Cohomology of the moduli stack of algebraic vector bundles

Let $\mathscr{V}\mathrm{ect}_n$ be the moduli stack of vector bundles of rank $n$ on schemes. We prove that, if $E$ is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies the projective bundle formula, then $E^*(\mathscr{V}\mathrm{ect}_{n,S})$ is freely generated by Chern classes $c_1,\dotsc,c_n$ over $E^*(S)$ for any scheme $S$. Examples include all multiplicative localizing invariants.

In algebraic topology, the cohomology of the classifying spaces of unitary groups is fundamental: for a complex oriented cohomology theory E and n ≥ 0, there is a canonical ring isomorphism where c 1 , . . ., c n are the universal Chern classes.The goal of this paper is to establish its algebraic counterpart.Examples of algebraic cohomology theories our results apply to are localizing invariants in the sense of [BGT13] such as algebraic K-theory and topological Hochschild homology, as well as non 1 -localized algebraic cobordism, which we define.In order to work in this generality, we develop a version of motivic homotopy theory and show that all localizing invariants are representable there.This would be the key computational step toward further study of algebraic K-theory and algebraic cobordism beyond 1 -homotopy invariance. 1  Let us start by discussing what the algebraic counterpart of complex oriented cohomology theories should be.To detect orientations, we take the viewpoint of transfers; see [Qui71,§1] for the relation between complex orientations, cobordism, and transfers in algebraic topology.On the algebraic side, it is shown in [EHKSY20] that the algebraic cobordism MGL is universal among 1 -local motivic spectra with finite quasismooth transfers.Taking this into account, we consider Zariski sheaves with finite quasi-smooth transfers on Date: March 6, 2023.The first author was support by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.The second author was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 896517.
In the sequel [AI23], the results and techniques obtained in this paper are applied to prove a universality of K-theory.Generalizations of the results in this paper are also established in op.cit.derived schemes, which we call sheaves with transfers for short, cf.Definition 1.1 and 1.6.In practice, we restrict to sheaves on the ∞-category Sch S of derived schemes of finite presentation over a qcqs derived scheme S. We remark that derived schemes are essential to formulate sheaves with finite quasi-smooth transfers.
Instead of the 1 -homotopy invariance as in Morel-Voevodsky's theory [MV99], our basic input is the projective bundle formula.We say that a sheaf E with transfers on Sch S satisfies projective bundle formula or is pbf-local if the map ) is an equivalence for every X ∈ Sch S and n ≥ 0, where ι * is the pushforward along a fixed linear embedding n−1 → n , which comes from the transfers of E. All localizing invariants satisfy projective bundle formula, while they do not satisfy 1 -homotopy invariance in general.
To state our main result, we introduce some notations.Let Shv tr (Sch S ) be the ∞-category of sheaves with transfers on Sch S (Definition 1.6), Shv tr pbf (Sch S ) its full subcategory spanned by pbf-local sheaves with transfers, and SH tr pbf (Sch S ) the ∞-category of spectrum objects in Shv tr pbf (Sch S ).Then SH tr pbf (Sch S ) has a canonical symmetric monoidal structure and an algebra object there is a sheaf of ring spectra which is equipped with transfers and satisfies projective bundle formula.For example, a multiplicative localizing invariant yields an ∞ -algebra in SH tr pbf (Sch S ) (Corollary 5.4).Let Vect n be the moduli stack of vector bundles of rank n.If E is a presheaf of spectra and X is an algebraic stack, then we write E * (X ) Theorem A (Corollary 4.7).Let S be a qcqs derived scheme and n ≥ 0. Let E be a homotopy commutative algebra in SH tr pbf (Sch S ).Then there is a canonical ring isomorphism The isomorphism is well-known for oriented 1 -local motivic ring spectra, cf.[NSØ09, Proposition 6.2].For algebraic K-theory, the isomorphism was known when the base is a regular scheme as a special case of the 1 -local motivic result.Topological Hochschild homology is not 1 -homotopy invariant, but it would be possible to prove the isomorphism directly by using the comparison with the de Rham-Witt complex; see [Tot18] for a related computation.Our theorem extends these to results on general localizing invariants over general bases and gives a unified proof.
