Fundamental classes in motivic homotopy theory

We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fulton's intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by D\'eglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.

In the coherent setting, abstract duality was realized through the adjunction of the exceptional functors (f ! , f ! ) (see [Har66,Chap. 5]). The concept of dualizing complex was pivotal: it was discovered soon after that Borel-Moore homology [BM60] can be described as homology with coefficients in the (topological) dualizing complex. The theory of ℓ-adic sheaves developed in SGA4 [SGA4] was the first complete incarnation of the six functors formalism, and for a long time the only one available in algebraic geometry. A key aspect of the six functor formalism that was highlighted in the seminar SGA5 [SGA5] is the absolute purity property. Stated in op. cit. as a conjecture, it was partially solved by Thomason [Tho84], and completely settled later by Gabber [Fuj02,ILO14].
More recently, Morel and Voevodsky introduced motivic homotopy theory [MV99,Voe98]. As in algebraic topology, the stable motivic homotopy category classifies cohomology theories which satisfy homotopy invariance with respect to the algebraic affine line A 1 . The stable motivic homotopy category also satisfies the six functors formalism (see [Ayo07]). Moreover, it satisfies a suitable universal property [Rob15] and contains the classical theories of algebraic geometry, such as Betti cohomology, etale ℓ-adic cohomology, algebraic de Rham cohomology in characteristic 0, and rigid cohomology in positive characteristic. It also incorporates newer theories such as motivic cohomology, algebraic Ktheory and algebraic cobordism. The latter-mentioned theories share the common property of being oriented, like their respective topological analogues, singular cohomology, complex K-theory, and complex cobordism. However, a salient feature of the motivic homotopy category is that it also contains theories which are not oriented, such as Chow-Witt groups [BM00a,Fas07,Fas08] and Milnor-Witt motivic cohomology [DF17b], Balmer's higher Witt groups [Bal99], hermitian K-theory (also called higher Grothendieck groups, [Hor05,PW19]), certain variants of algebraic cobordism [PW18], and the stable cohomotopy groups, represented by the motivic sphere spectrum.
The formalism of six operations gives rise to a great deal of structure at the level of cohomology and Borel-Moore homology groups. Parts of this structure were axiomatized by Bloch and Ogus [BO74], via their notion of Poincaré duality theory, and later through the bivariant theories of Fulton and MacPherson [FM81]. The key element of these axiomatizations was the notion of the fundamental class, which was used to express duality isomorphisms.
Main problematic. Our goal in this paper is to incorporate Fulton and MacPherson's ideas into stable motivic homotopy theory, thereby obtaining a universal bivariant theory. In order to treat oriented and non-oriented spectra in a single theory, we have to replace Tate twists, as used for example in the Bloch-Ogus axiomatic, by "Thom twists", i.e., twists with respect to vector bundles (or more generally, with respect to virtual vector bundles). Let us explain the justification for this idea.
Our first inspiration is Morel and Voevodsky's homotopy purity theorem, which asserts that, for smooth closed pairs (X, Z), the homotopy type of X with support in Z is isomorphic to Th Z (N Z X), the Thom space of the normal bundle of Z in X. Here the homotopy type of Th Z (N Z X) should be understood as the homotopy type of Z twisted by the vector bundle N Z X. Another motivation is Morel's work [Mor12] on computations of homotopy groups, in which a crucial role is played by the construction of good transfer maps for finite field extensions in the unstable homotopy category. In this work, twists are usually avoided but at the cost of choosing orientations. Similar constructions enter into play in Voevodsky's theory of framed correspondences [Voe01, GP18, EHK + 17], where the "framing" provides a chosen trivialization of the normal bundle. Finally, Calmès and Fasel have introduced the notion of MW-correspondences, based on Chow-Witt theory, where transfers do appear with a twist. These examples show the utmost importance of having good transfer or Gysin morphisms in A 1 -homotopy theory. The last indication which points out to our central construction is the extension obtained in [Jin16] of the finer operations of Fulton's intersection theory, such as refined Gysin morphisms, in the motivic Borel-Moore theory (see in particular [Jin16,Def. 3.1]). The translation becomes possible once one recognizes Borel-Moore motivic homology as a particular instance of bivariant theory.
Theorem (Theorem 3.3.2 and Proposition 3.3.4). For any smoothable lci 2 morphism f : X → Y , there exists a canonical class η f , called the fundamental class of f : where L f is the virtual tangent bundle of f (equivalently, the virtual bundle associated with the cotangent complex of f , which is perfect under our assumption).
These classes satisfy the following properties: (i) Associativity. Consider morphisms f : X → Y and g : Y → Z such that f , g, and g • f are smoothable and lci. Then one has: Further applications. We finally give several applications of our formalism. The first one is the possibility of extending Fulton's theory of refined Gysin morphism to an arbitrary ring spectrum (Definition 4.2.5). These new refined Gysin morphisms are used to define specialization maps in any representable theory, on the model of Fulton's definition of specialization for the Chow group. In fact, our specialization maps can be lifted to natural transformations of functors (see Paragraph 4.5.6 for details). Most interestingly, the theory can be applied to Chow-Witt groups and give specializations of quadratic cycles (see Example 4.5.5).
Note that the idea of refining classical formulas to the quadratic setting has been explored recently by many authors [Fas07,Fas09,Hoy15,KW19,Lev17b]. In this direction another application of the theory we develop is a motivic refinement of the classical Gauss-Bonnet formula [SGA5,VII 4.9]. Given a smooth proper S-scheme X, the categorical Euler characteristic χ cat (X/S) is the endomorphism of the motivic sphere spectrum S S given by the trace of the identity map of Σ ∞ + (X) ∈ SH (S). A simple application of our excess intersection formula then computes this invariant as the degree of the Euler class of the tangent bundle T X/S (see Theorem 4.6.1). This result is a generalization of a theorem of Levine [Lev17b, Thm. 1], which applies when S is the spectrum of a field and p : X → S is smooth projective. It also recovers the SL-oriented variant in [LR20, Theorem 1.5], used in op. cit. to prove a certain explicit formula for the quadratic Euler characteristic conjectured by Serre.
Related work and further developments. Bivariant theories represented by oriented motivic spectra were studied in detail in [Dég18a]. In that setting, the fundamental class of a regular closed immersion is given by a construction of Navarro [Nav18], which itself is based on a construction of Gabber in the setting ofétale cohomology [ILO14,Exp. XVI]. A similar construction to our fundamental class for closed immersions, in the context of equivariant stable A 1 -homotopy, has been developed independently in recent work of Levine [Lev17a].
An immediate but important consequence of our work here is that the cohomology theory represented by any motivic spectrum E admits a canonical structure of framed transfers; that is, it extends to a presheaf on the category of framed correspondences (see [Voe01,EHK + 17]). It is proven in [EHK + 17] that this structure can be used to recognize infinite P 1 -loop spaces, in the same way that E ∞ -structures can be used to recognize infinite loop spaces in topology (see also [GP18]). In [EHK + 20,DK20], framed transfers are applied to construct categories of finite E-correspondences, for any motivic spectrum E, together with canonical functors from the category of framed correspondences. As explained in op. cit., such functors play an important role in the yoga of motivic categories. Another application of the existence of framed transfers is a topological invariance statement for the motivic homotopy category, up to inverting the exponential characteristic of the base field [EK20].
An application of our theory of Euler classes can be found in [JY18], where it is used to give a characterization of the characteristic class of a motive.
Finally, our constructions can be extended to the setting of quasi-smooth morphisms in derived algebraic geometry. This yields a formalism of motivic virtual fundamental classes, see [Kha19].
Contents. In Section 2 we construct the bivariant theory and cohomology theory associated to a motivic ring spectrum, and study their basic properties. Following Fulton-MacPherson [FM81], we also introduce the abstract notion of orientations of morphisms in this setting; fundamental classes will be examples of orientations. We then show how any choice of orientation gives rise to a purity transformation.
The heart of the paper is Section 3, where we construct fundamental classes and verify their basic properties. In the case where f is smooth, the fundamental class comes from the purity theorem (see Definition 2.3.5). For the case of a regular closed immersion we use the technique of deformation to the normal cone. We then explain how to glue these to obtain a fundamental class for any quasi-projective lci morphism. Throughout this section, we restrict our attention to the bivariant theory represented by the motivic sphere spectrum.
