Modules over algebraic cobordism
Elden Elmanto, Marc Hoyois, Adeel A. Khan, Vladimir Sosnilo, Maria Yakerson
MMODULES OVER ALGEBRAIC COBORDISM
ELDEN ELMANTO, MARC HOYOIS, ADEEL A. KHAN, VLADIMIR SOSNILO,AND MARIA YAKERSON
Abstract.
We prove that the ∞ -category of MGL-modules over any scheme is equivalentto the ∞ -category of motivic spectra with finite syntomic transfers. Using the recognitionprinciple for infinite P -loop spaces, we deduce that very effective MGL-modules over aperfect field are equivalent to grouplike motivic spaces with finite syntomic transfers.Along the way, we describe any motivic Thom spectrum built from virtual vector bundlesof nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes withthe corresponding tangential structure. In particular, over a regular equicharacteristic base,we show that Ω ∞ P MGL is the A -homotopy type of the moduli stack of virtual finite flatlocal complete intersections, and that for n >
0, Ω ∞ P Σ n P MGL is the A -homotopy type ofthe moduli stack of finite quasi-smooth derived schemes of virtual dimension − n . Contents
1. Introduction 21.1. Modules over algebraic cobordism 21.2. Framed correspondences and the motivic recognition principle 21.3. Geometric models of motivic Thom spectra 31.4. Analogies with topology 41.5. Related work 51.6. Conventions and notation 51.7. Acknowledgments 62. Twisted framed correspondences 62.1. Stable vector bundles and K-theory 62.2. Presheaves of twisted framed correspondences 72.3. Descent and additivity 82.4. Comparison theorems 102.5. Base change 143. Geometric models of motivic Thom spectra 163.1. Thom spectra of vector bundles 163.2. Thom spectra of nonnegative virtual vector bundles 193.3. General nonnegative Thom spectra 213.4. Algebraic cobordism spectra 253.5. Hilbert scheme models 274. Modules over algebraic cobordism 294.1. Modules over MGL 294.2. Modules over MSL 314.3. Motivic cohomology as an MGL-module 31Appendix A. Functors left Kan extended from smooth algebras 32Appendix B. The ∞ -category of twisted framed correspondences 35References 38 Date : August 7, 2019.M.H. was partially supported by NSF grant DMS-1761718.M.Y. was supported by DFG - SPP 1786 “Homotopy Theory and Algebraic Geometry”. a r X i v : . [ m a t h . AG ] A ug E. ELMANTO, M. HOYOIS, A. A. KHAN, V. SOSNILO, AND M. YAKERSON Introduction
This article contains two main results: • a concrete description of the ∞ -category of modules over Voevodsky’s algebraic cobor-dism spectrum MGL (and its variants such as MSL); • a computation of the infinite P -loop spaces of effective motivic Thom spectra in termsof finite quasi-smooth derived schemes with tangential structure and cobordisms.Both results have an incarnation over arbitrary base schemes and take a more concrete formover perfect fields. We discuss them in more details in § § Modules over algebraic cobordism.
If S is a regular scheme over a field with resolutionsof singularities, there are well-known equivalences of ∞ -categories Mod H Z ( SH (S)) (cid:39) DM (S) , Mod H˜ Z ( SH (S)) (cid:39) (cid:103) DM (S) , expressing Voevodsky’s ∞ -category of motives DM (S) and its Milnor–Witt refinement (cid:103) DM (S)as ∞ -categories of modules over the motivic E ∞ -ring spectra H Z and H ˜ Z , which representmotivic cohomology and Milnor–Witt motivic cohomology; see [RØ08, CD15, BF18, EK19].These equivalences mean that a structure of H Z -module (resp. of H ˜ Z -module) on a motivicspectrum is equivalent to a structure of transfers in the sense of Voevodsky (resp. a structureof Milnor–Witt transfers in the sense of Calm`es and Fasel).Our main result, Theorem 4.1.3, gives an analogous description of modules over Voevodsky’salgebraic cobordism spectrum MGL: we construct an equivalence between MGL-modules andmotivic spectra with finite syntomic transfers :(1.1.1) Mod
MGL ( SH (S)) (cid:39) SH fsyn (S) . Notably, we do not need resolutions of singularities and are able to prove this over arbitraryschemes S. Furthermore, we obtain similar results for other motivic Thom spectra; for example,MSL-modules are equivalent to motivic spectra with transfers along finite syntomic morphismswith trivialized canonical sheaf (Theorem 4.2.1). It is worth noting that, even though bothsides of (1.1.1) only involve classical schemes, our construction of the equivalence uses derivedalgebraic geometry in an essential way.Over a perfect field k , we prove a cancellation theorem for finite syntomic correspondences.This allows us to refine (1.1.1) to an equivalence Mod
MGL ( SH veff ( k )) (cid:39) H fsyn ( k ) gp between very effective MGL-modules and grouplike motivic spaces with finite syntomic transfers(see Theorem 4.1.4).1.2.
Framed correspondences and the motivic recognition principle.
The startingpoint of the proof of (1.1.1) is a description of motivic Thom spectra in terms of framed cor-respondences . The notion of framed correspondence was introduced by Voevodsky [Voe01]and later developed by Garkusha, Panin, Ananyevskiy, and Neshitov [GP18a, GP18b, AGP18,GNP18]. Subsequently, a more flexible formalism of framed correspondences was developed bythe authors in [EHK + ∞ -category Corr fr (Sm S )whose objects are smooth S-schemes and whose morphisms are spans(1.2.1) ZX Y f g ODULES OVER ALGEBRAIC COBORDISM 3 with f finite syntomic, together with an equivalence L f (cid:39) L f is the cotangentcomplex of f and K(Z) is the algebraic K-theory space of Z. Starting with the ∞ -category Corr fr (Sm S ), we can define the symmetric monoidal ∞ -categories H fr (S) of framed motivicspaces and SH fr (S) of framed motivic spectra . The reconstruction theorem states that there isan equivalence(1.2.2) SH (S) (cid:39) SH fr (S)between motivic spectra and framed motivic spectra over any scheme S [Hoy18, Theorem 18].This equivalence can be regarded as the “sphere spectrum version” of the equivalences discussedin § Z S turns out to be the framed suspension spectrum of the constant sheaf Z [Hoy18,Theorem 21].Over a perfect field k , the motivic recognition principle states that the framed suspensionfunctor Σ ∞ T , fr : H fr ( k ) → SH fr ( k ) (cid:39) SH ( k )is fully faithful when restricted to grouplike objects, and that its essential image is the subcat-egory of very effective motivic spectra [EHK + ∞ T , fr Σ ∞ T , fr is computed as group completion on framed motivic spaces. Thus, if a motivic spec-trum E over k is shown to be the framed suspension spectrum of a framed motivic space X,then its infinite P -loop space Ω ∞ T E is the group completion X gp . This will be our strategy tocompute the infinite P -loop spaces of motivic Thom spectra.1.3. Geometric models of motivic Thom spectra.
The general notion of motivic Thomspectrum was introduced in [BH18, Section 16]. In particular, there is a motivic Thom spectrumM β associated with any natural transformation β : B → K, where K is the presheaf of K-theoryspaces on smooth schemes. For example, if β is the inclusion of the rank n summand of K-theory, then M β (cid:39) Σ n T MGL. The motivic spectrum M β is very effective when β lands in therank (cid:62) (cid:62) ⊂ K.In this paper, we show that the motivic Thom spectrum M β of any morphism β : B → K (cid:62) isthe framed suspension spectrum of an explicit framed motivic space. Before stating the generalresult more precisely, we mention some important special cases:(1) If Y is smooth over S and ξ ∈ K(Y) is a K-theory element of rank (cid:62)
0, the Thomspectrum Th Y / S ( ξ ) ∈ SH (S) is the framed suspension spectrum of the presheaf on Corr fr (Sm S ) sending X to the ∞ -groupoid of spans (1.2.1) where Z is a derived schemeand f is finite and quasi-smooth, together with an equivalence L f (cid:39) − g ∗ ( ξ ) in K(Z).(2) The algebraic cobordism spectrum MGL S is the framed suspension spectrum of themoduli stack FS yn S of finite syntomic S-schemes. More generally, for n (cid:62)
0, Σ n T MGL S is the framed suspension spectrum of the moduli stack FQS m n S of finite quasi-smoothderived S-schemes of relative virtual dimension − n .(3) The special linear algebraic cobordism spectrum MSL S is the framed suspension spec-trum of the moduli stack FS yn orS of finite syntomic S-schemes with trivialized canonicalsheaf. More generally, for n (cid:62)
0, Σ n T MSL S is the framed suspension spectrum of themoduli stack FQS m or ,n S of finite quasi-smooth derived S-schemes of relative virtual di-mension − n with trivialized canonical sheaf.We now explain the general paradigm. Given a natural transformation β : B → K of presheaveson smooth S-schemes, we define a β -structure on a morphism f : Z → X between smooth
E. ELMANTO, M. HOYOIS, A. A. KHAN, V. SOSNILO, AND M. YAKERSON
S-schemes to be a lift of its shifted cotangent complex to B(Z):BZ K. β − L f More generally, if f : Z → X is a morphism between quasi-smooth derived S-schemes, we definea β -structure on f to be a lift of − L f to ˜B(Z), where ˜B is the left Kan extension of B toquasi-smooth derived S-schemes (although left Kan extension is a rather abstract procedure,it turns out that ˜B admits a concrete description for every B of interest). Then, for any β : B → K (cid:62) , we show that the motivic Thom spectrum M β ∈ SH (S) is the framed suspensionspectrum of the moduli stack FQS m β S of finite quasi-smooth derived S-schemes with β -structure(Theorem 3.3.10).Over a perfect field k , this result becomes much more concrete. Indeed, using the motivicrecognition principle, we deduce that Ω ∞ T M β is the motivic homotopy type of the group com-pletion of the moduli stack FQS m βk (Corollary 3.3.12). Even better, the group completion isredundant if β lands in the positive-rank summand of K-theory.The case of the algebraic cobordism spectrum MGL k deserves more elaboration. For itsinfinite P -loop space, we obtain an equivalenceΩ ∞ T MGL k (cid:39) L zar L A FS yn gp k of E ∞ -ring spaces with framed transfers. For n >
0, we obtain equivalences of FS yn k -modulesΩ ∞ T Σ n T MGL k (cid:39) L nis L A FQS m nk (see Corollary 3.4.2). Finally, using an algebraic version of Whitney’s embedding theorem forfinite schemes, we can replace the moduli stack FS yn k by the Hilbert scheme Hilb flci ( A ∞ k ) offinite local complete intersections in A ∞ k , which is a smooth ind-scheme and a commutativemonoid up to A -homotopy (see Theorem 3.5.2):Ω ∞ T MGL (cid:39) L zar (L A Hilb flci ( A ∞ k )) gp . Analogies with topology.
A cobordism between two compact smooth manifolds M andN is a smooth manifold W with a proper morphism W → R whose fibers over 0 and 1 areidentified with M and N; see for example [Qui71, § A -path in the moduli stack of properquasi-smooth schemes. As in topology, one can also consider moduli stacks of schemes withsome stable tangential structure (see § A -localization of such a moduli stack is thenanalogous to a cobordism space of structured manifolds (a space in which points are manifolds,paths are cobordisms, homotopies between paths are cobordisms between cobordisms, etc.).From this perspective, our computation of Ω ∞ T M β for β of rank 0 is similar to the followingcomputation of Galatius, Madsen, Tillmann, and Weiss in topology [GMTW09]: given a mor-phism of spaces β : B → BO, the infinite loop space of the Thom spectrum M β is the cobordismspace of zero-dimensional compact smooth manifolds with β -structured stable normal bundle.The only essential difference is that in topology these cobordism spaces are already grouplike,due to the existence of nontrivial cobordisms to the empty manifold. In FS yn βk , however, theempty scheme is not A -homotopic to any nonempty scheme, so group completion is necessary.To our knowledge, when β is of positive rank, the topological analog of our computation is notrecorded in the literature. One expects for example an identification, for M a smooth manifold ODULES OVER ALGEBRAIC COBORDISM 5 and n >
0, of the mapping space Maps(M , Ω ∞ Σ n MU) with the cobordism space of complex-oriented submanifolds of M of codimension n . In the other direction, it is an interesting problemto extend our computation of Ω ∞ T M β to β of negative rank, where the topological story suggestsa relationship with a moduli stack of proper quasi-smooth schemes of positive dimension.Finally, we note that our description of MGL-modules can be understood as a more coherentversion of Quillen’s geometric universal property of MU [Qui71, Proposition 1.10].1.5. Related work.
Similar computations of motivic Thom spectra in terms of framed cor-respondences were obtained independently by Garkusha and Neshitov [GN18]. Our approachdiffers in that we work with tangentially framed correspondences as defined in [EHK + + ∞ -category, we are able to make much more structuredcomputations, which are crucial for describing ∞ -categories of modules over motivic Thomspectra. Our notion of motivic Thom spectrum is also strictly more general than the one in[GN18].The fact that MGL-cohomology groups admit finite syntomic transfers is well known (see[Pan09] for the case of finite transfers between smooth schemes and [D´eg18] for the generalcase). They are also an essential feature in the algebraic cobordism theory of Levine and Morel[LM07]. Such transfers were further constructed by Navarro [Nav16] on E-cohomology groupsfor any MGL-module E, and D´eglise, Jin, and Khan [DJK18] showed that these transfers existat the level of spaces. Our main result implies that E-cohomology spaces admit coherent finitesyntomic transfers, and that this structure even characterizes MGL-modules.In [LS16], Lowrey and Sch¨urg give a presentation of the algebraic cobordism groups Ω n (X) (cid:39) MGL n,n (X) with projective quasi-smooth derived X-schemes as generators (for X smooth overa field of characteristic zero). Our results give a comparable presentation for n (cid:62) ∞ -groupoid (Ω ∞ T Σ n T MGL)(X), which holds also in positive characteristic but is onlyZariski-local. We hope that there is a common generalization of both results, namely, a globaldescription of the sheaves Ω ∞ T Σ n T MGL, for all n ∈ Z and in arbitrary characteristic, in termsof quasi-smooth derived schemes.1.6. Conventions and notation.
This paper is a continuation of [EHK + op. cit. In particular, PSh(C) denotes the ∞ -category of presheaves onan ∞ -category C, and PSh Σ (C) ⊂ PSh(C) is the full subcategory of presheaves that transformfinite sums into finite products (called Σ -presheaves for short). We denote by L zar and L nis theZariski and Nisnevich sheafification functors, by L A the (naive) A -localization functor, andby L mot the motivic localization functor.In addition, we use derived algebraic geometry throughout the paper, following [TV08,Chapter 2.2] and [Lur18, Chapter 25]. By a derived commutative ring we mean an object ofCAlg ∆ = PSh Σ (Poly), where Poly is the category of polynomial rings Z [ x , . . . , x n ] for n (cid:62) ∆R the ∞ -category (CAlg ∆ ) R / and by CAlg smR ⊂ CAlg ∆R the full subcategory of smooth R-algebras. If Ris discrete, we further denote by CAlg ♥ R ⊂ CAlg ∆R the full subcategory of discrete R-algebras,which is a 1-category containing CAlg smR .We write dAff for the ∞ -category of derived affine schemes and dSch for that of derivedschemes. If X is a derived scheme, we denote by X cl its underlying classical scheme. Everymorphism f : Y → X in dSch admits a cotangent complex L f ∈ QCoh (Y); we say that f is quasi-smooth if it is locally of finite presentation and L f is perfect of Tor-amplitude (cid:54) E. ELMANTO, M. HOYOIS, A. A. KHAN, V. SOSNILO, AND M. YAKERSON
Throughout this paper, S denotes a fixed base scheme, arbitrary unless otherwise specified.1.7.
