The zeroth P^1-stable homotopy sheaf of a motivic space

We establish a kind of"degree zero Freudenthal Gm-suspension theorem"in motivic homotopy theory. From this we deduce results about the conservativity of the P^1-stabilization functor. In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy invariant sheaf in terms of the Rost--Schmid complex. This establishes the main conjecture of [BY18], which easily implies the aforementioned results.


Introduction
After recalling some preliminaries in §2, this article has two main sections of very different flavor. In §3 we establish a technical result about Rost-Schmid complexes of strictly homotopy invariant sheaves. Then in §4 we draw applications to the stabilization problem in motivic homotopy theory. We now describe these two main sections in reverse order, and then we sketch their relation. For more background and motivation, the reader may wish to consult the introduction of [BY20].
1.1. P 1 -stabilization in motivic homotopy theory. Motivic homotopy theory is the universal homotopy theory of smooth algebraic varieties, say over a field k. It is built by freely adjoining homotopy colimits to the category of smooth k-varieties, and then enforcing Nisnevich descent and making A 1 contractible [MV99]. Write Spc(k) * for the pointed version of this theory. 1 This is a symmetric monoidal category (the monoidal operation being given by the smash product), and every pointed smooth variety defines an object in it. Given a pointed motivic space X ∈ Spc(k) * , the classical homotopy groups upgrade to homotopy sheaves 2 π i (X ).
The Riemann sphere P 1 := (P 1 , 1) ∈ Spc(k) * plays a similar role to the ordinary sphere in classical topology. Stable motivic homotopy theory is concerned with the category obtained by making Σ P 1 := ∧P 1 into an equivalence. It is this context in which algebraic cycles and motivic cohomology naturally appear.
Date: January 12, 2022. 1 We think of this as an ∞-category, but no information will be lost for the purposes of this introduction by just considering its homotopy 1-category.
2 I.e. Nisnevich sheaves on the site of smooth k-varieties.
We can take a more pedestrian approach. The functor Σ P 1 has a right adjoint Ω P 1 , and there is a directed diagram of endofunctors of Spc(k) * id → Ω P 1 Σ P 1 =: Q 1 → Ω 2 P 1 Σ 2 P 1 =: Q 2 → · · · → Ω n P 1 Σ n P 1 =: Q n → . . . ; denote by Q its homotopy colimit. Then QX is the P 1 -stabilization of X , and the homotopy sheaves of QX are called the P 1 -stable homotopy sheaves of X .
A simple form of our main application of our technical result is as follows. It is reminiscent of the fact that for an ordinary space X, the sequence of sets {π 0 Ω i Σ i X} i≥0 is given by π 0 X, F π 0 X, Z(π 0 X), Z(π 0 X), . . . , where for a pointed set A, F A denotes the free group on A (with identity given by the base point), 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 ≥ 3 (if char(k) = 0, n = 2 is also allowed). Then for X ∈ Spc(k) * , the canonical map π 0 Q n X → π 0 QX is an isomorphism.
Example 1.2. Morel's computations [Mor12,Corollary 6.43] imply that for X = S 0 , already π 0 Q 2 S 0 ≃ GW ≃ π 0 QS 0 . Our result shows that this stabilization is not special to S 0 , except that our results are not strong enough to establish stabilization at Q 2 , only at Q 3 . See also Remark 4.3.
In particular Σ P 1 and all of its iterates, and also Σ ∞ P 1 , are conservative on the same subcategory. Proof. This is an immediate consequence of Corollary 4.15 and e.g. [WW17,Corollary 2.23].
The results in §4 are stronger than the sample given above; in fact they are stated in terms of the stabilization functor from S 1 -spectra to P 1 -spectra. The reader is encouraged to skip to this section directly. Our main results in the form of Corollary 4.9, Theorem 4.14 and Corollary 4.15 can be understood without reading the rest of the article (except perhaps for taking a glance at §4.1, where some notation is introduced).

1.2.
Pullbacks and the Rost-Schmid complex. The results sketched above are obtained by combining the main results of [BY20] with a technical result that we describe now. Essentially, this establishes [BY20, Conjecture 6.10] (for n ≥ 3); all our applications are a consequence of this and were already anticipated when writing [BY20].