Theorem A follows from a comparison of the "motivic" homotopy type of Vect n and that of the infinite grassmannian Gr n .The cohomology of the latter is rather simple to calculate and we get Theorem A. The comparison is stated as follows.Let γ * be the left adjoint of the forgetful functor Shv tr (Sch S ) → Shv(Sch S ) and L mot the localization functor enforcing projective bundle formula on sheaves with transfers.Then: Theorem B (Theorem 3.1).For every qcqs derived scheme S and n ≥ 0, the canonical map is an equivalence.
We believe that our methods would give new insights into the theory of motives beyond 1 -homotopy invariance.Over the past few years, there have been several attempts to study non 1 -local motivic phenomena as in [Bin20, BPØ20, KMSY21].However, all of them have technical limitations; for example, one cannot expect that general localizing invariants are representable there (since their categories are -linear), while they are representable in our category.In addition to localizing invariants, we are pursing more general theories such as algebraic cobordism.The initial algebra object in SH tr pbf (Sch S ) should be regarded as the "periodic algebraic cobordism" PMGL, which has not been constructed without 1 -localization.In fact, we have a modification so that the initial algebra object represents the non-periodic algebraic cobordism MGL.Then Voevodsky's cobordism should be recovered as the 1 -localization L 1 MGL and Annala's cobordism in [Ann21] should be recovered as π 0 MGL.We hope to discuss these comparisons elsewhere.
We conclude this introduction with a brief outline of this paper.In Section 1, we build setups for sheaves with transfers which satisfy projective bundle formula.In Section 2, we prove some basic properties of pbflocal sheaves with transfers in terms of Euler classes.In Section 3, we prove a key technical lemma for the proof of Theorem B and prove the comparison for n = 1.In Section 4, we develop a theory of Chern classes and use it for computing the cohomology of grassmannians.Then we complete the proof of Theorem B and get Theorem A as its corollary.In Section 5, we prove that every localizing invariant is representable in SH tr pbf (Sch S ) so that a multiplicative localizing invariant yields an ∞ -algebra in SH tr pbf (Sch S ).Convention.We use the language of ∞-categories as set out in [Lur17a,Lur17b].We refer to [Lur18] for the theory of derived schemes.We assume that all derived schemes are qcqs (quasi-compact and quasiseparated).We say that a morphism X → Y of derived schemes is quasi-smooth if it is of finite presentation and the cotangent complex L X /Y is of Tor-amplitude ≥ 1, cf. [KR19, §2].A vector bundle on a derived scheme X is a locally free quasi-coherent module of finite rank on X .Our treatment of sheaves with transfers are strongly influenced by the theory of framed correspondences as developed in [EHKSY21], from which we adopt some notations.

SHEAVES WITH TRANSFERS AND PROJECTIVE BUNDLE FORMULA
Fix a qcqs derived scheme S. Let Sch S be the ∞-category of derived schemes of finite presentation over S and Corr(Sch S ) the ∞-category of correspondences/spans in Sch S , cf. [BH21, Appendix C].By the assumption of finite presentation, the ∞-categories Sch S and Corr(Sch S ) are essentially small.Recall that an object of Corr(Sch S ) is an object of Sch S and a morphism from Let Corr fqsm (Sch S ) be the subcategory of Corr(Sch S ) spanned by morphisms whose left span is finite and quasi-smooth.
One can think of f * as a Gysin morphism.
Example 1.4.We write FQSm S for the presheaf with transfers on Sch S represented by S, which is by definition the moduli stack of finite quasi-smooth derived schemes.
The canonical functor γ: Sch S → Corr fqsm (Sch S ) preserves finite coproducts and Corr fqsm (Sch S ) is canonically endowed with a symmetric monoidal structure for which γ is symmetric monoidal.It follows that γ induces an adjunction γ * : PSh Σ (Sch S ) ⇄ PSh tr Σ (Sch S ): γ * and γ * is symmetric monoidal.Here, we endow PSh Σ (Sch S ) and PSh tr Σ (Sch S ) with the unique symmetric monoidal structures for which the Yoneda embeddings are symmetric monoidal and the tensor products are compatible with small colimits, cf.[Lur17b, Proposition 4.8.1.10].We note that the projection formula is automatic for presheaves with transfers.
Lemma 1.5 (Projection formula).Let E be a presheaf with transfers on Sch S and f : Y → X a finite quasismooth morphism in Sch S .Then the following diagrams commute where E ⊗ E denotes the tensor product in PSh tr Σ (Sch S ).
Proof.Since the tensor products commute with colimits, we may assume that E is representable by a derived scheme in Sch S .Then the commutativity is straightforward to check.
