The zeroth P^1-stable homotopy sheaf of a motivic space
aa r X i v : . [ m a t h . K T ] J u l THE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE TOM BACHMANN
Abstract.
We establish a kind of “degree zero Freudenthal G m -suspension theorem” in motivic ho-motopy theory. From this we deduce results about the conservativity of the P -stabilization functor.In order to establish these results, we show how to compute certain pullbacks in the cohomology ofa strictly homotopy invariant sheaf in terms of the Rost–Schmid complex. This establishes the mainconjecture of [BY18], which easily implies the aforementioned results. Contents
1. Introduction 11.1. P -stabilization in motivic homotopy theory 11.2. Pullbacks and the Rost–Schmid complex 21.3. Acknowledgements 31.4. Notation and conventions 32. Preliminaries 32.1. Strictly homotopy invariant sheaves 32.2. Cousin and Rost–Schmid complexes 43. A “formula” for closed pullback 54. Applications 114.1. Notation and hypotheses 114.2. The heart of SH S ( k )( d ) 124.3. Canonical resolutions 14References 151. Introduction
After recalling some preliminaries in §
2, this article has two main sections of very different flavor. In § § P -stabilization in motivic homotopy theory. Motivic homotopy theory is the universal ho-motopy theory of smooth algebraic varieties, say over a field k . It is built by freely adjoining homotopycolimits to the category of smooth k -varieties, and then enforcing Nisnevich descent and making A con-tractible [MV99]. Write S pc( k ) ∗ for the pointed version of this theory. This is a symmetric monoidalcategory (the monoidal operation being given by the smash product), and every pointed smooth varietydefines an object in it. Given a pointed motivic space
X ∈ S pc( k ) ∗ , the classical homotopy groups upgrade to homotopy sheaves π i ( X ).The Riemann sphere P := ( P , ∈ S pc( k ) ∗ plays a similar role to the ordinary sphere in classicaltopology. Stable motivic homotopy theory is concerned with the category obtained by making Σ P := ∧ P into an equivalence. It is this context in which algebraic cycles and motivic cohomology naturally appear.We can take a more pedestrian approach. The functor Σ P has a right adjoint Ω P , and there is a directeddiagram of endofunctors of S pc( k ) ∗ id → Ω P Σ P =: Q → Ω P Σ P =: Q → · · · → Ω n P Σ n P =: Q n → . . . ; Date : July 16, 2020. We think of this as an ∞ -category, but no information will be lost for the purposes of this introduction by justconsidering its homotopy 1-category. I.e. Nisnevich sheaves on the site of smooth k -varieties. denote by Q its homotopy colimit. Then Q X is the P -stabilization of X , and the homotopy sheaves of Q X are called the P -stable homotopy sheaves of X .A simple form of our main application of our technical result is as follows. It is reminiscent of thefact that for an ordinary space X , the sequence of sets { π Ω i Σ i X } i ≥ is given by π X , F π X , Z ( π X ), Z ( π X ), . . . , where for a pointed set A , F A denotes the free group on A (with identity given by the basepoint), and Z ( A ) denotes the free abelian group on A (with 0 given by the base point). Theorem 1.1.
Let k be a perfect field and n ≥ (if char ( k ) = 0 , n = 2 is also allowed). Then for X ∈ S pc( k ) ∗ , the canonical map π Q n X → π Q X is an isomorphism.Proof. This is an immediate consequence of Corollary 4.9 and e.g. Morel’s Hurewicz theorem [Mor12,Theorem 6.37]. (cid:3)
Example . Morel’s computations [Mor12, Corollary 6.43] imply that for X = S , already π Q S ≃ GW ≃ π QS . Our result shows that this stabilization is not special to S , except that our results arenot strong enough to establish stabilization at Q , only at Q . See also Remark 4.3.We also obtain some conservativity results; here is a simple form. It is similar to the fact thatstabilization is conservative on simply connected topological spaces. Write S pc( k ) ∗ ( n ) ⊂ S pc( k ) ∗ forthe subcategory generated under homotopy colimits by objects of the form X + ∧ G ∧ nm with X ∈ Sm k (and G m := ( A \ , X + := X ` ∗ ). Denote by S pc( k ) ∗ , ≥ ⊂ S pc( k ) ∗ the subcategory of A -simplyconnected spaces. Theorem 1.3.
Let k be perfect, and put n = 1 if char ( k ) = 0 and n = 3 if char ( k ) > . Then thestabilization functor Q : S pc( k ) ∗ , ≥ ∩ S pc( k ) ∗ ( n ) → S pc( k ) ∗ is conservative (i.e., detects equivalences). In particular Σ P and all of its iterates, and also Σ ∞ P , are conservative on the same subcategory. Proof.
This is an immediate consequence of Corollary 4.15 and e.g. [WW17, Corollary 2.23]. (cid:3)
The results in § S -spectra to P -spectra. The reader is encouraged to skip to this sectiondirectly. Our main results in the form of Corollary 4.9, Theorem 4.14 and Corollary 4.15 can be under-stood without reading the rest of the article (except perhaps for taking a glance at § Pullbacks and the Rost–Schmid complex.
The results sketched above are obtained by combin-ing the main results of [BY18] with a technical result that we describe now. Essentially, this establishes[BY18, Conjecture 6.10] (for n ≥ M be a strictly homotopy invariant sheaf (see § X a smooth variety. Morel has proved [Mor12, Corollary 5.43] thatthere is a very convenient complex, known as the Rost–Schmid complex C ∗ ( X, M ), which can be used tocompute the Nisnevich cohomology H ∗ ( X, M ). This complex has the special property that C n ( X, M )only depends on the n -fold contraction M − n , and similarly so does the boundary map C n ( X, M ) → C n +1 ( X, M ). Let Z ⊂ X have codimension ≥ d . An obvious modification C ∗ Z ( X, M ) of C ∗ ( X, M ) canbe used to compute H ∗ Z ( X, M ); by construction one has C nZ ( X, M ) = 0 for n < d . It follows that thegroup H dZ ( X, M ) only depends on M − d (in fact this holds for all groups H ∗ Z ( X, M ), but we are mostinterested in the lowest one). Now let f : Y → X be a morphism of smooth varieties with f − ( Z ) alsoof codimension ≥ d on Y . Then the pullback map(1) f ∗ : H dZ ( X, M ) → H df − ( Z ) ( Y, M )is a morphism of abelian groups, both of which only depend on M − d .It is not difficult to show (using the results of [BY18]; see the proof of Theorem 4.6 for details) that[BY18, Conjecture 6.10] is equivalent to the statement that the morphism (1) also only depends on M − d ,in an appropriate sense. The main result of this article (Theorem 3.1) states that this is true. In fact, our original plan for [BY18] was to establish [BY18, Conjecture 6.10] (and hence the results in §
4) by provingthat f ∗ only depends on M − d . This turned out to be more difficult than we had anticipated. HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 3 We establish this by adapting an argument of Levine, using a variant of Gabber’s presentation lemmato set up an induction on d . (The case d = 0 holds tautologically.)1.3. Acknowledgements.
It is my pleasure to thank Maria Yakerson, Marc Levine and Mike Hopkinsfor fruitful discussions about these problems.I would also like to thank Marc Hoyois and Maria Yakerson for comments on a draft of this article.1.4.
Notation and conventions.
We fix throughout a field k . All non-trivial results will require k tobe perfect.Given a presheaf M on the category of smooth varieties over k , and an essentially smooth k -scheme X , we denote by M ( X ) the evaluation at X of the canonical extension of M to pro-(smooth schemes),into which the category of essentially smooth schemes embeds by [Gro67, Proposition 8.13.5]. In otherwords, if X = lim i X i is a cofiltered limit of smooth k -schemes with affine transition maps, then M ( X ) =colim i M ( X i ) (and this is known to be independent of the presentation of X ).Given a scheme X and a point x ∈ X , we identify x and Spec ( k ( x )). In particular, if X is smooth and k is perfect (so that x is essentially smooth), then we write M ( x ) for what is often denoted M ( k ( x )).Given a scheme X and d ≥
0, we write X ( d ) for the set of points of X of codimension d on X ; inother words if x ∈ X then x ∈ X ( d ) if and only if dim X x = d , where X x denotes the localization of X in x . For example, X (0) is the set of generic points of X .For a regular immersion Y ֒ → X , we denote by N Y/X the normal bundle and by ω Y/X = det N ∨ Y/X thedeterminant of the conormal bundle. More generally for any morphism Y → X such that the cotangentcomplex L Y/X is perfect we write ω Y/X = det L Y/X .2.