Let M be a strictly homotopy invariant sheaf (see §2 for this and related notions, and a more complete account of the following sketch) and X a smooth variety. Morel has proved [Mor12, Corollary 5.43] that there is a very convenient complex, known as the Rost-Schmid complex C * (X, M ), which can be used to compute 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 ) can be used to compute H * Z (X, M ); by construction one has C n Z (X, M ) = 0 for n < d. It follows that the group H d Z (X, M ) only depends on M −d (in fact this holds for all groups H * Z (X, M ), but we are most interested in the lowest one). Now let f : Y → X be a morphism of smooth varieties with f −1 (Z) also of codimension ≥ d on Y . Then the pullback map (1) is a morphism of abelian groups, both of which only depend on M −d .
It is not difficult to show (using the results of [BY20]; see the proof of Theorem 4.6 for details) that [BY20, Conjecture 6.10] is equivalent to the statement that the morphism (1) also only depends on M −d , in an appropriate sense. 3 The main result of this article (Theorem 3.1) states that this is true.
We establish this by adapting an argument of Levine, using a variant of Gabber's presentation lemma to set up an induction on d. (The case d = 0 holds tautologically.) 1.3. From pullbacks to stabilization. This article brings to conclusion a program started in [BY20]. There we developed the following strategy for establishing stabilization results such as Theorem 1.1. First we note that S 1 -stabilization is well-understood and behaves largely as in topology; thus it suffices to prove the analogous result for motivic S 1 -spectra. (For detailed definitions of this and the following notions, see §4.1.) Write SH S 1 (k)(d) ⊂ SH S 1 (k) for the localizing subcategory generated by d-fold G m -suspensions, and similarly SH(k) eff (d) ⊂ SH(k) for the localizing subcategory generated by the image of SH S 1 (k)(d). These categories afford t-structures induced by the canonical generating sets, and hence the stabilization functor SH S 1 (k)(d) → SH(k) eff (d) is right-t-exact. One finds that in order to prove stabilization results, it will be enough to show that the induced functor on hearts SH S 1 (k)(d) ♥ → SH(k) eff (d) ♥ is an equivalence. Since the right hand category is by now well-understood, let us focus on the left hand side. It is not difficult to show that the functor of d-fold G m -loops In other words, we may think of objects of SH S 1 (k)(d) ♥ as strictly homotopy invariant sheaves with extra structure. One way of phrasing the main result of [BY20] is (see e.g. [BY20,Remark 4.17]) that this extra structure is precisely the data of closed pullbacks on cohomology with support in codimension d. These are precisely the kinds of maps that we show only depend on M −d "in an appropriate sense". To be more specific, the appropriate sense is that M −d is a so-called sheaf with A 1 -transfers (see Remark 3.2 for details), and the pullback only depends on this additional structure.
All of this more or less implies 4 that SH S 1 (k)(d) ♥ is equivalent to the full subcategory of the category of sheaves with A 1 -transfers on objects of the form M −d . It follows from [BY20,Theorem 5.19] that for d big enough, this subcategory is equivalent to SH(k) eff (d) ♥ , as desired.
1.4. Acknowledgements. It is my pleasure to thank Maria Yakerson, Marc Levine and Mike Hopkins for fruitful discussions about these problems.
I would also like to thank Marc Hoyois and Maria Yakerson for comments on a draft of this article.
1.5. Notation and conventions. We fix throughout a field k. All non-trivial results will require k to be 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 other words, 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; in other 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 the determinant of the conormal bundle. More generally for any morphism Y → X such that the cotangent complex L Y /X is perfect we write ω Y /X = det L Y /X .

Preliminaries
We recall some well-known results from motivic homotopy theory.
3 In fact, our original plan for [BY20] was to establish [BY20, Conjecture 6.10] (and hence the results in §4) by proving that f * only depends on M −d . This turned out to be more difficult than we had anticipated. 4 Making these arguments precise requires some further effort; for this reason in §4.2 we follow a slightly different strategy.
2.1. Strictly homotopy invariant sheaves. We write Sm k for the category of smooth k-schemes. We make it into a site by endowing it with the Nisnevich topology [Nis89]. This is the only topology we shall 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 1 × X, M ) is an isomorphism. We denote the category of strictly homotopy invariant sheaves by HI(k).
Example 2.1. For a commutative ring A, denote by GW (A) its Grothendieck-Witt ring, i.e. the additive group 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 strictly homotopy invariant (combine [OP99, Theorem A] and [Mor12, § §2,3]).