Finally in Section 4, we return to the setting of the bivariant theory represented by any motivic ring spectrum. We show how the fundamental class gives rise to Gysin homomorphisms. We prove the excess intersection formula in this setting (Proposition 4.2.2). We also discuss the purity transformation, the absolute purity property, and duality isomorphisms (identifying bivariant groups with certain cohomology groups). We then import some further constructions from Fulton's intersection theory, including refined Gysin maps and specialization maps. Finally, we conclude with a proof of the motivic Gauss-Bonnet formula mentioned above.
Conventions. The following conventions are in place throughout the paper: (1) All schemes in this paper are assumed to be quasi-compact and quasi-separated.
(2) The term s-morphism is an abbreviation for "separated morphism of finite type". Similarly, an s-scheme over S is an S-scheme whose structural morphism is an s-morphism.
(3) We write A 1 for the affine line over Spec(Z) and G m for the complement of the origin. For a scheme X, we write A 1 X and G m X for A 1 × X and G m × X, respectively. (4) We follow [SGA6, Exps. VII-VIII] for our conventions on regular closed immersions and lci morphisms. Recall that if X and Z are regular schemes, or are both smooth over some base S, then any closed immersion Z → X is regular. Given a regular closed immersion i : Z → X, we write N i , N Z X, or occasionally N (X, Z), for its normal bundle. Recall that a morphism of schemes X → S is lci (= a local complete intersection) if it admits, Zariski-locally on the source, a factorization X i − → Y p − → S, where p is smooth and i is a regular closed immersion. If it is also smoothable, then it admits such a factorization globally. This is for example the case if f is quasi-projective (in the sense that it factors through an immersion into some projective space P n S ). (5) A cartesian square of schemes In this case we also say that p is transverse to f . Recall that if p or f is flat, then this condition is automatic. (6) We will make use of the language of stable ∞-categories [HA]. Given a stable ∞-category C , we write Maps C (X, Y ) for the mapping spectrum of any two objects X and Y . We write Hom C (X, Y ) or simply [X, Y ] for the abelian group of connected components π 0 Maps C (X, Y ). (7) Given a topological S 1 -spectrum E, we write x ∈ E to mean that x is a point in the infinite loop space Ω ∞ (X) (or an object in the corresponding ∞-groupoid).
hosted by the FRIAS. F. Jin is partially supported by the DFG Priority Programme SPP 1786 Project "Motivic filtrations over Dedekind domains".

Bivariant theories and cohomology theories
2.1. The six operations. Given any (quasi-compact quasi-separated) scheme S, we write SH (S) for the stable ∞-category of motivic spectra, as in [Hoy15,Appendix C] or [Kha16]. When S is noetherian and of finite dimension, then the homotopy category of SH (S) is equivalent, as a triangulated category, to the stable A 1 -homotopy category originally constructed by Voevodsky [Voe98]. As S varies, these categories are equipped with the formalism of Grothendieck's six operations [Ayo07,CD19]. In this subsection we briefly recall this formalism, and its ∞-categorical refinement as constructed in [Kha16,Chap. 2] (see also [Rob14] for another approach).
2.1.1. First, the stable presentable ∞-category SH (S) is symmetric monoidal, and we denote the monoidal product and monoidal unit by ⊗ and S S , respectively. It also admits internal hom objects Hom(E, F) ∈ SH (S) for all E, F ∈ SH (S). For any morphism of schemes f : T → S, we have a pair of adjoint functors f * : SH (S) → SH (T ), f * : SH (T ) → SH (S), called the functors of inverse and direct image along f , respectively. If f is an s-morphism 5 , i.e., a separated morphism of finite type, then there is another pair of adjoint functors called the functors of exceptional direct and inverse image along f , respectively. Each of these operations is 2-functorial.
(2) There is a natural transformation f ! → f * which is invertible when f is proper.
(3) There is an invertible natural transformation f * → f ! when f is an open immersion.
(4) The operation of exceptional direct image f ! satisfies a projection formula against inverse image. That is, there is a canonical isomorphism for any s-morphism f : T → S and any E ∈ SH (S), F ∈ SH (T ). (5) The operation f ! satisfies base change against inverse images g * , and similarly f ! satisfies base change against direct images g * . That is, for any cartesian square where f and g are s-morphisms, there are canonical isomorphisms All the above data are subject to a homotopy coherent system of compatibilities (see [Kha16, Chap. 2, Sect. 5]).
2.1.3. The A 1 -homotopy invariance property of SH is encoded in terms of the six operations as follows. For a scheme S and any vector bundle π : E → S, the functor π * : SH (S) → SH (E) is fully faithful. In particular, the unit Id → π * π * is invertible.
5 Using Zariski descent, the operations (f ! , f ! ) can be extended to the case where f is locally of finite type; assuming this extension, the reader can globally redefine the term "s-morphism" as "locally of finite type morphism".
2.1.4. Given a locally free sheaf E of finite rank over S, let E = Spec S (S(E)) denote the associated vector bundle 6 . There is an auto-equivalence called the E-suspension functor, with inverse denoted Σ −E . This functor is compatible with the monoidal product ⊗ via a projection formula that provides canonical isomorphisms Σ E (E) ≃ E ⊗ Σ E (S S ) for any E ∈ SH (S). It is also compatible with the other operations in the sense that we have canonical isomorphisms The motivic spectrum Σ E (S S ) ∈ SH (S) is (the suspension spectrum of) the Thom space of E, and is denoted Th Given a motivic spectrum E ∈ SH (S), we denote by E(n) ∈ SH (S) the motivic spectrum 2.1.5. Let Vect(S) denote the groupoid of locally free sheaves on S of finite rank, and Pic(SH (S)) the ∞-groupoid of ⊗-invertible objects in SH (S). The assignment E → Th S (E) determines a map of presheaves of ∞-groupoids Th : Vect → Pic(SH ).
Moreover, if K denotes the presheaf sending S to its Thomason-Trobaugh K-theory space K(S), this extends to a map of E ∞ -groups Th : K → Pic(SH ), see [BH17,Subsect. 16.2]. In particular, any perfect complex E on S defines a K-theory class 7 E ∈ K(S) and thus an auto-equivalence Σ E : SH (S) → SH (S) and a Thom space Th S (E) ∈ SH (S). The formulas (2.1.4.a) also extend. Moreover, any exact triangle E ′ → E → E ′′ of perfect complexes induces canonically a path E ≃ E ′ + E ′′ in the space K(S), hence also identifications 2.1.7. If f is a smooth s-morphism, then by Ayoub's purity theorem, there is a canonical isomorphism of functors where T f is the relative tangent bundle. It follows in particular that f * admits a left adjoint which satisfies base change and projection formulas against inverse images g * . If f isétale then we get an isomorphism p f : f * ≃ f ! , generalizing Paragraph 2.1.2(3). 6 Throughout the text we will generally not distinguish between a locally free sheaf E and its associated vector bundle E. Thus for example we will also write Σ E instead of Σ E , or similarly E ∈ K(S) instead of E ∈ K(S) (see Paragraph 2.1.5 below). It should always be clear from the context what is intended.
7 An abuse of language we will commit often is to say "K-theory class" when it would be more precise to say "point of the K-theory space".
2.1.8. Let f : X → Y be a closed immersion of s-schemes over S. Suppose that X and Y are smooth over S, with structural morphisms p : X → S and q : Y → S. Then by the relative purity theorem of Morel-Voevodsky, there exist isomorphisms of functors where N X Y denotes the normal bundle.
Many further compatibilities can be derived from the ones already listed. A few that will be especially useful in this paper are as follows: 2.1.9. Given a cartesian square as in (2.1.2.a) where f and g are s-morphisms, the base change formula (Paragraph 2.1.2(5)) induces a natural transformation It follows from the purity theorem (Paragraph 2.1.7) that if f or p is smooth, then Ex * ! is invertible. For example if i : Z → S is a closed immersion, then the cartesian square gives rise to a canonical natural transformation 2.1.10. For any s-morphism f : X → S and any pair of motivic spectra E, F ∈ SH (S), there is a canonical morphism 2.2. Bivariant theories. In this subsection we construct the bivariant theory represented by a motivic spectrum, and state its main properties. In fact, bivariant theory is only one of the "four theories" associated to a motivic spectrum (cf. [VSF00, Chap. 4, Sect. 9]). For sake of completeness we define them all now: Definition 2.2.1. Let S be a scheme and E ∈ SH (S) a motivic spectrum.
