THE HOMOTOPY CONIVEAU TOWER

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1 THE HOMOTOPY CONIVEAU TOWER MARC LEVINE Abstract. We examine the homotopy coniveau tower for a general cohomology theory on smooth k-schemes an give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In aition, the homotopy coniveau tower agrees with Voevosky s slice tower for S 1 -spectra, giving a proof of a connecteness conjecture of Voevosky. The homotopy coniveau tower construction extens to a tower of functors on the Morel-Voevosky stable homotopy category, an we ientify this P 1 -stable homotopy coniveau tower with Voevosky s slice tower for P 1 -spectra. We also show that the 0th layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general P 1 -spectrum the structure of a moule over motivic cohomology. This recovers an extens results of Voevosky on the 0th layer of the slice filtration, an yiels a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence. Contents 0. Introuction 3 1. Spaces, spectra an homotopy categories Presheaves of spaces Spectra Notation Nisnevic moel structure A 1 -local moel structure Simplicial spectra 9 2. The homotopy coniveau tower The construction Mathematics Subject Classification. Primary 14C25, 19E15; Seconary 19E08 14F42, 55P42. Key wors an phrases. Bloch-Lichtenbaum spectral sequence, algebraic cycles, Morel-Voevosky stable homotopy category, slice filtration. The author gratefully acknowleges the support of the Humbolt Founation through the Wolfgang Paul Program, an support of the NSF via grants DMS an DMS

2 2 MARC LEVINE 2.2. First properties Localization Stable homology of spectra The localization theorem The e-looping theorem Functoriality an Chow s moving lemma Functoriality The purity theorem Generalize cycles The semi-local Some vanishing theorems The main result Computations Well-connecte theories Cycles Well-connecteness The case of K-theory Bloch motivic cohomology The slice tower for S 1 -spectra The slice tower The splitting The proof of Theorem An S 1 -connecteness result The connectivity conjecture The P 1 -stable theory P 1 -spectra Moel structure an homotopy categories The stable homotopy coniveau tower The slice tower in SH(k) The proof of Theorem The sphere spectrum an the HZ-moule structure The funamental class of a system The reverse cycle map The extension to a map in SH(k) The cycle class map σ 0 1 an HZ The motivic Atiyah-Hirzebruch spectral sequence Proucts HZ-moules an DM The spectral sequence 63 References 65

3 THE HOMOTOPY CONIVEAU TOWER 3 0. Introuction The original purpose of this paper was to give an alternative argument for the technical unerpinnings of the papers [2, 4], in which the constructruction of a spectral sequence from motivic cohomology to K-theory is given. As in the metho use by Suslin [19] to analyze the Grayson spectral sequence, we rely on localization properties of the relevant spectra. Having one this, it becomes clear that the metho applies more generally to a presheaf of spectra on smooth schemes over a perfect base fiel k, satisfying certain conitions. We therefore give a general iscussion for a presheaf of spectra E : Sm/k op Spt which is homotopy invariant an satisfies Nisnevic excision. For such a functor, an an X in Sm/k, we construct the homotopy coniveau tower (0.0.1)... E (p+1) (X, ) E (p) (X, )... E (0) (X, ) E(X) where the E (p) (X, ) are simplicial spectra with n-simplices the limit of the spectra with support E W (X n ), where W is a close subset of coimension p in goo position. This is just the evient extension of the tower use by Frielaner-Suslin in [4]. One can consier this tower as the algebraic analog of the one in topology forme by applying a cohomology theory to the skeletal filtration of a CW complex. The main objects of our stuy are the layers E (p/p+1) (X, ) in this tower. We have iscusse the functoriality of the E (p) (X, ) an the tower (0.0.1) in [11]. The metho use is a variation on the classical Chow s moving lemma, aie by Gabber s approach to moving lemmas. The results of [11] replace the total spectra E (p) (X, ) with functors E (p) : Sm/k op Spt, an similarly for the tower (0.0.1) an layers E (p/p+1) ; these results require only that E is homotopy invariant an satisfies Nisnevic excision. The homotopy invariance of the E (p) (X, ) is also verifie in [11]. The methos of [10] allows us to prove a localization property for the homotopy coniveau tower. We iterate the functors E (p), an thereby fin a simple escription of the layers. In this, a crucial role is playe by the 0th layers (Ω p P 1 E) (0/1) of the p-fol P 1 -loop space Ω p P 1 E, where (Ω P 1E)(X) := fib[e(x P 1 ) E(X )]. Inee, for W smooth, the restriction map E (0/1) (W ) E (0/1) (k(w ))