Definition 1.6.A sheaf with transfers on Sch S is a presheaf E with transfers such that γ * E is a sheaf on the Zariski site Sch S,Zar .We write Shv tr (Sch S ) for the full subcategory of PSh tr Σ (Sch S ) spanned by sheaves with transfers.
Definition 1.7.We say that a (pre)sheaf E with transfers on Sch S satisfies projective bundle formula or is pbf-local if, for every X ∈ Sch S and n ≥ 0, the map ) is an equivalence.We write Shv tr pbf (Sch S ) for the full subcategory of PSh tr Σ (Sch S ) spanned by pbf-local sheaves with transfers.
Remark 1.8.If E is a pbf-local presheaf with transfers, then for any X -points a, b : Remark 1.9.Every pbf-local sheaf with transfers has its values in grouplike ∞ -spaces.We give a proof in the next section, cf.Corollary 2.9.Remark 1.11.A pbf-local sheaf E of spectra with transfers can be identified with a presheaf of spectra on Corr fqsm (Sch S ) such that Ω ∞−n E is a pbf-local sheaf with transfers for every n ≥ 0.
There is an adjunction and the left adjoint B ∞ mot is called the infinite bar construction.2Then SH tr pbf (Sch S ) admits a unique symmetric monoidal structure for which the infinite bar construction is symmetric monoidal by [GGN15, Theorem 5.1].
Example 1.12.Every localizing invariant in the sense of [BGT13] naturally defines a pbf-local sheaf of spectra with transfers on Sch S for each qcqs derived scheme S.Moreover, if a localizing invariant is multiplicative, then it defines an ∞ -algebra in SH tr pbf (Sch S ).We give a proof in Section 5.
Base changes.Let f : T → S be a morphism of derived schemes.Then the base change functor Sch S → Sch T induces adjunctions It is straightforward to see that the left adjoints f * are symmetric monoidal and compatible with the localization functor L mot and the infinite bar construction B ∞ mot .
Approximation of the motivic localization.The assignments n → ∆n and n → ∆n ∞ form semi-cosimplicial objects in the category of schemes in a standard way, and the map ι : ∆n Lemma 1.13.Let E be a presheaf with transfers.Then the canonical map E → L mot E factors as where |(−)| denotes the geometric realization.
Proof.Note that if E is pbf-local then there is a canonical equivalence from which the desired factorization is immediate.

EULER CLASSES
Definition 2.1.We define PMGL S := L mot FQSm S , which is a unit object in the symmetric monoidal ∞category Shv tr pbf (Sch S ).
Remark 2.2.PMGL S is a version of the periodic algebraic cobordism.Comparison with the existing theories will be discussed elsewhere.See [EHKSY20, Corollary 3.4.2]for the similar description of the 1 -local algebraic cobordism.
Definition 2.3.Let be a vector bundle on a derived scheme X ∈ Sch S .We define the Euler class e( ) ∈ π 0 PMGL S (X ) by e( ) := s * s * (1 X ), where s denotes the zero section of the total space of s : X → ( ) := Spec(Sym( ∨ )).
Remark 2.4.The Euler classes are compatible with base changes, i.e., for a morphism f : Y → X in Sch S , we have f * e( ) = e( f * ).Since every pbf-local sheaf E with transfers is a module over PMGL S , the Euler class of a vector bundle on X yields an endomorphism of E(X ) up to homotopies.
Let a be a global section of a vector bundle on X .Then the derived vanishing locus V a of a is a derived scheme defined by the cartesian square By definition, we have e( ) = j 0 * (1 V 0 ).Lemma 2.5.Let be a vector bundle on a derived scheme X ∈ Sch S .Then, for any global section a of , we have j a * (1 V a ) = e( ).In particular, if admits a nowhere vanishing global section, then e( ) = 0.
Proof.Let V be the derived vanishing locus of the global section at 0 of the twisting sheaf (1) on 1 X , and let j : Lemma 2.6.Let E be a pbf-local sheaf with transfers on Sch S .Let be a vector bundle of rank r ≥ 1 on a derived scheme X ∈ Sch S and ξ the Euler class of the canonical line bundle (1) on ( ).Then the morphism r−1 i=0 is an equivalence.