Acknowledgments.
We would like to thank Andrei Druzhinin, Joachim Jelisiejew, JacobLurie, Akhil Mathew, Fabien Morel, and David Rydh for many useful discussions that helpedus at various stages in the writing of this paper.We would also like to express our gratitude to Grigory Garkusha, Ivan Panin, AlexeyAnanyevskiy, and Alexander Neshitov, who realized Voevodsky’s ideas about framed correspon-dences in a series of groundbreaking articles. Our present work would not have been possiblewithout theirs.The final stages of this work were supported by the National Science Foundation undergrant DMS-1440140, while the first two authors were in residence at the Mathematical SciencesResearch Institute in Berkeley, California, during the “Derived Algebraic Geometry” programin spring 2019.Finally, this paper was completed during a two-week stay at the Institute for Advanced Studyin Princeton in July 2019. We thank the Institute for excellent working conditions.2.
Twisted framed correspondences
Recall that a framed correspondence from X to Y is a spanZX Y f g where f is finite syntomic, together with an equivalence L f (cid:39) + ξ ∈ K(Y) of rank 0, one can consider a “twisted” version of this definition byinstead requiring an equivalence L f + g ∗ ( ξ ) (cid:39) ξ ∈ K(Y) of rank (cid:62) § § frS (Y , ξ ), h nfrS (Y , ξ ), and h efrS (Y , ξ ) of ξ -twisted tangentially framed, normally framed, and equationally framed correspondences, andin § § frS (Y , ξ ), h nfrS (Y , ξ ), and h efrS (Y , ξ ) are motivically equivalent. Most proofs in § § + § frS (Y , ξ ) to smooth S-schemes is compatible, up to motivic equivalence, with any basechange S (cid:48) → S; this is a key technical result that will allow us to establish our main results overarbitrary base schemes (rather than just over fields).2.1.
Stable vector bundles and K-theory.
For X a derived scheme, we denote by Vect(X)the ∞ -groupoid of finite locally free sheaves on X. We define the ∞ -groupoid sVect(X) as thecolimit of the sequence Vect(X) ⊕ O X −−−→ Vect(X) ⊕ O X −−−→ Vect(X) → · · · . Thus, an object of sVect(X) is a pair ( E , m ) where E ∈ Vect(X) and m (cid:62)
0, which we canthink of as the formal difference E − O m X ; the rank of such a pair is rk( E ) − m . Two pairs( E , m ) and ( E (cid:48) , m (cid:48) ) are equivalent if they have the same rank and there exists n (cid:62) E ⊕ O m (cid:48) ⊕ O n (cid:39) E (cid:48) ⊕ O m ⊕ O n . ODULES OVER ALGEBRAIC COBORDISM 7
Note that sVect(X) is a Vect(X)-module and the canonical map Vect(X) → K(X) factorsthrough sVect(X).For R a derived commutative ring, sVect(R) is noncanonically a sum of copies of BGL(R).Indeed, if ξ = (E , m ) ∈ sVect(R), we haveΩ ξ (sVect(R)) = colim n Aut R (E ⊕ R n ) . If we choose an equivalence E ⊕ F (cid:39) R r , computing the colimit of the sequenceAut R (E) → Aut R (E ⊕ F) → Aut R (E ⊕ F ⊕ E) → · · · in two ways yields an equivalence colim n Aut R (E ⊕ R n ) (cid:39) colim n Aut R (R n ) = GL(R). Lemma 2.1.1.
Let R be a derived commutative ring. Then the canonical map sVect(R) → K(R) is a plus construction in the sense of Quillen, i.e., it is the universal map that kills thecommutator subgroup of π (sVect(R) , ξ ) (cid:39) GL( π R) for all ξ ∈ sVect(R) . In particular, it isacyclic.Proof. This is an instance of the McDuff–Segal group completion theorem. We refer to [RW13,Theorem 1.1] or [Nik17, Theorem 9] for modern treatments. (cid:3)
Remark 2.1.2.
Similarly, if Vect SL (R) denotes the monoidal groupoid of finite locally free R-modules with trivialized determinant and K SL (R) is its group completion, then sVect SL (R) → K SL (R) is a plus construction.2.2. Presheaves of twisted framed correspondences.
Let X and Y be derived S-schemesand let ξ ∈ K(Y) be of rank (cid:62)
0. The ∞ -groupoid of ξ -twisted framed correspondences from Xto Y over S is defined by:h frS (Y , ξ )(X) = ZX Y f g + f finite quasi-smooth L f (cid:39) − g ∗ ( ξ ) in K(Z) . When ξ = 0 and X , Y ∈ Sch S , this definition recovers the notion of tangentially framed corre-spondence considered in [EHK + § Lemma 2.2.1.
Let f : Z → X be a quasi-finite quasi-smooth morphism of derived schemes ofrelative virtual dimension . Then f is flat. In particular, if X is classical, then Z is classical.Proof. Since f is flat if and only if Z cl (cid:39) Z × X X cl and f cl is flat, we can assume X classical.The question is local on Z, so we can assume X affine and Z cut out by exactly n equations in A n X (since f has relative virtual dimension 0). Since f is quasi-finite, we see that f cl : Z cl → Xis a relative global complete intersection, hence syntomic [Stacks, Tag 00SW]. In particular, thedefining equations of Z in A n X locally form a regular sequence, whence Z cl = Z [KR19, 2.3.2]. (cid:3) Note that h frS (Y , ξ ) is a presheaf on dSch S . In fact, it is a Σ-presheaf on the ∞ -category Corr fr (dSch S ) of framed correspondences (see Appendix B). It is also covariantly functorial inthe pair (Y , ξ ).To relate ξ -twisted framed correspondences with motivic homotopy theory, we will also needto study two auxiliary versions of twisted framed correspondences, namely an “equationallyframed” version and a “normally framed” version. E. ELMANTO, M. HOYOIS, A. A. KHAN, V. SOSNILO, AND M. YAKERSON
Let X , Y ∈ dSch S and let ξ = ( E , m ) ∈ sVect(Y) be of rank (cid:62)
0. We define:h efrS (Y , ξ )(X) = colim n →∞ ZX Y f g + f finite i : Z → A n X closed immersion over X ϕ : ( A n X ) h Z → A n − m × V ( E ) ϕ − (0 × Y) (cid:39) Z, ϕ | Z (cid:39) g , h nfrS (Y , ξ )(X) = colim n →∞ ZX Y f g + f finite quasi-smooth i : Z → A n X closed immersion over X N i (cid:39) O n − m Z ⊕ g ∗ ( E ) . More precisely, for each m , the right-hand sides are functors Vect(Y) (cid:62) m → PSh(dSch S ) in thevariable E . As m varies, these functors fit together in a cone over the sequenceVect(Y) ⊕ O Y −−−→ Vect (cid:62) (Y) ⊕ O Y −−−→ Vect (cid:62) (Y) → · · · , which induces sVect (cid:62) (Y) → PSh(dSch S ). Here, the notation X h Z for X a derived scheme andZ ⊂ X a closed subset refers to the pro-object of ´etale neighborhoods of Z in X, see [EHK + V ( E ) = Spec(Sym( E )) is the vector bundle over Y associated with E .For ξ = 0 and X , Y ∈ Sch S , these definitions recover the notions of equationally framedand normally framed correspondences from [EHK + op. cit. , we will not discuss the “level n ” versions of h efr and h nfr , for simplicity.However, it is clear that many of the results below hold at finite level (a notable exception isProposition 2.3.6).There are forgetful maps h efrS (Y , ξ ) −→ h nfrS (Y , ξ ) −→ h frS (Y , ξ )in PSh Σ (dSch S ). Note that h efrS (Y , ξ )(X) is discrete (i.e., a set) when X and Y are classical, since ϕ determines g as well as the derived closed subscheme Z. On the other hand, h nfrS (Y , ξ )(X) isusually not discrete when rk ξ (cid:62) Remark 2.2.2.
Let X , Y ∈ dSch S and ξ = ( E , m ) ∈ sVect (cid:62) (Y). Then there is a naturalisomorphismh efrS (Y , ξ )(X) (cid:39) colim n →∞ Maps (cid:18) X + ∧ ( P ) ∧ n , L nis (cid:18) V ( O n − m Y ⊕ E ) V ( O n − m Y ⊕ E ) − (cid:19)(cid:19) . If X and Y are classical this is a special case of [EHK + + ⊂ Y is a closed subset, thenY (cid:48) (cid:55)→ (Y (cid:48) ) h Z is an ´etale cosheaf on ´etale Y-schemes; by topological invariance of the ´etale topos,this follows from the underived statement.2.3. Descent and additivity.
We say that a presheaf F : dSch opS → Spc satisfies closed gluing if F ( ∅ ) (cid:39) ∗ and if for any diagram of closed immersions X ← (cid:45) Z (cid:44) → Y in dSch S , the canonicalmap F (X (cid:116) Z Y) → F (X) × F (Z) F (Y)is an equivalence.We say that a presheaf F : dSch opS → Spc is finitary if, for every cofiltered diagram (X α ) ofqcqs derived schemes with affine transition maps, the canonical mapcolim α F (X α ) → F (lim α X α )is an equivalence. ODULES OVER ALGEBRAIC COBORDISM 9
Proposition 2.3.1.
Let Y ∈ dSch S and let ξ ∈ K(Y) be of rank (cid:62) . (i) h frS (Y , ξ ) is a Nisnevich sheaf on qcqs derived schemes. (ii) If Y is locally finitely presented over S , then h frS (Y , ξ ) is finitary. (iii) Let R (cid:44) → Y be a Nisnevich (resp. ´etale) covering sieve generated by a single map. Then h frS (R , ξ ) → h frS (Y , ξ ) is a Nisnevich (resp. ´etale) equivalence.Proof. (i) and (ii) are clear (using the corresponding properties of K-theory). Let us prove(iii). Since h frS (R , ξ ) → h frS (Y , ξ ) is a monomorphism, it suffices to show that it is a Nisnevich(resp. ´etale) effective epimorphism. Refining the sieve R if necessary, we can assume that itis generated by a single ´etale map Y (cid:48) → Y. Let X ∈ dSch S , f : Z → X a finite quasi-smoothmorphism, and g : Z → Y an S-morphism. It suffices to show that there is a Nisnevich (resp.´etale) covering of X where g factors through R. Since f is finite, the sieve on X consistingof all maps X (cid:48) → X such that X (cid:48) × X Z → Z → Y factors through Y (cid:48) is covering in theNisnevich (resp. ´etale) topology. Indeed, if X (cid:48) is local and henselian (resp. strictly henselian),then (X (cid:48) × X Z) cl is a finite sum of henselian (resp. strictly henselian) local schemes, so the ´etalemap X (cid:48) × X Z × Y Y (cid:48) → X (cid:48) × X Z has a section. (cid:3)
Proposition 2.3.2.
Let Y , . . . , Y k ∈ dSch S , let ξ ∈ K(Y (cid:116) · · · (cid:116) Y k ) have rank (cid:62) , and let ξ i be the restriction of ξ to Y i . Then the canonical map h frS (Y (cid:116) · · · (cid:116) Y k , ξ ) → h frS (Y , ξ ) × · · · × h frS (Y k , ξ k ) is an equivalence.Proof. Clear. (cid:3)
Proposition 2.3.3.
Let Y ∈ dSch S and ξ ∈ sVect (cid:62) (Y) . (i) h efrS (Y , ξ ) is a sheaf for the quasi-compact ´etale topology. (ii) h efrS (Y , ξ ) satisfies closed gluing. (iii) If Y is locally finitely presented over S , then h efrS (Y , ξ ) is finitary. (iv) Let R (cid:44) → Y be a Nisnevich (resp. ´etale) covering sieve generated by a single map. Then h efrS (R , ξ ) → h efrS (Y , ξ ) is a Nisnevich (resp. ´etale) equivalence.Proof. The proof of each point is exactly the same as the corresponding point of [EHK + efrS (Y , ξ ) from Remark 2.2.2. (cid:3) Lemma 2.3.4.
The presheaf dSch opS → Spc , X (cid:55)→ { quasi-smooth derived X -schemes } satisfies closed gluing.Proof. The assertion without the quasi-smoothness condition follows from [Lur18, Theorem16.2.0.1] (as in the proof of [Lur18, Theorem 16.3.0.1], which is the spectral analog). It remainsto prove the following: if Y is a derived scheme and f : Y → X (cid:116) X X is a morphism whosebase change f ∗ : Y ∗ → X ∗ is quasi-smooth for each ∗ ∈ { , , } , then f is quasi-smooth. Wehave that f is locally finitely presented by [Lur18, Proposition 16.3.2.1(3)]. It remains to showthat the cotangent complex L f is perfect and has Tor-amplitude (cid:54)
1. Since the cotangentcomplex is stable under base change, the pullback of L f to X ∗ is L f ∗ . The claim now followsfrom [Lur18, Proposition 16.2.3.1(3,7)]. (cid:3) Proposition 2.3.5.
Let Y ∈ dSch S and ξ ∈ sVect (cid:62) (Y) . (i) h nfrS (Y , ξ ) satisfies Nisnevich excision. (ii) h nfrS (Y , ξ ) satisfies closed gluing. (iii) If Y is locally finitely presented over S , then h nfrS (Y , ξ ) is finitary. (iv) Let R (cid:44) → Y be a Nisnevich (resp. ´etale) covering sieve generated by a single map. Then h nfrS (R , ξ ) → h nfrS (Y , ξ ) is a Nisnevich (resp. ´etale) equivalence.Proof. (i) By definition, the presheaf h nfrS (Y , ξ ) is a filtered colimit over n of its level n versions,which are clearly sheaves for the fpqc topology. We conclude since the property of Nisnevichexcision is preserved by filtered colimits.(ii) This follows from the closed gluing property for connective quasi-coherent sheaves [Lur18,Theorem 16.2.0.1] and Lemma 2.3.4.(iii) Clear.(iv) The proof is identical to that of Proposition 2.3.1(iii). (cid:3) Proposition 2.3.6.
Let Y , . . . , Y k ∈ dSch S , let ξ ∈ sVect(Y (cid:116) · · · (cid:116) Y k ) be of rank (cid:62) , andlet ξ i be the restriction of ξ to Y i . Then the canonical maps h efrS (Y (cid:116) · · · (cid:116) Y k , ξ ) → h efrS (Y , ξ ) × · · · × h efrS (Y k , ξ k )h nfrS (Y (cid:116) · · · (cid:116) Y k , ξ ) → h nfrS (Y , ξ ) × · · · × h nfrS (Y k , ξ k ) are A -equivalences.Proof. Same as the proof of [EHK + (cid:3) It follows from Proposition 2.3.6 that the presheaves L A h efrS (Y , ξ ) and L A h nfrS (Y , ξ ) havecanonical structures of E ∞ -objects (cf. [EHK + Comparison theorems. If f : A → B is a morphism of derived commutative rings, thereis a canonical B-linear map (cid:15) f : cofib( f ) ⊗ A B → L f , called the Hurewicz map associated with f [Lur18, § f is connective (i.e., if Spec( f )is a closed immersion), then (cid:15) f is 2-connective [Lur18, Proposition 25.3.6.1]. In particular, if i : Z → X is a closed immersion between derived affine schemes and I is the fiber of i ∗ : O (X) → O (Z), we have a 1-connective map I ⊗ O (X) O (Z) → L i [ −
1] = N i . Lemma 2.4.1.