Preliminaries
We recall some well-known results from motivic homotopy theory.2.1.
Strictly homotopy invariant sheaves.
We write Sm k for the category of smooth k -schemes. Wemake it into a site by endowing it with the Nisnevich topology [Nis89]. This is the only topology weshall use; all cohomology will be with respect to it. Unless noted otherwise, by a (pre)sheaf we mean a(pre)sheaf of abelian groups on Sm k .Recall that a sheaf M is called strictly homotopy invariant if, for all X ∈ Sm k , the canonical map H ∗ ( X, M ) → H ∗ ( A × X, M ) is an isomorphism. We denote the category of strictly homotopy invariantsheaves by HI ( k ). Example . For a commutative ring A , denote by GW ( A ) its Grothendieck-Witt ring, i.e. the additivegroup completion of the semiring of isometry classes of non-degenerate, symmetric bilinear forms on A [MH73]. Write GW for the associated Nisnevich sheaf on Sm k . Then GW turns out to be strictlyhomotopy invariant (combine [OP99, Theorem A] and [Mor12, §§ Remark . As mentioned in the introduction, there exists a universal homotopy theory built outof (pointed) smooth varieties by enforcing A -homotopy invariance and Nisnevich descent [MV99]; wedenote it by S pc( k ) ∗ . By construction, for M ∈ HI ( k ), the Eilenberg-MacLane spaces K ( M, i ) defineobjects in S pc( k ) ∗ . In this way, results about S pc( k ) ∗ translate into properties of the cohomology ofstrictly homotopy invariant sheaves; we will use this correspondence freely in the sequel.2.1.1. Unramifiedness.
Let X ∈ Sm k be connected and ∅ 6 = U ⊂ X be open. Then for M ∈ HI ( k ), thecanonical map M ( X ) → M ( U ) is an injection [Mor05, Lemma 6.4.4]. It follows that if ξ ∈ X is thegeneric point, then M ( X ) ֒ → M ( ξ ).2.1.2. Contractions.
For a presheaf M , write M − for the presheaf given by M − ( X ) = ker ( M (( A \ × X ) i ∗ −→ M ( X )) and M − n for the n -fold iteration of this construction. Here i : X → ( A \ × X denotes the inclusion at 1 ∈ A . Pullback along the structure map splits i ∗ and hence M − is a summandof the internal mapping object Hom( A \ , M ). It follows that M − n is a ((strictly) homotopy invariant)(pre)sheaf if M is.2.1.3. GW -module structure. Let X ∈ Sm k and u ∈ O ( X ) × . Multiplication by u defines an endo-morphism of ( A \ × X and hence of Hom( A \ , M )( X ); passing to the summand we obtain h u i : M − ( X ) → M − ( X ). Suppose that M ∈ HI ( k ). Since the map Z [ O × ] → GW , u
7→ h u i issurjective on fields, unramifiedness implies that this construction extends in at most one way to a GW -module structure on M − . It turns out that this GW -module structure always exists [Mor12, Lemma3.49]. TOM BACHMANN
Twisting.
Given a line bundle L on X ∈ Sm k , write L × for the sheaf of non-vanishing sections.For M ∈ HI ( k ) and d > M − d ( X, L ) = H ( X, M − d × O × L × ); here the action of O × on M − d isvia O × → GW and the action on L × is given by multiplication. Note that since h u i = h u − i we have M − d ( X, L ) ≃ M − d ( X, L − ).2.1.5. Thom spaces.
For X ∈ Sm k and V a vector bundle on X of rank d , we have T h ( V ) := V /V \ X ∈S pc( k ) ∗ . For M ∈ HI ( k ), there are canonical isomorphisms [Mor12, Lemma 5.35][ V /V \ X , K ( M, d )] S pc( k ) ∗ ≃ H d X ( V, M ) ≃ M − d ( X, det V ) . Homotopy purity.
Let X ∈ Sm k , U ⊂ X open with reduced closed complement Z = X \ U alsosmooth. Then in S pc( k ) ∗ there is a canonical equivalence [MV99, § X/X \ Z ≃ T h ( N Z/X ) . Boundary maps.
Let X ∈ Sm k and x ∈ X ( d ) . Then X x is an essentially smooth scheme withclosed point x . Homotopy purity supplies us with the collapse sequence X x → X x /X x \ x ≃ T h ( N x/X ) ∂ −→ Σ( X x \ x ) . Pullback along ∂ induces the boundary map in the long exact sequence of cohomology with support. Wemost commonly use the case where d = 1. Then X x \ x = ξ where ξ is the generic point of X (specializingto x ), and the boundary map takes the familiar form ∂ : M ( ξ ) → M − ( x, ω x/X ) . Monogeneic transfers.
Let k be perfect and K/k be a finitely generated field extension, whence X = Spec ( K ) is an essentially smooth scheme. Let K ( x ) /K be a finite, monogeneic field extension. Weare supplied with an embedding X ′ = Spec ( K ( x )) x ֒ −→ A X ⊂ P X and thus homotopy purity provides uswith a collapse map P X → P X / P X \ X ′ ≃ T h ( N X ′ / A X ) ≃ T h ( O X ′ );here the normal bundle is canonically trivialized by the minimal polynomial of x . Pullback along thiscollapse map induces the monogeneic transfer [Mor12, p.99] τ x : M − ( K ( x )) → M − ( K ) . Slightly more generally, suppose that z ∈ P K is any closed point. Then we have the transfer maptr z : M − ( z, ω z/ P K ) ≃ H z ( P K , M ) → H ( P K , M ) ≃ M − ( K ) . This contains no new information: if z ∈ A K then tr z coincides up to isomorphism with τ z , and the onlyother case is z = ∞ which is a rational point, and so tr z is isomorphic to the identity.2.2. Cousin and Rost–Schmid complexes.
Let M be a sheaf of abelian groups on X . The cohomol-ogy of M on X can be computed using the coniveau spectral sequence ; see e.g. [CTHK97, § E page one finds the so-called Cousin complex [CTHK97, (1.3)](2) M ( X ) → M x ∈ X (0) M ( x ) → M x ∈ X (1) H x ( X, M ) → · · · → M x ∈ X ( d ) H dx ( X, M ) =: C d ( X, M ) → · · · . Here(3) H dx ( X, M ) = colim V ∋ x H d { x }∩ V ( V, X ) , where the colimit runs over open neighborhoods of x . The boundary maps in (2) are induced by certainboundary maps in long exact sequences of cohomology with support.Now suppose that M ∈ HI ( k ). By the Bloch–Ogus–Gabber theorem [CTHK97, Theorem 6.2.1], theCousin complex (2) is then exact when viewed as a complex of sheaves (i.e. for X local). Since it consistsof flasque sheafes, it can thus be used to compute the Zariski cohomology of M . The terms also turn outto be Nisnevich-acyclic (see [CTHK97, Theorem 8.3.1] or [Mor12, Lemma 5.42]), and hence the Cousincomplex computes the Nisnevich cohomology of M as well (which thus turns out to coincide with theZariski cohomology). Here and in the sequel we view essentially smooth schemes as defining pro-objects in S pc( k ). Morel calls this the geometric transfer . HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 5 Remark . The Cousin complex can also be used to compute cohomology with support in a closedsubscheme Z ; just replace C d ( X, M ) = M x ∈ X ( d ) H dx ( X, M ) by C dZ ( X, M ) = M x ∈ X ( d ) ∩ Z H dx ( X, M ) . This holds since the resolving sheaves are flasque.Now let k be perfect. We would like to make this complex more explicit. As a first step, homotopypurity (see §§ H dx ( X, M ) ≃ M − d ( x, ω x/X ) . The boundary maps can also be identified. We only use the following weak form of this result.