Proof.Since the question is local on X , we may assume that the is trivial, i.e., = r X .We prove by induction on r.The case r = 1 is trivial, and let r ≥ 2. Fix a linear embedding j : r−2 X → r−1 X , which is given by x ∈ H 0 ( r−1 X , (1)).Then we have a cartesian diagram and j * (1) = ξ by Lemma 2.5.It follows that the diagram commutes.The right vertical arrow is an equivalence since E is pbf-local.Therefore, the result follows from the induction hypothesis.
Corollary 2.7.Let E be a homotopy commutative algebra in SH tr pbf (Sch S ), i.e., a commutative algebra object in the homotopy category.Then there is a canonical ring isomorphism Proof.By Lemma 2.6, it remains to show that ξ n+1 = 0 in π 0 PMGL S ( n X ).By the projection formula (Lemma 1.5), ξ n+1 is the image of the unit by the Gysin morphism along the inclusion of the derived intersection of (n + 1)-copies of a hyperplane in n , but this is empty and thus ξ n+1 = 0.
Lemma 2.8 (Bass fundamental theorem).Let E be a pbf-local sheaf with transfers on Sch S .Then, for every X ∈ Sch S and i ≥ 0, there is a split exact sequence where the multiplication by ν gives a splitting π i E(X ) → π i+1 E(D + (T 0 T 1 ) X ).

Proof. Consider the diagram
The vertical sequence is exact since E is a sheaf and the horizontal sequence exhibits π i E( ∆1 X ) as a direct sum of two copies of π i E(X ) by Lemma 2.6.The boundary map ∂ factors through the left summand as indicated since the right diagonal map is injective.Now the result follows from a simple diagram chase.
Corollary 2.9.Every pbf-local sheaf with transfers on Sch S has its values in grouplike ∞ -spaces.
Proof.It suffices to show that if E is a pbf-local sheaf with transfers on Sch S then π 0 E(X ) is a group for every X ∈ Sch S .Since π 0 E(X ) is a quotient of π 1 E(D + (T 0 T 1 ) X ) by Lemma 2.8, it is a group.

MODULI STACK OF VECTOR BUNDLES
In this section, we study the homotopy type of the moduli stack of vector bundles.We start by recalling the construction of the moduli stack of vector bundles.Let Sch be the ∞-category of all derived schemes.Then there is a functor QCoh : Sch op → Cat ∞ classifying all quasi-coherent modules, which is constructed as in [Lur18, §6.2.2],where Cat ∞ denotes the ∞-category of possibly large ∞-categories.For a nonnegative integer n, let QCoh lfree n be the subfunctor of QCoh spanned by locally free quasi-coherent modules of rank n.Then the moduli stack Vect n of vector bundles of rank n is the functor defined as the composite where (−) ∼ is the functor taking the maximal subgroupoids of ∞-categories.We write Pic := Vect 1 , which is by definition the Picard stack.
The presheaf Vect n is a fpqc sheaf since so is the presheaf QCoh classifying quasi-coherent modules by [Lur18, Proposition 6.2.3.1] and the property being locally free of rank n is local for the fpqc topology by [Lur18, Proposition 2.9.1.4].The presheaf Vect n is finitary in the sense that it carries filtered limits of qcqs derived schemes with affine transition maps to colimits by [Lur18, Corollary 4.5.1.10].For a qcqs derived scheme S, we write Vect n,S for the restriction of Vect n to Sch S .Then Vect n,S is compatible with base changes, i.e., for every morphism f : T → S of derived schemes, the map f * Vect n,S → Vect n,T is a Zariski local equivalence, cf.[EHKSY20, Proposition A.0.4]For non-negative integers n and N , the n-th grassmannian Gr n ( N ) of N classifies all quotients N ։ , where is a vector bundle of rank n.The projection N +1 → N discarding the last factor induces an immersion Gr n ( N ) → Gr n ( N +1 ).Let Gr n := colim N Gr n ( N ) and regard it as an ind-scheme.We write ∞ := Gr 1 , which is the infinite projective space.
Theorem 3.1.For every qcqs derived scheme S and n ≥ 0, the canonical map L mot γ * Gr n,S → L mot γ * Vect n,S is an equivalence.
The proof is completed in the next section.In this section, we prove a key technical lemma (Lemma 3.3) and prove the equivalence for n = 1.Note that we may assume that S = Spec( ) to prove Theorem 3.1 since both sides commute with base changes.In the rest of this section, we work over the ∞-category Sch := Sch Spec( ) unless otherwise stated.