Let Z X Z X i i be a Cartesian square of derived affine schemes, where all arrows are closed immersions. Let I and I be the fibers of the maps i ∗ : O (X) → O (Z) and i ∗ : O (X ) → O (Z ) . Then the canonicalmap I → I × N i N i is connective. ODULES OVER ALGEBRAIC COBORDISM 11
Proof.
We factor this map as follows: I α −→ I × I ⊗ O (X0) O (Z ) ( I ⊗ O (X) O (Z)) β −→ I × N i N i By [Lur18, Proposition 25.3.6.1], the canonical maps I ⊗ O (X) O (Z) → N i and I ⊗ O (X ) O (Z ) → N i are 1-connective. Being a base change of the latter, the projection ( I ⊗ O (X ) O (Z )) × N i N i → N i is 1-connective. It follows that the map I ⊗ O (X) O (Z) → ( I ⊗ O (X ) O (Z )) × N i N i is connective, hence that its base change β is connective. It remains to show that α is connective.Let T be the closed subscheme of X obtained by gluing Z and X along Z . We can factor α as I → I ⊗ O (X) O (T) → I × I ⊗ O (X0) O (Z ) ( I ⊗ O (X) O (Z)) . The first map is clearly connective. Since I (cid:39) I ⊗ O (X) O (X ), the second map is an equivalenceby Milnor patching [Lur18, Theorem 16.2.0.1]. Thus, α is connective. (cid:3) Proposition 2.4.2.
Let X , Y ∈ dSch S , let ξ ∈ sVect(Y) be of rank (cid:62) , and let X ⊂ X be aclosed subscheme. Suppose that X is affine and that Y admits an ´etale map to an affine bundleover S . Then the map h efrS (Y , ξ )(X) → h efrS (Y , ξ )(X ) × h nfrS (Y ,ξ )(X ) h nfrS (Y , ξ )(X) is an effective epimorphism.Proof. Write ξ = ( E , m ). An element in the right-hand side consists of: • a span X f ←− Z g −→ Y with f finite quasi-smooth; • a closed immersion i : Z → A n X over X with an equivalence τ : N i (cid:39) O n − m Z ⊕ g ∗ ( E ); • an equational ξ -framing of the induced span X f ←− Z g −→ Y:Z i −→ A n X ← U ϕ −→ V ( O n − m Y ⊕ E ) , α : Z (cid:39) ϕ − (Y) , where i is the pullback of i , U → A n X is an affine ´etale neighborhood of Z , and ϕ extends g ; • an identification of the equivalence N i (cid:39) O n − m Z ⊕ g ∗ ( E ) induced by τ with that inducedby α .The goal is to construct an equational ξ -framing ( ϕ, α ) of X f ←− Z g −→ Y that simultaneouslyinduces the normal framing τ and the equational ξ -framing ( ϕ , α ). Using [EHK + of Z in A n X to an ´etale neighborhood U of Z in A n X . Refining U if necessary, we can assume that U is affine (by [EHK + h : U → Y be the composition of ϕ : U → V ( O n − m Y ⊕ E ) and the projection V ( O n − m Y ⊕ E ) → Y. We first construct a simultaneous extension h : U → Y of h : U → Y and g : Z → Y.Suppose first that Y is an affine bundle over S. Since U is affine, U × S Y → U is a vector bundleover U. It follows that the restriction mapMaps S (U , Y) → Maps S (U , Y) × Maps S (Z , Y) Maps S (Z , Y) (cid:39) Maps S (U (cid:116) Z Z , Y)is surjective, so the desired extension exists. In general, let p : Y → A be an ´etale map where Ais an affine bundle over S. By the previous case, there exists an S-morphism U → A extending p ◦ h and p ◦ g . Then the ´etale map U × A Y → U has a section over U (cid:116) Z Z, so there existsan affine open subset U (cid:48) ⊂ U × A Y that is an ´etale neighborhood of U (cid:116) Z Z in U. We cantherefore replace U by U (cid:48) , and the projection U (cid:48) → Y gives the desired extension.It remains to construct a Y-morphism ϕ : U → V ( O n − m Y ⊕ E ) extending ϕ and an equivalence α : Z (cid:39) ϕ − (Y) lifting α such that the induced trivialization N i (cid:39) O n − m Z ⊕ g ∗ ( E ) is equivalentto τ . Recall that Y-morphisms U → V ( O n − m Y ⊕ E ) correspond to morphisms of O U -modules O n − m U ⊕ h ∗ ( E ) → O U . Let I and I be the fibers of the restrictions map O (U) → O (Z) and O (U ) → O (Z ). By Lemma 2.4.1, the morphism of O (U)-modules I → I × N i N i is connective. Since O (U) n − m ⊕ h ∗ ( E ) is a projective object in connective O (U)-modules [Lur17a,Proposition 7.2.2.7], the morphism O (U) n − m ⊕ h ∗ ( E ) → I × N i N i induced by α and τ liftsto a morphism O (U) n − m ⊕ h ∗ ( E ) → I . This defines a Y-morphism ϕ : U → V ( O n − m Y ⊕ E ) extending ϕ together with a factorizationof Z → U through ϕ − (Y), i.e., a U-morphism α : Z → ϕ − (Y). By construction, α lifts α and induces the equivalence τ on conormal sheaves; since both Z and ϕ − (Y) are regularlyembedded in U, α is an ´etale closed immersion. Thus, there exists a function a on U such that α induces an equivalence Z (cid:39) ϕ − (Y) ∩ U a . Replacing U by U a concludes the proof. (cid:3) Corollary 2.4.3.
Suppose that Y ∈ Sm S is a finite sum of schemes admitting ´etale maps toaffine bundles over S and let ξ ∈ sVect (cid:62) (Y) . Then the map L A h efrS (Y , ξ ) → L A h nfrS (Y , ξ ) is an equivalence on derived affine schemes. In particular, it induces an equivalence L zar L A h efrS (Y , ξ ) (cid:39) L zar L A h nfrS (Y , ξ ) . Proof.
By Proposition 2.3.6, we can assume that Y admits an ´etale map to an affine bundleover S. By Proposition 2.4.2, for every n (cid:62)
0, the maph efrS (Y , ξ ) A n → h efrS (Y , ξ ) ∂ A n × h nfrS (Y ,ξ ) ∂ A n h nfrS (Y , ξ ) A n is surjective on affines. By Propositions 2.3.3(ii) and 2.3.5(ii), both h efrS (Y , ξ ) and h nfrS (Y , ξ )satisfy closed gluing. It follows that the maph efrS (Y , ξ ) A • → h nfrS (Y , ξ ) A • is a trivial Kan fibration of simplicial spaces when evaluated on any affine scheme, and weconclude using [Lur18, Theorem A.5.3.1]. (cid:3) Corollary 2.4.4.
Let Y be a smooth S -scheme and ξ ∈ sVect(Y) of rank (cid:62) . Then the map h efrS (Y , ξ ) → h nfrS (Y , ξ ) in PSh Σ (dSch S ) is a motivic equivalence.Proof. If Y is the filtered colimit of quasi-compact open subschemes, then on qcqs derivedschemes the presheaves h efrS (Y , ξ ) and h nfrS (Y , ξ ) are filtered colimits of the corresponding sub-presheaves, so we can assume Y quasi-compact. Let { U , . . . , U k } be an open cover of Y byS-schemes admitting ´etale maps to affine bundles over S. The map U (cid:116)· · ·(cid:116) U k → Y is a Zariskicovering map; by Propositions 2.3.3(iv) and 2.3.5(iv), L nis h efrS ( − , ξ ) and L nis h nfrS ( − , ξ ) preservethe colimit of its ˇCech nerve. Thus, we can assume that Y is a sum of schemes admitting ´etalemaps to affine bundles over S. Then the claim follows from Corollary 2.4.3. (cid:3) For Z → X a finite morphism of derived schemes, we denote by Emb X (Z , A n X ) the space ofclosed immersions Z → A n X over X (note that this is not a discrete space in general, becauseclosed immersions of derived schemes are not monomorphisms). We letEmb X (Z , A ∞ X ) = colim n →∞ Emb X (Z , A n X ) . ODULES OVER ALGEBRAIC COBORDISM 13
Proposition 2.4.5.
Let X be a derived affine scheme, Z → X a finite morphism, X → X aclosed immersion, and Z = Z × X X . Suppose that (X ) cl → X cl is finitely presented. Then thepullback map Emb X (Z , A ∞ X ) → Emb X (Z , A ∞ X ) is an effective epimorphism.Proof. Let i : Z → A n X be a closed immersion over X , given by n functions g , . . . , g n onZ . Let g (cid:48) , . . . , g (cid:48) n be lifts of these functions to Z. Note thatfib( O (Z) → O (Z )) (cid:39) fib( O (X) → O (X )) ⊗ O (X) O (Z) . Since Z → X is finite and (X ) cl → X cl is finitely presented, the O (X)-module π fib( O (Z) → O (Z )) is finitely generated; let h , . . . , h m ∈ O (Z) be the images of a finite set of generators.Then the n + m functions g (cid:48) , . . . , g (cid:48) n , h , . . . , h m define a closed immersion i : Z → A n + m X overX whose pullback to X is equivalent to i in Emb X (Z , A ∞ X ). (cid:3) Corollary 2.4.6.
Let X be a derived affine scheme and Z → X a finite morphism. Then thepresheaf dAff opX → Spc , X (cid:48) (cid:55)→ Emb X (cid:48) (Z × X X (cid:48) , A ∞ X (cid:48) ) is A -contractible.Proof. This follows from Proposition 2.4.5 as in [EHK + (cid:3) Let Z → X be a finite quasi-smooth morphism of derived schemes of relative virtual codi-mension c . Any closed immersion i : Z → A n X over X is then quasi-smooth, hence has a finitelocally free conormal sheaf N i = L i [ − X (Z , A n X ) → Vect n + c (Z) , i (cid:55)→ N i . Taking the colimit over n , we get a morphismEmb X (Z , A ∞ X ) → sVect c (Z) ⊂ sVect(Z) . We denote by Emb ξ X (Z , A ∞ X ) its fiber over ξ ∈ sVect(Z). Note that there is a commutativesquare Emb X (Z , A ∞ X ) sVect(Z) ∗ K(Z) , − L f inducing a canonical map Emb ξ X (Z , A ∞ X ) → Maps
K(Z) ( ξ, − L f )on the horizontal fibers over ξ . Proposition 2.4.7.
Let f : Z → X be a finite quasi-smooth morphism of derived affine schemesand let ξ ∈ sVect(Z) . Then the morphism of simplicial spaces Emb ξ X × A • (Z × A • , A ∞ X × A • ) → Maps
K(Z × A • ) ( ξ, − L f ) induces an equivalence on geometric realizations.Proof. Let us write X • = X × A • and Z • = Z × A • for simplicity. Recall that the givenmorphism comes from a natural transformation of Cartesian squaresEmb ξ X • (Z • , A ∞ X • ) Emb X • (Z • , A ∞ X • ) ∗ sVect(Z • ) ξ −→ Maps
K(Z • ) ( ξ, − L f ) ∗∗ K(Z • ) . − L f ξ We consider two cases. If [ ξ ] (cid:54) = [ − L f ] in π K(Z), then Maps
K(Z) ( ξ, − L f ) is empty and theresult holds trivially. Suppose that [ ξ ] = [ − L f ] in π K(Z). Then − L f lives in the connectedcomponent K(Z) (cid:104) ξ (cid:105) ⊂ K(Z) containing ξ . Since Z is affine and [ L f ] = [ O n Z ] − [ N i ] for any closedimmersion i : Z → A n X over X, the conormal sheaf N i is stably isomorphic to ξ . It follows thatthe map Emb X (Z , A ∞ X ) → sVect(Z) lands in the component sVect(Z) (cid:104) ξ (cid:105) ⊂ sVect(Z) containing ξ . We may therefore rewrite the above Cartesian squares as follows:Emb ξ X • (Z • , A ∞ X • ) Emb X • (Z • , A ∞ X • ) ∗ sVect(Z • ) (cid:104) ξ (cid:105) ξ −→ Maps
K(Z • ) ( ξ, − L f ) ∗∗ K(Z • ) (cid:104) ξ (cid:105) . − L f ξ Recall that sVect(Z • ) (cid:104) ξ (cid:105) is equivalent to BGL(Z • ). The map sVect(Z • ) (cid:104) ξ (cid:105) → K(Z • ) (cid:104) ξ (cid:105) betweenthe lower right corners is acyclic in each degree by Lemma 2.1.1. It follows that its geometricrealization is an equivalence, since it is an acyclic map whose domain has abelian fundamentalgroups. The map between the upper right corners also induces an equivalence on geometricrealizations by Corollary 2.4.6. Since the lower right corners are degreewise connected, it followsfrom [Lur17a, Lemma 5.5.6.17] that geometric realization preserves these Cartesian squares, andwe obtain the desired equivalence on the upper left corners. (cid:3) Corollary 2.4.8.
Let Y ∈ dSch S and let ξ ∈ sVect (cid:62) (Y) . Then the map L A h nfrS (Y , ξ ) → L A h frS (Y , ξ ) is an equivalence on derived affine schemes. In particular, it induces an equivalence L zar L A h nfrS (Y , ξ ) (cid:39) L zar L A h frS (Y , ξ ) . Proof.
This follows from Proposition 2.4.7 using [EHK + (cid:3) Combining Corollaries 2.4.4 and 2.4.8, we obtain:
Theorem 2.4.9.
Let Y be a smooth S -scheme and ξ ∈ sVect(Y) of rank (cid:62) . Then the maps h efrS (Y , ξ ) → h nfrS (Y , ξ ) → h frS (Y , ξ ) in PSh Σ (dSch S ) are motivic equivalences. Base change.Lemma 2.5.1.
Let X be a derived affine scheme, X → X a closed immersion, and Z anaffine quasi-smooth X -scheme. Then there exists an affine quasi-smooth X -scheme Z such that Z × X X (cid:39) Z .Proof. Choose a smooth affine X-scheme V and a closed immersion Z → V = V × X X overX (for example, V = A n X ). The conormal sheaf N of the immersion Z → V is finite locallyfree. By [Gru72, Corollaire I.7], replacing V by an ´etale neighborhood of Z if necessary, thereexists a finite locally free sheaf E on V lifting N ; let E be its pullback to V . Let I be the fiberof O V → O Z . Recall that there is a canonical surjection (cid:15) : I → N in QCoh cn (V ) (see § E and E are projective in their respective ∞ -categories of connective quasi-coherent ODULES OVER ALGEBRAIC COBORDISM 15 sheaves, we can find successive lifts
E O V E N I O V . ϕϕ (cid:15) ι By Nakayama’s lemma, the morphism ϕ : E → I is surjective in a neighborhood of Z in V ;hence, the quasi-smooth closed subscheme of V defined by ι ◦ ϕ (i.e., the zero locus of thecorresponding section of the vector bundle V ( E ) → V ) has the form Z (cid:116) K. Replacing Vby an affine open neighborhood of Z if necessary, we can assume K = ∅ . Let Z ⊂ V be thequasi-smooth closed subscheme defined by ϕ . By construction, Z × X X is the quasi-smoothclosed subscheme of V defined by ι ◦ ϕ , which is Z . (cid:3) Let X be a derived affine scheme and Z ⊂ X a closed subscheme. We say that the pair(X , Z) is henselian if the underlying classical pair (X cl , Z cl ) is henselian. By the topologicalinvariance of the ´etale site, (X , Z) is henselian if and only if, for every ´etale affine X-scheme Y,the restriction map Maps X (X , Y) → Maps X (Z , Y)is an effective epimorphism.