Theorem 2.4 (Morel [Mor12], Corollary 5.44) . Let M ∈ HI ( k ) , X ∈ Sm k and d ≥ . Then the boundarymap C d ( X, M ) → C d +1 ( X, M ) in the Cousin complex only depends on the sheaf M − d , together with (if d > ) its structure as a GW -module from § In fact, Morel proves this result by identifying the Cousin complex with another complex called the
Rost–Schmid complex (which has the same terms but a priori different boundary maps). In other wordsthe boundary map in the Cousin complex admits an explicit formula, involving only the codimension 1boundary of § § In the sequel, we will not distinguish between the Cousin and Rost–Schmid complexes.3.
A “formula” for closed pullback
In this section we establish our main result.
Theorem 3.1.
Let k be a perfect field, M ∈ HI ( k ) , f : Y → X ∈ Sm k , d ≥ , Z ⊂ X closed ofcodimension ≥ d such that f − ( Z ) ⊂ Y is also of codimension ≥ d . Then the map f ∗ : H dZ ( X, M ) → H df − ( Z ) ( Y, M ) only depends on M − d ∈ HI ( k ) , together with its GW -module structure and transfers along monogeneicfield extensions (in the sense of §§ In the sequel, we shall say “depends only on M − d ” to mean what is asserted in the theorem, i.e.“depends only on M − d as a GW -module with transfers”. To make sense of the statement, recall thatthe group H dZ ( Y, M ) depends only on M − d , as is seen from the Rost–Schmid resolution of M (i.e. usingRemark 2.3 and Theorem 2.4). Remark . Note that the Rost–Schmid complex is functorial in smooth morphisms in an obvious way,so that the theorem is clear e.g. for f an open immersion. We will often use this in conjunction with theobservation (which follows e.g. from the form of the Rost–Schmid complex) that if Z has codimension ≥ d on X , then H dZ ( X, M ) ֒ → M z H dz ( X z , M ) , where the sum is over the (finitely many) generic points of Z of codimension d on X .If the support is smooth and the intersection is transverse, all is well. Lemma 3.3.
Suppose that both Z and f − ( Z ) (with its induced scheme structure as a pullback) aresmooth. Then f ∗ : H dZ ( X, M ) → H df − ( Z ) ( Y, M ) only depends on M − d .Proof. Since Z is smooth (and so is X ), we may write Z = Z ` Z where all components of Z havecodimension precisely d on X , and all components of Z have codimension > d . Consider the commutativediagram H dZ ( X, M ) f ∗ −−−−→ H df − ( Z ) ( Y, M ) x x H dZ ( X, M ) f ∗ −−−−→ H df − ( Z ) ( Y, M ) . The monogeneic transfer on M − d is extra structure, determined by the presentation of M − d as a contraction. But theboundary C d → C d +1 involves the monogeneic transfer only on M − d − , and so indeed only depends on the sheaf M − d . TOM BACHMANN
Here the vertical maps are extension of support, and hence only depend on M − d . Moreover by construc-tion the left hand vertical map is an isomorphism. We may thus replace Z by Z , i.e. assume that allcomponents of Z have codimension precisely d on X .Let η be a generic point of f − ( Z ) (necessarily of codimension d on Y ). Shrinking X around f ( η ) usingRemark 3.2, we may assume that the normal bundle N Z/X is trivial. Since f : ( Y, f − ( Z )) → ( X, Z ) isa morphism of smooth closed pairs [Hoy17, § Y /Y \ f − ( Z ) → X/X \ Z is equivalent to T h ( g ), where g : N f − ( Z ) /Y → N Z/X is the map induced by f . Our assumptions on codimension implythat f ∗ N Z/X ≃ N f − ( Z ) /Y , whence g ≃ f | f − ( Z ) × id A d . Pullback along T h ( g ) ≃ ( f | f − ( Z ) ) + ∧ T d thusonly depends on M − d , and the result follows. (cid:3) Recall that for (sets, say, and hence presheaves of sets) A ⊂ X, B ⊂ Y , we have a canonical isomor-phism(4) X/A ∧ Y /B ≃ X × Y / ( A × Y ∪ X × B ) . Construction 3.4.
Let Z ⊂ X × P be closed with image Z ′ in X . Applying the isomorphism (4) with Y = P , A = X \ Z ′ , B = ∅ we obtain the following equivalence X × P / ( X × P \ Z ′ × P ) ≃ ( X/X \ Z ′ ) ∧ P . Together with the stable splitting P ≃ ∨ P and extension of support this induces a maptr Z : H dZ ( X × P , M ) → H dZ ′ × P ( X × P , M ) → H d − Z ′ ( X, M − ) . Remark . This construction is clearly functorial in X . Lemma 3.6.
In the above notation, suppose that Z ⊂ X × P has codimension ≥ d (so that Z ′ ⊂ X hascodimension ≥ d − ). Then tr Z only depends on M − d .Proof. The transfer is given by pullback along the collapse map P X / P X \ P Z ′ → P X / P X \ Z . Remarks3.5 and 3.2 imply that the problem is local on X around generic points of Z ′ of codimension d −
1; wemay thus assume that Z ′ is smooth [Stacks, Tag 0B8X] and N Z ′ /X is trivial. Lemma 3.8 below identifiesthe transfer with the collapse map N P Z ′ / P X /N P Z ′ / P X \ P Z ′ → N P Z ′ / P X /N P Z ′ / P X \ Z By assumption N P Z ′ / P X is trivial of rank d −
1, so this map identifies with A d − P Z ′ / A d − P Z ′ \ P Z ′ → A d − P Z ′ / A d − P Z ′ \ Z. Applying isomorphism (4) with X = A d − , Y = P Z ′ , A = A d − \ B = ∅ or B = P Z ′ \ Z ,this identifies with A d − / A d − \ ∧ P Z ′ + id ∧ t −−−→ A d − / A d − \ ∧ P Z ′ / P Z ′ \ Z. (Use that ( A d − \ × P Z ′ = A d − P Z ′ \ P Z ′ and ( A d − \ × P Z ′ ∪ A d − × ( P Z ′ \ Z ) = A d − P Z ′ \ Z .) Pullbackalong t is the monogeneic transfer for Z/Z ′ , essentially by definition (see § (cid:3) Remark . The above proof shows that, on the level of the Rost–Schmid complex, the map tr Z is givenas follows. For z ∈ Z of codimension e in X × P and with image z ′ of codimension e − X , the mapis given in components by C e ( X × P , M ) ⊃ M − e ( z, ω z/X × P ) ≃ M − e ( z, ω z/ P z ′ ⊗ ω P z ′ / P X ) tr −→ M − e ( z ′ , ω z ′ /X ) ⊂ C e − ( X, M − ) . Here tr is the monogeneic transfer coming from the embedding z ∈ P z ′ .We used above the following form of the homotopy purity equivalence. Lemma 3.8.
Let Z ⊂ Y ⊂ X be closed immersions with X, Y smooth. Then the collapse map X/ ( X \ Y ) → X/ ( X \ Z ) is canonically homotopic to the collapse map N Y/X /N Y/X \ Y → N Y/X /N Y/X \ Z. HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 7 Proof.
Write X ′ = X \ Z and Y ′ = Y \ Z . We can write the collapse map as XX \ Y → X/X \ YX ′ /X ′ \ Y ′ . Since ( X ′ , Y ′ ) → ( X, Y ) is a morphism of smooth closed pairs, it is compatible with purity equivalences[Hoy17, after proof of Theorem 3.23], and so the collapse map identifies with N Y/X N Y/X \ Y → N Y/X / ( N Y/X \ Y ) N Y ′ /X ′ / ( N Y ′ /X ′ \ Y ′ ) ≃ N Y/X ( N Y/X \ Y ) ∪ N Y ′ /X ′ = N Y/X N Y/X \ Z ;see also [Hoy17, top of p. 24]. This is the desired result. (cid:3) Lemma 3.9.