(i) Bivariant theory. For any s-morphism p : X → S and any K-theory class v ∈ K(X), we define the v-twisted bivariant spectrum of X over S as the mapping spectrum We also write for each integer n ∈ Z. (ii) Cohomology theory. For any morphism p : X → S and any v ∈ K(X), we define the v-twisted cohomology spectrum of X over S as the mapping spectrum We also write for each integer n ∈ Z.
(iii) Bivariant theory with proper support (or homology). For any s-morphism p : X → S and any Ktheory class v ∈ K(X), we define the spectrum of v-twisted bivariant theory with proper support of X over S as the mapping spectrum (iv) Cohomology with proper support. For any s-morphism p : X → S and any K-theory class v ∈ K(X), we define the spectrum of v-twisted cohomology with proper support of X over S as the mapping spectrum Remark 2.2.2. Note that we have canonical identifications for any s-scheme X over S and v ∈ K(X).
Remark 2.2.3. The bivariant groups E * (X/S, * ) were previously called Borel-Moore homology groups in [Dég18a]. This terminology is justified when S is the spectrum of a field, and coincides with that of [VSF00, Chap. 4, Sect. 9]. However, in the case where S is an arbitrary scheme, and especially singular, the homology groups E * (X/S, * ) are no longer given by the cohomology with coefficients in a dualizing object, which is a characteristic property of the original theory of Borel and Moore. For that reason we find the terminology "bivariant" more suitable.
Remark 2.2.4. Given a morphism f : T → S, we can consider the inverse image E T = f * (E) ∈ SH (T ) and the associated bivariant theory E T (−/T, * ) over T . When there is no risk of confusion, we will usually abuse notation by writing E(−/T, * ) = E T (−/T, * ).
Remark 2.2.5. Note that any isomorphism v ≃ w in K(X) induces an isomorphism of bivariant spectra E(X/S, v) ≃ E(X/S, w) and of cohomology spectra E(X, v) ≃ E(X, w). More precisely, the assignments v → E(X/S, v) and v → E(X, v) are functors on K(X) (viewed as an ∞-groupoid).
Notation 2.2.6. In the notation E(X/S, v) and E(X, v), we will sometimes implicitly regard v as a class in K(X) even if it is actually defined over some deeper base. For example we will write E(X/S, v) = E(X/S, f * (v)) where f : X → S and v ∈ K(S).
2.2.7. The bivariant theory represented by a motivic spectrum E ∈ SH (S) satisfies the following axioms, which are K-graded and spectrum-level refinements of the axioms of Fulton and MacPherson [FM81]: (1) Base change. For any cartesian square there is a canonical base change map This is induced by the natural transformation where we have used the base change formula (Paragraph 2.1.2(5)) and the formula (2.1.4.a).
(2) Proper covariance. For any proper morphism f : X → Y of s-schemes over S, there is a direct image map . This covariance is induced by the unit map f ! f ! → Id and the identification f ! ≃ f * as f is proper (Paragraph 2.1.2(2)).
(3)Étale contravariance. For anyétale s-morphism f : X → Y of s-schemes over S, there is an inverse image map . This contravariance is induced by the purity isomorphism p f : f ! ≃ f * (Paragraph 2.1.7). (4) Product. If E is equipped with a multiplication map µ E : E ⊗ E → E, then for s-morphisms p : X → S and q : Y → X, and any K-theory classes v ∈ K(X) and w ∈ K(Y ), there is a map Given maps y : Th Y (w)[m] → q ! E X and x : Th X (v)[n] → p ! E S , the product y.x is defined as follows: These structures satisfy the usual properties stated by Fulton and MacPherson (functoriality, base change formula both with respect to base change andétale contravariance, compatibility with pullbacks and projection formulas; see [Dég18a, 1.2.8] for the precise formulation).
Remark 2.2.8. One of the main objectives of this paper concerns the extension of contravariant functoriality frométale morphisms to smoothable lci morphisms. This will be achieved in Theorem 4.2.1.
Remark 2.2.9. Note that a particular case of the product of Paragraph 2.2.7(4) is the cap-product: The localization theorem (Paragraph 2.1.6) gives the following direct corollary: Proposition 2.2.10. Let i : Z → X be a closed immersion of s-schemes over S, with quasi-compact complementary open immersion j : U → X. Then there exists, for any e ∈ K(X), a canonical exact triangle of spectra called the localization triangle. Moreover, this triangle is natural with respect to the contravariance in S, the contravariance in X/S forétale S-morphisms, and the covariance in X/S for proper Smorphisms (see parts (1)-(3) of Paragraph 2.2.7).
A special case of the naturality in Proposition 2.2.10 is the following: for any e ∈ K(X). Herej,ĩ denote the obvious closed immersions obtained by restriction, andĩ ′ ,j ′ the complementary open immersions.
Consider cartesian squares of s-schemes over S: such that i and k are closed immersions with quasi-compact complementary open immersions i ′ and k ′ , respectively. For any π ∈ E(Y /X, e ′ )[r] with e ′ ∈ K(Y ), r ∈ Z, set: Then the following diagram of localization triangles is commutative: Proof. The left-hand square commutes by the projection formula. The right-hand square commutes since products are compatible with base change.
If this multiplication is unital, resp. associative, resp. commutative, then the bivariant theory represented by E inherits the same property. This is in particular the case when E is equipped with an E ∞ -ring structure. For example, assume that the multiplication is commutative in the sense that it is further equipped with a commutative diagram in SH (S): where τ is the isomorphism swapping the two factors. Given s-schemes p : X → S and q : Y → S and Then there is an identification 2.2.14. For any E ∈ SH (S) and v ∈ K(S), the functor X → E(X/S, v) satisfies descent with respect to Nisnevich squares and abstract blow-up squares (hence satisfies cdh descent), on the category of s-schemes over S.

2.3.
Orientations and systems of fundamental classes. Following Fulton-MacPherson, we now introduce the notion of orientation of a morphism f . As we recall in the next subsection, any choice of orientation gives rise to a Gysin map in bivariant theory (Paragraph 2.4.1). The fundamental classes we construct in Section 3 will be examples of orientations. For simplicity, throughout this discussion we will restrict our attention to the bivariant theory represented by the sphere spectrum S: for any s-morphism p : X → S and any v ∈ K(X). We will refer to this simply as bivariant A 1 -theory.
and e f ∈ K(X). When there is no risk of confusion, we write simply η f instead of (η f , e f ).
Remark 2.3.3. The above use of the term "orientation" is taken from [FM81]. We warn the reader however that it is unrelated to the notion of "oriented motivic spectrum" (see Definition 4.4.1).
This defines a canonical orientation η f ∈ H(X/S, T f ). It will be useful to have the following description of the purity isomorphism p f (cf. [CD19, Def. 2.4.25, Cor. 2.4.37]). We begin by considering the cartesian square ∆: , modulo the tautological identification between T f and the normal bundle N δ . The exchange transformation Ex * ! is invertible because f is smooth (see Paragraph 2.1.9).
Definition 2.3.6. Let S be a scheme and let C be a class of morphisms between s-schemes over S. A system of fundamental classes for C consists of the following data: and an isomorphism η C f ≃ 1 in H(S/S, e f ) ≃ H(S/S, 0). (iii) Associativity formula. Let f : X → Y and g : Y → Z be morphisms in C such that the composite g • f is also in C . Then there are identifications in K(X) and We say that a system of fundamental classes (η C f ) f is stable under transverse base change if it is equipped with the following further data: (iv) Transverse base change formula. For any tor-independent cartesian square Remark 2.3.8. Let S be a scheme and let C be a class of morphisms between s-schemes over S.
Suppose (η f ) f is a system of fundamental classes for C as in Definition 2.3.6. For any morphisms f : and isomorphisms of Thom spaces Example 2.3.9. It follows from [Ayo07, 1.7.3] that the family of orientations η f for f smooth (Definition 2.3.5) forms a system of fundamental classes for the class of smooth s-morphisms. Moreover, this system is stable under (arbitrary) base change: explicit homotopies ∆ * (η f ) ≃ η g as in Definition 2.3.6(iv) are provided by the deformation to the normal cone space, as in the proof of [Dég18a, Lem. 2.3.13] (where the right-hand square (3) can be ignored).