Lemma 2.5.2.
Let (X , Z) be a henselian pair of derived affine schemes. Then the inducedmorphism of ∞ -groupoids Vect(X) → Vect(Z) is -connective.Proof. The morphism Vect(X cl ) → Vect(Z cl ) is 0-connective by [Gru72, Corollaire I.7]. If P , Q ∈ Vect n (X cl ), isomorphisms P (cid:39) Q are sections of a GL n -torsor over X cl , which is in particular asmooth affine X cl -scheme. Using [Gru72, Th´eor`eme I.8], this implies that Vect(X cl ) → Vect(Z cl )is 1-connective. It remains to observe that Vect(X) → Vect(X cl ) is 2-connective, because π Hom R (P , Q) (cid:39) Hom π (R) ( π (P) , π (Q)) when P is a finite locally free R-module. (cid:3) Lemma 2.5.3.
Let
S = Spec R be an affine scheme, Y a smooth S -scheme with an ´etale mapto a vector bundle over S , and ξ ∈ sVect (cid:62) (Y) . Then the functor h nfrS (Y , ξ ) : CAlg ∆R → Spc isleft Kan extended from
CAlg smR .Proof.
We check conditions (1)–(3) of Proposition A.0.1. Condition (1) holds by Proposi-tion 2.3.5(iii), and condition (3) is a special case of closed gluing (Proposition 2.3.5(ii)). Let(X , X ) be a henselian pair of derived affine R-schemes, f : Z → X a finite quasi-smoothmorphism, i : Z → A n X a closed immersion over X , g : Z → Y an S-morphism, and τ anequivalence N i (cid:39) O n − m Z ⊕ g ∗ ( E ), where ξ = ( E , m ). We have to construct a lift of this datafrom X to X. Since h nfrS (Y , ξ ) is finitary (Proposition 2.3.5(iii)) and (X , X ) is henselian forany X containing X [Stacks, Tag 0DYD], we can assume that X → X is finitely presented.By Lemma 2.5.1, there exists an affine quasi-smooth lift f : Z → X of f . Moreover, by [Lur18,Corollary B.3.3.6], we have Z = Z (cid:48) (cid:116) Z (cid:48)(cid:48) where Z (cid:48) → X is finite and Z (cid:48)(cid:48) → X does not hit X ;thus we can assume f finite. By Proposition 2.4.5, increasing n if necessary, we can also lift i to a closed immersion i : Z → A n X over X. By assumption, there exists an ´etale map h : Y → Vwhere V is a vector bundle over S. Then the restriction mapMaps S (Z , V) → Maps S (Z , V)is an effective epimorphism (since Z is affine), so the composite h ◦ g lifts to a map Z → V. Theprojection Y × V Z → Z has a section over Z , hence over Z since h is affine ´etale and (Z , Z ) ishenselian [Gro67, Proposition 18.5.6(i)]. If g is the composite Z → Y × V Z → Y, then g extends g . Finally, since (Z , Z ) is henselian, we can lift τ to an equivalence N i (cid:39) O n − m Z ⊕ g ∗ ( E ) byLemma 2.5.2. (cid:3) Theorem 2.5.4.
Let f : S (cid:48) → S be a morphism of schemes, Y a smooth S -scheme, and ξ ∈ K(Y) of rank (cid:62) . Then the canonical map f ∗ (h frS (Y , ξ ) | Sm S ) → h frS (cid:48) (Y S (cid:48) , ξ S (cid:48) ) | Sm S (cid:48) is a motivic equivalence.Proof. This is obvious if f is smooth, so we may assume S and S (cid:48) affine. Note that if wedo not restrict these presheaves to smooth schemes, this map is obviously an equivalence. Ittherefore suffices to show that L mot h frS (Y , ξ ), viewed as a presheaf on affine S-schemes, is themotivic localization of a colimit of presheaves represented by smooth affine S-schemes. If Yis the filtered colimit of quasi-compact open subschemes, then h frS (Y , ξ ) is the filtered colimitof the corresponding subpresheaves, so we can assume Y quasi-compact. Then there exists afinite open cover { U , . . . , U k } of Y by S-schemes admitting ´etale maps to vector bundles overS. The map U (cid:116) · · · (cid:116) U k → Y is a Zariski covering map; by Proposition 2.3.1(iii), L nis h frS ( − , ξ )preserves the colimit of its ˇCech nerve. Together with Proposition 2.3.2, we can assume thatY admits an ´etale map to a vector bundle over S. In this case, we know from Lemma 2.5.3that h nfrS (Y , ξ ) is left Kan extended from smooth affine S-schemes. Since we have a motivicequivalence h nfrS (Y , ξ ) → h frS (Y , ξ ) by Theorem 2.4.9, we are done. (cid:3) Geometric models of motivic Thom spectra
The main result of this section, Theorem 3.3.10, identifies the motivic Thom spectrum M β of any β : B → K (cid:62) with the framed suspension spectrum of a concrete framed motivic space,namely the moduli stack of finite quasi-smooth schemes with β -structure. We obtain this resultin several steps: • In § • In § Y / S ( ξ ) where Y is asmooth S-scheme and ξ ∈ K(Y) has rank (cid:62) • Finally, in § β -structure, we recall the formalism ofmotivic Thom spectra, and we deduce the general theorem.At each step we also obtain a computation of the infinite P -loop spaces of these Thom spectraover perfect fields, using the motivic recognition principle. In § § Thom spectra of vector bundles.
Let (Sm S ) / Vect → Sm S denote the Cartesian fibrationclassified by Vect : Sm opS → Spc. An object of (Sm S ) / Vect is a pair (Y , E ) where Y is a smoothS-scheme and E is a finite locally free sheaf on Y, and a morphism (Y , E ) → (Y (cid:48) , E ) is a pair( f, ϕ ) where f : Y → Y (cid:48) is an S-morphism and ϕ : E (cid:39) f ∗ ( E (cid:48) ) is an isomorphism in QCoh (Y).Similarly, we denote by (Sm S ) / K (cid:62) the ∞ -category of pairs (Y , ξ ) where Y is a smooth S-schemeand ξ ∈ K(Y) is of rank (cid:62) ODULES OVER ALGEBRAIC COBORDISM 17
Since Vect and K (cid:62) are presheaves of symmetric monoidal ∞ -categories (under direct sum),both (Sm S ) / Vect and (Sm S ) K (cid:62) acquire symmetric monoidal structures, where(Y , ξ ) ⊗ (Y , ξ ) = (Y × S Y , π ∗ ( ξ ) ⊕ π ∗ ( ξ )) . The assignment (Y , ξ ) (cid:55)→ h frS (Y , ξ ) is a right-lax symmetric monoidal functor (Sm S ) / K (cid:62) → PSh Σ ( Corr fr (Sm S )) (see Appendix B). Construction 3.1.1.
Let (Y , E ) ∈ (Sm S ) / Vect and let V × ( E ) ⊂ V ( E ) denote the complementof the zero section of the vector bundle V ( E ). We define a morphismΘ Y / S , E : h frS ( V ( E ) / V × ( E )) → h frS (Y , E )in PSh Σ ( Corr fr (Sm S )) as follows. Let z : Y (cid:44) → V ( E ) be the zero section. Then z is a regularclosed immersion with a canonical equivalence L z (cid:39) E [1], whence an equivalence τ : L z (cid:39) − E in K(Y). The span Y V ( E ) Y z id together with the equivalence τ defines a canonical element of h frS (Y , E )( V ( E )). Moreover, itsimage in h frS (Y , E )( V × ( E )) is the empty correspondence, which is the zero element. This definesthe desired map Θ Y / S , E . The morphism Θ Y / S , E is clearly natural and symmetric monoidal inthe pair (Y , E ) ∈ (Sm S ) / Vect .We now consider the diagram of symmetric monoidal ∞ -categories(3.1.2) (Sm S ) / Vect
PSh Σ (Sm S ) ∗ H (S) ∗ SH (S)(Sm S ) / K (cid:62) PSh Σ ( Corr fr (Sm S )) H fr (S) SH fr (S), Th γ ∗ L mot Θ γ ∗ Σ ∞ T (cid:39) γ ∗ h fr L mot Σ ∞ T , fr where: • Th sends (Y , E ) to the quotient V ( E ) / V × ( E ); • h fr sends (Y , ξ ) to the the presheaf h frS (Y , ξ ); • Θ is the natural transformation with components Θ Y / S , E .Note that Th and h fr are only right-lax symmetric monoidal, but all the other functors inthis diagram are strictly symmetric monoidal (and Th becomes strictly monoidal after Zariskisheafification). The fact that γ ∗ : SH (S) → SH fr (S) is an equivalence was proved in [Hoy18,Theorem 18].Our goal in this subsection is to prove the following theorem: Theorem 3.1.3.
Let S be a scheme, Y a smooth S -scheme, and E a finite locally free sheafon Y . Then Σ ∞ T , fr Θ Y / S , E is an equivalence. In other words, the boundary of (3.1.2) is strictlycommutative. One of the main inputs is the following theorem of Garkusha–Neshitov–Panin:
Theorem 3.1.4 (Garkusha–Neshitov–Panin) . Let k be an infinite field, Y a smooth separated k -scheme of finite type, and n (cid:62) . Then the canonical map h efr k ( A n Y / ( A n Y − → h efr k (Y , O n ) of presheaves on Sm k is a motivic equivalence. Here, the left-hand side uses the formal extension of h efr k to PSh Σ (Sm k ) ∗ (see [EHK + , U , ϕ, g ) fromX to A n Y , where g = ( g , g ) : U → A n × Y, to the correspondence (Z ∩ g − (0) , U , ( ϕ, g ) , g ). Proof of Theorem 3.1.4.
Modulo the notation, this follows from the level 0 part of [GNP18,Theorem 1.1], which assumes that k is an infinite perfect field. The result was generalized byDruzhinin in [Dru18], where it is made clear that it holds as stated here over any infinite field(the perfectness assumption only being needed to ensure that L mot = L nis L A when n > (cid:3) Lemma 3.1.5.
Let k be a field and let F ∈ PSh Σ , A ( Corr fr (Sm k )) . Suppose that L nis ( F K ) is agrouplike presheaf of E ∞ -spaces on Sm K for some separable algebraic field extension K /k . Then L nis F is grouplike.Proof. Let X be the henselization of a point in a smooth k -scheme and let α ∈ π ( F (X)). Notethat X K is a finite sum of henselian local schemes. By the assumption and a continuity argument,there exists a finite separable extension k (cid:48) /k such that the image of α in π ( F (X k (cid:48) )) has anadditive inverse β . By [EHK + ϕ from Spec k to Spec k (cid:48) in Corr fr (Sm k ) such that ϕ ∗ ( α k (cid:48) ) = d (cid:15) α , where d = [ k (cid:48) : k ]. Hence, d (cid:15) α + ϕ ∗ ( β ) = 0.Since 1 is a summand of d (cid:15) , this implies that α has an additive inverse. (cid:3) Lemma 3.1.6.
Let k be a field, Y a smooth k -scheme, E a finite locally free sheaf on Y of rank (cid:62) , and ξ ∈ K(Y) of rank (cid:62) . Then the Nisnevich sheaves L nis L A h fr k ( V ( E ) / V × ( E )) and L nis L A h fr k (Y , ξ ) on Sm k are grouplike. If k is infinite, they are connected. In fact, these sheaves are connected even if k is finite, see Remark 3.2.3 below. Proof.
By Lemma 3.1.5, it suffices to prove the last statement. We must show that any sectionof h fr k ( V ( E ) / V × ( E )) or h fr k (Y , ξ ) over a henselian local scheme X is A -homotopic to 0. We canassume that the finite X-scheme in such a section is connected and hence has a unique closedpoint. Thus, we can shrink Y so that it admits an ´etale map to an affine space and so that E and ξ are trivial. By Corollaries 2.4.3 and 2.4.8, it then suffices to show that the sheavesL nis L A h efr k ( A n Y / ( A n Y − nis L A h efr k (Y , O n )are connected when n (cid:62)
1. This is precisely [GNP18, Lemma A.1]. (cid:3)
Proposition 3.1.7.
Let k be a field, Y a smooth k -scheme, and E a finite locally free sheaf on Y . Then the map L mot Θ Y /k, E is an equivalence in H fr ( k ) .Proof. As in the proof of Theorem 2.5.4, we can assume Y separated of finite type and E trivial.We can also assume E of rank (cid:62)
1, since the statement is tautological when E = 0. If k isinfinite, the result follows by combining Theorems 3.1.4 and 2.4.9, noting that the squareh efr k ( A n Y / ( A n Y − efr k (Y , O n )h fr k ( A n Y / ( A n Y − fr k (Y , O n ) Θ is commutative. In light of Lemma 3.1.6, the result for k finite follows immediately from[EHK + (cid:3) Proof of Theorem 3.1.3.
The source and target of Σ ∞ T , fr L mot Θ Y / S , E are both stable under basechange: this is obvious for the source, and for the target it follows from Theorem 2.5.4. Thequestion is in particular local on S, so we can assume S qcqs. We can also assume Y qcqs ODULES OVER ALGEBRAIC COBORDISM 19 as in the proof of Theorem 2.5.4. By Noetherian approximation, we can then assume S offinite type over Spec Z . In this case, equivalences in SH (S) are detected pointwise on S [BH18,Proposition B.3], so we can assume that S is the spectrum of a field. Now the claim followsfrom Proposition 3.1.7. (cid:3) Thom spectra of nonnegative virtual vector bundles.Theorem 3.2.1.