Let X be (essentially) smooth, i : Y ֒ → X closed of codimension with Y essentiallysmooth, Z ⊂ X × P of codimension ≥ d such that W := ( Y × P ) ∩ Z also has codimension ≥ d on Y × P . Write Z ′ , W ′ for the images of Z and W in X, Y , respectively. Let η , . . . , η r be the genericpoints of W of codimension d . Suppose further that Z → Z ′ is quasi-finite and W → W ′ is birational at η . Then i ∗ : H dZ ( X × P , M ) → H dW ( Y × P , M ) only depends on M − d , the map i ∗ : H d − Z ′ ( X, M − ) → H d − W ′ ( Y, M − ) and the maps i ∗ η j : H dZ (( X × P ) η j , M ) → H dW (( Y × P ) η j , M ) , for j > . In particular, the map i ∗ does not depend on i ∗ . Proof.
By Remarks 3.5 and 3.7, we have a commutative diagram H dZ ( X × P , M ) H dW ( Y × P , M ) L i C dη i ( Y × P , M ) H d − Z ′ ( X, M − ) H d − W ′ ( Y, M − ) L η ∈ W ′ ∩ Y ( d − C d − η ( Y, M − ) . i ∗ tr Z tr W tr W i ∗ By Lemma 3.6, the vertical maps only depend on M − d , and it follows from Remark 3.7 and our assump-tion that W → W ′ is birational at η that the right hand vertical map is injective on the componentcorresponding to η . Let a ∈ H dZ ( X × P , M ). Write i ∗ ( a ) = b + · · · + b r , where b i ∈ C dη i ( Y × P , M ). For j >
1, we know i ∗ η j , hence we know b j and thus we know tr W ( b j ). Since we know the bottom horizontalmap, we know i ∗ tr Z ( a ) = tr W ( i ∗ ( a )). Consequently we know tr W ( b ) = tr W ( i ∗ a ) − P j> tr W ( b j ), andhence b . This concludes the proof. (cid:3) Example . If d = 1, then the map i ∗ : H d − Z ′ ( X, M − ) ⊂ M − ( X ) → H d − W ′ ( Y, M − ) ⊂ M − ( Y )clearly only depends on M − , as desired. Example . If Z is smooth and transverse to Y at η j for j >
1, then i ∗ η j only depends on M − d byLemma 3.3, as desired.The following is the key reduction. It is an adaptation of [Lev10, Lemma 7.2]. Lemma 3.12.
Let
X, Y be (essentially) smooth, i : Y ֒ → X closed of codimension , Z ⊂ X ofcodimension ≥ d such that W = Y ∩ Z is of codimension ≥ d in Y . Then i ∗ : H dZ ( X, M ) → H dW ( Y, M ) only depends on M − d .Proof. By a continuity argument, we may assume that X is smooth over k .Using Remark 3.2, we may shrink X around a generic point of W . We may thus assume that W is smooth over k , and connected. Pullback along the smooth map X × A → X yields an understoodisomorphism H dZ ( X, M ) → H dZ × A ( X × A , M ), functorial in X . It hence suffices to understand pullbackalong i × A . Let w = Spec ( F ) be a generic point of W × A of codimension d on Y × A . Then w lies overthe generic point of A [Stacks, Tag 0CC1]. We may thus (using Remark 3.2 again) pass to the genericfiber over Spec ( k ( t )) ∈ A ; essentially we have base changed the entire problem to k ( t ) /k . Let us denotethe base change of X by X , and so on. Since Z is geometrically reduced over k [Stacks, Tag 020I], itsbase change Z is geometrically reduced over k ( t ) [Stacks, Tag 0384]. Lemma 3.13 below supplies us TOM BACHMANN with an ´etale neighborhood X → X of w and a smooth map X → W such that Z → X → W isgenerically smooth and Y → W is smooth. Let X = X × W { w } . Our base changes are illustrated inthe following diagram X −−−−→ X −−−−→ X = X ⊗ k ( t ) −−−−→ X × A −−−−→ X y y w −−−−→ W = W ⊗ k ( t ) . By construction X → X is a pro-(´etale neighborhood) of w ∈ X ⊗ k ( t ), and so (using again Remark3.2) we may replace X ⊗ k ( t ) by X .With these preparatory constructions out of the way, we rename X to X , Y to Y and Z to Z . We nowhave a smooth map X → Spec ( F ), where F is infinite (since it contains k ( t )), Y → Spec ( F ) is smoothand Z → Spec ( F ) is generically smooth. Also W = { w } is an F -rational point of X , dim X = d + 1,dim Y = d and dim Z = 1. Shrinking X if necessary may assume that Y ⊂ X is principal, say cut outby f ∈ O ( X ), that every component of Z meets w , that Z is smooth away from w , and that X is affine.Lemma 3.15 below supplies us with ¯ u : Z → A with ¯ u ( w ) = 0, ¯ uf : Z → A finite and ¯ u having nodouble roots. Pick u ∈ O ( X ) reducing to ¯ u ∈ O ( Z ). Then φ := uf : X → A is finite when restrictedto Z , satisfies φ ( w ) = 0, and we claim that φ is smooth at all points of φ − (0) ∩ Z . Note that byconstruction ( u, f ) have no common root on Z ( w being the only root of f on Z ), and neither do ( u, du )( u not having double roots on Z ) or ( f, df ) ( Z ( f ) being smooth). It follows that d ( uf ) = udf + f du doesnot vanish at points p ∈ Z with ( uf )( p ) = 0, proving the claim.We may thus apply Lemma 3.16 below to obtain φ , . . . , φ d +1 : X → A such that φ = ( φ , . . . , φ d +1 ) : X → A d +1 is ´etale at all points of φ − (0) ∩ Z , has φ ( w ) = 0, and there exists an open neighborhood0 ∈ U ⊂ A dF such that Z U φ −→ A U is a closed immersion (in fact U = U × A d − ). The non-´etale locus of φ meets Z in finitely many points (namely a closed subset not containing w , and hence no componentof the curve Z ), none of which map to 0 under φ . Shrinking U further, we may thus assume that Z U is contained in the ´etale locus V of φ . Since Z U ≃ φ ( Z U ) → φ − ( φ ( Z U )) ∩ V is a section of a separatedunramified morphism it is clopen [Stacks, Tag 024T], i.e. we have φ − ( φ ( Z U )) ∩ V = Z U ` Z ′ with Z ′ closed in V .Let D = D ( u ); this is an open neighborhood of w in X . Note that Z ( uf ) ∩ D = Y ∩ D . Let X ′ = ( φ − ( A U ) ∩ V ) \ Z ′ ,U = U ∩ ( { } × A d − ) and Y ′ = φ − ( A U ) ∩ X ′ = Z ( uf ) ∩ X ′ . We have a commutative diagram of schemes X i ←−−−− Y ←−−−− Y ∩ D ∩ X ′ x (cid:13)(cid:13)(cid:13) X ′ ←−−−− Y ′ ←−−−− Y ′ ∩ D φ y ψ y A U j ←−−−− A U . Here j is the canonical closed immersion, and ψ is the restriction (i.e. base change) of φ . In particular ψ is ´etale and Y ′ is smooth. By construction, φ is an ´etale neighborhood of Z U , and Z U φ −→ A U → U isfinite. There is an induced commutative diagram H dZ ( X, M ) i ∗ −−−−→ H dW ( Y, M ) −−−−→ ≃ H dW ( Y ∩ D ∩ X ′ , M ) o y (cid:13)(cid:13)(cid:13) H dZ U ( X ′ , M ) −−−−→ H dZ U ∩ Y ′ ( Y ′ , M ) o −−−−→ H dW ( Y ′ ∩ D, M ) φ ∗ x ≃ ψ ∗ x ≃ H dZ U ( A U , M ) j ∗ −−−−→ H dZ U ∩ Z ( uf ) ( A U , M ) . We need to understand the top left hand horizontal map. All the labelled isomorphisms are pullbackalong ´etale maps, and isomorphisms by excision. The two maps labelled o are also pullback along ´etalemorphisms (in fact open immersions), and hence understood. HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 9 We have thus reduced to understanding j ∗ ; we rename Z U to Z and Z U ∩ Z ( uf ) to V . Since Z isfinite over U , it remains closed in P U , and hence by a further excision argument it suffices to understand¯ j ∗ : H dZ ( P U , M ) → H dV ( P U , M ) . We have V = { w, z , . . . , z r } , and each z i is a smooth point of Z (since w is the only singular point of Z ). Moreover z i is a smooth point of V , since z i is a simple root of u (since by construction u has nodouble roots on Z ). By Lemma 3.3, the pullback j ∗ s : H dZ (( P U ) z s , M ) → H dV (( P U ) z s , M )only depends on M − d . Thus applying Lemma 3.9, it suffices to understand k ∗ : H d − Z ′ ( U, M − ) → H d − V ′ ( U , M − );here Z ′ and V ′ are the images of Z and V in U . If d = 1 we are done by Example 3.10. The general case(i.e. d >
1) now follows by induction (i.e. restart the argument with (
U, U , Z ′ ) in place of ( X, Y, Z )). (cid:3)
Lemma 3.13.