Example 2.3.10. In Section 3, we will extend Example 2.3.9 to the class of smoothable lci smorphisms. Recall from [Ill06] that an lci morphism f : X → S admits a perfect cotangent complex is the class of the normal bundle. Every smoothable lci morphism f factors through a regular closed immersion i followed by a smooth morphism p, and such a factorization induces an identification The fundamental class of a smoothable lci s-morphism f : X → S will then be an orientation in H(X/S, L f ) (see Theorem 3.3.2).
We finish with a discussion of strong orientations and duality isomorphisms. (1) We say that (η f , e f ) is strong if for any v ∈ K(X), cap-product with η f induces an isomorphism In that case, we refer to γ η f as the duality isomorphism associated with the strong orientation η f . ( Remark 2.3.12. It follows immediately from the construction of the cap product that a universally strong orientation is strong. Example 2.3.13. If f is smooth, then the orientation η f of Definition 2.3.5 is universally strong by the purity theorem (Paragraph 2.1.7).
The following lemma explains the terminology "universally strong".
Lemma 2.3.14. Let f : X → S be an s-morphism. Let (η f , e f ) be a universally strong orientation for f . Then for any cartesian square is an isomorphism, as claimed.

Gysin maps.
Definition 2.4.1. Let f : X → Y be a morphism of s-schemes over S. Then any orientation η f ∈ H(X/Y, e f ) gives rise to a Gysin map: x → η f .x using the product in bivariant A 1 -theory, for all e ∈ K(Y ). When the orientation η f is clear, we simply put: f ! = η ! f . Proposition 2.4.2. Let S be a scheme and let C be a class of morphisms between s-schemes over S.
Suppose (η f ) f is a system of fundamental classes for C as in Definition 2.3.6. (i) Functoriality. Let f and g be morphisms in C such that the composite g • f is also in C . Then for every e ∈ K(Z) there is an induced identification Suppose given a tor-independent cartesian square of s-schemes over S of the form where f and g are in C , and u and v are proper. Then for every e ∈ K(Y ) there is an induced identification Example 2.4.3. If f is a smooth s-morphism, then the fundamental class η f (Definition 2.3.5) gives rise to canonical Gysin maps This extends the contravariant functoriality frométale morphisms to smooth morphisms.
Lemma 2.4.4. Let X be an s-scheme over S and let p : E → X be a vector bundle. Then the tangent bundle T p is identified with p −1 (E) and the Gysin map is invertible.
Proof. In view of the construction of the Gysin map, the claim follows directly from the facts that the morphism η p : Remark 2.4.6. The Thom isomorphism satisfies the properties of compatibility with base change and with direct sums (that is, φ E⊕F/X ≃ φ E⊕F/F •φ F/X for vector bundles E and F over X). These follow respectively from the compatibility of the Gysin morphisms p ! with base change and with composition.
We conclude this subsection by recording the naturality of the localization sequences (Proposition 2.2.10) with respect to Gysin maps of smooth morphisms.
Proof. The right-hand square commutes by the associativity property of fundamental classes of smooth morphisms (Example 2.3.9). The left-hand square commutes by the naturality of the relative purity isomorphism of Morel-Voevodsky [Hoy15,Prop. A.4], in view of the construction of the fundamental classes η f and η g (Example 2.3.4).
2.5. Purity transformations. The notion of orientation seen in the preceding subsection is part of our twisted version of the bivariant formalism of Fulton and MacPherson. We state in this subsection a variant, or a companion, of this notion in the spirit of Grothendieck's six functors formalism.
2.5.1. Let us fix an s-morphism f : X → S, and an orientation (η f , e f ). According to our definitions, the class η f ∈ H(X/S, e f ) can be seen as a morphism in SH (X): This gives rise to a natural transformation to the orientation η f , defined as the following composite: where the exchange transformation Ex ! * ⊗ is as in Paragraph 2.1.10.
(i) Suppose f is smooth and consider the canonical orientation η f of Definition 2.3.5. It follows by construction that in this case the associated purity transformation p(η f ) is nothing else than the purity isomorphism p f (2.3.4.a). In particular, p(η f ) is an isomorphism in this case. (ii) Note that the datum of an orientation (η f , e f ) and that of the associated purity transformation p(η f ) are essentially interchangeable. Indeed we recover η f by evaluating p(η f ) at the unit object S S .
2.5.3. Consider the notation of the previous definition. Then one associates to p(η f ), using the adjunction properties, two natural transformations: The first (resp. second) natural transformation will be called the trace map (resp. co-trace map) associated with the orientation η f , following the classical usage in the literature. These two maps are functorial incarnations of the Gysin map defined earlier (Paragraph 2.4.1), as we will see later (see Paragraph 4.3.3).
The notion of a system of fundamental classes (Definition 2.3.6) was introduced to reflect the functoriality of Gysin morphisms. For completeness, we now formulate the analogous functoriality property for the associated purity transformations.
Proposition 2.5.4. Let S be a scheme and let C be a class of morphisms between s-schemes over S.
Suppose (η f ) f is a system of fundamental classes for C as in Definition 2.3.6.
Functoriality. Let f and g be morphisms in C such that the composite g • f is also in C . Then there is a commutative square where the left-hand vertical isomorphism is from Remark 2.3.8, and the lower horizontal arrow is the horizontal composition of the 2-morphisms p(η f ) and p(η g ).
(ii) Transverse base change. Assume that the system (η f ) f is stable under transverse base change. Suppose given a tor-independent cartesian square of the form where f and g are in C . Then there are commutative squares of natural transformations: If u and v are s-morphisms, then there are also commutative squares Proof. Claim (i) follows from axioms (ii) and (iii) of Definition 2.3.6. In claim (ii), commutativity of (a) follows directly from axiom (iv) of Definition 2.3.6. The square (b) can be derived from (a) by applying u * on the left and v * on the right, and using the naturality of the unit and counit of the adjunctions (u * , u * ) and (v * , v * ), respectively. Similarly, commutativity of square (c) will follow similarly from (d). For (d), we may unravel the definition of the purity transformation (Paragraph 2.5.1) to write the square as follows: Observing that p g (S) = p g (v * (S)), we can use square (a) to decompose the left-hand middle arrow into a natural transformation induced by p f (S) and an exchange transformation. The commutativity of the resulting square is then a formal exercise.
Remark 2.5.5. Suppose that the class C contains all identity morphisms and is closed under composition, and let S C denote the subcategory of the category of schemes S whose morphisms all belong to C . At the level of homotopy categories, Proposition 2.5.4(i) implies that the assignment f → p(η f ) defines a natural transformation of contravariant pseudofunctors p : Ho(SH ) e * → Ho(SH ) ! on the category S C , where the notation is as follows: • T ri denotes the (2, 1)-category of large triangulated categories, triangulated functors, and invertible triangulated natural transformations.
• Ho(SH ) ! is the pseudofunctor (S C ) op → T ri , given by the assignments • Similarly Ho(SH ) e * is the pseudofunctor (S C ) op → T ri given by the assignments The expected enhancement to a natural transformation at the level of ∞-categories requires further work that we do not undertake in this paper.
We can also reformulate the transverse base change property (Proposition 2.5.4(ii)) in terms of the (co)trace maps.
Corollary 2.5.6. Under the assumptions of Proposition 2.5.4(ii), the following diagrams commute:

Construction of fundamental classes
3.1. Euler classes. Before proceeding to our construction of fundamental classes, we begin with a discussion of Euler classes in the setting of bivariant theories. Our basic definition is very simple and can be formulated unstably.
3.1.1. Let X be a scheme and E be a vector bundle over X. Recall that the Thom space Th X (E) ∈ SH (X) is in fact the suspension spectrum of a pointed motivic space in H • (X). Moreover, the latter can be modelled by the pointed Nisnevich sheaf of sets where E × is complement of the zero section. Note that Thom spaces are functorial with respect to monomorphisms of vector bundles. That is, if ν : F → E is a monomorphism of vector bundles over X, one gets a canonical morphism of pointed sheaves: ν * : Th X (F ) → Th X (E).