Let Y be a smooth S -scheme and ξ ∈ K(Y) of rank (cid:62) . Then there is anequivalence Th Y / S ( ξ ) (cid:39) Σ ∞ T , fr h frS (Y , ξ ) in SH (S) (cid:39) SH fr (S) , natural and symmetric monoidal in (Y , ξ ) .Proof. We consider the following restriction of the diagram (3.1.2):Vect(S) PSh Σ (Sm S ) ∗ H (S) ∗ SH (S)K (cid:62) (S) PSh Σ ( Corr fr (Sm S )) H fr (S) SH fr (S). Th γ ∗ L mot Θ γ ∗ Σ ∞ T (cid:39) γ ∗ h fr L mot Σ ∞ T , fr By Theorem 3.1.3, the boundary of this diagram is strictly commutative. The composite ofthe top row is symmetric monoidal and lands in Pic( SH (S)), hence it extends uniquely to asymmetric monoidal functor Vect(S) gp → SH (S). The composite of the bottom row is right-laxsymmetric monoidal, and we claim that it is in fact strictly symmetric monoidal, i.e., that for ξ, η ∈ K (cid:62) (S) the structural mapΣ ∞ T , fr h frS (S , ξ ) ⊗ Σ ∞ T , fr h frS (S , η ) → Σ ∞ T , fr h frS (S , ξ + η ) . is an equivalence. Indeed, this assertion is local on S, so we can assume that ξ and η arefinite locally free sheaves on S, in which case the claim follows from the commutativity ofthe diagram. Similarly, the composite of the bottom row lands in Pic( SH fr (S)), as can bechecked locally on S. Thus, the bottom row extends uniquely to a symmetric monoidal functorK (cid:62) (S) gp (cid:39) K(S) → SH fr (S), and we have an induced commutative diagramVect(S) gp SH (S)K(S) SH fr (S). Th (cid:39) γ ∗ h fr As a functor of S, the top horizontal map factors uniquely through K(S), giving the motivicJ-homomorphism K(S) → SH (S). Hence, we obtain the identificationTh S / S ( ξ ) (cid:39) Σ ∞ T , fr h frS (S , ξ )for ξ ∈ K (cid:62) (S). To complete the proof, we must show that for f : Y → S smooth and ξ ∈ K (cid:62) (Y)the canonical map f (cid:93) Σ ∞ T , fr h frY (Y , ξ ) → Σ ∞ T , fr h frS (Y , ξ )is an equivalence. The assertion is local on Y (by Propositions 2.3.1(iii) and 2.3.2), so we canassume that ξ is a finite locally free sheaf. In this case, by Theorem 3.1.3, this map is identifiedwith the canonical map f (cid:93) Th Y / Y ( ξ ) → Th Y / S ( ξ ), which is an equivalence. (cid:3) Corollary 3.2.2.
Let k be a perfect field, Y a smooth k -scheme, and ξ ∈ K(Y) of rank (cid:62) .Then there is an equivalence Ω ∞ T , fr Th Y /k ( ξ ) (cid:39) L zar L A h fr k (Y , ξ ) gp in H fr ( k ) , natural and symmetric monoidal in (Y , ξ ) . Moreover, if the rank of ξ is (cid:62) , then L nis L A h fr k (Y , ξ ) is already grouplike. Proof.
The first statement follows from Theorem 3.2.1, the fact that the functorΣ ∞ T , fr : H fr ( k ) gp → SH fr ( k )is fully faithful [EHK + mot can be computed as L zar L A on PSh Σ ( Corr fr (Sm k )) gp [EHK + ξ has rank (cid:62)
1, then L nis L A h fr k (Y , ξ ) is grouplike by Lemma 3.1.6. (cid:3) Remark 3.2.3.
In the setting of Corollary 3.2.2, if ξ has rank (cid:62) n , then L nis L A h fr k (Y , ξ ) is n -connective (as a Nisnevich sheaf). This is obvious if n = 0. If n (cid:62)
1, then it is grouplikeand hence equivalent to Ω ∞ T Th Y /k ( ξ ), which is n -connective by Morel’s stable A -connectivitytheorem. Corollary 3.2.4.
Let k be a perfect field, Y a smooth k -scheme, and ξ ∈ sVect(Y) of rank (cid:62) .Then there are equivalences Ω ∞ T Th Y /k ( ξ ) (cid:39) L zar (L A h nfr k (Y , ξ )) gp (cid:39) L zar (L A h efr k (Y , ξ )) gp , natural in (Y , ξ ) .Proof. The first equivalence follows by combining Corollaries 3.2.2 and 2.4.8. To deduce thesecond equivalence from Corollary 2.4.4, it is enough to show thatL zar (L A h efr k (Y , ξ )) gp (cid:39) (L mot h efr k (Y , ξ )) gp . This follows from [EHK + A h efr k (Y , ξ ) is naturally a presheaf on Corr efr ∗ (Sm k ), and for any X ∈ Sm k the endomorphism σ ∗ X of (L A h efr k (Y , ξ ))(X) is homotopicto the identity by [Yak19, Lemma 3.1.4]. (cid:3) Remark 3.2.5.
One can give a more direct proof of a less structured version of Corollary 3.2.2if ξ = ( E , m ) ∈ sVect (cid:62) (Y). Voevodsky’s lemma (Remark 2.2.2) provides a mapL mot h efr k (Y , ξ ) gp → Ω m T Ω ∞ T Σ ∞ T ( V ( E ) / V × ( E )) . To show that it is an equivalence, we can assume that ξ = O n Y . In this case, it follows from[EHK + ξ = 0). Note that there is no hope of obtaining the monoidal equivalence ofCorollary 3.2.2 in this way, because sVect(Y) has no monoidal structure. Corollary 3.2.6.
Let S be pro-smooth over a field. For every ξ ∈ K(S) of rank (cid:62) , there isan equivalence Ω ∞ T , fr Σ ξ S (cid:39) L zar L A h frS (S , ξ ) gp in H fr (S) . Moreover, if ξ has rank (cid:62) , we can replace the group completion on the right-handside by Nisnevich sheafification.Proof. By Theorem 3.2.1, we have an equivalenceΣ ξ S (cid:39) Σ ∞ T , fr h frS (S , ξ )for any scheme S. By adjunction, we get a mapL zar L A h frS (S , ξ ) gp → Ω ∞ T , fr Σ ξ T S . To prove that it is an equivalence when S is pro-smooth over a field, we can assume ξ trivialsince the question is local on S. We are then reduced to the case of a perfect field, which followsfrom Corollary 3.2.2. (cid:3) ODULES OVER ALGEBRAIC COBORDISM 21
General nonnegative Thom spectra.Definition 3.3.1.
Let S be a scheme. A stable tangential structure over S is a morphism β : B → K in PSh Σ (dSch S ). We say that β has rank n if β lands in the rank n summand ofK-theory. Given a quasi-smooth morphism f : Z → X with Z ∈ dSch S , a β -structure on f is alift of − L f to B(Z).Given a stable tangential structure β over S, we denote by FQS m β S : dSch opS → Spc the modulistack of β -structured finite quasi-smooth schemes over S: FQS m β S (X) = { β -structured finite quasi-smooth derived X-schemes } . Example 3.3.2.
Let ι n be the inclusion of the rank n summand of K. A ι n -structure on aquasi-smooth morphism f is simply the property that f has relative virtual dimension − n . Wewill also write FQS m n S = FQS m ι n S for the moduli stack of finite quasi-smooth schemes of dimension − n . If n = 0, this is themoduli stack FS yn S of finite syntomic schemes (which is a smooth quasi-separated algebraicstack over S), by Lemma 2.2.1. Example 3.3.3. If β : ∗ → K is the zero section, then a β -structure on a quasi-smooth mor-phism f : Z → X is an equivalence L f (cid:39) framing of f .The presheaf FQS m β S coincides with the presheaf FS yn frS considered in [EHK + Example 3.3.4. If β is the fiber of det : K → Pic, then a β -structure on a quasi-smoothmorphism f : Z → X is an equivalence det( L f ) (cid:39) O Z . We call this structure an orientation of f , and we write FQS m orS = FQS m β S for the moduli stack of oriented finite quasi-smooth schemes. We note that det − ( O ) coincides(on qcqs derived schemes) with the presheaf K SL from Example A.0.6, defined as the rightKan extension from derived affine schemes of the group completion of the monoidal groupoidVect SL of locally free sheaves with trivialized determinant. Indeed, on derived affine schemes,sVect SL → K SL is a plus construction (see Remark 2.1.2), and in particular it is acyclic. Themap sVect SL → det − ( O ) is also acyclic, being a pullback of sVect → K. It follows that K SL → det − ( O ) is an acyclic map, whence an equivalence since the source has abelian fundamentalgroups. Example 3.3.5. If β is the fiber of the motivic J-homomorphism K → Pic( SH ), then a β -structure on a quasi-smooth morphism f : Z → X is an equivalence Th Z ( L f ) (cid:39) Z in SH (Z) (cid:39) SH (Z cl ) (see [Kha16] for the extension of SH ( − ) to derived schemes). Example 3.3.6.
Let Y ∈ dSch S and ξ ∈ K (cid:62) (Y). If β : Y → K classifies ξ , then FQS m β S =h frS (Y , ξ ) as defined in § β a stable tangential structure over S, we formally have FQS m β S (cid:39) colim Y ∈ dSch S b ∈ B(Y) h frS (Y , β ( b ))in PSh(dSch S ), where the colimit is indexed by the source of the Cartesian fibration classifiedby B : dSch opS → Spc. Note that this ∞ -category has finite sums (because B is a Σ-presheaf)and hence is sifted, so that the above formula is also valid in PSh Σ ( Corr fr (dSch S )). Proposition 3.3.7.
Let S be a scheme. The functor PSh Σ (dSch S ) / K (cid:62) → PSh Σ ( Corr fr (dSch S )) , β (cid:55)→ FQS m β S , preserves Nisnevich equivalences and ´etale equivalences. Proof.
Note that this functor preserves sifted colimits. By [Lur17b, Corollary 5.1.6.12], there isa canonical equivalence PSh Σ (dSch S ) / K (cid:62) (cid:39) PSh Σ ((dSch S ) / K (cid:62) ). By [BH18, Lemma 2.10], ittherefore suffices to prove the following: for any Y ∈ dSch S , any ξ ∈ K (cid:62) (Y), and any Nisnevich(resp. ´etale) covering sieve R (cid:44) → Y, the induced map h frS (L Σ R , ξ ) → h frS (Y , ξ ) is a Nisnevich (resp.´etale) equivalence. If R is a finitely generated sieve, this follows from Proposition 2.3.1(iii)since L Σ R is a sieve generated by a single map. If Y is quasi-compact, then R admits a finitelygenerated refinement, so we are done in this case. In general, write Y as a filtered colimitof quasi-compact open subschemes Y α ⊂ Y and let R α = R × Y Y α . Then the canonicalmap colim α h frS (Y α , ξ ) → h frS (Y , ξ ) is an equivalence on quasi-compact derived schemes, andin particular it is a Nisnevich equivalence. Similarly, colim α h frS (L Σ R α , ξ ) → h frS (L Σ R , ξ ) is aNisnevich equivalence, and we conclude by 2-out-of-3. (cid:3) The following somewhat technical definition plays a crucial role in the sequel.
Definition 3.3.8.
A stable tangential structure β : B → K over S is called smooth if the counitmap ˜B → B is a Nisnevich equivalence, where ˜B is the left Kan extension of B | Sm S to dSch S . Lemma 3.3.9.
Let S be a scheme and β : B → K a stable tangential structure over S . Supposethat B is left Kan extended along SmAff T ⊂ dAff T for every T in some affine Nisnevich coverof S . Then β is smooth.Proof. Consider the square of adjunctionsPSh nis (Sm S ) PSh nis (dSch S )PSh nis (SmAff T ) PSh nis (dAff T ). LKEres resresLKERKE res RKE
It is easy to show that the square of right adjoints commutes, because the inclusions SmAff T ⊂ Sm T and dAff T ⊂ dSch T induce equivalences of Nisnevich ∞ -topoi. Hence, the square of leftadjoints commutes as well. This shows that the counit map ˜B → B is a Nisnevich equivalencewhen restricted to dAff T for all T in the cover, hence it is a Nisnevich equivalence. (cid:3) In Appendix A, we provide many examples of stable tangential structures satisfying theassumption of Lemma 3.3.9, which are therefore smooth. In particular, K-theory itself has thisproperty. More generally, if X is a smooth algebraic stack over S with quasi-affine diagonal andwith a structure of E -monoid over Vect, and if I ⊂ Z is any subset, then the stable tangentialstructure X gp × L Σ Z L Σ I → K is smooth (by Corollary A.0.5 and Lemma A.0.7). For example,the stable tangential structures in Examples 3.3.2, 3.3.3, and 3.3.4 are smooth, while the onein Example 3.3.6 is smooth if and only if Y is smooth.We briefly recall the formalism of motivic Thom spectra from [BH18, Section 16]. To amorphism β : B → Pic( SH ) in PSh(Sm S ) one can associate a Thom spectrum M β ∈ SH (S).As in topology, it is given by a formal colimit construction:M β = colim f : Y → S smooth b ∈ B(Y) f (cid:93) β ( b ) . Moreover, it has good multiplicative properties: one has a symmetric monoidal functorM : PSh(Sm S ) / Pic( SH ) → SH (S) . In particular, if β is an E n -morphism, then M β is an E n -ring spectrum.Recall also that the motivic J-homomorphism is a morphism of E ∞ -spacesK(S) → Pic( SH (S)) , ξ (cid:55)→ Th S / S ( ξ ) , ODULES OVER ALGEBRAIC COBORDISM 23 natural in S. Restricting M along the J-homomorphism, we obtain a symmetric monoidalfunctor M : PSh(Sm S ) / K → SH (S)(which factors through H (S) / K , see [BH18, Remark 16.11 and Proposition 16.9]). For n ∈ Z ,the shifted algebraic cobordism spectrum Σ n T MGL S ∈ SH (S) is the Thom spectrum of therestriction of the J-homomorphism to the rank n summand of K-theory [BH18, Theorem 16.13]. Theorem 3.3.10.
Let S be a scheme and β : B → K (cid:62) a smooth stable tangential structureover S . Then there is an equivalence M β (cid:39) Σ ∞ T , fr FQS m β S in SH (S) (cid:39) SH fr (S) , natural and symmetric monoidal in β . In particular, if β is E n for some (cid:54) n (cid:54) ∞ , then this is an equivalence of E n -ring spectra. Remark 3.3.11.
In the statement of Theorem 3.3.10, M β depends only on the restrictionof β to Sm S . If we start with a morphism β : B → K (cid:62) in PSh Σ (Sm S ), we can alwaysapply the theorem with β : B → K (cid:62) the left Kan extension of β to obtain an equivalenceM β (cid:39) Σ ∞ T , fr FQS m β S . (Note that B is a Σ-presheaf on dSch S , because for X , . . . , X k ∈ dSch S the sum functor (cid:81) i (Sm S ) X i / → (Sm S ) (cid:96) i X i / is left adjoint, hence coinitial.) Proof.
This is a formal consequence of Theorem 3.2.1. We haveM β = colim Y ∈ Sm S b ∈ B(Y) Th Y / S ( β ( b )) (cid:39) colim Y ∈ Sm S b ∈ B(Y) Σ ∞ T , fr h frS (Y , β ( b )) , We therefore want to show that, when β is smooth, the mapcolim Y ∈ Sm S b ∈ B(Y) Σ ∞ T , fr h frS (Y , β ( b )) → colim Y ∈ dSch S b ∈ B(Y) Σ ∞ T , fr h frS (Y , β ( b ))induced by the inclusion (Sm S ) / B ⊂ (dSch S ) / B is an equivalence. By Proposition 3.3.7, we mayas well assume that B is the left Kan extension of B | Sm S . In that case, we claim that theinclusion (Sm S ) / B ⊂ (dSch S ) / B is right adjoint and hence cofinal. Indeed, we haveB(Y) = colim Y (cid:48) ∈ Sm S Y → Y (cid:48) B(Y (cid:48) ) , hence (dSch S ) / B (cid:39) C / B ◦ d where C ⊂ Fun(∆ , dSch S ) is the full subcategory of morphismswhose codomain is smooth. Forgetting the domain is then left adjoint to the above inclusion. (cid:3) Corollary 3.3.12.