Let X be a smooth scheme over an infinite field K , W ⊂ X, Y ⊂ X smooth closedsubschemes, W ⊂ Z ⊂ X with Z ⊂ X closed and Z geometrically reduced over K (but not necessarilysmooth). Let w ∈ W ∩ Y such that dim w W ≤ dim w Y . There exists an ´etale neighborhood X ′ → X of w together with a smooth morphism X ′ → W such that Z × X X ′ → W is generically smooth (that is,the smooth locus is dense in the source) and Y × X X ′ → W is smooth.Proof. We modify [D´e07, Corollary 5.11]. Shrinking X around w , we may assume that X is affineand there exist f , . . . , f d ∈ O ( X ) such that W = Z ( f , . . . , f d ) and W has codimension everywhereexactly d in X . Let { z , . . . , z r } ⊂ Z be a choice of smooth point in every component of Z , which existbecause Z is geometrically reduced [Stacks, Tag 056V]. Let dim X = d + n . We claim that there exist g , . . . , g n ∈ O ( X ) such that dg , . . . , dg n are linearly independent (over the respective residue fields) inΩ w W, Ω w Y and Ω z i Z for every i ; we shall prove this at the end. It follows that df , . . . , df d , dg , . . . , dg n are linearly independent in Ω w X . Consider the map F = ( f , . . . , f d , g , . . . , g n ) : X → A d + n . Let p : A d + n → A n be the projection to the last n coordinates. By [Gro67, 17.11.1], F is smooth at w , pF | W is smooth at w , pF | Y is smooth at w , and pF | Z is smooth at z i . In particular pF | Z is genericallysmooth. Shrinking X further around w , we may assume that F, pF | W and pF | Y are smooth (whencethe former two are ´etale), and of course pF | Z remains generically smooth. Applying the construction of[D´e07, § § j −−−−→ P −−−−→ A dW −−−−→ W q y y pF | W y X F −−−−→ A d + n p −−−−→ A n . Here both squares are cartesian by definition, j is an open immersion and qj is an ´etale neighborhoodof W (by construction of Ω). Since p, F and j are smooth so is Ω → W . By construction Z → X → A n is generically smooth, thus so Z × X P → W , and hence also Z × X Ω → W . Similarly Y → X → A n issmooth and hence so is Y × X Ω → W . Setting X ′ = Ω, the result follows.It remains to prove the claim. Embed X into A N , and let g , . . . , g n be linear projections. ByLemma 3.14 below applied to ( A N ) ∗ ⊗ K K ( w ) ⊂ (Ω A N ) ⊗ K K ( w ) → Ω w W , dg , . . . , dg n will be linearlyindependent in Ω w W for general g j , and similarly for Ω w Y and Ω z i Z . The result follows. (cid:3) For completeness, we include a proof of the following elementary fact.
Lemma 3.14.
Let K ′ /K be a finite field extension. Let V be a finite dimensional K -vector space, V ′ a K ′ -vector space of dimension ≥ n , and V K ′ → V ′ a surjection. Write A ( V ) for the associatedvariety over K (isomorphic to A dim VK ). There is a non-empty open subset U of A ( V ) n such that any ( v , . . . , v n ) ∈ U ( K ) have K ′ -linearly independent images in V ′ . In particular, if F is infinite, then n general elements of V are linearly independent in V ′ .Proof. Replacing V ′ by a quotient of dimension n , we may assume that dim K ′ V ′ = n . There is a map D : A ( V ′ ) n → A K ′ such that n elements of V ′ are linearly independent if and only if their image under D is non-zero (pick a basis of V ′ and let D be the determinant). By adjunction the composite A ( V ) nK ′ → A ( V ′ ) n D −→ A K ′ defines a map D ′ : A ( V ) n → R ( A K ′ ), where R denotes the Weil restriction along K ′ /K (see e.g. [BLR90, § n elements of V have image in V ′ linearly independent over K ′ if and only if their image under D ′ is non-zero. Since D is not the zero map neither is D ′ , and hence U = D ′− ( R ( A K ′ ) \ (cid:3) Lemma 3.15.
Let Z be an affine curve over the field F , smooth away from a rational point w , and let f : Z → A be nowhere constant. There exists u : Z → A such that u ( w ) = 0 and f u : Z → A is finite.If F is infinite, it can be arranged that u has no double zeros.Proof. Let ¯ Z be a compactification of Z which is smooth away from w , and ¯ Z \ Z = { z , . . . , z r } . Since Z is smooth at infinity, any map Z → A extends to ¯ Z → P . It suffices to find u i : Z → A such thatord z i ( u i ) + ord z i ( f ) <
0, for every i . Then u = P i u e i i for suitably big e i will satisfy the same condition,but for all i at once. Now uf : ¯ Z → P is proper and ¯ Z \ Z ⊃ ( uf ) − ( ∞ ) ⊃ { z , . . . , z r } , which impliesthat uf : Z → A is finite (being proper and affine). If u ( w ) = 0 then replace u by u + 1; the first claimfollows. Replacing u by u n + g for suitable g and n large, we may assume that du has only finitely manyzeros. Then u + c for general c has no double zeros (away from w ) and satisfies u ( w ) = 0, so that if F isinfinite we may arrange the second claim.Let ˜¯ Z be the normalization of ¯ Z , and ˜ Z ⊂ ˜¯ Z the open subset over Z , i.e. the normalization of Z . Notethat ˜¯ Z → ¯ Z is an isomorphism near z i . By Riemann–Roch, we can find v : ˜¯ Z → P with an arbitrarilylarge pole at z i , no poles away from z i , so in particular no poles on ˜ Z . Since ˜ Z → Z is integral, thereexists an equation v n + a v n − + · · · + a n = 0, with a i : Z → A . It follows that (at least) one of the a i must have a pole at z i (at least) as large as v . This concludes the proof. (cid:3) We used above the following variant of the second part of Gabber’s Lemma [Gab94, Lemma 3.1(b)];our proof is heavily inspired by Gabber’s.
Lemma 3.16.
Let F be an infinite field, X smooth and affine of dimension d over F , Z ⊂ X closed, w ∈ Z a rational point, e < d , φ ′ = ( φ , . . . , φ e ) : X → A e such that φ ′ ( w ) = 0 , φ ′ | Z : Z → A e is finiteand φ ′ is smooth at all points of φ ′− (0) ∩ Z .Then there exist φ e +1 , . . . , φ d : X → A such that φ = ( φ , . . . , φ d ) : X → A d is ´etale at all points of φ ′− (0) ∩ Z , φ ( w ) = 0 and there exists an open neighborhood ∈ W ⊂ A e such that Z W → A d − eW is aclosed immersion. Note that the new map φ is also finite when restricted to Z . Proof.
