Then the category hCorr R (Sch S ) acquires a symmetric monoidal structure given on objects by X ⊗ Y = X × S Y . On morphisms, one uses the external pairing The compatibility between this pairing and composition of finite R-correspondences uses the commutativity of R. Furthermore, the functor M R : hCorr R (Sm S ) → Mod R (hSH(S)) admits a canonical symmetric monoidal structure. We omit the somewhat tedious details.
Remark 4.1.13. If R is an E n+1 -ring spectrum (1 n ∞), we expect an E nmonoidal structure on the ∞-category Corr R (Sm S ) and on the functor M R : Corr R (Sm S ) → Mod R (SH(S)) (see Remark 4.1.11).
4.1.8. We can express finite R-correspondences in terms of twisted cohomology with support. Let Z ⊂ X × S Y be a reduced subscheme that is finite and universally open over X, and assume that Y is smooth over S. As explained in 2.1.21, there is a canonical equivalence where π Y : X × S Y → Y is the projection. This equivalence is moreover natural in Z by 2.1.21 (2), so that Alternatively, by (4.2) and the continuity of K-theory, Corr KGL S (X, Y ) is the K-theory space of the stable ∞-category of perfect complexes on X × S Y supported on a subscheme finite and equidimensional over X.
4.1.9. We observe that the notion of finite R-correspondence between smooth S-schemes depends only on the very effective cover of R (in the sense of Spitzweck-Østvaer [45]). Proof. It will suffice to show that the map is an equivalence for every X ∈ Sm S and every finite smoothable morphism Z → X. By standard limit arguments, we can reduce to the case S = S 0 . By (2.10), it is enough to show that the map (f 0 R) Z (V, ξ) → R Z (V, ξ) is an equivalence for every V ∈ Sm S , ξ ∈ K(V ) of rank r, and Z ⊂ V fiberwise of codimension r. Since the question is local on V , we can assume that ξ is pulled back from S, so that and similarly forf 0 R. Sincef r Σ ξ R Σ ξf 0 R, it remains to show that Σ ∞ T (V /V − Z) is very r-effective. By [6, Proposition B.3] and the assumptions on S, we may assume that S is the spectrum of a perfect field. In this case, Z admits a finite stratification by smooth schemes and the result is easily proved by induction using the purity isomorphism.
4.1.10. In case k is a field and R = HZ or R = HZ, the hypothetical ∞-category Corr R (Sm k ) happens to be a 1-category, that is, it is equivalent to its homotopy category hCorr R (Sm k ). This is a special case of the following proposition. We refer to [4, § 3] for the definition of the effective homotopy t-structure.
Proposition 4.1.20. Let k be a perfect field and let R ∈ SH(k) be a motivic spectrum in the heart of the effective homotopy t-structure. For any essentially smooth k-scheme S and X, Y ∈ Sm S , the ∞-groupoid Corr R S (X, Y ) is discrete.
Proof. Using the description of Corr R S (X, Y ) given in 4.1.14, it suffices to show that the ∞-groupoid is discrete for any Z ⊂ X × S Y finite over X. By a standard limit argument, we can assume that S is smooth over k. The result then follows from Lemma 4.1.21.
Lemma 4.1.21. Let k be a perfect field, V a smooth k-scheme, ξ ∈ K(V ), and Z ⊂ V a closed subscheme of codimension rk ξ. Let R ∈ SH(k) be a motivic spectrum in the heart of the effective homotopy t-structure. Then the ∞-groupoid R Z (V, ξ) is discrete.
Proof. Suppose first that Z is smooth. We then have the purity equivalence (2.7) Let ζ = ξ − N Z/X . By the assumption on ξ, we have rk ζ 0. The assumption on R means that R is right orthogonal to SH eff 1 (k). Since Z is smooth, R Z ∈ SH(Z) is right orthogonal to SH eff 1 (Z), hence so is Σ ζ R Z , because Σ −ζ is a right t-exact endomorphism of SH eff (Z). It follows at once that R(Z, ζ) = Maps SH(Z) ( If Z is an arbitrary closed subscheme, we can assume that it is reduced since cohomology with support only depends on Z red . We will prove the claim by induction on the dimension of Z. If Z is empty, then R Z (V, ξ) is contractible. Otherwise, since k is perfect, Z is generically smooth, so there is a reduced closed subscheme Z 1 ⊂ Z of strictly smaller dimension such that Z − Z 1 is smooth. By (2.8), we have a fiber sequence (of grouplike E ∞ -spaces) By the induction hypothesis, ξ) is also discrete.  Spec k) is discrete for all X ∈ Sm k , that is,f 0 R belongs to the heart of the effective homotopy t-structure. Thus, the hypothetical ∞-category Corr R (Sm k ) is a 1-category if and only iff 0 R ∈ SH eff (k) ♥ , in which case R is necessarily an algebra over π eff 0 (1) HZ. In particular, for more general R, there is no hope to recover R-modules from the 1-category hCorr R (Sm k ). 4.1.11. It will be useful to have a description of the composition in hCorr R (Sm S ) in terms of cohomology with support. Suppose that X, Y, T are smooth S-schemes and Z ⊂ X × S Y and Z ⊂ Y × S T are reduced subschemes that are finite and universally open over X and Y , respectively. We will refer to the diagram where μ is the cup product and p XT ! is the Gysin map (2.11). More succinctly, We have the following comparison with the pairing θ BM defined in 4.1.3. Proof. Recall from 2.1.21 that we can factor the cup product as follows: where i : Z × T → X × Y × T . In the following diagram, the left column is θ and the right column is θ BM : The three rectangles commute by 2.1.21 (1), (3) and (2), respectively.