Remark 2.2. As mentioned in the introduction, there exists a universal homotopy theory built out of (pointed) smooth varieties by enforcing A 1 -homotopy invariance and Nisnevich descent [MV99]; we denote it by Spc(k) * . By construction, for M ∈ HI(k), the Eilenberg-MacLane spaces K(M, i) define objects in Spc(k) * . In this way, results about Spc(k) * translate into properties of the cohomology of strictly homotopy invariant sheaves. For example given X ∈ Sm k and Z ⊂ X closed we have an isomorphism ). We will use this correspondence freely in the sequel. Example 2.3. We have H 1 (P 1 K , M ) ≃ M −1 (K), for any finitely generated separable field extension K/k. Indeed we can cover P 1 K by two copies of , so the claim follows from the Mayer-Vietoris sequence for this covering.
2.1.3. GW -module structure. Let X ∈ Sm k and u ∈ O(X) × . Multiplication by u defines an endomorphism of (A 1 \ 0) × X and hence of Hom(A 1 \ 0, M )(X); passing to the summand we obtain u : surjective on fields, unramifiedness implies that this construction extends in at most one way to a GWmodule structure on M −1 . It turns out that this GW -module structure always exists [Mor12, Lemma 3.49].
2.1.4. Twisting. Given a line bundle L on X ∈ Sm k , write L × for the sheaf of non-vanishing sections.
2.1.6. Homotopy purity. Let X ∈ Sm k , U ⊂ X open with reduced closed complement Z = X \ U also smooth. Then in Spc(k) * there is a canonical equivalence [MV99, §3 Theorem 2.23] 2.1.7. Boundary maps. Let X ∈ Sm k and x ∈ X (d) . Then X x is an essentially smooth scheme with closed point x. Homotopy purity supplies us with the collapse sequence 5 Pullback along ∂ induces the boundary map in the long exact sequence of cohomology with support. We most commonly use the case where d = 1. Then X x \ x = ξ where ξ is the generic point of X (specializing to x), and the boundary map takes the familiar form 2.1.8. 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. We are supplied with an embedding X ′ = Spec(K(x)) x ֒− → A 1 X ⊂ P 1 X and thus homotopy purity provides us with a collapse map P 1 ; here the normal bundle is canonically trivialized by the minimal polynomial of x. Pullback along this collapse map induces the monogeneic transfer 6 [Mor12, p.99] Slightly more generally, suppose that z ∈ P 1 K is any closed point. Then we have the transfer map tr z : . This contains no new information: if z ∈ A 1 K then tr z coincides up to isomorphism with τ z , and the only other case is z = ∞ which is a rational point, and so tr z is isomorphic to the identity.

Cousin and Rost-Schmid complexes.
Let M be a sheaf of abelian groups on X. The cohomology of M on X can be computed using the coniveau spectral sequence; see e.g. [CTHK97,§1]. On the zero line of the E 1 page one finds the so-called Cousin complex , where the colimit runs over open neighborhoods of x. The boundary maps in (2) are induced by certain boundary 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], the Cousin complex (2) is then exact when viewed as a complex of sheaves (i.e. for X local). Since it consists of flasque sheafes, it can thus be used to compute the Zariski cohomology of M . The terms also turn out to be Nisnevich-acyclic (see [CTHK97,Theorem 8.3.1] or [Mor12, Lemma 5.42]), and hence the Cousin complex computes the Nisnevich cohomology of M as well (which thus turns out to coincide with the Zariski cohomology).
Remark 2.4. The Cousin complex can also be used to compute cohomology with support in a closed subscheme Z; just replace 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, homotopy purity (see § §2.1.5,2.1.6) allows us to identify the groups (3) in the Cousin complex more explicitly as . Indeed by generic smoothness, shrinking V if necessary we may assume that Z := {x} ∩ V is smooth of codimension d on V , and then ; the claim now follows by taking colimits. The boundary maps of the Cousin complex can also be identified. We only use the following weak form of this result. 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 words the boundary map in the Cousin complex admits an explicit formula, involving only the codimension 1 boundary of §2.1.7 and (composites of) the monogeneic transfer of §2.1.8. 7 In the sequel, we will not distinguish between the Cousin and Rost-Schmid complexes.