4 4 MARC LEVINE is a weak equivalence, which enables us to exten the functor E (0/1) to all k-schemes as a locally constant sheaf for the Zariski topology. One can then ientify E (p/p+1) (X) with the simplicial spectrum E (p/p+1) s.l. (X, ) having n-simplices E (p/p+1) s.l. (X, n) = x X (p) (n) (Ω p P 1 E) (0/1) (k(x)). Here X (p) (n) is the set of coimension p points of X n, with closure in goo position. Once one has this escription of the layers, it is easy to compute the layers for K-theory, as one can easily show that Ω p P K = K an 1 that K (0/1) (F ) is canonically a K(K 0 (F ), 0) = K(Z, 0) for F a fiel. This gives a irect ientification of K (p/p+1) s.l. (X, n) with Bloch s higher cycle group z p (X, n), an thus the weak equivalence K (p/p+1) (X, ) with z p (X, ). The natural setting for the homotopy coniveau tower is in the Morel- Voevosky homotopy category of S 1 spectra, SH S 1(k). This is essentially the category of presheaves of spectra on Sm/k, localize with respect to the Nisnevic topology an A 1 weak equivalence. Voevosky [22] constructs the slice tower... f n+1 E f n E... f 0 E = E for E SH S 1(k), where f n E E is universal for maps of the form Σ P 1 F E, in analogy with the classical Postnikov tower for spectra. Our main result in this irection is the ientification of this S 1 -slice tower with our homotopy coniveau tower (for k an infinite perfect fiel). The localization properties of the homotopy coniveau tower enable us to show how the truncation functors f n commute with the P 1 -loops functor: f n+1 Ω P 1 = Ω P 1 f n. This in turn proves a connecteness conjecture of Voevosky: Conjecture Let Σ P 1 SH S 1(k) be the localizing subcategory of SH S 1(k) generate by the presheaves Σ P 1 F, F SH S 1(k). If E is in Σ P 1 SH S 1(k), then Ω P 1Σ P 1E is also in Σ P 1 SH S 1(k). After this, we turn to the P 1 -stable theory. Let SH(k) be the Morel- Voevosky stable homotopy category of P 1 -spectra over k. There is an infinite P 1 -suspension functor Σ P : SH 1 S1(k) SH(k)

5 THE HOMOTOPY CONIVEAU TOWER 5 with essential image enote SH eff (k). tower Voevosky efines the slice... f p+1 E f p E... f 0 E f 1 E... E. in SH(k), where the map f E E is universal for maps F E with F in Σ P 1 SH eff (k). The localization property for the E (p) allows one to efine a P 1 - spectrum φ p E for a P 1 -spectrum E := (E 0, E 1,...), by the formula (φ p E) n := E (n+p) n. This gives the stable homotopy coniveau tower... φ p+1 E φ p E... φ 0 E φ 1 E... E. We show that φ E is in Σ P 1 SH eff (k), an that the canonical map φ E f E is an isomorphism, thus ientifying the slice tower with the stable homotopy coniveau tower. Finally, we compute of the 0th layer σ 0 in the homotopy coniveau tower for the motivic sphere spectrum 1, assuming that the basefiel k is perfect. The iea here is that the cycle-like escription of E (p/p+1) s.l. (X, ) enables one to efine a reverse cycle map rev : HZ σ 0 1. It is then rather easy to show that rev inuces a weak equivalence after applying the 0th layer functor σ 0 again. However, since motivic cohomology is alreay the 0th layer of K-theory, applying σ 0 leaves HZ unchange, an similarly for σ 0 1, giving the esire weak equivalence HZ σ 0 1. The analogous statement for the slice filtration in characteristic zero has been proven by Voevosky [24] by a ifferent metho. In any case, for a P 1 -spectrum E, each layer σ p E is a HZ-moule. Assuming that Voevosky s slice tower has a lifting to a natural tower in Jarine s category of motivic symmetric spectra [8], work of Roenigs- Ostvar implies that the layers s n E in the slice tower for E are naturally the Eilenberg-Maclane spectra of objects of the category of motives over k, DM(k). We thus have the objects π µ p E of DM(k) whose Eilenberg-Maclane P 1 -spectrum satisfies H((π µ p E)[p]) = σ p E.