Let k be a perfect field and β : B → K (cid:62) a smooth stable tangential structureover k . Then there is an equivalence Ω ∞ T , fr M β (cid:39) L zar L A ( FQS m βk ) gp in H fr ( k ) , natural and symmetric monoidal in β . Moreover, if β has rank (cid:62) , L nis L A FQS m βk is already grouplike.Proof. The first statement follows from Theorem 3.3.10 as in the proof of Corollary 3.2.2. Let˜ β : ˜B → K (cid:62) be the left Kan extension of β | Sm k . Since β is smooth, the map FQS m ˜ βk → FQS m βk is a Nisnevich equivalence by Proposition 3.3.7. After applying L A , it remains an effectiveepimorphism on Nisnevich stalks. To prove that L nis L A FQS m βk is grouplike when β has rank (cid:62)
1, we may therefore replace β by ˜ β and assume that B is left Kan extended along Sm k ⊂ dSch k .In this case, we have an equivalence FQS m βk (cid:39) colim Y ∈ Sm k b ∈ B(Y) h fr k (Y , β ( b )) . Since L A preserves colimits as an endofunctor of PSh(Sm k ), we haveL nis L A FQS m βk (cid:39) L nis colim Y ∈ Sm k b ∈ B(Y) L A h fr k (Y , β ( b )) (cid:39) L nis colim Y ∈ Sm k b ∈ B(Y) L nis L A h fr k (Y , β ( b )) . Since this colimit is sifted, it can be computed in PSh Σ ( Corr fr (Sm k )). By Corollary 3.2.2, eachsheaf L nis L A h fr k (Y , β ( b )) is grouplike, so we conclude using the fact that grouplike objects arestable under colimits. (cid:3) Corollary 3.3.13.
Let k be a perfect field and β : B → K (cid:62) a smooth stable tangential structureover k . For any n (cid:62) , there is an equivalence Ω n T L mot FQS m β + nk (cid:39) L mot ( FQS m βk ) gp in PSh Σ (Sm k ) .Proof. This follows immediately from Corollary 3.3.12 since M( β + n ) (cid:39) Σ n T M β . (cid:3) Remark 3.3.14.
The equivalence of Theorem 3.3.10 admits a conditional description in termsof virtual fundamental classes as follows. For β : B → K a smooth stable tangential structureover S, let
PQS m β S denote the moduli stack of proper quasi-smooth S-schemes with β -structure.There should exist a canonical morphismfc β : PQS m β S → Ω ∞ T , fr M β in PSh Σ ( Corr fr (Sm S )) sending f : Z → X to the image by the Gysin transfer f ! : M β (Z , L f ) → M β (X) of the Thom class t β ( − L f ) ∈ M β (Z , L f ) determined by the lift of − L f to B(Z). Assum-ing the existence of fc β with evident naturality and multiplicativity properties, one can showthat the equivalence of Theorem 3.3.10 is adjoint to fc β | FQS m β S . Indeed, it is enough to provethis when β is the class of a finite locally free sheaf E on a smooth S-scheme Y. In this case, itis easy to check that the compositeh frS ( V ( E ) / V × ( E )) Θ Y / S , E −−−−−→ h frS (Y , E ) fc E −−→ Ω ∞ T , fr Σ ∞ T ( V ( E ) / V × ( E ))is adjoint to the identity.This can partially be made precise for β of rank 0. Gysin transfers for regular closed immer-sions between classical schemes are constructed in [DJK18]. Using the canonical factorizationZ (cid:44) → V ( f ∗ O Z ) → X of a finite syntomic morphism f : Z → X, one can construct a morphismfc β : FS yn β S → Ω ∞ T M β in CMon(PSh Σ (Sm S )), which is natural in β (cf. [EHK + § β = 0 ∈ K(Y) for some smooth S-scheme Y,the assertion that fc β is induced by the equivalence of Theorem 3.3.10 is nevertheless verifiedby [EHK + Remark 3.3.15.
For β a smooth stable tangential structure of rank 0 over a perfect field k ,one can obtain more directly an equivalence Ω ∞ T M β (cid:39) L zar L A ( FS yn βk ) gp using the morphismfc β : FS yn βk → Ω ∞ T M β from Remark 3.3.14. By [EHK + + gp β isa motivic equivalence when β = 0 ∈ K(Y) for some smooth k -scheme Y. It follows that fc gp β is amotivic equivalence for any β ∈ K(Y) of rank 0, since the question is local on Y. Finally, sincefc β is natural in β and Ω ∞ T preserves sifted colimits [EHK + gp β is a motivic equivalence for any β of rank 0. However, this approach does not sufficeto understand MGL-modules over perfect fields (as in Theorem 4.1.4 below), because we do notyet know if the morphism fc β can be made symmetric monoidal in β , nor if it can be promotedto a morphism of presheaves with framed transfers. ODULES OVER ALGEBRAIC COBORDISM 25
Question 3.3.16.
Following the discussion in Remark 3.3.14, it is natural to ask the followingquestions:(1) For β a smooth stable tangential structure of rank (cid:62) k , is theinclusion FQS m βk ⊂ PQS m βk a motivic equivalence after group completion?(2) For β a smooth stable tangential structure of arbitrary rank over k , is there an equiva-lence Ω ∞ T M β (cid:39) L mot ( PQS m βk ) gp ?Regarding (1), Remark 3.3.14 implies that L mot ( FQS m βk ) gp is a direct factor of L mot ( PQS m βk ) gp .Moreover, for β = id K , one can show that the A -localization L A PQS m k is already grouplike(the proof will appear elsewhere). An affirmative answer to (1) would therefore imply thatL mot PQS m k is the group completion of L mot FS yn k .3.4. Algebraic cobordism spectra.Theorem 3.4.1.
Let S be a scheme. (i) There is an equivalence of E ∞ -ring spectra MGL S (cid:39) Σ ∞ T , fr FS yn S in SH (S) (cid:39) SH fr (S) . (ii) For every n (cid:62) , there is an equivalence of MGL S -modules Σ n T MGL S (cid:39) Σ ∞ T , fr FQS m n S in SH (S) (cid:39) SH fr (S) . (iii) There is an equivalence of E ∞ -ring spectra (cid:95) n (cid:62) Σ n T MGL S (cid:39) Σ ∞ T , fr FQS m S in SH (S) (cid:39) SH fr (S) .Proof. These are instances of Theorem 3.3.10, where β is the inclusion of the rank n summandof K-theory (for (i) and (ii)) or the identity map K (cid:62) → K (cid:62) (for (iii)). Indeed, these summandsof K-theory satisfy the assumption of Lemma 3.3.9 by Example A.0.8. (cid:3) Corollary 3.4.2.
Let S be a pro-smooth scheme over a field. (i) There is an equivalence of E ∞ -ring spaces Ω ∞ T , fr MGL S (cid:39) L zar L A ( FS yn S ) gp in H fr (S) . (ii) For every n (cid:62) , there are equivalences of FS yn S -modules Ω ∞ T , fr Σ n T MGL S (cid:39) L zar L A ( FQS m n S ) gp (cid:39) L nis L A FQS m n S in H fr (S) . (iii) There is an equivalence of E ∞ -ring spaces Ω ∞ T , fr (cid:95) n (cid:62) Σ n T MGL S (cid:39) L zar L A ( FQS m S ) gp in H fr (S) . Proof.
When S is the spectrum of a perfect field, these statements are instances of Corol-lary 3.3.12. In general, we can choose a pro-smooth morphism f : S → Spec k where k is aperfect field, and the results over k pull back to the results over S. (cid:3) Let us also spell out the specializations of Theorem 3.3.10 and Corollary 3.3.12 to the smoothstable tangential structure of Example 3.3.4.
Theorem 3.4.3.
Let S be a scheme. (i) There is an equivalence of E ∞ -ring spectra MSL S (cid:39) Σ ∞ T , fr FS yn orS in SH (S) (cid:39) SH fr (S) . (ii) For every n (cid:62) , there is an equivalence of MSL S -modules Σ n T MSL S (cid:39) Σ ∞ T , fr FQS m or ,n S in SH (S) (cid:39) SH fr (S) . (iii) There is an equivalence of E -ring spectra (cid:95) n (cid:62) Σ n T MSL S (cid:39) Σ ∞ T , fr FQS m orS restricting to an equivalence of E ∞ -ring spectra (cid:95) n (cid:62) Σ n T MSL S (cid:39) Σ ∞ T , fr FQS m or , evS in SH (S) (cid:39) SH fr (S) . Corollary 3.4.4.
Let S be a pro-smooth scheme over a field. (i) There is an equivalence of E ∞ -ring spaces Ω ∞ T , fr MSL S (cid:39) L zar L A ( FS yn orS ) gp in H fr (S) . (ii) For every n (cid:62) , there are equivalences of FS yn orS -modules Ω ∞ T , fr Σ n T MSL S (cid:39) L zar L A ( FQS m or ,n S ) gp (cid:39) L nis L A FQS m or ,n S in H fr (S) . (iii) There is an equivalence of E -ring spaces Ω ∞ T , fr (cid:95) n (cid:62) Σ n T MSL S (cid:39) L zar L A ( FQS m orS ) gp restricting to an equivalence of E ∞ -ring spaces Ω ∞ T , fr (cid:95) n (cid:62) Σ n T MSL S (cid:39) L zar L A ( FQS m or , evS ) gp in H fr (S) . ODULES OVER ALGEBRAIC COBORDISM 27
Hilbert scheme models.
Using the A -contractibility of the space of embeddings of afinite scheme into A ∞ (Corollary 2.4.6), we can recast our models for Ω ∞ T MGL and Ω ∞ T MSL(and others) in terms of Hilbert schemes, at the cost of losing the identification of the framedtransfers and of the multiplicative structures.Let X be an S-scheme. We define the functor Hilb fqs (X / S) : Sch opS → Spc byHilb fqs (X / S)(T) = { closed immersions Z → X T such that Z → T is finite and quasi-smooth } , and we denote by Hilb fqs ,n (X / S) the subfunctor where Z → T has relative virtual dimension − n (which is contractible unless n (cid:62) fqs , (X / S) = Hilb flci (X / S) . In particular, if X is smooth and quasi-projective over S, then Hilb fqs , (X / S) is representableby a smooth S-scheme (see [EHK + fqs ( A ∞ S / S) = colim n →∞ Hilb fqs ( A n S / S) , and similarly for Hilb fqs ,n ( A ∞ S / S).
Lemma 3.5.1.
Let S be a scheme. Then the forgetful map Hilb fqs ( A ∞ S / S) → FQS m S is a universal A -equivalence on affine schemes (i.e., any pullback of this map in PSh(Sch S ) isan A -equivalence on affine schemes).Proof. Let f : T → S and let Z ∈ FQS m S (T). Form the Cartesian squareP Z THilb fqs ( A ∞ S / S) FQS m S . Z By universality of colimits, it suffices to show that P Z → T is an A -equivalence on affineschemes. By inspection, P Z is f (cid:93) of the presheaf T (cid:48) (cid:55)→ Emb T (cid:48) (Z × T T (cid:48) , A ∞ T (cid:48) ) on Sch T , and thelatter is A -contractible on affine schemes by Corollary 2.4.6. (cid:3) Theorem 3.5.2.
Suppose S is pro-smooth over a field. (i) There is an equivalence Ω ∞ T MGL S (cid:39) L zar (L A Hilb flci ( A ∞ S / S)) gp . (ii) For every n (cid:62) , there are equivalences Ω ∞ T Σ n T MGL S (cid:39) L zar (L A Hilb fqs ,n ( A ∞ S / S)) gp (cid:39) L nis L A Hilb fqs ,n ( A ∞ S / S) . Proof.
This follows immediately from Corollary 3.4.2 and Lemma 3.5.1. (cid:3)
Define the functor Hilb or ,n (X / S) : Sch opS → Spc and the forgetful mapHilb or ,n (X / S) → Hilb fqs ,n (X / S)so that the fiber over Z ∈ Hilb fqs ,n (X / S)(T) is the ∞ -groupoid of equivalences det( L Z / T ) (cid:39) O Z .In other words, Hilb or ,n (X / S) is the Weil restriction of the G m -torsor Isom(det( L Z ) , O Z ) overthe universal Z . In particular, if X is smooth and quasi-projective over S, then Hilb or , (X / S) isrepresentable by a smooth S-scheme.
Theorem 3.5.3.
Suppose S is pro-smooth over a field. (i) There is an equivalence Ω ∞ T MSL S (cid:39) L zar (L A Hilb or , ( A ∞ S / S)) gp . (ii) For every n (cid:62) , there are equivalences Ω ∞ T Σ n T MSL S (cid:39) L zar (L A Hilb or ,n ( A ∞ S / S)) gp (cid:39) L nis L A Hilb or ,n ( A ∞ S / S) . Proof.
This follows immediately from Corollary 3.4.4 and Lemma 3.5.1, noting thatHilb or ,n ( A ∞ S / S) (cid:39) Hilb fqs ( A ∞ S / S) × FQS m S FQS m or ,n S . (cid:3) For any smooth stable tangential structure β : B → K (cid:62) , Lemma 3.5.1 gives a description ofΩ ∞ T M β in terms of the functor classifying derived subschemes Z of A ∞ with some structure onthe image of the shifted cotangent complex L Z [ −
1] in K-theory. However, it is perhaps morenatural to classify derived subschemes Z of A ∞ with some structure on the conormal sheaf N Z / A ∞ ∈ sVect(Z). If β : B → sVect (cid:62) is a morphism in PSh Σ (dSch S ), we define the functorHilb β ( A ∞ S / S) : Sch opS → Spc by the Cartesian squaresB(Z) × sVect(Z) { N Z / A ∞ T } THilb β ( A ∞ S / S)(T) Hilb fqs ( A ∞ S / S). Z Lemma 3.5.4.
Let S be a scheme and β : B → sVect (cid:62) a stable tangential structure over S .Then the forgetful map Hilb β ( A ∞ S / S) → FQS m β S is an A -equivalence on affine schemes.Proof. This map is the colimit of the mapsh nfrS (Y , β ( b )) → h frS (Y , β ( b ))over Y ∈ dSch S and b ∈ B(Y), which are A -equivalences on affine schemes by Corollary 2.4.8. (cid:3) Theorem 3.5.5.
Let k be a perfect field and β : B → sVect (cid:62) a smooth stable tangentialstructure over k . Then there is an equivalence Ω ∞ T M β (cid:39) L zar (L A Hilb β ( A ∞ k /k )) gp . Moreover, if β has rank (cid:62) , L nis L A Hilb β ( A ∞ k /k ) is already grouplike.Proof. This follows immediately from Lemma 3.5.4 and Corollary 3.3.12. (cid:3)
On can recover Theorems 3.5.2 and 3.5.3 from Theorem 3.5.5 using the motivic equivalencessVect → K and sVect SL → K SL and the fact that the functor M : PSh(Sm S ) / K → SH (S) invertsmotivic equivalences [BH18, Proposition 16.9]. ODULES OVER ALGEBRAIC COBORDISM 29 Modules over algebraic cobordism
In this section, we show that modules over motivic Thom ring spectra can be describedas motivic spectra with certain transfers. We first treat the case of MGL in § § § Z , which is an MGL-module, asa motivic spectrum with finite syntomic transfers: it is the suspension spectrum of the constantsheaf Z equipped with canonical finite syntomic transfers.It is worth pointing out that, although the theorems in this section do not involve anyderived algebraic geometry, their proofs use derived algebraic geometry in an essential way (viaSection 3).4.1. Modules over MGL.