A "formula" for closed pullback
In this section we establish our main result.
Then the map , together with its GW -module structure and transfers along monogeneic field extensions (in the sense of § §2.1.3,2.1.8).
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".
Remark 3.2. Let us make precise the notion that "f * only depends on M −d ".
For this first recall from [BY20, §5.1] the notion of a presheaf with A 1 -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 every finite monogeneic extension K(x)/K a transfer τ x : F (K(x)) → F (K). A morphism of presheaves with A 1 -transfers is a morphisms of sheaves which commutes with the GW (K)-module structures and the transfers. If M ∈ HI(k) and d ≥ 1, then M −d acquires the structure of a presheaf with A 1 -transfers (see § §2.1.3,2.1.8 or [BY20, Example 5.2]). Now suppose given M, N ∈ HI(k) and an isomorphism and similarly for Y ), by identifying the Rost-Schmid resolutions with support in Z (i.e. using Remark 2.4 and Theorem 2.5; this does not even depend on the identification of the transfers). The theorem asserts that this isomorphism is compatible with the pullback f * (and for this we crucially need the compatibility of the transfers).
Remark 3.3. 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 the observation (which follows e.g. from the form of the Rost-Schmid complex) that if Z has codimension ≥ d on X, then 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 3.4. Suppose that both Z and f −1 (Z) (with its induced scheme structure as a pullback) are smooth. Then f * : Since Z is smooth (and so is X), we may write Z = Z 0 Z 1 where all components of Z 0 have codimension precisely d on X, and all components of Z 1 have codimension > d. Consider the commutative diagram Here the vertical maps are extension of support, and hence only depend on M −d . Moreover by construction the left hand vertical map is an isomorphism. We may thus replace Z by Z 0 , i.e. assume that all components of Z have codimension precisely d on X. Recall from Remark 2.2 that the pullback f * : M ). Let η be a generic point of f −1 (Z) (necessarily of codimension d on Y ). Shrinking X around f (η) using Remark 3.3, we may assume that the normal bundle N Z/X is trivial. Since f : (Y, f −1 (Z)) → (X, Z) is a morphism of smooth closed pairs [Hoy17,§3.5 Hoy17, Theorem 3.23]. Our assumptions on codimension imply that Recall that for (sets, say, and hence presheaves of sets) A ⊂ X, B ⊂ Y , we have a canonical isomorphism Construction 3.5. Let Z ⊂ X × P 1 be closed with image Z ′ in X. Applying the isomorphism (4) with Y = P 1 , A = X \ Z ′ , B = ∅ we obtain the following equivalence Together with the stable splitting P 1 + ≃ ½ ∨ P 1 and extension of support this induces a map . Remark 3.6. This construction is clearly functorial in X.
Lemma 3.7. In the above notation, suppose that Z ⊂ X × P 1 has codimension ≥ d (so that Z ′ ⊂ X has codimension ≥ d − 1). Then tr Z only depends on M −d .
Proof. The transfer is given by pullback along the collapse map P 1 X /P 1 X \ P 1 Z ′ → P 1 X /P 1 X \ Z. Remarks 3.6 and 3.3 imply that the problem is local on X around generic points of Z ′ of codimension d − 1; we may thus assume that Z ′ is smooth [Stacks, Tag 0B8X] and N Z ′ /X is trivial. Lemma 3.9 below identifies the transfer with the collapse map Applying isomorphism (4) with X = A d−1 , Y = P 1 Z ′ , A = A d−1 \0 and respectively B = ∅ or B = P 1 Z ′ \Z, this identifies with along t is the monogeneic transfer for Z/Z ′ , essentially by definition (see §2.1.8). The result follows.
Remark 3.8. The above proof shows that, on the level of the Rost-Schmid complex, the map tr Z is given as follows. For z ∈ Z of codimension e in X × P 1 and with image z ′ of codimension e − 1 in X, the map is given in components by Here tr is the monogeneic transfer coming from the embedding z ∈ P 1 z ′ . We used above the following form of the homotopy purity equivalence.
Lemma 3.9. Let Z ⊂ Y ⊂ X be closed immersions with X, Y smooth. Then the collapse map is canonically homotopic to the collapse map Proof. Write X ′ = X \ Z and Y ′ = Y \ Z. We can write the collapse map as 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 see also [Hoy17,top of p. 24]. This is the desired result.