Proof.Consider the universal line bundle univ on Pic.Then the total space ( univ ) is defined as the stack classifying line bundles with a global section, i.e., ( univ )(X ) is the space of all maps X → with being a line bundle on X .We have a map s : Pic → ( univ ) classifying zero sections, which can be expressed as a colimit of quasi-smooth closed immersions.Hence, the Gysin morphism s * is well-defined and the Euler class of univ is defined by ξ := s * s * (1) ∈ π 0 PMGL(Pic).We note that PMGL(Pic) is complete with respect to ξ since it is a limit of PMGL( ×n m ) where ξ = 0 and a limit of complete modules is complete.Then we have a commutative diagram The left vertical map is well-defined because of the ξ-completeness of PMGL(Pic).The diagonal arrow is an equivalence by Lemma 2.6.In particular, the canonical map γ * ∞ → L mot γ * ∞ lifts to a map γ * Pic → L mot γ * ∞ up to homotopies, which gives a desired left inverse.
The next goal is to construct a right inverse of L mot γ * Gr n → L mot γ * Vect n .In order to do that, we pursue the idea of closed gluing as in [EHKSY21, §A.2], which provides a method to show that some map is an 1 -homotopy equivalence.See [HJNTY20, Proposition 4.7], where it is used to show that the canonical map Gr n → Vect n is an 1 -homotopy equivalence on affine schemes.Adopting this to our situation, we can try to solve the lifting problem However, it will turn out soon that this is impossible, mainly because the sheaf cohomology of ∆ * is nontrivial.To fix this, note that for each l ≥ 1 the twisting sheaves (l) on ∆ * define a point of the semisimplicial sheaf Pic( ∆• ).Then we prove the following.

Lemma 3.3. Let k be a non-negative integer.
(A) The diagram Proof.Note that the ∞-category Shv(Sch ) of Zariski sheaves is a hypercomplete ∞-topos since each derived scheme X ∈ Sch is assumed to be of finite presentation over .
(A) Let F be the semi-simplicial object in Shv(Sch ) defined by the pullback square We show that the induced map |F | → |Vect n | on the geometric realization is an equivalence in Shv(Sch ).By [Lur18, Theorem A.5.3.1], it suffices to show that the map F → Vect n is a trivial Kan fibration, i.e., the map is an effective epimorphism for every m ≥ 0, see [op.cit., A.5.1.6]for the notation.This map is identified with the map and it is an equivalence for m > k, and thus we may assume that m ≤ k.Then, since Gr n and Vect n satisfy closed gluing by [Lur18, Example 17.3.1.3],the map is further identified with for a given surjection α.Note that the bottom row is a fiber sequence.Since H 1 ( ∆m A , (k − m) ⊕n ) = 0, there exists a lift α ′ as indicated.Furthermore, since (k − m) is globally generated, we can add extra sections to ensure that the lift α ′ is surjective.This completes the proof of (A).
(B) For each m ≥ 0, we have a commutative diagram of semi-simplicial objects Hence, it suffices to show that the top horizontal composition is homotopic to the canonical map.Note that, for any point * ∈ ∆m not meeting ∆m ∞ , the pullback along the inclusion i : { * } → ∆m induces an inverse of p * on the top.Since i * (k + 1) is trivial, the assertion follows.
Corollary 3.4.For every n, k ≥ 0, the diagram where the rightmost horizontal arrows come from the canonical map |cosk k X | → |cosk k X | ≤k ≃ |X | ≤k for a semi-simplicial space X .Hence, we get a desired lift.
Corollary 3.5.For every qcqs derived scheme S, the canonical map Proof.We may assume S = Spec( ).By Corollary 3.4, we have a commutative diagram The map φ k is characterized as a unique map which makes the diagram commutative since the right vertical map has a left inverse by Lemma 3.2.In particular, these maps are assembled into a map Since the ∞-topos Shv(X Zar ) has finite homotopy dimension for each derived scheme X ∈ Sch , we have an equivalence lim k L Zar (γ 3 Since the canonical map has a left inverse by Lemma 3.2, we conclude that it is an equivalence.Corollary 3.6.Let E be a homotopy commutative algebra in SH tr pbf (Sch S ).Then there is a canonical ring isomorphism where t is the Euler class of the universal line bundle.
Proof.This is immediate from Corollary 2.7 and Corollary 3.5.