Definition 3.1.2. Let E be a vector bundle over a scheme X, and s be its zero section. We can regard s as a monomorphism of vector bundles s : X → E. We define the Euler class e(E) of E/X as the induced map in H • (X): It is easy to see that Euler classes commute with base change: Lemma 3.1. 4. For any morphism f : Y → X and any vector bundle E over X, the following diagram is commutative: Proof. This follows from the fact that the functor f * commutes with cokernels, and the base change of the zero section of E is the zero section of f −1 (E).
3.1.5. By construction, the Thom space fits into a cofiber sequence in H • (X): By A 1 -homotopy invariance, the projection map E → X induces an isomorphism in H • (X), whose inverse is induced by the zero section s : X → E. It follows from our construction that the following diagram commutes: ≀ s * Definition 3.1.6. For any vector bundle E over X, the Euler cofiber sequence is the cofiber sequence The Euler cofiber sequence immediately yields the following characteristic property of Euler classes: Proposition 3.1.7. Let E be a vector bundle over X. Any nowhere vanishing section s of E → X induces a null-homotopy of the morphism X + → Th X (E) corresponding to the Euler class e(E). In particular, s induces an identification e(E) ≃ 0 in H(X, E ).
Proof. Such a section s induces a section of the morphism (E × ) + → X + in the Euler cofiber sequence.
Corollary 3.1.8. Let E be a vector bundle over X. If E contains the trivial line bundle A 1 X as a direct summand, then there is an identification e(E) ≃ 0 in H(X, E ).
3.1.9. Suppose we have an exact sequence of vector bundles in fact exists already unstably: in H • (X). This is clear when the sequence (3.1.9.a) is split, and one reduces to this case by pulling back to the Hom X (E ′′ , E ′ )-torsor of splittings of the sequence (which is fully faithful by A 1 -invariance).
Lemma 3.1.10. Given an exact sequence of vector bundles as in (3.1.9.a), the following diagram is commutative: Proof. Argue as above to reduce to the case where (3.1.9.a) is split.
The additivity property of Euler classes is then a direct corollary: Proposition 3.1.11. Given an exact sequence of vector bundles as in (3.1.9.a), the following diagram is commutative: Proof. Compose the diagram of Lemma 3.1.10 with the map e(E ′ ) : X + → Th X (E ′ ) (on the left).
3.2. Fundamental classes: regular closed immersions. In this section we construct the fundamental class of a regular closed immersion and demonstrate its expected properties. Before proceeding, we make a brief digression to consider a certain preliminary construction.
3.2.1. Let X be a scheme and consider the diagram For any e ∈ K(X), we have a commutative diagram using the identifications T π ≃ 1 in K(G m X), Tπ ≃ 1 in K(A 1 X). (Here, as usual, 1 ∈ K(X) denotes the unit, i.e., the class A 1 X , for any scheme X.) The right-hand square consists of Gysin maps and commutes by Example 2.3.9; moreover, the morphismπ ! is invertible (Lemma 2.4.4). In the left-hand square, the morphism γ ηπ is the duality isomorphism (Definition 2.3.11) associated to the strong orientation (Example 2.3.13) of π, and the square evidently commutes by construction of the morphisms involved. Since the left vertical arrow π * admits a retraction s * 1 , given by the inverse image by the unit section s 1 : X → G m X in cohomology, we also get a canonical retraction ν t of the right vertical arrow j ! . in SH (G m X); that is, γ t is multiplication by {t} ∈ H(G m X/X, 0)[−1]. By construction, {t} is stable under arbitrary base changes. If X is an s-scheme over some base S, we will abuse notation and write γ t also for the map γ t : H(X/S, e) → H(G m X/S, e)[−1], given again by the assignment x → {t}.x.

Consider now the localization triangle
We now proceed to the construction of the fundamental class.

3.2.3.
Let X be an S-scheme and i : Z → X a regular closed immersion. We write D Z X or D(X, Z) for the (affine) deformation space B Z×0 (X × A 1 ) − B Z×0 (X × 0), as defined by Verdier (denoted M (Z/X) in [Ver76,§2]); here B Z X denotes the blow-up of X along Z. This fits into a diagram of tor-independent cartesian squares for any e ∈ K(X).
Definition 3.2.4. With notation as above, the specialization to the normal cone map associated to i is the composite where γ t is the map constructed in Paragraph 3.2.2.
Definition 3.2.5. The fundamental class η i ∈ H(Z/X, − N Z X ) associated to the regular closed immersion i is the image of 1 ∈ H(X/X, 0) by the composite where φ NZ X/Z is the Thom isomorphism of p : N Z X → Z (Definition 2.4.5). In other words, η i = φ NZ X/Z σ Z/X (1) .
(i) By definition of the Thom isomorphism, the fundamental class η i is determined uniquely by the property that there is a canonical identification in SH (Z). (iv) Let us assume that X is an s-scheme over a base S. Then the orientation (η i , − N i ) gives rise to a Gysin map (Definition 2.4.1): It follows from the definitions that this map can also be described as the composite: therefore comparing our construction with that of Verdier [Ver76]. Note also that σ Z/X ≃ p ! i ! so that the Gysin map and the specialization map uniquely determine each other. (v) One can describe the map η i more concretely as follows. Let us recall the deformation diagram: First the map {t} : S GmX [1] → π ! (S X ) corresponds by adjunction and after one desuspension to a map: Then one gets the following composite map: where the last isomorphism uses the purity isomorphism p p . Using the identification S D ≃ r * (S X ) and the adjunction (r * , r * ) we deduce a map: where the last isomorphism follows from the A 1 -homotopy invariance of SH (Paragraph 2.1.3). The latter composite is nothing else than the morphism Th Z (N Z X) ⊗ η i .

3.2.7.
Consider a cartesian square where i and k are regular closed immersions Then we get a morphism of deformation spaces D T Y → D Z X and similarly a morphism of vector bundles: where ν is in general a monomorphism of vector bundles (i.e. the codimension of T in Y can be strictly smaller than that of Z in X: there is excess of intersection). We put ξ = q −1 N Z X/N T Y , the excess intersection bundle.
Proposition 3.2.8 (Excess intersection formula). With notation as above, we have a canonical isomorphism is the Euler class of ξ (Definition 3.1.2 and Remark 3.1.3).
Proof. Let us put D ′ T Y = D Z X × X Y and N ′ T Y = q −1 N Z X. Then we get the following commutative diagram of schemes, in which each square is cartesian: Therefore, one gets the following commutative diagram: Here the right-hand arrow labelled ∆ * is the change of base map (Paragraph 2.2.7(1)); we have abused notation by also writing ∆ * for the two analogous maps on the left and middle (induced by the obvious cartesian squares). Square (1) (resp. (2)) is commutative because of the naturality of localization triangles with respect to the proper covariance (resp. base change). Square (3) is commutative by definition of ν * , and square (4) by compatibility of Thom isomorphisms with respect to base change. Now observe that the image of {t} ∈ H(G m X/X, 0)[−1] by the counter-clockwise composite in the above diagram is nothing else than the class ∆ * (η i ) ∈ H(T /Y, − N ′ T Y ). Similarly the image by the clockwise composite is the class ν * (η k ) ∈ H(T /Y, − N ′ T Y ). We conclude using Lemma 3.1.10.
Example 3.2.9. We get the following usual applications of the preceding formula.
(i) Transverse base change formula. If we assume that p is transverse to i, then ν is an isomorphism and the excess bundle vanishes. Thus we get a canonical identification ∆ * (η i ) ≃ η k . (ii) Self-intersection formula. If we apply the formula to the self-intersection square Remark 3.2.10. If X is a scheme, E is a vector bundle over X and s 0 : X → E is the zero section, the self-intersection formula (3.2.9.a) applied to s 0 says that we can recover the Euler class of E from the fundamental class of s 0 by base change along the self-intersection square. That is, where ∆ denotes the self-intersection square associated to s 0 . More generally, if s : X → E is an arbitrary section, consider the cartesian square We will now state good properties of our constructions of orientations for regular closed immersions, culminating in the associativity formula. Note that all these formulas will be subsumed once we will get our final construction.
3.2.11. First consider a cartesian square of S-schemes: such that i is a regular closed immersion and f is smooth. The isomorphisms of vector bundles Lemma 3.2.12. With notation as above, one has the commutative square Proof. It suffices to show that both squares in the following diagram commute: In fact, the commutativity of (1) (where we have denoted the canonical functions of D Z X and D T Y by the same letter t) is obvious, and (2) follows from Proposition 2.4.7.