From framed correspondences to finite R-correspondences
Let R be a homotopy associative ring spectrum. We will construct a canonical functor where Corr fr (Sch S ) is the ∞-category of framed correspondences constructed in [20].

For S-schemes X and Y , we define a map
as follows. A framed correspondence from X to Y is given by a span where f is finite syntomic, together with a trivialization τ ∈ Maps K(Z) (0, L f ).
Since the morphism f is finite syntomic, it has a fundamental class . We will also denote by Applying the trivialization τ , we get an element τ * (η f ) ∈ R BM (Z/X).
The map (f, g): Z → X × S Y is finite; we denote by V ⊂ X × S Y its reduced image. Note that V is finite and universally open over X. Using the proper pushforward in Borel-Moore homology, we obtain (f, g) * (τ * (η f )) ∈ R BM (V /X). This construction defines a map Taking the colimit over the groupoid of finite syntomic spans from X to Y , we obtain the desired map (4.3). It is clear that Φ R (γ(f )) = Γ R (f ) for any S-morphism f , and in particular Φ R preserves identity morphisms. Let α = (Z, f, g, τ ) ∈ Corr fr S (X, Y ) and β = (Z , h, s, τ ) ∈ Corr fr S (Y, T ) be framed correspondences, where τ ∈ Maps K(Z) (0, L f ) and τ ∈ Maps K(Z ) (0, L h ), and form the composite 2-span The composition β • α is then given by We want to compare Φ R (β) • Φ R (α) and Φ R (β • α). We first note the following equations between fundamental classes: Here the first equality is the stability of fundamental classes under tor-independent base change, the second holds by definition of σ, and the last is the associativity of fundamental classes [17,Definition 2.3.6]. Let V ⊂ X × S Y be the image of Z and V ⊂ Y × S T the image of Z . We now consider the following diagram in which all parallelograms are Cartesian: For any z ∈ R BM (Z/X) and z ∈ R BM (Z /Y ), we have the following equivalences in R BM (V /X), where the parenthetical justifications refer to [14, 1.2.8]: Plugging in z = τ * η f and z = τ * η h and pushing forward the result to R BM (V • V /X) gives the desired equivalence To see that Φ R is indeed an hSpc-enriched functor, we must show that this equivalence is natural in the pair (α, β) ∈ Corr fr S (X, Y ) × Corr fr S (Y, Z). This is essentially obvious from the construction, using the functoriality of fundamental classes discussed in 2.2.7 and 2.2.8.
The following corollary is a variant of [19,  Proof. By definition of these functors, we have given isomorphisms M R Φ R (X) R ⊗ Σ ∞ T X + R ⊗ γ * Σ ∞ T,fr X for every X ∈ Sm S . Moreover, when R is homotopy commutative, these isomorphisms trivially intertwine the monoidal structures of these functors. It thus remains to show that the following square commutes for every X, Y ∈ Sm S : By Theorem 3.3.10, the top horizontal map in (4.4) is the fundamental transfer tr η . Let ϕ = (Z, f, g, τ ) be a framed correspondence from X to Y , and let V ⊂ X × S Y be the reduced image of Z: In the following diagram, the top row is tr η (ϕ) (in the form described in Remark 3.1.4), while the bottom row is M R Φ R (ϕ): The square involving α commutes by definition of α. The commutativity of the boundary of this diagram witnesses the commutativity of the square (4.4).
Remark 4.2.9. If R is A ∞ , we can replace the lower right corner in Proposition 4.2.7 by hMod R (SH(S)). Continuing Remarks 4.1.7, 4.1.11 and 4.1.13, we moreover expect that this square can be promoted to a commuting square of ∞-categories, and of E n -monoidal ∞-categories if R is E n+1 .