Let
X ֒ → A N with w mapped to 0. We claim that general linear projections φ e +1 , . . . , φ d havethe desired properties. They vanish on w by definition.In order for φ to be smooth at some point x ∈ φ ′− (0) ∩ Z , we need only ensure that dφ , . . . , dφ d ∈ Ω x X are linearly independent [Gro67, 17.11.1]. Since φ ′ is smooth at x , the dφ , . . . , dφ e are linearlyindependent at x , and then the other dφ i are linearly independent, for general φ i ; this follows fromLemma 3.14 applied to V ′ = Ω x X/ h dφ , . . . , dφ e i . Since φ ′− (0) ∩ Z is a finite set of points, the ´etalenessclaim holds for general φ i .It remains to prove the claim about the closed immersion. Note that by Nakayama’s lemma, if f : X → Y is a morphism of affine S -schemes with X finite over S , S noetherian, and there exists s ∈ S such that f s : X s → Y s is a closed immersion, then there exists an open neighborhood U of s such that f U : X U → Y U is a closed immersion. Let ψ : Z → A d be the restriction of φ , which we view as amorphism over S = A e via φ ′ (and the projection A d → A e to the first e coordinates). It is thus enoughto show that ψ : φ ′− (0) ∩ Z → A d − e is a closed immersion (for general φ i ) Since φ is ´etale at all pointsof φ ′− (0) ∩ Z (for general φ i ) and Z → X is a closed immersion, ψ is unramified at all points above 0and so ψ is unramified (for general φ i ). Since φ : Z → A d is finite so is ψ ; in fact φ ′− (0) ∩ Z is finiteover F . By [Stacks, Tags 04XV and 01S4], a morphism is a closed immersion if and only if it is proper,unramified and radicial; we already know that ψ is finite (hence proper) and unramified. Being radicialis fpqc local on the target [Stacks, Tag 02KW], so may be checked after geometric base change. In otherwords (using that φ ′− (0) ∩ Z is finite over F ) we need the φ i to separate a finite number of specifiedgeometric points. This clearly holds for general φ i . (cid:3) Proof of Theorem 3.1.
If the theorem holds for composable maps f and g , then it holds for f g . Given f : Y → X , we factor it as Y i −→ Γ f p −→ X ;here Γ f is the graph of f . Then i is a regular immersion and p is smooth. It follows that p − ( Z ) ⊂ Γ f has codimension ≥ d (see e.g. [Gro67, Corollary 6.1.4]). Hence it suffices to prove the result for i, p HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 11 separately; i.e. we may assume that f is either a regular immersion or a smooth morphism. The case ofsmooth maps was already explained in Remark 3.2, so assume that f : Y ֒ → X is a regular immersion.As usual we may assume that Z ∩ Y = { z } is a closed point and X is local. It follows (e.g. from [Stacks,Tag 00NQ]) that there exists a sequence of codimension 1 embeddings of essentially smooth schemes Y = Y d ֒ → Y d +1 ֒ → · · · ֒ → Y n = X. Let Z i = Z ∩ Y i . Then dim Z i ≥ dim Z i +1 − Z n = n − d = dim Z d + n − d so that dim Z i = i − d for all i . It follows that we may prove the result for each Y i ֒ → Y i +1 separately;this is Lemma 3.12. (cid:3) Applications
After introducing some notation in § SH S ( k )( d ) (for d ≥
3) in § § S -spectrum by infinite P -loop spectra arising from the adjunction SH S ( k ) ⇆ SH ( k ) and deduce someconservativity results.In this section we will freely use the language of ∞ -categories as set out in [Lur17b, Lur17a]. We alsoassume that k is perfect; we will restate this assumption with the most important results only.4.1. Notation and hypotheses.
We write SH S ( k ) for the category of motivic S -spectra [Mor03, § k ), and SH ( k ) = SH S ( k )[ G ∧− m ]for the category of motivic spectra [Mor03, § k ∗ Σ ∞ S −−→ SH S ( k ) σ ∞ −−→ SH ( k ) eff ⊂ SH ( k ) , and we denote by ω ∞ the right adjoint of σ ∞ . Here SH ( k ) eff is the localizing subcategory generated bythe image of σ ∞ .There are localizing subcategories SH S ( k ) ⊃ SH S ( k )(1) ⊃ · · · ⊃ SH S ( k )( d ) ⊃ . . . ;here SH S ( k )( d ) is generated by Σ ∞ S X + ∧ G ∧ dm for X ∈ Sm k . The inclusion SH S ( k )( d ) ⊂ SH S ( k ) hasa right adjoint which we denote by f d . There are canonical cofiber sequences f d +1 → f d → s d definingthe functors s d . There is a similar filtration of SH ( k ) eff , given by SH ( k ) eff ( d ) := SH ( k ) eff ∧ G ∧ dm , andthe right adjoints (respectively cofibers) are again denoted by f d (respectively s d ). See [Lev08, Voe02]or [BY18, § SH S ( k ) has a t -structure with non-negative part generated by Σ ∞ S X + for X ∈ Sm k ; itsheart canonically identifies with HI ( k ) [Mor03, Lemma 4.3.7(2)]. We denote by E ≥ , E ≤ and π E thetruncations and homotopy sheaves, respectively. The categories SH S ( k )( d ), SH ( k ), SH ( k ) eff ( d ) haverelated t -structures, with non-negative parts generated by X + ∧ G ∧ dm .Recall from [BY18, § A -transfers. This is just a presheaf F on Sm k together with for every finitely generated field K/k a GW ( K )-module structure on F ( K ), and for everyfinite monogeneic extension K ( x ) /K a transfer τ x : F ( K ( x )) → F ( K ). The category SH ( k ) eff ♥ embedsfully faithfully into the category of presheaves with A -transfers [BY18, Corollary 5.17] (morphisms inthis category are given by morphisms of presheaves compatible with the GW -module structures andtransfers). Given a presheaf with A -transfers M , we say that the transfers extend to framed transfers if M is in the essential image of this embedding. Recall also that Morel has shown that if M ∈ HI ( k )and d >
0, then M − d canonically extends to a presheaf with A -transfers (see § Definition 4.1.
Let k be a perfect field and d > M ∈ HI ( k ). We shall say that hypothesis T d ( M ) holds if the canonical A -transfers on M − d extend to framed transfers. We shall say that hypothesis T d ( k ) holds if T d ( M ) holds for all M ∈ HI ( k ).(2) We shall say that hypothesis S d ( k ) holds if for any E ∈ SH S ( k ) and i ∈ Z the spectrum f d π i s d E is in the essential image of ω ∞ : SH ( k ) → SH S ( k ). Remark . If k is perfect then T d ( k ) holds for any d ≥
3, and if char ( k ) = 0 then T ( k ) also holds[BY18, Theorem 5.19]. We abuse notation somewhat and view this as a functor SH S ( k ) → SH S ( k ). Remark . We speculate that T ( k ) holds for any perfect field. Remark . If f : Spec ( l ) → Spec ( k ) is an algebraic extension (automatically separable) and M ∈ HI ( k )such that T d ( M ) holds, then also T d ( f ∗ M ) holds. This is obvious for f finite, and the general case followsby continuity and essentially smooth base change. Theorem 4.5 (Levine [Lev10]) . Let char ( k ) = 0 . Then S d ( k ) holds for any d > .Proof. This is essentially [Lev10, Theorem 2]; we just have to show that s p,n E ≃ f n Σ p + n π p + n s n E. By definition [Lev10, main construction (9.2)], s p,n E ( X ) is the realization of (a rectification of) thesimplicial object Σ p π p ( s n E ) ( n ) ( X, • ); here ( s n E ) ( n ) ( X, • ) is the homotopy coniveau tower model of f n s n E ≃ s n E , and π p just means taking the p -th Eilenberg-MacLane spectrum of the (levelwise) ordinaryspectrum ( s n E ) ( n ) ( X, • ).The map ( s n E ) ≥ n + p → s n E of spectral sheaves induces a map α p : (( s n E ) ≥ p + n ) ( n ) ( − , • ) → ( s n E ) ( n ) ( − , • )of simplicial spectral presheaves. I claim that α p induces an isomorphism on π i for i ≥ p , and that thesource has π i = 0 for i < p . This yields an equivalence(( s n E ) ≥ p + n ) ( n ) ( − , • ) ≃ τ ≥ p ( s n E ) ( n ) ( − , • ) , where τ ≥ p just means levelwise truncation of the simplicial presheaf of spectra. Taking cofibers we obtain(Σ p + n π p + n s n E ) ( n ) ( − , • ) ≃ Σ p π p ( s n E ) ( n ) ( − , • ) , which is what we set out to prove (using [Lev08, Theorem 7.1.1]).It is hence enough to prove the claim. Thus let X ∈ Sm k and W ⊂ A mX have codimension ≥ n . Thedefinition of the homotopy coniveau tower (recalled e.g. in [Lev10, § H − iW ( A mX , s n E ) ≃ H − iW ( A mX , ( s n E ) ≥ n + p ) for i ≥ p and H − iW ( A mX , ( s n E ) ≥ n + p ) = 0 for i < p. Considering the (strongly convergent) descent spectral sequence (for F ∈ SH S ( k )) H pW ( A mX , π q F ) ⇒ H p − qW ( A mX , F ) , for this it suffices to show that for j ∈ Z we have H iW ( A mX , π j s n E ) = 0 for i = n. We can compute this cohomology group using the Rost–Schmid resolution; since W has codimension ≥ n the vanishing follows from the observation that π j ( s n E ) − i = 0 for i > n , which holds since Ω i G m s n E ≃ i > n ), by definition. (cid:3) The heart of SH S ( k )( d ) . Consider the adjunction σ ∞ : SH S ( k ) ⇆ SH ( k ) : ω ∞ . Then σ ∞ ( SH S ( k )( d )) ⊂ SH ( k ) eff ( d ) and σ ∞ ( SH S ( k )( d ) ≥ ) ⊂ SH ( k ) eff ( d ) ≥ , for any d ≥
0. More-over it follows from [BY18, Lemmas 6.1(2) and 6.2(1,2)] that ω ∞ ( SH ( k ) eff ( d )) ⊂ SH S ( k )( d ) and ω ∞ ( SH ( k ) eff ( d ) ≥ ) ⊂ SH S ( k )( d ) ≥ . This implies that there is an induced adjunction π d σ ∞ : SH S ( k )( d ) ♥ ⇆ SH ( k ) eff ( d ) ♥ : ω ∞ , where π d denotes the truncation functor in the t -structure on SH ( k ) eff ( d ). Theorem 4.6.