Lemma 3.10. Let X be (essentially) smooth, i : Y ֒→ X closed of codimension 1 with Y essentially smooth, Z ⊂ X × P 1 of codimension ≥ d such that W := (Y × P 1 ) ∩ Z also has codimension ≥ d on Y × P 1 . Write Z ′ , W ′ for the images of Z and W in X, Y , respectively. Let η 1 , . . . , η r be the generic points of W of codimension d. Suppose further that Z → Z ′ is quasi-finite and W → W ′ is birational at η 1 . Then In particular, the map i * does not depend on i * 1 .
Proof. By Remarks 3.6 and 3.8, we have a commutative diagram By Lemma 3.7, the vertical maps only depend on M −d , and it follows from Remark 3.8 and our assumption that W → W ′ is birational at η 1 that the right hand vertical map is injective on the component corresponding to η 1 . Let a ∈ H d Z (X × P 1 , M ). Write i * (a) = b 1 + · · ·+ b r , where b i ∈ C d ηi (Y × P 1 , M ). For j > 1, we know i * ηj , hence we know b j and thus we know tr W (b j ). Since we know the bottom horizontal map, we know i * tr Z (a) = tr W (i * (a)). Consequently we know tr W (b 1 ) = tr W (i * a) − j>1 tr W (b j ), and hence b 1 . This concludes the proof.
Example 3.11. If d = 1, then the map i * : Example 3.12. If Z is smooth and transverse to Y at η j for j > 1, then i * ηj only depends on M −d by Lemma 3.4, as desired.
The following is the key reduction. It is an adaptation of [Lev10, Lemma 7.2].
Proof. By a continuity argument, we may assume that X is smooth over k.
Using Remark 3.3, 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 1 → X yields an understood isomorphism H d Z (X, M ) → H d Z×A 1 (X × A 1 , M ), functorial in X. It hence suffices to understand pullback along i×A 1 . Let w = Spec(F ) be a generic point of W ×A 1 of codimension d on Y ×A 1 . Then w lies over the generic point of A 1 [Stacks, Tag 0CC1]. We may thus (using Remark 3.3 again) pass to the generic fiber over Spec(k(t)) ∈ A 1 ; essentially we have base changed the entire problem to k(t)/k. Let us denote the base change of X by X 1 , and so on. Since Z is geometrically reduced over k [Stacks, Tag 020I], its base change Z 1 is geometrically reduced over k(t) [Stacks, Tag 0384]. Lemma 3.14 below supplies us with anétale neighborhood X 2 → X 1 of w and a smooth map X 2 → W 1 such that Z 2 → X 2 → W 1 is generically smooth and Y 2 → W 1 is smooth. Let X 3 = X 2 × W1 {w}. Our base changes are illustrated in the following diagram By construction X 3 → X 1 is a pro-(étale neighborhood) of w ∈ X ⊗ k(t), and so (using again Remark 3.3) we may replace X ⊗ k(t) by X 3 . With these preparatory constructions out of the way, we rename X 3 to X, Y 3 to Y and Z 3 to Z. We now have a smooth map X → Spec(F ), where F is infinite (since it contains k(t)), Y → Spec(F ) is smooth and 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 out by f ∈ O(X), that every component of Z meets w, that Z is smooth away from w, and that X is affine.
Lemma 3.16 below supplies us withū : Z → A 1 withū(w) = 0,ūf : Z → A 1 finite andū having no double roots. Pick u ∈ O(X) reducing toū ∈ O(Z). Then φ 1 := uf : X → A 1 is finite when restricted to Z, satisfies φ 1 (w) = 0, and we claim that φ 1 is smooth at all points of φ −1 1 (0) ∩ Z. Note that by construction (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 does not vanish at points p ∈ Z with (uf )(p) = 0, proving the claim.
We may thus apply Lemma 3.17 below to obtain φ 2 , . . . , φ d+1 : . The non-étale locus of φ meets Z in finitely many points (namely a closed subset not containing w, and hence no component of the curve Z), none of which map to 0 under φ 1 . Shrinking U further, we may thus assume that We have a commutative diagram of schemes Here j is the canonical closed immersion, and ψ is the restriction (i.e. base change) of φ. In particular ψ isétale and Y ′ is smooth. By construction, φ is anétale neighborhood of Z U , and . We need to understand the top left hand horizontal map. All the labelled isomorphisms are pullback alongétale maps, and isomorphisms by excision. The two maps labelled o are also pullback alongétale morphisms (in fact open immersions), and hence understood.