6 6 MARC LEVINE For E : Sm/k Spt the 0th S 1 -spectrum of E an X Sm/k, the spectral sequence associate to the homotopy coniveau tower can be expresse as E p,q 2 := H p (X, π qe) µ = Ê p q(x), wheere Ê is the completion of E with respect to the homotopy coniveau tower. Uner certain connectivity properties of E, one has E = Ê. Using the bi-grae homotopy groups of a P 1 -spectrum, this gives the weight-shifte spectral sequence E p,q 2 = H p (X, (π µ qe) Z(b)) = Ê p+q,b (X). For the K-theory P 1 -spectrum K := (K, K,...), our computation of the layers K (p/p+1) gives π µ p K = Z(p)[p]; p = 0, ±1, ±2,..., an we recover the Bloch-Lichtenbaum, Frielaner-Suslin spectral sequence E p,q 2 := H p q (X, Z( q)) = K p q (X). We have ha a great eal of help in eveloping the techniques that went in to this paper. Fabien Morel playe a crucial role in numerous iscussions on the A 1 -stable homotopy category an relate topics. Conversations with Bruno Kahn, Jens Hornbostel an Marco Schlichting were very helpful, as were the lectures of Bjorn Dunas an Vlaamir Voevosky at the Sophus Lie Summer Workshop in A 1 -stable homotopy theory. I woul also like to thank Paul Arne Ostvar for his etaile comments on an earlier version of this manuscript. Finally, I am very grateful to the Humbolt Founation an the Universitität Essen for their support of this reseach, especially my colleagues Hélène Esnault an Eckart Viehweg.

7 THE HOMOTOPY CONIVEAU TOWER 7 1. Spaces, spectra an homotopy categories 1.1. Presheaves of spaces. Let Spc enote the category of spaces, i.e., simplicial sets, an Spc the category of pointe simplicial sets. For a category C, we have the category Spc(C) of presheaves of spaces on C, an Spc (C) of presheaves of pointe spaces. We give Spc an Spc the stanar moel structures: cofibrations are monomorphisms, weak equivalences are weak equivalences on the geometric realization, an fibrations are etemine by the RLP with respect to trivial cofibrations; the fibrations are then exactly the Kan fibrations. We let A enote the geometric realization, an [A, B] the homotopy classes of (pointe) maps A B. We give Spc(C) an Spc (C) the moel structure of functor categories escribe by Bousfiel-Kan [3]. That is, the cofibrations an weak equivalences are the pointwise ones, an the fibrations are etermine by the RLP with respect to trivial cofibrations. We let HSpc(C) an HSpc (C) enote the associate homotopy cateogies. Note that Spc(C) an Spc (C) inherit operations from Spc an Spc, for instance limits an colimits. In particular, in Spc (C) we have wege prouct A B := A B/A B. We also have the inclusions Spc Spc(C), Spc Spc(C) as constant presheaves, giving us the suspension functor on Spc (C), ΣA := S 1 A, an the inclusion + : Spc(C) Spc(C) by aing a isjoint base-point. These operations pass to the homotopy category Spectra. Let Spt enote the category of spectra. To fix ieas, a spectrum will be a sequence of pointe spaces E 0, E 1,... together with maps of pointe simplicial sets ɛ n : S 1 E n E n+1. Maps of spectra are maps of the unerlying spaces which are compatible with the attaching maps ɛ n. The stable homotopy groups π s n(e) are efine by π s n(e) := lim m [Sm+n, E m ]. The category Spt has the following moel structure: Cofibrations are maps f : E F such that E 0 F 0 is a cofibration, an for each n 0, the map E n+1 S 1 E n S 1 F n F n+1 is a cofibration. Weak equivalences are the stable weak equivalences, i.e., maps f : E F which inuce an isomorphism on π s n for all n. Fibrations are characterize by having the RLP with respect to trivial cofibrations. Let Spt(C) be the category of presheaves of spectra on C. We use the following moel structure on Spt(C) (see [7]): Cofibrations an