Let k be a perfect field such that T d ( k ) holds. Then the functor ω ∞ : SH ( k ) eff ( d ) ♥ →SH S ( k )( d ) ♥ is an equivalence of categories. This establishes [BY18, Conjecture 6.10] (for n = d ). HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 13 Proof.
The functor is fully faithful by [BY18, Theorem 6.9]; it hence suffices to prove essential surjectivity.We shall prove the following more precise statement: if M ∈ HI ( k ) and T d ( M ) holds, then M is in theessential image of ω ∞ .We first prove this assuming that k is infinite. We have π i ( M ) − d = 0 for i = 0 [BY18, Lemma6.2(3)] and hence the canonical map M → f d π M is an equivalence (indeed it induces an equivalenceon π i ( − ) − d for every i , and this detects equivalence in SH S ( k )( d ) by [BY18, Lemma 6.1(1)]). Byassumption, the A -transfers on π ( M ) − d extend to framed transfers; there hence exists ˜ M ∈ SH ( k ) eff ♥ such that ω ∞ ( ˜ M ) − d ≃ π ( M ) − d as presheaves with A -transfers. By Lemma 4.7 below (this is wherewe use the assumption that k is infinite), this implies that f d ω ∞ ( ˜ M ) ≃ f d π ( M ). It follows from [Lev08,Theorem 9.0.3] that f d commutes with ω ∞ ; we thus find that M ≃ f d π M ≃ f d ω ∞ ( ˜ M ) ≃ ω ∞ f d ˜ M .
The claim is thus proved for k infinite.Now let k be finite and M ∈ HI ( k ) such that T d ( M ) holds. Since ω ∞ is fully faithful, M is in theessential image of ω ∞ if and only if the canonical map M → ω ∞ σ ∞♥ M is an isomorphism. The functors ω ∞ and σ ∞♥ commute with essentially smooth base change. Let f : Spec ( l ) → Spec ( k ) be an infinitealgebraic p -extension of k , for some prime p . Using Lemma 4.8 below we reduce to proving that f ∗ M is in the essential image of ω ∞ . By Remark 4.4, T d ( f ∗ M ) holds, and thus we are reduced to what wasalready established.This concludes the proof. (cid:3) Lemma 4.7.
Let k be an infinite perfect field, M, N ∈ HI ( k ) and d > . Suppose that T d ( M ) holds.Any isomorphism M − d ≃ N − d respecting the A -transfers yields an equivalence f d M ≃ f d N .Proof. We have f d M ≃ M ( d ) [Lev08, Theorem 7.1.1] (this is where we use the assumption that k is infinite). As explained in [BY18, Remark 4.17], the (truncated) BLRS complex of M provides amodel of M ( d ) which only depends on M − d as a presheaf of GW -modules together with the maps f ∗ : H dZ ( X, M ) → H df − ( Z ) ( Y, M ) for f : Y → X ∈ Sm k , Z ⊂ X closed of codimension ≥ d such that f − ( Z ) also has codimension ≥ d . (In order to apply this remark, we need to know that M − d has framedtransfers (see e.g. [BY18, Proposition 4.14]); this is the only reason for assuming T d ( k ).) Theorem 3.1shows that f ∗ only depends on M − d as a presheaf with A -transfers. The result follows. (cid:3) Lemma 4.8.
Let k be a perfect field and k p /k (respectively k q /k ) a separable algebraic p -extension(respectively q -extension), for primes p = q . Let α : E → F ∈ SH S ( k )( d ) such that T d ( M ) holds forany homotopy sheaf M of E or F . If the image of α in SH S ( k p )( d ) × SH S ( k q )( d ) is an equivalence,then so is α .Proof. It suffices to prove that Ω d G m ( α ) is an equivalence [BY18, Lemma 6.1(1)], i.e. that π i ( α ) − d is anisomorphism for all i . By assumption this is a morphism between sheaves admitting framed transfers, andby [BY18, Corollary 5.17] the morphism preserves the transfers. The result thus follows from [EHK + d G m ). (cid:3) The following is our degree zero G m -Freudenthal theorem. Corollary 4.9.
Suppose that T d ( k ) holds.(1) Let E ∈ SH S ( k ) ≥ ∩ SH S ( k )( d ) . Then π ( E ) ≃ π ( ω ∞ σ ∞ E ) . (2) Let E ∈ SH S ( k ) ≥ . Then π ( ω ∞ σ ∞ E ) ≃ π ( G ∧ dm ∧ E ) − d . In particular this holds for d ≥ if k is perfect, and for d ≥ if char ( k ) = 0 .Proof. (1) We have E ∈ SH S ( k )( d ) ≥ [BY18, Lemma 6.2(3)]. Write π d for the homotopy object in the t -structure on SH S ( k )( d ). Then since σ ∞ , ω ∞ are both right- t -exact [BY18, Lemma 6.2(1,2)] we learnfrom Theorem 4.6 that ( ∗ ) π d E ≃ π d ω ∞ σ ∞ E . Since SH S ( k )( d ) ≥ ⊂ SH S ( k ) ≥ (by construction), wehave π π d ≃ π (when applied to objects in SH S ( k )( d ) ≥ ), and hence the result follows by applying π to ( ∗ ). (2) We have π ( ω ∞ σ ∞ E ) ≃ π ( ω ∞ Ω d G m Σ d G m σ ∞ E ) ≃ π (Ω d G m ω ∞ σ ∞ Σ d G m E ) ≃ π ( ω ∞ σ ∞ G ∧ dm ∧ E ) − d (1) ≃ π ( G ∧ dm ∧ E ) − d . (cid:3) Canonical resolutions.
Given any adjunction F : C ⇆ D : G of ∞ -categories, there is a monadstructure on GF [Lur17a, Proposition 4.7.4.3]. Hence for E ∈ C there is a canonical “triple resolution” E → E • , where E • denotes a cosimplicial object with E n = ( GF ) ◦ ( n +1) ( E ). In more detail, by definitionof a monad, GF promotes to an E -algebra in Fun( C , C ) under the composition monoidal structure,and then [MNN17, Construction 2.7] yields an augmented cosimplical object in Fun( C , C ); the tripleresolution is obtained by applying this cosimplicial endofunctor to E . This also makes it clear that thetriple resolution is functorial in E .Applying this to the stabilization adjunction σ ∞ : SH S ( k ) ⇆ SH ( k ) : ω ∞ , we obtain the canonicalresolution E → E ∧ := lim ∆ [( ω ∞ σ ∞ ) ◦ ( • +1) E ] , functorially in E . Definition 4.10.