We have thus reduced to understanding j * ; we rename Z U to Z and Z U ∩ Z(uf ) to V . Since Z is finite over U , it remains closed in P 1 U , and hence by a further excision argument it suffices to understand . We have V = {w, z 1 , . . . , 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 no double roots on Z). By Lemma 3.4, the pullback U0 ) zs , M ) only depends on M −d . Thus applying Lemma 3.10, it suffices to understand ; here Z ′ and V ′ are the images of Z and V in U . If d = 1 we are done by Example 3.11. The general case (i.e. d > 1) now follows by induction (i.e. restart the argument with (U, U 0 , Z ′ ) in place of (X, Y, Z)).
Lemma 3.14. Let X be a smooth scheme over an infinite field K, W ⊂ X, Y ⊂ X smooth closed subschemes, W ⊂ Z ⊂ X with Z ⊂ X closed and Z geometrically reduced over K (but not necessarily smooth). Let w ∈ W ∩ Y such that dim w W ≤ dim w Y . There exists anétale 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 [Dé07, Corollary 5.11]. Shrinking X around w, we may assume that X is affine and there exist f 1 , . . . , f d ∈ O(X) such that W = Z(f 1 , . . . , f d ) and W has codimension everywhere exactly d in X. Let {z 1 , . . . , z r } ⊂ Z be a choice of smooth point in every component of Z, which exist because Z is geometrically reduced [Stacks, Tag 056V]. Let dim X = d + n. We claim that there exist g 1 , . . . , g n ∈ O(X) such that dg 1 , . . . , dg n are linearly independent (over the respective residue fields) in Ω w W, Ω w Y and Ω zi Z for every i; we shall prove this at the end. It follows that df 1 , . . . , df d , dg 1 , . . . , dg n are linearly independent in Ω w X. Consider the map F = (f 1 , . . . , f d , g 1 , . . . , 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 generically smooth. Shrinking X further around w, we may assume that F, pF | W and pF | Y are smooth (whence the former two areétale), and of course pF | Z remains generically smooth. Applying the construction of [Dé07, §5.5, §5.9], we obtain a commutative diagram as follows Here both squares are cartesian by definition, j is an open immersion and qj is anétale neighborhood of 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 is smooth 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 1 , . . . , g n be linear projections. By Lemma 3.15 below applied to (A N ) * ⊗ K K(w) ⊂ (ΩA N ) ⊗ K K(w) → Ω w W , dg 1 , . . . , dg n will be linearly independent in Ω w W for general g j , and similarly for Ω w Y and Ω zi Z. The result follows.
For completeness, we include a proof of the following elementary fact.
Lemma 3.15. 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 associated variety over K (isomorphic to A dim V K ). There is a non-empty open subset U of A(V ) n such that any (v 1 , . . . , 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 1 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 where R denotes the Weil restriction along K ′ /K (see e.g. [BLR90, §7.6]). By construction 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 ′−1 (R(A 1 K ′ ) \ 0) is the desired non-empty open subset.
The last statement follows since non-empty open subsets of affine space over an infinite field have rational points.
Lemma 3.16. Let Z be an affine curve over the field F , smooth away from a rational point w, and let f : Z → A 1 be nowhere constant. There exists u : Z → A 1 such that u(w) = 0 and f u : Z → A 1 is finite. If F is infinite, it can be arranged that u has no double zeros.
Proof. LetZ be a compactification of Z which is smooth away from w, andZ \ Z = {z 1 , . . . , z r }. Since Z is smooth at infinity, any map Z → A 1 extends toZ → P 1 . It suffices to find u i : Z → A 1 such that ord zi (u i ) + ord zi (f ) < 0, for every i. Then u = i u ei i for suitably big e i will satisfy the same condition, but for all i at once. Now uf :Z → P 1 is proper andZ \ Z ⊃ (uf ) −1 (∞) ⊃ {z 1 , . . . , z r }, which implies that uf : Z → A 1 is finite (being proper and affine). If u(w) = 0 then replace u by u + 1; the first claim follows. Replacing u by u n + g for suitable g and n large, we may assume that du has only finitely many zeros. Then u + c for general c has no double zeros (away from w) and satisfies u(w) = 0, so that if F is infinite we may arrange the second claim.