CHERN CLASSES AND FORMAL GROUP LAWS
Definition 4.1.Let be a vector bundle of rank r ≥ 1 on a derived scheme X ∈ Sch S .We let c 0 ( ) = 1.For 1 ≤ i ≤ r, we define the i-th Chern class c i ( ) ∈ π 0 PMGL S (X ) to be unique elements which satisfy the formula in π 0 PMGL S ( ( )), cf.Lemma 2.6.We write c( ) := r i=0 c i ( )t i , which is the total Chern class.Let m: Pic × Pic → Pic be the map classifying the tensor products of line bundles.Consider the induced map and let f univ be the image of c 1 ( univ ) in where the isomorphism is by Corollary 3.6.Then f univ is a formal group law over π 0 PMGL S (S).Moreover, the formal inverse power series of t in π 0 PMGL S (S)[[t]] ≃ π 0 PMGL S (Pic S ) corresponds to the first Chern class of −1 univ .The next two lemmas are the standard properties of Chern classes and formal group laws.
Lemma 4.3.Suppose that X ∈ Sch S is noetherian and admits an ample line bundle.Then the first Chern classes of line bundles on X are nilpotent in π 0 PMGL S (X ) and we have an equality for every pair of line bundles , ′ on X .
Proof.If is a globally generated line bundle, then it is generated by a finite number of global sections since X is noetherian.Then it is immediate from the definition of the Euler class that e( ) = c 1 ( ) is nilpotent.It follows that the map induced by the map X → Pic classifying factors through π 0 PMGL S (X )[t]/t m for some m > 0. In particular, c 1 ( −1 ), which is the image of the formal inverse of t, is also nilpotent.
Since X admits an ample line bundle, every line bundle on X can be written as 1 ⊗ −1 2 for some globally generated line bundles 1 and 2 .Since we have seen that c 1 ( 1 ) and c 1 ( −1 2 ) are nilpotent, the map , y) m for some m > 0. Since c 1 ( ) is the image of f univ , it is nilpotent.Then the last claim is an immediate consequence of the construction.Lemma 4.4.Suppose that X ∈ Sch S is noetherian and admits an ample line bundle.Let be a vector bundle of rank r ≥ 1 on X .Then the following hold in π 0 PMGL S (X ): (1 + c 1 ( i )t).
Proof.Firstly, (iv) follows from (iii) by taking the pullback of to the derived scheme representing full flags of .Similarly, (ii) follows from (iii) and Lemma 4.3.We need a splitting trick for the other assertions.Suppose we are given a fiber sequence of vector bundles on X .Let t 0 , t 1 be homogeneous coordinates of 1 , and let ˜ be the cofiber of the map of vector bundles on 1 × X .Then i * (i) It is obvious for line bundles.For the general case, we may assume that is a direct sum of line bundles r i=1 i by the splitting trick.Since Sym( i ) = i Sym( i ), we have e( ) = i e( i ), and thus the assertion follows from (iii).
(iii) We may assume that = r i=1 i .Consider the universal quotient → (1) on ( ).The induced map i → (1) gives a global section s i of −1 i (1), and let D i ⊂ ( ) be the derived vanishing locus of s i .Proof of Theorem 3.1.We may assume that the base scheme is S = Spec( ).We first prove that the canonical map admits a left inverse.Let univ be the universal vector bundle of rank n over Vect n .Then the projective space ( univ ) is defined as the stack classifying quotients univ ։ with being a line bundle.Then the Euler class ξ of the universal quotient bundle on ( univ ) is defined as in Lemma 3.2 and the map is an isomorphism for any pbf-local sheaf E with transfers by Lemma 2.6.Hence, we can define the Chern classes c i of univ by the formula in Definition 4.1.Then we get a commutative diagram and the diagonal arrow is an equivalence by Lemma 4.5.Hence, the canonical map γ * Gr n → L mot γ * Gr n lifts to a map γ * Vect n → L mot γ * Gr n , which gives a desired left inverse.The rest of the proof is identical to that of Corollary 3.5.By Corollary 3.4, we have a commutative diagram The map φ k is characterized as a unique map which makes the diagram commutative since the right vertical map has a left inverse.In particular, these maps are assembled into a map which induces a right inverse of the canonical map L mot γ * Gr n → L mot γ * Vect n . 4Since we have seen that the canonical map has a left inverse, we conclude that it is an equivalence.