8 8 MARC LEVINE weak equivalences are given pointwise, an fibrations are characterize by having the RLP with respect to trivial cofibrations. We enote the associate homotopy category by HSpt(C). We write SH for the homotopy category of Spt Notation. For a scheme B, Sm/B is the category of smooth separate B-schemes of finite type. For a morphism f : Y X in Sm/B, an E Spt(B), we write E(X/Y ) for the homotopy fiber of f : E(X) E(Y ). Similarly, if (Z, z : B Z) is a pointe B-scheme, E(Z (X/Y )) is the homotopy fiber of z : E(Z X/Z Y ) E(X/Y ). In case j : U X is an open immersion with close complement W X, we write E W (X) for E(X/U). We write X + for X Spec k/spec k. Given an E Spt(B) an a Y B Sm/B, we let E (Y ) enote the presheaf E (Y ) (Z) := E(Z B Y ) Nisnevic moel structure. Fix a noetherian base-scheme B an let C be a subcategory of Sm/B with the same objects as Sm/B an containing all the smooth B-morphisms. In particular, the Nisnevic topology is efine on C. For a point x X, with X C, an E = (E 0, E 1,...) a presheaf of spectra on C, the stalk of E at x, E x, is the spectrum (E 0x, E 1x,...), where E nx is the stalk (in the Nisnevic topology) of the presheaf of spaces E n at x. Let Spt Nis (C) enote the moel category with the same unerlying category an cofibrations as Spt(C), where a map f : E F is a weak equivalence if fx : Ẽ x F x is a weak equivalence in Spt for all x X C, an the fibrations are characterize by having the right lifting property (RLP) with respect to trival cofibrations. We let HSpt Nis (C) enote the associate homotopy category an write Spt Nis (B) = Spt Nis (Sm/B), SH s (B) := HSpt Nis (Sm/B). For etails, we refer the reaer to [7] A 1 -local moel structure. One imposes the relation of A 1 -weak equivalence in SH s (B) by means of Bousfiel localization applie to the moel category Spt Nis (B). An object E Spt(B) is calle A 1 -local if the map E E (A1 ) inuce by the projections Y A 1 Y is a weak equivalence in Spt Nis (B). A map f : F F in Spt Nis (B) is an A 1 -weak equivalance if f : Hom SHs(B)(F, E) Hom SHs(X)(F, E) is an isomorphism for all A 1 -local E. Spt S 1(B) is the moel category with the same unerlying category an cofibrations as Spt Nis (B), the weak equivalences being the A 1 -weak equivalences, an the fibrations

9 THE HOMOTOPY CONIVEAU TOWER 9 etermine by the RLP with respect to trivial cofibrations. The fact that this is inee a moel category is iscusse in [8]. We refer to Spt S 1(B) as the category of S 1 -spectra over B. The homotopy category HSpt S 1(B) is enote SH S 1(B) Simplicial spectra. For a spectrum E, we have the Postnikov tower... τ N E τ N 1 E... E with τ N E E the N 1-connecte cover of E, i.e., τ N E E is an isomorphism on homotopy groups π n for n N, an π n (τ N E) = 0 for n < N. One can make this tower functorial in E, so we can apply the construction τ N to functors E : C Spt. We have the category Or with objects the finite orere sets [n] := {0 <... < n}, n = 0, 1,..., an maps orer-preserving maps of sets. Let Or N be the full subcategory with objects [n], 0 n N. Let E : Or op Spt be a simplicial spectrum. We have the N- truncate simplicial spectrum E N : Or op N Spt, the associate total spectrum E N, an the tower of spectra (1.6.1) E 0... E N... E Since taking the total spectrum commutes with filtere colimits, we have the natural weak equivalences hocolim M E M E hocolim N,M τ N E M hocolim N τ N E. When the context makes the meaning clear, we will often omit the separate notation for the total spectrum, an freely pass between a simplicial spectrum an its associate total spectrum. 2. The homotopy coniveau tower 2.1. The construction. We fix a noetherian base scheme S, separate an of finite Krull imension. We have the cosimplicial scheme, with r = Spec (Z[t 0,..., t r ]/ j t j 1).

10 10 MARC LEVINE The vertices of r are the close subschemes vi r efine by t i = 1, t j = 0 for j i. A face of r is a close subscheme efine by equations of the form t i1 =... = t is = 0. Let E Spt(S) be a presheaf of spectra. For X in Sm/S with close subscheme W an open complement j : X \ W X, we have the homotopy fiber E W (X) of j : E(X) E(X \ W ). If we have a chain of close subsets W W X, we have a natural map i W,W : E W (X) E W (X) an a natural weak equivalence (2.1.1) cofib(i W,W : E W (X) E W (X)) E W \W (X \ W ). Here cofib means homotopy cofiber in the category of spectra. For X in Sm/S, we let S (p) X (r) enote the set of close subsets W of X r such that coim X F (W (X F )) p for all faces F of r. Clearly, sening r to S (p) X (r) efines a simplicial set S (p) X ( ). We let X(p) (r) be the set of coimension p points x of X r with closure x S (p) X (r). We let E (p) (X, r) enote the (filtere) limit E (p) (X, r) = hocolim W S (p) X (r) E W (X r ). Sening r to E (p) (X, r) efines a simplicial spectrum E (p) (X, ). Since (r) is a subset of S (p) (r), we have the tower of simplicial spectra S (p+1) X X (2.1.2)... E (p+1) (X, ) E (p) (X, )... E (0) (X, ), which we call the homotopy coniveau tower. We let E (p/p+1) (X, ) enote the cofiber of the map E (p+1) (X, ) E (p) (X, ). Two properties of E that we shall often require are: A1. E is homotopy invariant: For each X in Sm/S, the map p : E(X) E(X A 1 ) is a weak equivalence. A2. E satisfies Nisnevic excision: Let f : X X be an étale morphism in Sm/S, an W X a close subset. Let W = f 1 (W ), an suppose that f restricts to an isomorphism W W. Then f : E W (X) E W (X ) is a weak equivalence. Instea of A2, we will occasionally require the weaker conition: A2. E satisfies Zariski excision: Let j : U X be an open immersion in Sm/S, an W X a close subset containe in U. Then j : E W (X) E W (U) is a weak equivalence.