LetZ be the normalization ofZ, andZ ⊂Z the open subset over Z, i.e. the normalization of Z. Note thatZ →Z is an isomorphism near z i . By Riemann-Roch, we can find v :Z → P 1 with an arbitrarily large pole at z i , no poles away from z i , so in particular no poles onZ. SinceZ → Z is integral, there exists an equation v n + a 1 v n−1 + · · · + a n = 0, with a i : Z → A 1 . 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.
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 3.17. Let F be an infinite field, X smooth and affine of dimension Then there exist φ e+1 , . . . , φ d : 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 have the desired properties. They vanish on w by definition.
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 a morphism over S = A e via φ ′ (and the projection A d → A e to the first e coordinates). It is thus enough to show that ψ 0 : φ ′−1 (0) ∩ Z → A d−e is a closed immersion (for general φ i ) Since φ isétale at all points of φ ′−1 (0) ∩ Z (for general φ i ) and Z → X is a closed immersion, ψ is unramified at all points above 0 and so ψ 0 is unramified (for general φ i ). Since φ : Z → A d is finite so is ψ; in fact φ ′−1 (0) ∩ Z is finite over 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 0 is finite (hence proper) and unramified. Being radicial is fpqc local on the target [Stacks, Tag 02KW], so may be checked after geometric base change. In other words (using that φ ′−1 (0) ∩ Z is finite over F ) we need the φ i to separate a finite number of specified geometric points. This clearly holds for general φ i .
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 i : Y → Γ f is the graph of f . Then i is a regular immersion and p is smooth. It follows that p −1 (Z) ⊂ Γ f has codimension ≥ d (see e.g. [Gro67, Corollary 6.1.4]). Hence it suffices to prove the result for i, p separately; i.e. we may assume that f is either a regular immersion or a smooth morphism. The case of smooth maps was already explained in Remark 3.3, so assume that f : Y ֒→ X is a regular immersion. As usual we may localize in a generic point of Z ∩ Y of codimension d on Y ; hence we may assume that Z ∩ Y = {z} is a closed point, X is local and dim Y = d. It follows (e.g. from [Stacks,Tag 00NQ]) that there exists a sequence of codimension 1 embeddings of essentially smooth schemes It follows that we may prove the result for each Y i ֒→ Y i+1 separately; this is Lemma 3.13.

Applications
After introducing some notation in §4.1, we identify the heart of SH S 1 (k)(d) (for d ≥ 3) in §4.2. This establishes [BY20, Conjecture 6.10]. Finally in §4.3 we study convergence of the resolution of an S 1 -spectrum by infinite P 1 -loop spectra arising from the adjunction SH S 1 (k) ⇆ SH(k) and deduce some conservativity results.
We also assume that k is perfect; we will restate this assumption with the most important results only.
4.1. Notation and hypotheses. We write SH S 1 (k) for the category of motivic S 1 -spectra [Mor03, §4] (i.e. the motivic localization of the category of spectral presheaves on Sm k ), and SH(k) = SH S 1 (k)[G ∧−1 m ] for the category of motivic spectra [Mor03,§5]. We use the following notation for the stabilization functors and we denote by ω ∞ the right adjoint of σ ∞ . Here SH(k) eff is the localizing subcategory generated by the image of σ ∞ . There are localizing subcategories here SH S 1 (k)(d) is generated by Σ ∞ S 1 X + ∧ G ∧d m for X ∈ Sm k . The inclusion SH S 1 (k)(d) ⊂ SH S 1 (k) has a right adjoint which we denote by f d . 8 There are canonical cofiber sequences f d+1 → f d → s d defining the functors s d . There is a similar filtration of SH(k) eff , given by SH(k) eff (d) := SH(k) eff ∧ G ∧d m , and the right adjoints (respectively cofibers) are again denoted by f d (respectively s d ). See [Lev08,Voe02] or [BY20, §6.1] for more details on these functors.