Lemma 3.2.13. With notation as above, one has a canonical identification Proof. Consider the following diagram: The right-hand square commutes by the associativity formula for Gysin morphisms associated with smooth morphisms (Example 2.3.9). Furthermore, the horizontal arrows p ! NZ X/Z and p ! NT Y /T are invertible (Lemma 2.4.4), so it suffices to show that the composite square commutes. But the upper and lower composites are the respective specialization maps σ Z/X and σ T /Y , so we conclude by Lemma 3.2.12.
3.2.14. Next we consider a commutative diagram of schemes: such that i is a closed immersion and p, q are smooth s-morphisms. In this situation, the canonical exact sequence of vector bundles over Z gives rise to an identification (3.2.14.a) in K(Z).
Lemma 3.2.15. With notation as above, one has a canonical identification Proof. Consider the cartesian square where s is the zero section and π is the composite map N Z X The commutativity of square (1) is obvious, that of (2) follows from Proposition 2.4.7 applied to the cartesian square ∆, and that of (3) follows from the associativity of Gysin morphisms associated with smooth morphisms (Example 2.3.9).
Before proceeding, we draw out some corollaries of the previous lemma.
Corollary 3.2.16. Consider the assumptions of Paragraph 3.2.14. Then the orientation η i is universally strong (Definition 2.3.11). In other words, the morphism η i : Proof. This follows from the previous lemma and the fact that the maps η p and η q are isomorphisms (Definition 2.3.5). is precisely the Thom isomorphism (Definition 2.4.5). This follows from Corollary 3.2.17. In cohomological terms, we also get the Thom isomorphism: 3.2.19. We now proceed towards the formulation of the associativity formula for the fundamental classes of two composable regular closed immersions Recall that there is a short exact sequence There is also a canonical isomorphism of vector bundles over Z; we will abuse notation and write N for both. We will make use of the double deformation space (cf. [Ros96,§10]) That is, D is the deformation space associated to D Z X| Y → D Z X, which is a regular closed immersion because the cartesian square is tor-independent. Indeed, this can be checked locally on X, so we can assume that Z → Y → X is a (transverse) base change of {0} → A m → A n for some 0 ≤ m ≤ n. Since the deformation space is stable under transverse base change (as is the question of tor-independence), we may reduce to the latter situation, in which case D Z X → X is just the projection A n × A 1 → A n , which is transverse to any morphism. Note that D is a scheme over X × A 2 ; we write s and t for the first and second projections to A 1 , respectively. Set The fibres of D over various subschemes of A 2 are summarized in the following table.
Lemma 3.2.20. Under the assumptions and notation of Paragraph 3.2.19, the diagram H(X/X, e) commutes for every e ∈ K(X).
Proof. By construction of the specialization maps, the square in question factors as in the following diagram: Some remarks on the notation are in order. First of all we have omitted the symbol × in the diagram.
We have also used exponents s and t to indicate that G s m , resp. G t m , is viewed as a subset of the s-axis, resp. t-axis, in A 2 . Finally, we have written γ u for multiplication with the class σ π ∈ H(G u m /Z, 0)[−1] with u ∈ {s, t}. Now observe that squares (2) and (3) we deduce that square (4) is also anti-commutative, whence the claim.
Theorem 3.2.21. Let S be a scheme. Then there exists a system of fundamental classes (η i ) i (Definition 2.3.6) on the class of regular closed immersions between s-schemes over S, satisfying the following properties: (i) For every regular closed immersion i : Z → X, the orientation η i ∈ H(Z/X, − N Z X ) is the fundamental class defined in Definition 3.2.5. (ii) The system is stable under transverse base change (Definition 2.3.6(iv)).
Proof. According to Definition 2.3.6 we must give the following data: Fundamental classes. For any regular closed immersion i : Z → X, we take the orientation (η i , e i ) with e i = − N Z X ∈ K(Z) and η i ∈ H(Z/X, − N Z X ) as in Definition 3.2.5. (ii) Normalisation. If i = Id S for a scheme S, then N S S = S and the specialization map and Thom isomorphism are both the identity maps on H(S/S, 0), so we have a canonical identification η i ≃ 1. (iii) Associativity formula. Given regular closed immersions i : Y → X and k : Z → Y , the composite k•i is again a regular closed immersion and we have an identification N Z X ≃ k * N Y X + N Z Y in K(Z). We obtain a canonical identification η k .η i ≃ η i•k in H(Z/X, − N Z X ) from the following commutative diagram H(X/X, 0) by evaluating at 1 ∈ H(X/X, 0), since the maps p ! NZ Y /Z and p ! N/NZ Y are invertible (Lemma 2.4.4). Note that each square is indeed commutative: (1) Apply Lemma 3.2.20.
(2) Apply Lemma 3.2.12 to the cartesian square This square factors into two triangles: The upper-right triangle commutes by construction of N Z (i) ! , the Gysin map associated to N Z (i) : N Z Y → N Z X. The lower-left triangle commutes by Lemma 3.2.15 applied to the commutative diagram: (4) Apply the associativity of Gysin morphisms associated with smooth morphisms (Example 2.3.9). (iv) Transverse base change formula. Suppose given a tor-independent cartesian square where i is a regular closed immersion. Then k is also a regular closed immersion and there is a canonical identification of vector bundles N T Y ≃ q −1 (N Z X). The canonical identification ∆ * (η i ) ≃ η k comes then from Example 3.2.9(i).
3.3. Fundamental classes: general case. In this subsection we conclude our main construction by gluing the system of fundamental classes defined on the class of smooth morphisms (Definition 2.3.5) together with the system defined on the class of regular closed immersions in the previous subsection (Theorem 3.2.21).
Proof. The diagram in question factors as follows: as claimed.
We are now ready to state the main theorem, defining a system of fundamental classes on the class of smoothable lci morphisms: Theorem 3.3.2. Let S be a scheme. Then there exists a system of fundamental classes (η f ) f (Definition 2.3.6) on the class of smoothable lci s-morphisms between s-schemes over S, satisfying the following properties: The restriction of the system (η f ) f to the class of smooth s-morphisms coincides with the system of Example 2.3.9. (ii) The restriction of the system (η f ) f to the class of regular closed immersions coincides with the system of Theorem 3.2.21. (iii) The system is stable under transverse base change (Definition 2.3.6(iv)).
Proof. To define the system (η f ) f , we must give the following data (see Definition 2.3.6): Fundamental classes. Given a smoothable lci s-morphism f : X → S, we may choose a factorization through a regular closed immersion i : X → Y and a smooth s-morphism p : Y → S. We define the fundamental class η f = η i .η p ∈ H(X/S, L f ). Note that, given another factorization through some i ′ : X → Y ′ and p ′ : Y → S, we obtain a canonical identification η i .η p ≃ η i ′ .η p ′ by applying Lemma 3.3.1 to the diagram Normalisation. If f = Id S for a scheme S, then we choose the trivial factorization f = Id S • Id S and the normalization properties of Example 2.3.9 and Theorem 3.2.21 give a canonical identification η f ≃ 1.1 ≃ 1.
(iii) Associativity formula. If f : X → Y and g : Y → Z are two smoothable lci s-morphisms, consider the commutative diagram where the i k 's are closed immersions and the p k 's are smooth morphisms, and the square is cartesian. By Example 2.3.9, Lemma 3.2.13 and Theorem 3.2.21 we have identifications (iv) Transverse base change formula. Suppose given a tor-independent cartesian square where f and g are smoothable lci s-morphisms. Choosing a factorization f = p • i, where i is a regular closed immersion and p is a smooth s-morphism, there is an induced factorization of the square ∆: where k is a regular closed immersion and q is a smooth s-morphism. Now by above and by the transverse base change properties of Example 2.3.9 and Theorem 3.2.21, we have identifications 3.3.3. In fact, the transverse base change property of Theorem 3.3.2(iii) is a special case of an excess intersection formula generalizing Proposition 3.2.8. Consider a cartesian square where f and g are smootable lci morphisms. Factor f = p • i as a closed immersion followed by a smooth morphism and consider the diagram of cartesian squares where k and i are regular closed immersions and q and p are smooth. By 3.2.7 there is a canonical monomorphism of Y -vector bundles N Y Q ν − → v −1 N X P . We let ξ be the quotient bundle.