4
As in the proof of Corollary 3.5, it is not clear if the induced map is indeed a right inverse.This gap has been fixed in [AI23], and the result here remains true.
Corollary 4.7.Let E be a homotopy commutative algebra in SH tr pbf (Sch S ).Then there is a canonical ring isomorphism where c i is the i-th Chern class of the universal vector bundle of rank n.
Proof.This follows from Theorem 3.1 and Corollary 4.6.

REPRESENTABILITY OF LOCALIZING INVARIANTS
is an equivalence, where f * := F ( f ) and f * is a right adjoint of f * .

We remark that the functor
Proof.We equip ⊗ with the vertical marking, i.e., a morphism f in ⊗ is marked if and only if it lies over the identity 〈n〉 → 〈n〉 for some n ≥ 0 and f is a product of marked morphisms in .It suffices to show that the functor F : ⊗ → Cat ∞ is right bivariant with collar change; then we can apply Macpherson's extension theorem [Mac20, Theorem 3.8.1]and the extension is lax cartesian since ⊗ → coCorr( ) ⊗ is essentially surjective.We have to show that, for a cocartesian square (Y 1 , . . ., Y m ) g f / / (X 1 , . . ., X m ) in ⊗ with f being a product of marked morphisms, the base change map g * f * → f ′ * g ′ * is an equivalence.This is clear if g is an inert morphism and thus we may assume that g is active.Then, by induction, the problem is reduced to the right adjointability of cocartesian squares of the form where δ is the codiagonal map.The square is cocartesian if and only if X ≃ X 1 ⊔ Y X 2 , and in this case the base change map δ 2 ), where f ′ i is the canonical map X i → X .Then it is an equivalence by the projection formula.
Proof.By [Lur17b, Proposition 2.4.1.7],the ∞-category Fun lax ( ⊗ , Cat ∞ ) of lax cartesian structures is equivalent to the ∞-category Fun lax ( ⊗ , Cat × ∞ ) of lax symmetric monoidal functors.Under this equivalence, a lax symmetric monoidal functor F : ⊗ → Cat × ∞ factors through the subcategory Cat ∞ (K) ⊗ constructed in [Lur17b, Notation 4.8.1.2],where we take K to be the set of all finite simplicial sets, if and only if the ∞-category F (X ) admits finite colimits for each X ∈ and F satisfies the condition (ii).Furthermore, since Cat ex ∞ is a reflective subcategory of Cat ∞ (K) and the left adjoint is symmetric monoidal, we conclude that F uniquely factors thought a lax symmetric monoidal functor ⊗ → (Cat ex ∞ ) ⊗ .
Corollary 5.4.Let E be a multiplicative localizing invariant and S a qcqs derived scheme.Then the associated pbf-local sheaf E of spectra with transfers is promoted to an ∞ -algebra in SH tr pbf (Sch S ).
Proof.We apply Lemma 5.2 to = Sch op and F = Perf.Then Perf satisfies projection formula by [Lur18, Remark 3.4.2.6], and thus we get a lax cartesian structure Perf † : (Corr fqsm (Sch) op ) ⊗ → Cat ∞ .This extension satisfies the condition of Lemma 5.3, and thus Perf † lifts to a lax symmetric monoidal functor to (Cat ex ∞ ) ⊗ .Then we get a lax symmetric monoidal functor We can regard E ⊗ as an ∞ -algebra in the symmetric monoidal ∞-category Fun(Corr fqsm (Sch S ) op , Sp) ⊗ of presheaves of spectra and the Day convolution products.Since the localization functor L mot is symmetric monoidal and E ⊗ underlies the pbf-local sheaf E of spectra with transfers, we conclude that E = L mot E ⊗ is an ∞ -algebra in SH tr pbf (Sch S ).
The ∞-category Shv tr pbf (Sch S ) is an accessible localization of PSh tr Σ (Sch S ) by [Lur17a, Proposition 5.5.4.15].We denote the localization functor by L mot : PSh tr Σ (Sch S ) → Shv tr pbf (Sch S ).Then Shv tr pbf (Sch S ) admits a unique symmetric monoidal structure for which the localization functor L mot is symmetric monoidal by [Lur17b, Proposition 4.1.7.4].Definition 1.10.A pbf-local sheaf of spectra with transfers on Sch S is a spectrum object in the ∞-category Shv tr pbf (Sch S ) in the sense of [Lur17b, §1.4.2].We write SH tr pbf (Sch S ) for the ∞-category of spectrum objects in Shv tr pbf (Sch S ).