Let
Corr fsyn (Sm S ) denote the symmetric monoidal (2 , f g where f is finite syntomic. Let H fsyn (S) denote the full subcategory of PSh Σ ( Corr fsyn (Sm S ))spanned by the A -invariant Nisnevich sheaves, and let SH fsyn (S) be the symmetric monoidal ∞ -category of T -spectra in H fsyn (S). We have the usual adjunctionΣ ∞ T , fsyn : H fsyn (S) (cid:29) SH fsyn (S) : Ω ∞ T , fsyn . The symmetric monoidal forgetful functor (cid:15) : Corr fr (Sm S ) → Corr fsyn (Sm S )(see [EHK + (cid:15) ∗ : H fr (S) → H fsyn (S) and (cid:15) ∗ : SH fr (S) → SH fsyn (S) . For clarity, we will denote the tensor products in H fr (S) and SH fr (S) by ⊗ fr and the ones in H fsyn (S) and SH fsyn (S) by ⊗ fsyn .We denote by h fsynS (X) the presheaf on Corr fsyn (Sm S ) represented by X ∈ Sm S . Lemma 4.1.1.
The forgetful functor (cid:15) ∗ : H fsyn (S) → H fr (S) is a strict H fr (S) -module functor.In other words, for any A ∈ H fr (S) and B ∈ H fsyn (S) , the canonical map A ⊗ fr (cid:15) ∗ (B) → (cid:15) ∗ ( (cid:15) ∗ (A) ⊗ fsyn B) is an equivalence.Proof. Since (cid:15) ∗ preserves colimits, we can assume that A = h frS (X) and B = h fsynS (Y) for somesmooth S-schemes X and Y. Since the stable tangential structure ι is smooth, we have byProposition 3.3.7 a Nisnevich equivalencecolim (Z ,ξ ) h frS (Y × S Z , π ∗ Z ( ξ )) → h fsynS (Y) , where the colimit is over all Z ∈ Sm S and ξ ∈ K(Z) of rank 0. Hence, it suffices to show thatthe map h frS (X) ⊗ fr h frS (Y × S Z , π ∗ Z ( ξ )) → h frS (X × S Y × S Z , π ∗ Z ( ξ ))is a motivic equivalence for all such pairs (Z , ξ ). Since the question is local on Z (by Propositions2.3.1(iii) and 2.3.2), we can assume ξ = 0, in which case it is obvious. (cid:3) Lemma 4.1.2. Σ ∞ T , fr (cid:15) ∗ (cid:39) (cid:15) ∗ Σ ∞ T , fsyn . Proof.
By Lemma 4.1.1, the T -stable adjunction (cid:15) ∗ : SH fr (S) (cid:29) SH fsyn (S) : (cid:15) ∗ is obtained from the unstable one by extending scalars along Σ ∞ T , fr : H fr (S) → SH fr (S). Thisimmediately implies the result. (cid:3) Theorem 4.1.3.
Let S be a scheme. There is an equivalence of symmetric monoidal ∞ -categories Mod
MGL ( SH (S)) (cid:39) SH fsyn (S) , natural in S and compatible with the forgetful functors to SH (S) .Proof. By Theorem 3.4.1(i), we have an equivalence of motivic E ∞ -ring spectraMGL S (cid:39) Σ ∞ T , fr h fsynS (S) . By Lemma 4.1.2, the right-hand side is (cid:15) ∗ Σ ∞ T , fsyn h fsynS (S), which means that MGL S is the imageof the unit by the forgetful functor SH fsyn (S) → SH (S). We therefore obtain an adjunction Mod
MGL ( SH (S)) SH fsyn (S) ΦΨ where Φ is symmetric monoidal and Ψ is conservative. It remains to show that the unit mapMGL S ⊗ Σ ∞ T Y + → ΨΦ(MGL S ⊗ Σ ∞ T Y + ) (cid:39) ΨΣ ∞ T , fsyn h fsynS (Y)is an equivalence for every Y ∈ Sm S . By Lemma 4.1.2 again, this map is Σ ∞ T , fr of the maph fsynS (S) ⊗ fr h frS (Y) → h fsynS (Y) , which is an equivalence by Lemma 4.1.1. (cid:3) Theorem 4.1.4.
Let k be a perfect field. (i) There is an equivalence of symmetric monoidal ∞ -categories Mod
MGL ( SH veff ( k )) (cid:39) H fsyn ( k ) gp under SH veff ( k ) (cid:39) H fr ( k ) gp . (ii) There is an equivalence of symmetric monoidal ∞ -categories Mod
MGL ( SH eff ( k )) (cid:39) SH S , fsyn ( k ) under SH eff ( k ) (cid:39) SH S , fr ( k ) .Proof. The proof of (i) is exactly the same as that of Theorem 4.1.3, using Corollary 3.4.2instead of Theorem 3.4.1. We obtain (ii) from (i) by stabilizing. (cid:3)
As a corollary, we obtain a cancellation theorem for A -invariant sheaves with finite syntomictransfers over perfect fields: Corollary 4.1.5.
Let k be a perfect field. Then the ∞ -category H fsyn ( k ) gp is prestable and thefunctor Σ G : H fsyn ( k ) gp → H fsyn ( k ) gp is fully faithful. ODULES OVER ALGEBRAIC COBORDISM 31
Modules over MSL.
We have completely analogous results for MSL instead of MGL.Consider the symmetric monoidal (2 , Corr or (Sm S ) whose objects are smooth S-schemes and whose morphisms are spans ZX Y f g where f is finite syntomic together with an isomorphism ω f (cid:39) O Z . We can form as usual thesymmetric monoidal ∞ -categories H or (S) and SH or (S). The following results are proved in thesame way as the corresponding results from § Theorem 4.2.1.
Let S be a scheme. There is an equivalence of symmetric monoidal ∞ -categories Mod
MSL ( SH (S)) (cid:39) SH or (S) , natural in S and compatible with the forgetful functors to SH (S) . Theorem 4.2.2.
Let k be a perfect field. (i) There is an equivalence of symmetric monoidal ∞ -categories Mod
MSL ( SH veff ( k )) (cid:39) H or ( k ) gp under SH veff ( k ) (cid:39) H fr ( k ) gp . (ii) There is an equivalence of symmetric monoidal ∞ -categories Mod
MSL ( SH eff ( k )) (cid:39) SH S , or ( k ) under SH eff ( k ) (cid:39) SH S , fr ( k ) . Corollary 4.2.3.
Let k be a perfect field. Then the ∞ -category H or ( k ) gp is prestable and thefunctor Σ G : H or ( k ) gp → H or ( k ) gp is fully faithful. Remark 4.2.4.
There are analogs of the above results for any E smooth stable tangential struc-ture β of rank 0 over S. Indeed, one can construct an ∞ -category Corr β (Sm S ) of β -structuredfinite syntomic correspondences using the formalism of labeling functors from [EHK + § Corr fr (Sm S ). Then for S arbitrary and k a perfectfield, we have equivalences of ∞ -categories Mod M β ( SH (S)) (cid:39) SH β (S) , Mod M β ( SH veff ( k )) (cid:39) H β ( k ) gp , Mod M β ( SH eff ( k )) (cid:39) SH S ,β ( k ) , which are symmetric monoidal if β is E ∞ . Moreover, the ∞ -category H β ( k ) gp is prestable andthe functor Σ G : H β ( k ) gp → H β ( k ) gp is fully faithful.4.3. Motivic cohomology as an MGL-module.
For A a commutative monoid, let A S de-note the corresponding constant sheaf on Sm S , which is an A -invariant Nisnevich sheaf. Thesheaf A S has canonical finite locally free transfers [BH18, Proposition 13.13], and in particularfinite syntomic transfers.Let H Z S ∈ SH (S) be the motivic cohomology spectrum defined by Spitzweck [Spi18], which isan E ∞ -algebra in Mod
MGL ( SH (S)) [Spi18, Remark 10.2]. The following theorem is a refinementof [Hoy18, Theorem 21]: Theorem 4.3.1.
For any scheme S , there is an equivalence of E ∞ -algebras H Z S (cid:39) Σ ∞ T , fsyn Z S in Mod
MGL ( SH (S)) (cid:39) SH fsyn (S) . Proof.
We first note that the right-hand side is stable under base change, since Σ ∞ T , fr (cid:15) ∗ ( Z S ) is[Hoy18, Lemma 20], Σ ∞ T , fr (cid:15) ∗ (cid:39) (cid:15) ∗ Σ ∞ T , fsyn (Lemma 4.1.2), and (cid:15) ∗ commutes with base change(by Theorem 4.1.3). We can therefore assume that S is a Dedekind domain. In this case,Ω ∞ T , fsyn H Z S is the constant sheaf of rings Z S with some finite syntomic transfers. As shownin the proof of [Hoy18, Theorem 21], these transfers are the canonical ones for framed finitesyntomic correspondences. Since Z S is a discrete constant sheaf and every finite syntomicmorphism Z → X can be framed Zariski-locally on X, we deduce that Ω ∞ T , fsyn H Z S is Z S withits canonical finite syntomic transfers. By adjunction, we obtain a morphism of E ∞ -algebras ϕ S : Σ ∞ T , fsyn Z S → H Z S in Mod
MGL ( SH (S)), which is stable under base change. It thus suffices to show that ϕ S is anequivalence when S is the spectrum of a perfect field, but this follows from Theorem 4.1.4. (cid:3) Arguing as in [Hoy18, Corollary 22], we obtain the following corollary:
Corollary 4.3.2.
Let S be a scheme and A an abelian group (resp. a ring; a commutative ring).Then there is an equivalence of H Z S -modules (resp. of E - H Z S -algebras; of E ∞ - H Z S -algebras) HA S (cid:39) Σ ∞ T , fsyn A S in Mod
MGL ( SH (S)) (cid:39) SH fsyn (S) . Remark 4.3.3.
It follows from Theorem 4.3.1 that the canonical morphism of E ∞ -ring spectraMGL S → H Z S is Σ ∞ T , fsyn of the degree map deg : FS yn S → N S . Appendix A. Functors left Kan extended from smooth algebras
A surprising observation due to Bhatt and Lurie is that algebraic K-theory, as a functor oncommutative rings, is left Kan extended from smooth rings. In this appendix, we present ageneral criterion for a functor on commutative rings to be left Kan extended from smooth rings,which we learned from Akhil Mathew, and we apply it to deduce some variants of the result ofBhatt and Lurie that are relevant for the applications of Theorem 3.3.10.A morphism of derived commutative rings f : A → B is called a henselian surjection if π ( f )is surjective and ( π (A) , ker π ( f )) is a henselian pair. Proposition A.0.1 (Mathew) . Let R be a commutative ring (resp. a derived commutativering) and F : CAlg ♥ R → Spc (resp.
F : CAlg ∆R → Spc ) a functor. Suppose that: (1) F preserves filtered colimits; (2) for every henselian surjection A → B , the map F(A) → F(B) is an effective epimor-phism; (3) for every henselian surjections A → C ← B , the square F(A × C B) F(B)F(A) F(C) is Cartesian.Then F is left Kan extended from CAlg smR . Remark A.0.2.
Conditions (1) and (2) of Proposition A.0.1 are also necessary, since they holdwhen F = Maps R (S , − ) for some smooth R-algebra S [Gru72, Th´eor`eme I.8]. Condition (3), onthe other hand, is not (for example, it fails for K-theory). ODULES OVER ALGEBRAIC COBORDISM 33
Proof.
Let ˜F be the left Kan extension of F | CAlg smR . Then ˜F is a colimit of functors satisfyingconditions (1)–(3) and in particular it satisfies conditions (1) and (2). The canonical map˜F → F is an equivalence on smooth R-algebras, hence on ind-smooth R-algebras. For anyA ∈ CAlg ♥ R (resp. A ∈ CAlg ∆R ), we can inductively construct an augmented simplicial objectB such that B[ ∅ ] = A and, for each n (cid:62)
0, B[∆ n ] is ind-smooth and B[∆ n ] → B[ ∂ ∆ n ] is ahenselian surjection. To conclude, we prove that both ˜F and F send B to a colimit diagram.Since ˜F is a colimit of functors that satisfy (1)–(3), it will suffice to show that F(B) is acolimit diagram. Henselian surjections are stable under pullback, so the map B[L] → B[K] isa henselian surjection for any inclusion of finite simplicial sets K ⊂ L. In particular, by (2),F(B[L]) → F(B[K]) is an effective epimorphism.Let K be a finite nonsingular simplicial set. Then K can be built from ∅ and simplices ∆ n bya finite sequence of pushouts, which are transformed by B[ − ] into Cartesian squares of henseliansurjections. By (3), we deduce that F(B[K]) (cid:39) F(B)[K] for such K, since this is trivially truefor K = ∅ and K = ∆ n . Applying this to K = ∂ ∆ n , we conclude that F(B)[∆ n ] → F(B)[ ∂ ∆ n ]is an effective epimorphism, hence that F(B) is a colimit diagram. (cid:3) Example A.0.3. SH ω : CAlg ♥ R → ∞ -Cat is left Kan extended from CAlg smR (apply Proposi-tion A.0.1 to Fun(∆ n , SH ( − ) ω ) (cid:39) for n (cid:62) Proposition A.0.4 (Mathew) . Let R be a derived commutative ring and X a smooth algebraicstack over R with quasi-affine diagonal (e.g., a smooth quasi-separated algebraic space), viewedas a functor X : CAlg ∆R → Spc . Then X is left Kan extended from CAlg smR .Proof.
We check conditions (1)–(3) of Proposition A.0.1. Condition (1) holds because X islocally of finite presentation, and condition (3) holds because Spec(A × C B) is the pushoutof Spec(B) ← Spec(C) → Spec(A) in the ∞ -category of derived algebraic stacks. It remainsto check condition (2). Let A → B be a henselian surjection between derived commutativeR-algebras, and let A (cid:48) = π A × π B B be the relative 0-truncation of A over B. Since A → A (cid:48) induces an isomorphism on π and X is smooth and nilcomplete, the induced map X(A) → X(A (cid:48) )is an effective epimorphism [Lur18, Lemma 17.3.6.4 and Remark 17.3.9.2]. Moreover, by (3),X(A (cid:48) ) (cid:39) X( π A) × X( π B) X(B). Thus, it remains to show that X( π A) → X( π B) is an effectiveepimorphism. In other words, we can assume A discrete and B = A / I for some ideal I ⊂ A suchthat (A , I) is a henselian pair.Let f : Spec(A / I) → X be a morphism over R. We must show that f can be extendedto Spec(A), and we may as well assume that A = R. Since Spec(A / I) is quasi-compact, wecan replace X by a quasi-compact open substack and assume X finitely presented over A. Bycondition (1), we can also assume that I ⊂ A is a finitely generated ideal. Then the pair (A , I)is a filtered colimit of pairs that are henselizations of pairs of finite type over Z . By [Ryd15,Proposition B.2], we are reduced to the case where (A , I) is the henselization of a pair of finitetype over Z . In this case, the map A → A ∧ I is regular [Stacks, Tag 0AH5], hence is a filteredcolimit of smooth morphisms A → B α by Popescu [Stacks, Tag 07GC]. Since X is smooth overA, we can compatibly extend f to Spec(A / I n ) for all n . As X is now Noetherian with quasi-affine diagonal, we can extend f to Spec(A ∧ I ) by Grothendieck’s algebraization theorem (asgeneralized by Bhatt and Halpern-Leistner [BHL17, Corollary 1.5] or Lurie [Lur18, Corollary9.5.5.3]), hence to Spec(B α ) for some α . We are thus in the following situation:XSpec(A / I) Spec(B α ) Spec(A). f Since (A , I) is henselian and B α is a smooth A-algebra, the morphism Spec(B α ) → Spec(A)admits a section fixing Spec(A / I), so we are done. (cid:3)
Corollary A.0.5.