11 THE HOMOTOPY CONIVEAU TOWER 11 We introuce one final axiom to hanle the case of finite resiue fiels. Suppose we have a finite Galois extension k k with group G. Given E Spt(k), efine π G π E by π G π E(X) := E(X k ) G, where ( ) G enotes a functorial moel for the G homotopy fixe point spectrum. We have as well the natural transformation π : E π G π E. A3. Suppose that k is a finite fiel. Let k k be a finite Galois extension of k with group G. Then after inverting G, the natural transformation π : E π G π E is a weak equivalence. Remark It is shown in [11, Corollary 9.4.2] that E satisfies A3 if E is the 0-spectrum of some P 1 -Ω-spectrum E Spt P 1(k) (see 8). Definition Let X be in Sm/S. The weight-complete spectrum Ê(X) is Ê(X) = holim E (0/p) (X, ). p Proposition Take E Spt(S), where S is a noetherian scheme of finite Krull imension. (1) There is a weakly convergent spectral sequence E p,q 1 = π p q (E (p/p+1) (X, )) = Ê p q(x). (2) If E = τ N E for some N, then the above spectral sequence is strongly convergent an the canonical map E (0) (X, ) Ê(X) is a weak equivalence. (3) If E is homotopy invariant, the canonical map E(X) E (0) (X, ) is a weak equivalence. Proof. (1) The spectral sequence is constructe by the stanar process of linking the long exact sequences of homotopy groups arising from the homotopy cofiber sequences E (p+1) (X, ) E (p) (X, ) E (p/p+1) (X, ). The first assertion then follows from the general theory of homotopy limits (see [3]). For (2), suppose E = τ N E. We first show that the sequence is strongly convergent. By (2.1.1) an a limit argument, we have π m (E (p/p+1) (X, r)) = ( lim π m E W \W (X r \ W ) ), W W

12 12 MARC LEVINE where the limit is over W S (p+1) X (r), W S (p) X (r). It follows that π m (E (p/p+1) (X, r)) = 0 for m < N 1. From the tower (1.6.1), we thus have the the strongly convergent spectral sequence E a,b 1 = π a (E (p/p+1) (X, b)) = π a b (E (p/p+1) (X, )). Since S (p) X (r) = for p > im X + r, this implies that π p q E (p/p+1) (X, ) = 0 for p > p q + im X + N + 1, from which it follows that the spectral sequence (2.1.3) is strongly convergent. Similarly, it follows that the natural map E (0) (X, ) Ê(X) is a weak equivalence. For (3) the simplicial spectrum E (0) (X, ) is just the simplicial spectrum E(X ), i.e., r E(X r ). Since E is homotopy invariant, the natural map E(X) E(X ) is a weak equivalence, completing the proof. Corollary Take E Spt(S), where S is a noetherian scheme of finite Krull imension. If E is homotopy invariant an E = τ N E for some N, then the homotopy coniveau tower (2.1.2) yiels a strongly convergent spectral sequence (2.1.3) E p,q 1 = π p q (E (p/p+1) (X, )) = E p q (X). which we call the homotopy coniveau spectral sequence First properties. We give a list of elementary properties of the spectra E (p) (X, ) (1) Sening X to E (p) (X, ) is functorial for equi-imensional (e.g. flat) maps Y X in Sm/S. (2) The pull-back p 1 : E (p) (X, ) E (p) (X A 1, ) is a weak equivalence. The proof is the same as that for Bloch s cycle complexes, given in [1]. For etails, see [11, Theorem 3.3.5]. (3) Sening E to E (p) (X, ) is functorial in E. (4) The functor E E (p) (X, ) sens weak equivalences to weak equivalences, an sen homotopy (co)fiber sequences to homotopy (co)fiber sequences. Exactly the same properties hol for the layers E (p/p+r). 3. Localization We now show that the simplicial spectra E (p) (X, ) behave well with respect to localization.