Proposition 3.3.4. With notation and assumptions as above, there is an identification is the Euler class of ξ (Definition 3.1.2 and Remark 3.1.3).
This follows from Proposition 3.2.8 and the fact that fundamental classes for smooth morphisms are compatible with any base change (Example 2.3.9).

Main results and applications
4.1. Fundamental classes and Euler classes with coefficients. 4.1.1. Let S be a scheme and E ∈ SH (S) a motivic spectrum. Observe that the bivariant spectra E(X/S, v) of Definition 2.2.1 are natural in E. That is, given any morphism ϕ : E → F in SH (S), there is an induced map of spectra for every s-scheme X over S and every v ∈ K(X). Note that ϕ * is compatible with the various functorialities of bivariant theory (Paragraph 2.2.7). Also, if E and F are equipped with multiplications which commute with ϕ, then the induced map ϕ * preserves products (as defined in Paragraph 2.2.7(4)).  If E is unital, associative and commutative, then because ρ X/S is compatible with products and "change of base" maps, the associativity and base change formulas provided by Theorem 3.3.2 are immediately inherited by the fundamental classes of Definition 4.1.3. If we extend Definition 2.3.6 as indicated in Remark 2.3.7, then we can state more precisely: Theorem 4.1.4. Let E ∈ SH (S) be a motivic spectrum equipped with a unital associative commutative multiplication. Then there exists a system of fundamental classes (η E f ) f on the class of smoothable lci s-morphisms between s-schemes over S. This system is stable under transverse base change, and recovers the system of Theorem 3.3.2 in the case E = S S . Definition 4.1.5. Let E ∈ SH (S) be a unital motivic spectrum. Then for any scheme X and any vector bundle E/X, one defines the Euler class of E/X with coefficients in E, denoted as the image of the class e(E) ∈ H(X, E ) ≃ H(X/X, − E ) of Definition 3.1.2 by the A 1 -regulator map ρ X/S (4.1.2.a).
Remark 4.1.6. It is possible to define fundamental classes with coefficients in arbitrary motivic spectra E ∈ SH (S), without using any multiplicative or even unital structure. Indeed the constructions of Section 3 (where we only considered the case E = S S for simplicity) extend immediately to general E without any difficulty. Alternatively we can replace the use of the A 1 -regulator map above by using instead the module structure, i.e., the canonical action 9 which is defined by the same formula used to define products in Paragraph 2.2.7, except that the multiplication map µ E : E ⊗ E → E is replaced by the map S S ⊗ E → E encoding the structure of S S -module on E. As a third approach, Gysin maps with arbitrary coefficients can be obtained from the purity transformation (Paragraph 4.3.3). Theorem 4.2.1. Let E ∈ SH (S) be a motivic spectrum equipped with a unital associative commutative multiplication. Then for any smoothable lci s-morphism f : X → Y of s-schemes over S, and any v ∈ K(Y ), there is a Gysin map We also have an excess intersection formula with coefficients (generalizing Proposition 3.3.4): Proposition 4.2.2. Let E ∈ SH (S) be a motivic spectrum equipped with a unital associative commutative multiplication. Suppose given a cartesian square of s-schemes over S of the form where f and g are smoothable lci s-morphisms. Let ξ denote the excess bundle as in Paragraph 3.3.3. Then we have a canonical identification If u and v are proper, then we also have an identification , and for any v ∈ K(X) and identification Applying Corollary 4.2.3 to the zero section s : X → E of a vector bundle, we obtain the following formula to compute Euler classes in E 0 (X, E ):  We now introduce the notions of refined fundamental class and refined Gysin maps, following Fulton's treatment in intersection theory (cf. [Ful98,6.2]). Definition 4.2.5. Suppose given a cartesian square of s-schemes over S of the form where f is a smoothable lci s-morphism.
(i) The refined fundamental class of f , with respect to ∆ and with coefficients in E, is the class The refined Gysin map of f , with respect to ∆ and with coefficients in E, is the Gysin map associated to the orientation (η E ∆ , u * L f ) of g, in the sense of Definition 2.4.1. That is, it is the induced map of bivariant spectra for any e ∈ K(Y ′ ). We sometimes denote it also by g ! ∆ .
In terms of refined fundamental classes, we can reformulate the transverse base change and excess intersection formulas as follows: of maps E(Y ′ /S, u * (e)) → E(X/S, L f + e), for any e ∈ K(Y ). (ii) If g is also smoothable lci s-morphism, then we have an identification In particular the refined Gysin map g ! ∆ is identified with the composite γ e(ξ,E) • g ! , where the map γ e(ξ,E) is multiplication by e(ξ, E). (iii) If the square ∆ is tor-independent, so that in particular g is also smoothable lci s-morphism, then we have an identification In particular the refined Gysin map g ! ∆ is identified with the Gysin map g ! . Proof.
, as well as trace and cotrace maps (Paragraph 2.5.3): These natural transformations satisfy 2-functoriality and transverse base change properties as described in Proposition 2.5.4 and Corollary 2.5.6.  In particular, this is another (obviously equivalent) way to realize the Gysin maps considered in Theorem 4.2.1.
We now observe that the purity transformation can be defined for any motivic ∞-category of coefficients: 4.3.4. Let T be a motivic ∞-category of coefficients in the sense of [Kha16, Chap. 2, Def. 3.5.2], defined on the site S of (qcqs) schemes. That is, T is a presheaf of symmetric monoidal presentable ∞-categories on S satisfying certain axioms that guarantee (see [Kha16,Chap. 2,Cor. 4.2.3]) that T admits a full homotopy coherent formalism of six operations.
At this point we note that all the definitions and constructions in Sections 2 and 3 make sense in the setting of T (and not only SH ), as they only use the six operations. In particular: (1) One can define the four theories (Definition 2.2.1) in this setting. For example, the bivariant theory represented by any E ∈ T (S) is given by: where p : X → S is an s-morphism and v ∈ K(X). Here we have written ½ S ∈ T (S) for the monoidal unit, and Th X (−v, T ) for the Thom space 10 internal to T .
(2) We have fundamental classes η T f ∈ E(X/Y, − L f , T ) for any smoothable lci s-morphism f : X → Y of s-schemes over S, with coefficients in any E ∈ T (S) for arbitrary T . These again form a system of fundamental classes as in Theorem 3.3.2, satisfying stability under transverse base change.
(3) We have Gysin maps in bivariant theory with coefficients in any E ∈ T (S) (as well as in the other three theories) for arbitrary T . These Gysin maps are functorial, satisfy transverse base change and excess intersection formulas. (4) We have natural transformations for any v ∈ K(X) of virtual rank r, which are functorial and respect the E ∞ -group structure on K(X) up to a homotopy coherent system of compatibilities. Then for any smoothable lci s-morphism f of relative virtual dimension d, the purity transformation takes the form: [2d] → f ! and similarly for the trace and cotrace maps. Example 4.3.6. Let S = Spec(Z[1/ℓ]) for a prime ℓ, and let Λ be one of Z/ℓ n Z, Z ℓ , or Q ℓ . Then, as X varies over S-schemes, the stable ∞-category ofétale Λ-sheaves D(Xé t , Λ) defines a motivic ∞-category of coefficients (see [SGA4], [Eke90], [CD16], [LZ12]). In particular, we obtain purity transformations of the form (4.3.5.a) generalizing the previously known constructions. Definition 4.3.7. Let S be a scheme and f : X → S a smoothable lci s-morphism of S-schemes. Let T be a motivic ∞-category of coefficients. We say that E ∈ T (S) is f -pure if the canonical morphism Note in particular that, for T = SH , the orientation η f is universally strong (Definition 2.3.2) if and only if S S is f -pure. We have the following variant of Lemma 2.3.14: Lemma 4.3.9. Let T be a motivic ∞-category of coefficients. Suppose that f : X → S is a smoothable lci s-morphism and that E ∈ T (S) is an f -pure object. Then there are duality isomorphisms Recall that an ∞-category of coefficients T is called continuous if, whenever a scheme S can be written as the limit of a filtered diagram (S α ) α of schemes with affine dominant transition maps, then the canonical functor colim is an equivalence, where the colimit is taken in the ∞-category of presentable ∞-categories (and colimit-preserving functors).