Let R be a derived commutative ring and X a smooth algebraic stack over R with quasi-affine diagonal and a monoidal structure. Then the group completion X gp : CAlg ∆R → Spc is left Kan extended from
CAlg smR .Proof.
For every A ∈ CAlg ∆R , the ∞ -category (CAlg smR ) / A has finite coproducts and hence issifted. It follows that the forgetful functors Mon gp (Spc) → Mon(Spc) → Spc commute withleft Kan extension along the inclusion CAlg smR ⊂ CAlg ∆R (since they preserve sifted colimits).In particular it suffices to show that the functorX gp : CAlg ∆R → Mon gp (Spc)is left Kan extended from CAlg smR . This is the composition of X with the group completionfunctor Mon(Spc) → Mon gp (Spc), which preserves colimits. It therefore suffices to show thatX is left Kan extended from smooth R-algebras, which follows from Proposition A.0.4. (cid:3) Example A.0.6.
Corollary A.0.5 implies that the following functors CAlg ∆R → Spc are leftKan extended from CAlg smR :(1) algebraic K-theory K, which is the group completion of the stack of finite locally freesheaves;(2) oriented K-theory K SL , which is the group completion of the stack of finite locally freesheaves with trivialized determinant;(3) symplectic K-theory K Sp , which is the group completion of the stack of finite locallyfree sheaves (necessarily of even rank) with a nondegenerate alternating bilinear form;(4) orthogonal K-theory K O , which is the group completion of the stack of finite locallyfree sheaves with a nondegenerate quadratic form;(5) Grothendieck–Witt theory GW, which is the group completion of the stack of finitelocally free sheaves with a nondegenerate symmetric bilinear form.All these examples are presheaves of E ∞ -spaces, except K SL which is E . However, the evenrank summand K SLev ⊂ K SL is E ∞ . Indeed, K SL is the pullback K × Pic Z Z and the map Z → Pic Z is only E , while its restriction to 2 Z ⊂ Z is E ∞ . Lemma A.0.7.
Let F → H f ← G be a diagram in Fun(CAlg ∆R , Spc) . If F is left Kan extendedfrom CAlg smR and f is relatively representable by smooth affine schemes, then F × H G is leftKan extended from CAlg smR .Proof.
We have a square of adjunctionsPSh(SmAff R ) / G PSh(dAff R ) / G PSh(SmAff R ) / H PSh(dAff R ) / HLKE f ∗ f ∗ resLKE f ∗ res f ∗ and we wish to prove that the square of left adjoints commutes. Under the identificationPSh(C) / X (cid:39) PSh(C / X ) [Lur17b, Corollary 5.1.6.12] and the assumption on f , the functors f ∗ areprecomposition with the pullback functors f ∗ : (SmAff R ) / H → (SmAff R ) / G and f ∗ : (Aff R ) / H → (Aff R ) / G . It is then obvious that the square of right adjoints commutes. (cid:3) Example A.0.8.
Let F : CAlg ∆R → Spc be one of the functors from Example A.0.6. Thenthere is a rank map F → L Σ Z . For any subset I ⊂ Z , let F I ⊂ F be the subfunctor consistingof elements with ranks in I. Then F I is left Kan extended from CAlg smR . This follows fromLemma A.0.7, since L Σ I ⊂ L Σ Z is relatively representable by smooth affine schemes. ODULES OVER ALGEBRAIC COBORDISM 35
Remark A.0.9.
The proof of Proposition A.0.4 is quite nonelementary. For the algebraicstacks from Example A.0.6, it is possible to prove much more directly that they are left Kanextended from CAlg smR . Let us give such a proof for Vect itself. Since the rank of a vectorbundle on a derived affine scheme is bounded, we haveVect = colim n Vect (cid:54) n . Let Gr (cid:54) n = colim k →∞ h(Gr ( A k ) (cid:116) · · · (cid:116) Gr n ( A k )) , where h is the Yoneda embedding. The canonical map Gr (cid:54) n → Vect (cid:54) n is an effective epimor-phism of presheaves on dAff R , since every vector bundle on a derived affine scheme is generatedby its global sections. For every n (cid:54) k , choose a vector bundle torsor U n,k → Gr n ( A k ) whereU n,k is affine, and choose maps U n,k → U n,k +1 compatible with G r n ( A k ) → G r n ( A k +1 ). LetU (cid:54) n = colim k →∞ h(U ,k (cid:116) · · · (cid:116) U n,k ) . Then the map U (cid:54) n → Gr (cid:54) n is an effective epimorphism of presheaves on dAff R , because everyvector bundle torsor over a derived affine scheme admits a section. Thus, U (cid:54) n → Vect (cid:54) n is aneffective epimorphism, and hence Vect (cid:54) n is the colimit of the simplicial diagram · · · U (cid:54) n × Vect (cid:54) n U (cid:54) n U (cid:54) n .Since Vect (cid:54) n is an Artin stack with smooth and affine diagonal, each term in this simplicialobject is a filtered colimit of smooth affine R-schemes. Appendix B. The ∞ -category of twisted framed correspondences In this appendix, we construct a symmetric monoidal ∞ -category Corr L ((dSch S ) / K ) whoseobjects are pairs (X , ξ ) where X is a derived S-scheme and ξ ∈ K(X), and whose morphisms arespans Z(X , ξ ) (Y , η ) f g where L f is perfect together with an equivalence f ∗ ( ξ ) + L f (cid:39) g ∗ ( η ) in K(Z). We also constructsymmetric monoidal functors Corr fr (dSch S ) → Corr L ((dSch S ) / K ) , X (cid:55)→ (X , ,γ : (dSch S ) / K → Corr L ((dSch S ) / K ) , (X , ξ ) (cid:55)→ (X , ξ ) , where Corr fr (dSch S ) is the ∞ -category of framed correspondences constructed in [EHK + γ extends γ : dSch S → Corr fr (dSch S ). These constructions are used severaltimes in the paper. For example, the presheaf h frS (Y , ξ ) on Corr fr (dSch S ) is the restrictionof the presheaf represented by (Y , − ξ ) on the wide subcategory of Corr L ((dSch S ) / K ) whosemorphisms have a finite quasi-smooth left leg, and the right-lax symmetric monoidal structureon the functor (Y , ξ ) (cid:55)→ h frS (Y , ξ ) is induced by the symmetric monoidal functor γ .We will construct Corr L ((dSch S ) / K ) using the formalism of labeling functors developed in[EHK + Definition B.0.1.
Let X • be a simplicial ∞ -category. A Segal presheaf on X • is a functorF : (cid:90) ∆ op X op • → Spc such that, for every n (cid:62) σ ∈ X n , the mapF( σ ) → F( ρ ∗ ( σ )) × F( σ ) · · · × F( σ n − ) F( ρ ∗ n ( σ ))induced by the Segal maps ρ i : [1] → [ n ] is an equivalence. It is called reduced if F | X op0 iscontractible, and it is called complete if, for every v ∈ X , the map F( v ) → F( ι ∗ ( v )) inducedby the unique map ι : [1] → [0] is an equivalence (equivalently, if F sends cocartesian edges over∆ opsurj to equivalences).Note that a Segal presheaf in the sense of [EHK + ∞ -category and M and N two classes of morphisms in C that are closed undercomposition and under pullback along one another. Recall from [EHK + , M , N) a simplicial ∞ -category Φ • (C , M , N) ⊂ Fun((∆ • ) op , C),where Φ n (C , M , N) is the subcategory of Fun((∆ n ) op , C) whose objects are the functors send-ing every edge of (∆ n ) op to M and whose morphisms are the Cartesian transformations withcomponents in N. We now repeat [EHK + Definition B.0.2.
Let (C , M , N) be a triple. A labeling functor on (C , M , N) is a Segal presheafon Φ • (C , M , N).Given a triple with labeling functor (C , M , N; F) and n (cid:62)
0, we define the space
Corr F n (C , M , N)by applying the Grothendieck construction to the functor
Corr n (C , M , N) → Spcsending an n -span σ : (Σ n , Σ L n , Σ R n ) → (C , M , N) to the limit of the composite (cid:90) ∆ op Φ • (Σ n , Σ L n , Σ R n ) op σ −→ (cid:90) ∆ op Φ • (C , M , N) op F −→ Spc . As in [EHK + → Fun(∆ op , Spc) , (C , M , N; F) (cid:55)→
Corr F • (C , M , N) . Let us unpack the simplicial space
Corr F • (C , M , N) in degrees (cid:54) • Corr F0 (C , M , N) is the space of pairs (X , α ) where X ∈ C and α ∈ F(X); • Corr F1 (C , M , N) is the space of spans ZX Y f g where f ∈ M and g ∈ N, together with ϕ ∈ F( f ), β ∈ F(Y), and an equivalence δ ∗ ( ϕ ) (cid:39) g ∗ ( β ); • the degeneracy map s : Corr F0 (C , M , N) → Corr F1 (C , M , N) sends (X , α ) to the identityspan on X with ϕ = ι ∗ ( α ), β = α , and δ ∗ ι ∗ ( α ) (cid:39) α the canonical equivalence; • the face map d : Corr F1 (C , M , N) → Corr F0 (C , M , N) sends a span as above to (X , δ ∗ ( ϕ )); • the face map d : Corr F1 (C , M , N) → Corr F0 (C , M , N) sends a span as above to (Y , β ). Proposition B.0.3.
Let (C , M , N; F) be a triple with labeling functor. Then
Corr F • (C , M , N) is a Segal space. If F is complete, then Corr F • (C , M , N) is a complete Segal space.Proof. The proof of the first statement is exactly the same as the proof of [EHK + (cid:3) ODULES OVER ALGEBRAIC COBORDISM 37
Remark B.0.4.
One can show that the above construction subsumes Haugseng’s ∞ -categoriesof spans with local systems [Hau18, Definition 6.8]. Indeed, to a presheaf of (complete) Segalspaces F on an ∞ -category C with pullbacks, one can associate a (complete) labeling functorF on C such that Corr F • (C) is the Segal space of spans in C with local systems valued in F .Let us denote by “perf” the class of morphisms of derived schemes with perfect cotangentcomplex. We now seek to construct a labeling functor K on the pair (dSch , perf), such thatthe restriction of K to Φ (dSch , perf) op = dSch op is the K-theory presheaf K. Moreover, for an n -simplex σ = (X ← X ← · · · ← X n )in Φ n (dSch , perf), the first vertex map K ( σ ) → K(X ) should be an equivalence (in particular, K should be complete), and for 0 (cid:54) i (cid:54) n the i th vertex map K(X ) (cid:39) K ( σ ) → K(X i ) shouldbe ξ (cid:55)→ f ∗ i ( ξ ) + L f i where f i : X i → X .Let p : X → S be a coCartesian fibration classified by a functor S → ∞ -Cat pt , rex . In[EHK + ∞ -category Gap S ( n, X) of relative n -gappedobjects of X, which is equivalent to the full subcategory of Fun(∆ n , X) spanned by the functorssending 0 to a p -relative zero object (i.e., a zero object in its fiber). LetFilt S ( n, X) = Fun(∆ n , X) . The simplicial ∞ -category Filt S ( • , X) classifies a coCartesian fibration Filt S (X) → ∆ op . LetGap S (X) ⊂ Filt S (X) be the full subcategory on those functors ∆ n → X sending 0 to a p -relative zero object. Then Gap S (X) → ∆ op is a coCartesian fibration classified by Gap S ( • , X).Moreover, the inclusion Gap S (X) ⊂ Filt S (X) has a left adjoint preserving coCartesian edges.By straightening, it gives rise to a morphism of simplicial ∞ -categoriesFilt S ( • , X) → Gap S ( • , X)over Fun(∆ • , S), sending x → x → · · · → x n to the relative n -gapped object 0 p ( x ) → x /x → · · · → x n /x , where x i /x denotes a p -relative cofiber. Remark B.0.5.
When S = ∗ , the simplicial map Filt S ( • , X) → Gap S ( • , X) is the one con-structed by Barwick in [Bar16, Corollary 5.20.1]. However, for general S, our notion of “relative”is essentially different.We specialize to the coCartesian fibration p : Perf → dSch op . For σ : ∆ n → dSch op , letGap σ ( Perf ) be the fiber of the coCartesian fibration p ∗ : Gap dSch op ( n, Perf ) → Fun(∆ n , dSch op )over σ , and let Filt σ ( Perf ) be the fiber of the coCartesian fibration p ∗ : Filt dSch op ( n, Perf ) → Fun(∆ n , dSch op ) over σ . Then, by the additivity property of K-theory, we have canonicalequivalences K(Gap σ ( Perf )) (cid:39) K( σ ) × · · · × K( σ n ) , K(Filt σ ( Perf )) (cid:39) K( σ ) × K( σ ) × · · · × K( σ n ) , such that the map Filt σ ( Perf ) → Gap σ ( Perf ) induces the projection onto the last n factors(cf. [EHK + + dSch op ( • , Perf ) KGap dSch op ( • , Perf )Fun(∆ • , dSch op ). In [EHK + dSch op ( • , Perf )Φ • (dSch , perf) op Fun(∆ • , dSch op ). L Form the Cartesian square(B.0.6) P • KFilt dSch op ( • , Perf )Φ • (dSch , perf) op KGap dSch op ( • , Perf ). L By [Lur09, Lemma 1.4.14], the right vertical map is a simplicial coCartesian fibration in spaces,and hence the left vertical map induces a coCartesian fibration in spaces (cid:90) ∆ op P • → (cid:90) ∆ op Φ • (dSch , perf) op , which is classified by a functor K : (cid:90) ∆ op Φ • (dSch , perf) op → Spc . It is not difficult to show that K is a labeling functor on (dSch , perf) with the desired properties(cf. [EHK + Definition B.0.7.
Let S be a derived scheme. The ∞ -category Corr L ((dSch S ) / K ) is thecomplete Segal space Corr K • (dSch S , perf).The labeling functor fr : (cid:90) ∆ op Φ • (dSch , perf) op → Spcconstructed in [EHK + L further along thezero section Φ • (dSch , perf) op → KFilt dSch op ( • , Perf ), so there is a canonical natural transfor-mation fr → K of labeling functors on (dSch , perf), inducing a functor Corr fr (dSch S ) → Corr L ((dSch S ) / K ) , X (cid:55)→ (X , . The functor γ : (dSch S ) / K → Corr L ((dSch S ) / K )is simply the inclusion of the wide subcategory on those spans whose left leg is an equivalence.Finally, we can equip Corr L ((dSch S ) / K ) with a symmetric monoidal structure, where(X , ξ ) ⊗ (Y , η ) = (X × S Y , π ∗ X ( ξ ) + π ∗ Y ( η )) . To that end we must promote K to a symmetric monoidal labeling functor [EHK + + § ∗ → coCartsending I + to the coCartesian fibration Perf IS → (dSch opS ) I . References [AGP18] A. Ananyevskiy, G. Garkusha, and I. Panin,
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