13 THE HOMOTOPY CONIVEAU TOWER Stable homology of spectra. For a simplicial set S, we have the simplicial abelian group ZS, with n-simplices ZS n the free abelian group on S n. If S, is a pointe simplicial set, efine Z(S, ) n := ZS n /Z. Let E = {E n, φ n : ΣE n E n+1 } be a spectrum; we take the E n to be pointe simplicial sets, an the φ n to be maps of pointe simplicial sets. Form the spectrum ZE by taking (ZE) n = Z(E n, ), where Zφ n : Σ(ZE) n (ZE) n+1 is the map inuce by φ n, compose with the natural map Σ(ZE) n Z(ΣE n ). The natural maps E n ZE n give a natural map E ZE of spectra; one shows that this construction respects weak equivalence an taking homotopy cofibers. The stable homology H n (E) is efine by H n (E) = π n (ZE). Using the Dol-Thom theorem, one has the formula for H n (E) as H n (E) = lim Hn+m (E m ), where H is reuce homology an the maps in the limit are the composition H n+m (E n ) = H n+m+1 (ΣE n ) φn H n+m+1 (E n+1 ). The Hurewicz theorem for simplicial sets gives the following analogous result for spectra: Proposition Let E be a spectrum which is N-connecte for some N Z. Then π n (E) = 0 for all n if an only if H n (E) = 0 for all n. Proof. Since both π n an H n respect weak equivalence, an are compatible with suspension of spectra, we may assume that N 1, an that E is an Ω-spectrum, i.e., the natural maps E n ΩE n+1 are weak equivalences. Then π n (E) = π n+m (E m ) for all m. Suppose H n (E) = 0 for all n; we prove by inuction that π n+m (E m ) = 0 for all n an m. By assumption π N+m (E m ) = 0 for all m, with N 1. We may therefore procee by inuction on n to show that π n+m (E m ) = 0 for all n an m. Supposing that π n+m 1 (E m ) = 0 for all m, the Hurewizc theorem implies that the Hurewicz map π n+m (E m ) H n+m (E m ) is an isomorphism for all m, an one easily checks that the Hurewicz map is compatible with the limits efining H n an π n. Thus, the maps H n+m (E m ) H n+m+1 (E m+1 ) are isomorphisms for all m; since the limit is zero by assumption, this implies that H n+m (E m ) = 0 for all m, whence π n+m (E m ) = 0 for all m. The proof that π n (E) = 0 for all n implies H n (E) = 0 for all n is similar, but easier, an is left to the reaer.

14 14 MARC LEVINE 3.2. The localization theorem. Let X be smooth an essentially of finite type over S, an let j : U X be an open subscheme, with complement i : Z X. We let S (p) X,Z (r) enote the subset of S(p) X (r) consisting of those W containe in Z r. Let S (p) U/X (r) be the image of S (p) X (r) in S(p) U (r) uner (j i) 1. Taking the colimit of E W (X r ) over W S (p) X,Z (r) an varying r an p gives us the tower of simplicial spectra... E (p+1) Z (X, ) E (p) Z (X, )... E() Z (X, ) = E(0) Z (X, ), where is any integer satisfying coim X Z j for all irreucible components Z j of Z. Similarly, taking the colimit of the E W (U r ) over W S (p) U/X (r) for varying p an r gives the tower of simplicial spectra... E (p+1) (U X, ) E (p) (U X, )... E (0) (U X, ). We have as well the natural maps i : E (p) Z (X, r) E(p) (X, r), j! : E (p) (X, r) E (p) (U X, r), ι : E (p) (U X, r) E (p) (U, r), j : E (p) (X, r) E (p) (U, r), with j = ι j!. Let E (p/p+s) ( ) enote the cofiber of the maps E (p+s) ( ) E (p) ( ). Supposing that E satisfies Zariski excision, we have the homotopy fiber sequences E (p) Z i (X, r) E (p) (X, r) j! E (p) (U X, r) E (p/p+s) Z (X, r) i E (p/p+s) (X, r) j! E (p/p+s) (U X, r). These give the homotopy fiber sequences of simplicial spectra E (p) Z i (X, ) E (p) (X, ) j! E (p) (U X, ) E (p/p+s) Z (X, ) i E (p/p+s) (X, ) j! E (p/p+s) (U X, ) The localization theorem is Theorem Let E be in Spt(S). Suppose the base-scheme S is a scheme essentially of finite type over a semi-local DVR with infinite resiue fiels. Then the maps E (p) (U X, ) E (p) (U, ) E (p/p+s) (U X, ) E (p/p+s) (U, )