4.3.1.
Reminders on motivic cohomology. Over a Dedekind domain D, we will consider the motivic cohomology spectrum HZ ∈ SH(D) constructed by Spitzweck [44,Definition 4.27]. It is an oriented E ∞ -ring spectrum that represents Bloch-Levine motivic cohomology. In particular, for an essentially smooth D-scheme X and ξ ∈ K(X) of rank r, we have HZ(X, ξ) z r zar (X, * ), (4.5) where z r zar (X, * ) denotes the sheafification of Bloch's cycle complex z r (X, * ) with respect to the Zariski topology on Spec D. This identification is natural in X, where the functoriality of Bloch's cycle complex comes from Levine's moving lemma [34]. If Z ⊂ X is a closed subscheme of codimension c, the localization theorem [33,Theorem 1.7] implies that Recall that HZ belongs to the heart of the effective homotopy t-structure on SH(D) [6,Lemma 13.6]. Being the zeroth slice of the sphere spectrum [6,Theorem B.4], the spectrum HZ admits in fact a unique E ∞ -ring structure with given unit.
When D is a field, HZ coincides with Voevodsky's motivic cohomology spectrum, but this is not known in general. In this case, Bloch's cycle complex admits an E ∞ -ring structure compatible with the intersection of cycles [10, § 5], and the equivalence (4.5) is multiplicative.
Proof. Let η ∈ S be the generic point. Since f (Z) is flat over S, the pullback z 0 (f (Z)) → z 0 (f (Z) × S η) is an isomorphism, so we may assume that S is the spectrum of a field k. By limit arguments, we can assume k perfect and Y smooth over k. Replacing Y by an open subscheme, we can further assume that Z and f (Z) are smooth over k. Since the Gysin map is compatible with purity isomorphisms, we are reduced to the following claim: if L/K is a finite extension of finitely generated fields over k, the Gysin map Z HZ(Spec L, L L/K ) → HZ(Spec K) Z is multiplication by [L : K]. This is a special case of [ by 4.1.14. It follows from (4.6) that Hence, where the sum is taken over all integral closed subschemes of X × S Y that are finite and surjective over a component of X. In particular, Corr HZ (Sm S ) is a 1-category, and its mapping spaces are the same as in Voevodsky's category.
To compare the composition laws, we use the description of the composition in Corr HZ (Sm S ) via the pairing θ (Lemma 4.1.24). The composition in Cor S is defined in exactly the same way, except that it uses the intersection product and the pushforward of cycles instead of the cup product and the Gysin map in HZ-cohomology. We must therefore show that these constructions yield cycles with the same multiplicities. Since the generic points of the cycles involved lie over generic points of S, we can replace S by its generic points and hence assume that S is a field. In this case, the intersection product and the cup product agree because the isomorphism (4.5) is compatible with the multiplicative structures. The fact that the pushforwards agree is a special case of Lemma 4.3.5. Finally, the fact that the symmetric monoidal structures agree also follows from the multiplicativity of the isomorphism (4.5).  Proof. Note that Φ HZ and cyc send a framed correspondence to finite correspondences with the same support, so it suffices to compare their multiplicities. Since the generic points of their support lie over generic points of S and both functors are natural in S, this can be done assuming that S = Spec k for some field k, which can moreover be assumed perfect by passing to its perfection. In this situation, we prove the following more general uniqueness statement: if are symmetric monoidal functors that satisfy ϕ 1 | Sm k Γ HZ ϕ 2 | Sm k and send every framed correspondence (Z, f, g, τ ) to a finite correspondence with support (f, g)(Z), then ϕ 1 ϕ 2 . We have induced symmetric monoidal functors such that ϕ * 1 |SH(k) ϕ * 2 |SH(k). By the reconstruction theorem [20, Theorem 3.5.12] it follows that ϕ * 1 ϕ * 2 . To check that ϕ 1 ϕ 2 , it suffices to compare their effect on a framed correspondence α ∈ Corr fr k (η, Y ) with connected support, where η is the generic point of a smooth k-scheme. Since ϕ 1 (α) and ϕ 2 (α) are supported on a single point, their equality can be checked modulo rational equivalence, that is, in hDM(k), so we are done. defined in [22,Corollaire 10.4.3]. On the other hand, for any f , there is a sheaf-theoretic pullback which agrees with the pullback in HZ-cohomology (by the naturality of (4.8)).
Lemma 4.3.14. The isomorphism (4.9) is natural with respect to flat morphisms in Sm k .
Proof. It suffices to show that the canonical inclusion is natural with respect to flat morphisms. This is obvious because the flat pullback on C 0 is by definition the sum of the pullbacks in Milnor-Witt K-theory.