Recall that SH S 1 (k) has a t-structure with non-negative part generated by Σ ∞ S 1 X + for X ∈ Sm k ; its heart canonically identifies with HI(k) [Mor03, Lemma 4.3.7(2)]. We denote by E ≥0 , E ≤0 and π 0 E the truncations and homotopy sheaves, respectively. The categories SH S 1 (k)(d), SH(k), SH(k) eff (d) have related t-structures, with non-negative parts generated by X + ∧ G ∧d m . Recall from [BY20, §5.1] the notion of a presheaf with A 1 -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 every finite monogeneic extension K(x)/K a transfer τ x : F (K(x)) → F (K). The category SH(k) eff♥ embeds fully faithfully into the category of presheaves with A 1 -transfers [BY20, Corollary 5.17] (morphisms in this category are given by morphisms of presheaves compatible with the GW -module structures and transfers). Given a presheaf with A 1 -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 1 -transfers (see §2.1.8 or [BY20, Example 5.2]).
Definition 4.1. Let k be a perfect field and d > 0.
(1) Let M ∈ HI(k). We shall say that hypothesis T d (M ) holds if the canonical A 1 -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 1 (k) and i ∈ Z the spectrum f d π i s d E is in the essential image of ω ∞ : SH(k) → SH S 1 (k).
Proof. The functor is fully faithful by [BY20, 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 the essential image of ω ∞ . We first prove this assuming that k is infinite. We have π i (M ) −d = 0 for i = 0 [BY20, Lemma 6.2(3)] and hence the canonical map M → f d π 0 M is an equivalence (indeed it induces an equivalence on π i (−) −d for every i, and this detects equivalence in SH S 1 (k)(d) by [BY20, Lemma 6.1(1)]). By assumption, the A 1 -transfers on π 0 (M ) −d extend to framed transfers; there hence existsM ∈ SH(k) eff♥ such that ω ∞ (M ) −d ≃ π 0 (M ) −d as presheaves with A 1 -transfers. By Lemma 4.7 below (this is where we use the assumption that k is infinite), this implies that f d ω ∞ (M ) ≃ f d π 0 (M ). It follows from [Lev08, Theorem 9.0.3] that f d commutes with ω ∞ ; we thus find that 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 the essential 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 infinite algebraic 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 was already established.
This concludes the proof.
(2) We have 4.3. Canonical resolutions. In this section we will freely use the language of ∞-categories as set out in [Lur17b, Lur17a]. Given any adjunction F : C ⇆ D : G of ∞-categories, there is a monad structure 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 definition of a monad, GF promotes to an E 1 -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 triple resolution is obtained by applying this cosimplicial endofunctor to E. This also makes it clear that the triple resolution is functorial in E.
Definition 4.10. We shall say that the canonical resolution converges for E if the above morphism is an equivalence.
Example 4.11. The canonical resolution converges if E is in the essential image of ω ∞ , since then the cosimplicial object is split.
Example 4.12. Given a cofiber sequence E 1 → E 2 → E 3 , if the canonical resolution converges for any two of the three terms, then it converges for the third. This holds since all the functors involved are stable.
The following result clearly holds in much greater generality, but for simplicity we state it in our restricted context. Lemma 4.13. Let E ∈ SH S 1 (k) and suppose given a tower · · · → E 2 → E 1 → E 0 := E and a sequence n i ∈ Z such that (i) lim i n i = ∞, (ii) E i ∈ SH S 1 (k) ≥ni (for all i) and (iii) the canonical resolution 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 1 (k) by F (X) = f ib(X → X ∧ ). By construction this is a stable functor such that F (X) ≃ 0 if and only if the canonical resolution converges for X. Since F (cof (E i+1 → E i )) ≃ 0 for all i, we find that the tower is constant. In order to prove that F (E) = 0 it thus suffices to show that lim i F (E i ) ≃ 0. Commuting the limits, we find that it thus suffices to show that the inner limit over i vanishes. Since σ ∞ , ω ∞ are both right-t-exact [BY20, Lemma 6.2(1,2)], we have f ib(. . . ) ∈ SH S 1 (k) ≥ni−1 , and hence it is enough to show that if X i is a sequence of spectra with X i ∈ SH S 1 (k) ≥ni then lim i X i ≃ 0. Since SH S 1 (k) is generated as a localizing subcategory by objects of the form Σ ∞ S 1 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 1 U + , SH S 1 (k) ≥n ] = 0. This follows from the descent spectral sequence.
Theorem 4.14. Let k be a perfect field such that T d (k) holds.