We shall say that the canonical resolution converges for E if the above morphism isan equivalence. Example . The canonical resolution converges if E is in the essential image of ω ∞ , since then thecosimplicial object is split. Example . Given a cofiber sequence E → E → E , if the canonical resolution converges for anytwo of the three terms, then it converges for the third. This holds since all the functors involved arestable.The following result clearly holds in much greater generality, but for simplicity we state it in ourrestricted context. Lemma 4.13.
Let E ∈ SH S ( k ) and suppose given a tower · · · → E → E → E := E and a sequence n i ∈ Z such that (i) lim i n i = ∞ , (ii) E i ∈ SH S ( k ) ≥ n i (for all i ) and (iii) the canonicalresolution converges for cof ( E i +1 → E i ) (for all i ).Then the canonical resolution converges for E .Proof. Define an endofunctor F of SH S ( k ) by F ( X ) = f ib ( X → X ∧ ). By construction this is astable functor such that F ( X ) ≃ X . Since F ( cof ( E i +1 → E i )) ≃ i , we find that the tower · · · → F ( E ) → F ( E ) = F ( E )is constant. In order to prove that F ( E ) = 0 it thus suffices to show that lim i F ( E i ) ≃
0. Commutingthe limits, we find thatlim i F ( E i ) ≃ lim i ≥ ,n ∈ ∆ f ib ( E i → ( ω ∞ σ ∞ ) ◦ ( n +1) E i ) ≃ lim n ∈ ∆ lim i f ib ( E i → ( ω ∞ σ ∞ ) ◦ ( n +1) E i );it thus suffices to show that the inner limit over i vanishes. Since σ ∞ , ω ∞ are both right- t -exact [BY18,Lemma 6.2(1,2)], we have f ib ( . . . ) ∈ SH S ( k ) ≥ n i − , and hence it is enough to show that if X i is asequence of spectra with X i ∈ SH S ( k ) ≥ n i then lim i X i ≃
0. Since SH S ( k ) is generated as a localizingsubcategory by objects of the form Σ ∞ S U + for U ∈ Sm k , considering the Milnor exact sequence [GJ09,Proposition VI.2.15] it suffices to show: if n > dim U then [Σ ∞ S U + , SH S ( k ) ≥ n ] = 0. This follows fromthe descent spectral sequence. (cid:3) Theorem 4.14.
Let k be a perfect field such that T d ( k ) holds.(1) The canonical resolution converges for all E ∈ SH S ( k ) ≥ ∩ SH S ( k )( d ) . HE ZEROTH P -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE 15 (2) Suppose in addition that S j ( k ) holds for all ≤ j < d . Then the canonical resolution convergesfor all E ∈ SH S ( k ) ≥ ∩ SH S ( k )(1) .Proof. (1) Write τ d ≥ i for the truncation in the t -structure on SH S ( k )( d ). We have E ∈ SH S ( k )( d ) ≥ [BY18, Lemma 6.2(3)]. Apply Lemma 4.13 with E i = τ d ≥ i E and n i = i ; assumption (i) is clear, and (ii)holds by [BY18, Lemma 6.2(1)]. For (iii), it suffices by Example 4.11 to show that π di E is in the essentialimage of ω ∞ . This follows from Theorem 4.6.(2) By Example 4.12, (1) and induction, it suffices to show that the canonical resolution convergesfor s j E , for 1 ≤ j < d . Since f j : SH S ( k ) → SH S ( k ) is right- t -exact [BY18, Lemma 6.2], we have s j E ∈ SH S ( k ) ≥ . Apply Lemma 4.13 with E i = f j [( s j E ) ≥ i ] and n i = i . Assumption (i) is clear, (ii)holds by right- t -exactness of f j , and (iii) holds by definition of S j ( k ) and Example 4.11. (cid:3) Corollary 4.15. (1) Suppose that T d ( k ) holds. Then σ ∞ : SH S ( k ) ≥ ∩ SH S ( k )( d ) → SH ( k ) is conservative.(2) Suppose that additionally S j ( k ) holds, for ≤ j < d . Then σ ∞ : SH S ( k ) ≥ ∩ SH S ( k )(1) → SH ( k ) is conservative.In particular (1) holds for d = 3 if k is perfect, and (2) holds if char ( k ) = 0 .Proof. Clearly convergence of canonical resolutions implies conservativity (which is equivalent to detect-ing zero objects, by considering cofibers), so this is immediate from Theorem 4.14. (cid:3)
References [BLR90] S. Bosch, W. L¨utkebohmert, and M. Raynaud,
N´eron Models , Springer, 1990[BY18] T. Bachmann and M. Yakerson,
Towards conservativity of G m -stabilization , arXiv:1811.01541[CTHK97] J.-L. Colliot-Th´el`ene, R. Hoobler, and B. Kahn, The Bloch-Ogus-Gabber theorem , Fields Inst. Commun. (1997), pp. 31–94[D´e07] F. D´eglise, Finite correspondences and transfers over a regular base , Algebraic cycles and motives. Volume 1,Lecture Note Ser., vol. 343, London Math. Soc., 2007[EHK +
19] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, and M. Yakerson,
Motivic infinite loop spaces , 2019[Gab94] O. Gabber,
Gersten’s conjecture for some complexes of vanishing cycles , manuscripta mathematica (1994),no. 1, pp. 323–343[GJ09] P. G. Goerss and J. F. Jardine, Simplicial homotopy theory , Springer Science & Business Media, 2009[Gro67] A. Grothendieck, ´El´ements de g´eom´etrie alg´ebrique. IV: ´Etude locale des sch´emas et des morphismes desch´emas. R´edig´e avec la colloboration de Jean Dieudonn´e. , Publ. Math., Inst. Hautes ´Etud. Sci. (1967),pp. 1–361[Hoy17] M. Hoyois, The six operations in equivariant motivic homotopy theory , Adv. Math. (2017), pp. 197–279[Lev08] M. Levine,
The homotopy coniveau tower , J. Topol. (2008), no. 1, pp. 217–267[Lev10] , Slices and transfers , Doc. Math.
Extra Vol. (2010), pp. 393–443[Lur17a] J. Lurie,
Higher Algebra , September 2017, [Lur17b] ,
Higher Topos Theory , April 2017, [MH73] J. W. Milnor and D. Husemoller,
Symmetric bilinear forms , vol. 73, Springer, 1973[MNN17] A. Mathew, N. Naumann, and J. Noel,
Nilpotence and descent in equivariant stable homotopy theory , Advancesin Mathematics (2017), pp. 994–1084[Mor03] F. Morel,
An introduction to A -homotopy theory , ICTP Trieste Lecture Note Ser. 15 (2003), pp. 357–441[Mor05] , The stable A -connectivity theorems , K-theory (2005), no. 1, pp. 1–68[Mor12] , A -Algebraic Topology over a Field , Lecture Notes in Mathematics, Springer Berlin Heidelberg, 2012[MV99] F. Morel and V. Voevodsky, A -homotopy theory of schemes , Publ. Math. l.H.´E.S. (1999), no. 1, pp. 45–143[Nis89] Y. A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences inalgebraic K-theory , Algebraic K-theory: connections with geometry and topology, Springer, 1989, pp. 241–342[OP99] M. Ojanguren and I. Panin,
A purity theorem for the Witt group , AnnalesScientifiques de l’´Ecole Normale Sup´erieure (1999), no. 1, pp. 71 – 86, [Stacks] Stacks Project Authors, The Stacks Project , 2018, http://stacks.math.columbia.edu [Voe02] V. Voevodsky,
Open Problems in the Motivic Stable Homotopy Theory , I , International Press Conference onMotives, Polylogarithms and Hodge Theory, International Press, 2002[WW17] K. Wickelgren and B. Williams,
The Simplicial EHP Sequence in A -Algebraic Topology , 2017 E-mail address ::