15 are weak equivalences THE HOMOTOPY CONIVEAU TOWER 15 Proof. The secon weak equivalence follows from the first by taking cofibers. For the first map, this result follows by exactly the same metho as use in the proof of [10, Theorem 8.10]. Inee, to show that the map E (p) (U X, ) E (p) (U, ) is a weak equivalence, it suffices to prove the result with E (p) (, n) replace by τ m E (p) (, n) for all m. By the Hurewicz theorem (Proposition 3.1.1), it suffices to show that E (p) (U X, ) E (p) (U, ) is a homology isomorphism. This follows by applying [10, Theorem 8.2], just as in the proof of Theorem 8.10 (loc. cit.). For the reaer s convenience, we inclue a sketch of the argument. Let E = (E 0, E 1,...) be a spectrum. Using the Dol-Kan corresponence, we can ientify the stable homology spectrum ZE with the complex forme by taking the normalize complex of the simplicial abelian group Z(E n, ) an then taking the limit over the boning maps Z(E n, )[n] Z(E n+1, )[n + 1]. Abusing notation, we enote this complex also by ZE; for the remainer of the proof, ZE will mean the complex, not the spectrum. For W U r, we have the complex Z(τ m E W (U r )) computing the stable homology of τ m E W (U r ). Taking the limit of W S (p) (r) or in S(p) (r) gives us the complexes ZE(p) m (U, r) an ZE (p) U U/X m (U X, r), which compute the stable homology of τ m E (p) (U, r) an τ m E (p) (U X, r). For W N r=0u r, let W n U n be the union of (i g) 1 (W ), as g : n r runs over structure morphisms for the cosimplicial scheme. Using the usual alternating sum of the pullback by coface maps i δi r : U r U r+1, we form the ouble complex n Zτ m E Wn (U n ) an enote the associate total complex by ZEm W (U ). Thus the limit of the complexes ZEm W (U ) over W S (p) U (r) or in S(p) U/X (r), r = 1, 2,..., computes the stable homology of the simplicial spectra n τ m E (p) (U, n) an n τ m E (p) (U X, n). We enote the limits of these complexes by ZE m (p) (U) an ZE m (p) (U X ), respectively. It thus suffices to show that is a quasi-isomorphism for all m. ι Z : ZE (p) m (U X ) ZE (p) m (U)

16 16 MARC LEVINE For W S (p) U (r), W S (p) U/X (r), let ι W : ZE W m (U ) ZE (p) m (U) an ι X W : ZEW m (U ) ZE m (p) (U X ) be the canonical maps. Next, we construct another pair of complexes which approximate ZE m (p) (U) an ZE m (p) (U X ). For this, fix an integer N 0. Let N i N be the subscheme efine by t i = 0; for I {0,..., N} let N I be the face i I N i. For I J, let i J,I : N I N J be the inclusion. Let ZSm/S be the aitive category generate by Sm/S, i.e., for connecte X, Y, Hom ZSm/S (X, Y ) is the free abelian group on the set of morphisms Hom Sm/S (X, Y ), an isjoint union becomes irect sum. We will construct objects in the category of complexes C(ZSm/S). Form the complex ( N, N ) which is I, I =n N I in egree n, an with ifferential n : ( N, N ) n ( N, N ) n+1 given by n := I, I =n n I, where is the sum n I : N I J, J =n 1 N J n I := n j=1 i I\{ij },I, where I = (i 1,..., i n ), i 1 <... < i n. We also have the complex Z, which is n in egree n, with ifferential the usual alternating sum of cobounary maps. The ientity map on N extens to a map of complexes Φ N : Z ( N, N )[ N]; the maps in egree r < N are all ±i r. We can take the prouct of this construction with U, giving us the complex U ( N, N ) an the map of complexes Φ N : U Z U ( N, N )[ N] For W S (p) U (N), form the complex ZEW m (U ( N, N )) by taking I, I =n Zτ m E W N n (U N n ) in egree n, using the ifferentials in U ( N, N ) to form a ouble complex an then taking the total complex. We thus have the map of complexes Φ N W : ZE W m (U ( N, N ))[ N] ZE W (U ).