2.6].Over an animated ring A such that π 0 (A) is local, π 0 of the right hand side is identified with the set of homotopy equivalent classes of surjections ⊕∞ ∂ ∆m A → (k + 1) ⊕n | ∂ ∆m A .We have to show that such a surjection lifts to a surjection ⊕∞ ∆m A → (k +1) ⊕n up to homotopies.Consider the diagram admits a lift Zariski locally as indicated, where (−) ≤k denotes the pointwise k-truncation.Proof.By Lemma 3.3, we have a commutative diagram with a Zariski local lift as indicated

Remark 4. 2 .
The Chern classes are compatible with base changes since the Euler classes are.If is a line bundle, then c 1 ( ) = e( ).
is nilpotent for each i ≥ 1.Hence, π * E(Gr n ( N S )) is complete for the (c 1 , . . ., c n )-adic topology and the above map factors through π * E(S)[[c 1 , . . ., c n ]].Taking limits with respect N , we obtain a mapπ * E(S)[[c 1 , . . ., c n ]] → lim N π * E(Gr n ( N S )) ≃ π * E(Gr n,S ),which is injective since N I N = ∅ and induces the identity modulo (c 1 , . . ., c n ).Note that π * E(Gr n,S ) is separated for the (c 1 , . . ., c n )-adic topology since we have an injection(c 1 , . . ., c n ) m lim N π * E(Gr n ( N S )) → lim N (c 1 , . . ., c n ) m π * E(Gr n ( N S ))andthe right hand side is zero when m → ∞.Then the result follows from the following observation: if A is an I -adically complete ring and B is an extension ring of A such that A/I A = B/I B and that B is separated for the I -adic topology, then A = B. Indeed, B = A + I B = A + I (A + I B) = • • • , and thus the separatedness implies the claim.Now we can complete the proof of the main theorem.

Notation 1.3. Let
f : Y → X be a morphism in Sch S and E a presheaf with transfers on Sch S .We let !f = f : Y → X denote the morphism in Corr fqsm (Sch S ) and let f Examples of pbf-local sheaves of spectra with transfers are supplied by localizing invariants.Let Cat ex ∞ be the ∞-category of small stable ∞-categories and exact functors.A localizing invariant is a functor Cat ex ∞ → Sp which carries exact sequences in Cat ex ∞ to fiber sequences of spectra, cf.[LT19, Definition 1.2].Let Sch be the ∞-category of all qcqs derived schemes and suppose that it is marked with respect to finite quasi-smooth morphisms in the sense of [Mac20, Definition 2.6.1].Then the functor Perf : Sch op → Cat ex ∞ of perfect complexes is bivariant in the sense of [Mac20, §3.2] by [Lur18, Corollary 3.2.3.3,Theorem6.1.3.2].Hence, by [Mac20, Theorem 3.8.1], the functor Perf extends to a functor Perf † : Corr fqsm (Sch) op → Cat ex ∞ .We regard a localizing invariant as a pbf-local sheaf with transfers by Lemma 5.1.The next goal is to show that a multiplicative localizing invariant yields an ∞ -algebra in SH tr pbf (Sch S ).A multiplicative localizing invariant is a lax symmetric monoidal functor (Cat ex ∞ ) ⊗ → Sp ⊗ whose underlying functor Cat ex ∞ → Sp is a localizing invariant.Let ⊗ be a cocartesian symmetric monoidal ∞-category whose underlying ∞-category is equipped with a marking with collar change [Mac20, 3.1.3] 5.Let F : ⊗ → Cat ∞ be a lax cartesian structure [Lur17b, Definition 2.4.1.1].Assume that: (i) The underlying functor F : → Cat ∞ is right bivariant with collar change [Mac20, 3.2.5].(ii) F satisfies projection formula, i.e., for any marked morphism Lemma 5.1.If E is a localizing invariant and S a qcqs derived scheme, then the compositeCorr fqsm (Sch S ) op → Corr fqsm (Sch) op Perf † −−→ Cat ex ∞ E − → Spis a pbf-local sheaf of spectra with transfers on Sch S .Proof.It suffices to show that the composite E •Perf † satisfies Zariski descent and projective bundle formula.The descent essentially follows from the work [TT90], see also [LT19, Lemma A.1], and the projective bundle formula is satisfied by [Kha20, Theorem B].