Proposition 4.3.10. Let S be a scheme, T a motivic ∞-category of coefficients, and E ∈ T (S) an object. Let f : X → Y be a smoothable s-morphism of S-schemes and denote by p : X → S and q : Y → S the structure morphisms. Assume that one of the following conditions is satisfied: (i) X and Y are smooth over S. (ii) X and Y are regular, and S is the spectrum of a field k. The ∞-category of coefficients T is continuous. The object E is defined 11 over a perfect subfield of k.
Then the morphism f is lci, and q * E is f -pure.
Proof. Since f factors through a closed immersion and a smooth morphism, we may reduce to the case of closed immersions, using the associativity formula and the fact that p T p is invertible for p smooth. Moreover, in both cases f is automatically a regular closed immersion. The second case reduces to the first by using the continuity property of T together with Popescu's theorem [Swa98]. For the first case, the morphisms p : X → S and q : Y → S are smooth. By construction we have a commutative diagram where the left-hand vertical arrow is invertible by Lemma 3.2.15 and the fact that η p is an isomorphism for p smooth (Definition 2.3.5). Therefore it suffices to note that the morphism induced by the exchange transformation Ex ! * ⊗ (Paragraph 2.1.10) is invertible. After writing E X = p * (E) and E Y = q * (E), and using the purity isomorphisms p p : p * ≃ Σ −Tp p ! and p q : q * ≃ Σ −Tq q ! (Paragraph 2.1.7), this follows from the the ⊗-invertibility of Thom spaces.
The following definition first appears (as a conjecture) in the context ofétale cohomology in [SGA5, I, 3.1.4]. In our setting it was already introduced in [Dég18b,Dég18a]. The following could be regarded as a more precise formulation, though in fact it is not difficult to see that both definitions are equivalent (cf. [Dég18a,Prop. 4

.2.2]).
Definition 4.3.11. Let S be a scheme, T be a motivic ∞-category of coefficients, and E ∈ T (S) an object. We say that E satisfies absolute purity if the following condition holds: given any commutative triangle where f , p and q are s-morphisms, f is smoothable lci, and X and Y are regular, then the inverse image q * (E) ∈ T (Y ) is f -pure.

(i)
In view of the functoriality property of the purity transformation (Proposition 2.5.4), and by part (i) of Remark 4.3.8, it suffices to check the absolute purity property for diagrams as above where f is a closed immersion. (ii) If S is the spectrum of a field and T is continuous, then it follows from Proposition 4.3.10 that every E ∈ T (S) satisfies absolute purity (see [DFKJ20, Appendix C] for more details). (iii) Given the previous definition, the absolute purity property is stable under direct factors, extensions, tensor product with strongly dualizable objects (as in Remark 4.3.8). One also deduces from the projection formula that absolutely pure objects are stable under direct image p * for p smooth and proper.   for every s-scheme X over S and every class v ∈ K(X) of virtual rank r, which are functorial and respect the E ∞ -group structure on K(X) up to a homotopy coherent system of compatibilities. Here we write simply r ∈ K(X) for the class of the trivial bundle of rank r. which are related to the Gysin maps of Theorem 4.2.1 via a commutative diagram: Therefore, Theorem 4.1.4 gives in particular a homotopy coherent refinement of the construction of [Dég18a]. Note, by the way, that the diagram above gives a simple proof of the Grothendieck-Riemann-Roch formula (cf.    [PW19] 14 , for any regular S-scheme X, there exists a motivic ring spectrum BO X ∈ SH (X) that represents hermitian Ktheory of smooth X-schemes. In view of its geometric model (denoted by BO geom in op.cit.), BO is defined over S (in the sense that there are canonical isomorphisms BO X ≃ f * (BO S ) for every f : X → S). Note that for non-regular schemes, BO-cohomology is a homotopy invariant version of hermitian K-theory (on the model of [Cis13]), though this notion has not yet been introduced and worked out as far as we know.
The twisted bivariant theory associated with BO as above is new. The Gysin morphisms that one gets on BO-cohomology are also new, at least in the generality of arbitrary proper smoothable lci s-morphisms, between arbitrary schemes (possibly singular and not defined over a base field). In the case of regular schemes, our construction for some part of hermitian K-theory (namely, that which compares to Balmer's higher Witt groups) should be compared to that of [CH11]. This would require a similar discussion to that of Paragraph 4.4.3 as, according to Panin and Walter, hermitian K-theory has a special kind of orientation which allows to consider only twists by line bundles (see also the next example). We intend to come back to these questions in a future work. Note that the Gysin morphisms for Balmer-Witt groups agree with the construction in [Nen07] in the case of a closed immersion between smooth quasi-projective schemes over a field of characteristic different from 2.
Example 4.4.6. Higher Chow-Witt groups. Let k be a perfect field. Introduced by Barge and Morel, the theory of Chow-Witt groups was fully developed by Fasel [Fas07,Fas08]. More recently, the theory was extended to "higher Chow-Witt groups" in a series of works [CF14,DF17a,DF17b]. In particular, given any coefficient ring R, there exists a motivic ring spectrum H MW R in SH (k) called the Milnor-Witt spectrum (cf. [DF17b, 3.1.2]). We denote by H MW (X/k, v, R) (resp. H MW (X, v, R)) its associated bivariant theory (resp. cohomology).
For any smooth s-scheme X over k and any v ∈ K(X) of virtual rank r, one has a canonical isomorphism H 0 MW (X, v, R) ≃ CH r (X, det(v)) ⊗ R, 13 For the record, Grothendieck mentioned that such a direct proof of his formula, without going through a factorisation and the use of a blow-up, should exist. 14 In the case where S is the spectrum of a field of characteristic different from 2, then one can also take the ring spectrum constructed in [Hor05].
which is contravariantly functorial in X and covariantly functorial in v [DF17a, 4.2.6, 4.2.7]. In particular, the ring spectrum H MW is symplectically oriented in the sense of Panin and Walter [PW19]. When X is possibly non-smooth, the bivariant theory H MW 0 (X/k, v) can be computed by a Gersten complex with coefficients in the Milnor-Witt ring of the residue fields, so we can put: (X/k, v, R) = CH r (X, det(v)) ⊗ R and view this as the Chow-Witt group of the scheme X. Similarly, the groups H MW i (X/k, v) for i ≥ 0 can be viewed as the higher Chow-Witt groups. In fact, we have canonical maps where n is the rank of the virtual bundle v, which are functorial in X with respect to proper pushforward (resp. pullback along open immersions).
The construction of Theorem 4.2.1 gives Gysin maps on these higher Chow-Witt groups, for any smoothable lci s-morphisms. These Gysin maps are functorial and satisfy transverse base change and excess intersection formulas. Furthermore, the maps ϕ X are compatible with Gysin morphisms by construction. All in all, we get a robust bivariant theory of higher Chow-Witt groups.
Example 4.4.7. A 1 -homology. Let S be a scheme. Recall that for any commutative ring R, there is a motivic ring spectrum NR S ∈ SH (S) representing A 1 -homology with coefficients. This is nothing else than the R-linearization S S ⊗ R of the motivic sphere spectrum (see [CD19,5.3.35] for another description). It is clear that NR is stable under base change in the sense that there are tautological isomorphisms f * (NR S ) ≃ NR T for every morphism f : T → S.
By Theorem 4.2.1 we obtain Gysin morphisms for the associated bivariant theories and cohomologies. Note in particular that this gives a very general notion of transfer maps in cohomology, along arbitary finite lci morphisms, extending the definitions of Morel in [Mor12]. 15 4.5. Application: specializations. In this subsection we investigate two of the many applications of the theory of refined Gysin maps (Definition 4.2.5). Throughout this subsection, we fix a motivic ∞-category of coefficients T , a scheme S, and an object E ∈ T (S). 4.5.1. Let S be a scheme. For an s-scheme X over S, any section s : S → X which is a regular closed immersion, and any s-morphism p : Y → X, consider the cartesian square  In the case where S = Spec(k) is the spectrum of a field, we can take s : S → X to be any regular k-rational point.   [Ful98,Sect. 10.1] to higher Chow groups. 15 Morel defines transfer maps only for finite field extensions, but he works unstably. the commutativity of the square Th X (−T p ) δ ! (π 1 ) * (S X ) δ * (π 1 ) * (S X )