If Z ⊂ X is smooth of codimension c, comparing Rost-Schmid complexes yields an isomorphism Π: CH n Z (X, L) CH n−c (Z, L ⊗ det(N Z/X ) −1 ), called the purity isomorphism. Proof. We can reduce to the case of the zero section of a vector bundle using the functoriality of the Rost-Schmid complex for smooth morphisms (Lemma 4.3.14), Jouanolou devices, and etale neighborhoods (cf. [30,Lemma 3.22]). Thus let V = V(E) be a vector bundle over X ∈ Sm k . We must show that the following square commutes: The top horizontal map is now the identity map by [50,Lemma 2.2]. Each vertical map is the composition of the Thom isomorphism for HZ and the canonical map HZ → K MW * . Levine shows in [35,Proposition 3.7] that the purity isomorphism Π above is the Thom isomorphism of an SL-orientation on the cohomology theory represented by the motivic spectrum K MW * . By [41,Theorem 5.9], such an orientation is classified by a unital morphism of spectra MSL → K MW * . But since the unit map 1 → MSL is a π 0 -isomorphism, there is a unique such morphism. Therefore the map HZ → K MW * intertwines the respective Thom isomorphisms, which implies the commutativity of the above square. 4.3.6. Comparison of pushforwards in Chow-Witt theory. Let k be a perfect field, f : X → Y a morphism between smooth k-schemes, Z ⊂ X a closed subscheme such that the restriction of f to Z is finite, and L an invertible sheaf on Y . We recall the definition of the Proof. Since HZ is in the heart of the effective homotopy t-structure, Corr HZ (Sm S ) is a 1-category by Proposition 4.1.20. For smooth S-schemes X and Y , we have Corr HZ S (X, Y ) colim Z⊂X×S Y HZ Z (X × S Y, π * Y Ω Y /S ) by 4.1.14. It follows from (4.8) that In particular, the mapping spaces in Corr HZ (Sm S ) are the same as in Cor S . To compare the composition laws, we use the description of the composition in Corr HZ (Sm S ) via the pairing θ (Lemma 4.1.24). The composition in Cor S is defined in the same way, except that it uses the Calmès-Fasel pushforward instead of the Gysin map, but these are the same by Proposition 4.3.17. Finally, the symmetric monoidal structures agree by the multiplicativity of the isomorphisms (4.8). (see [20, 3.4.7]). The rest of this section will be devoted to the proof of the following comparison theorem.  [20, 3.4.7]. Then the following diagram commutes: 4.3.9. We briefly recall the construction of the functor α. On objects one has α(X) = X. Given an equationally framed correspondence c = (Z, U, ϕ, g) ∈ Corr efr,n k (X, Y ), we construct a finite MW-correspondence α(c) ∈ Cor k (X, Y ) as follows.
Theétale morphism u : U → A n X induces a trivialization ω U/X u * ω A n X /X O U . Denote by π : A n X → X the projection. Since Z is finite and equidimensional over X, the morphism (πu, g): U → X × Y sends Z to a closed subscheme T , which is finite and equidimensional over X. The finite MW-correspondence α(c) ∈ Cor k (X, Y ) is then the image of Z(ϕ) by the Calmès-Fasel pushforward (πu, g) * : CH n Z (U ) CH n Z (U, ω U/X ) −→ CH d T (X × Y, ω X×Y /X ), d = dim(Y ).
4.3.10. The first step in the proof of Theorem 4.3.21 is to recast the construction of Z(ϕ) as a Thom class.
We recall that for E a motivic ring spectrum, the Thom class of a locally free sheaf E on X is the image of 1 by the purity equivalence E(X) E X (V(E), E). By  Proof. It is enough to show that Z(ϕ i ) = ϕ * i (t 1 ), because Z(ϕ) = Z(ϕ 1 ) · . . . · Z(ϕ n ) and the Thom class is multiplicative with respect to direct sum of vector bundles. By Lemma 4.3.14, since ϕ i : U → A 1 k is flat, the pullback ϕ * i on Chow-Witt groups can be computed using Rost-Schmid complexes. The commutative square shows that ∂[ϕ i ] = ϕ * i (∂[id A 1 ]) in C 1 |ϕi| (U, K MW 1 ), hence that Z(ϕ i ) = ϕ * i (Z(id A 1 )) in CH 1 |ϕi| (U ). It remains to observe that Z(id A 1 ) = t 1 , since the residue map ∂ t : K MW 1 (k(t)) → GW (k) takes [t] to 1. 4.3.11. The following lemma shows that the flatness assumption in Lemma 4.3.24 is essentially vacuous.