17 One shows that Φ N W (see [10, Lemma 2.6]). THE HOMOTOPY CONIVEAU TOWER 17 inuces a homology isomorphism in egrees < N Take W S (p) U (N). The main result of [10], Theorem 1.9, gives a map of complexes an a egree -1 map with the following properties: (1) H W = Ψ W Φ N. (2) Write Ψ W as a sum Ψ W : U Z U ( N, N )[ N] H W : U Z U ( N, N )[ N] Φ W = N i=0 I,j I =i n ij ψ iji with ψ iji : N i N I = N i maps in Sm/S. Then ψ 1 iji (W N i) is in S (p) U/X (N i). (3) Write H W as a sum N H W = n ij H iji i=0 I,j I =i with H iji : N i+1 N I = N i maps in Sm/S. Then H 1 iji (W N i) is in S (p) U (N i+1). If W W N i is in S (p) U/X (N i), then H 1 iji (W ) is in S (p) U/X (N i + 1). Thus Ψ W inuces the map of complexes Ψ W : ZE W m (U ( N, N ))[ N] ZE (p) m (G, U X ) an H W gives a egree 1 map H W : ZE W m (U ( N, N ))[ N] ZE (p) m (U) with HW = ι Z Ψ W ι W Φ N W Furthermore, if W W is in S (p) U/X (N), then H W gives a egree 1 map with H X W : ZE W m (U ( N, N ))[ N] ZE (p) m (U X ) H X W = Ψ W ι X W ΦN W. Since Φ N W is a homology isomorphism in egrees < N an ZE(p) m (U) an ZE m (p) (U X ) are the limits of ZEm W (U ) an ZE W m (U ),

18 18 MARC LEVINE respectively, this shows that ι Z is a quasi-isomorphism, completing the proof. Corollary Let E be in Spt(k) satisfying Zariski excision; if k is finite, we assume in aition that E satisfies axiom A3. Let j : U X be an open immersion in Sm/k with complement i : Z X. Then the sequences of spectra E (p) Z i (X, ) E (p) (X, ) j E (p) (U, ) E (p/p+s) Z (X, ) i E (p/p+s) (X, ) j E (p/p+s) (U, ) exten canonically to istinguishe triangles in SH. Proof. If k is infinite, this follows irectly from the weak homotopy fiber sequences E (p) Z i (X, ) E (p) (X, ) j! E (p) (U X, ) E (p/p+s) Z (X, ) i E (p/p+s) (X, ) j! E (p/p+s) (U X, ) an Theorem For k finite, one uses A3 an the existence of infinite extensions of k of relatively prime power egree to reuce to the case of an infinite fiel The e-looping theorem. Let Sm//S be the subcategory of Sm/S with the same objects an with morphisms Hom Sm /S (Y, Y ) the smooth S-morphisms Y Y. Definition (1) For E Spt(S), efine the presheaf of spectra Ω T E on Sm/S by Ω T E(Y ) := E Y 0 (Y P 1 ). The same formula efines Ω T E in Spt(Sm//S) for E Spt(Sm//S). For E Spt(X), we efine the presheaf of spectra Ω P 1E on Sm/X by Ω T E(Y ) := fib[e(y P 1 ) i E(Y )] (2) For E Spt(S), efine the functor Ω P 1E by Ω P 1E(X) := E(P 1 X + ) = fib(e(x P 1 ) res E(X )). We use the same formula to efine Ω P 1E Spt(Sm//S) for E Spt(Sm//S).

19 THE HOMOTOPY CONIVEAU TOWER 19 Remarks (1) If E is homotopy invariant an satisfies Nisnevic excision, the same hols for Ω T E an Ω P 1E. (2) The commutative iagram E(X P 1 ) E(X P 1 ) res E(X (P 1 \ 0)) res res E(X ) gives us the homotopy fiber sequence Ω T E(X) Ω P 1E(X) fib(e(x (P 1 \ 0)) res E(X )). If Eis homotopy invariant, fib(e(x (P 1 \ 0)) res E(X )) is weakly contractible, hence the natural map Ω T E Ω P 1E is a weak equivalence. Besies the usual uses of localization (e.g., reucing problems to the case of fiels) the localization theorem tells us how to commute the operation E E (p) (X, ) with the T -loops functor E Ω T E. For W Y a close subset of some Y Sm/S, an for E Spt(Sm//S), the spectrum with support (Ω T E) W (Y ) is the iterate homotopy fiber over the iagram E(Y P 1 ) E((Y \ W ) P 1 ) E(Y (P 1 \ 0)) E((Y \ W ) (P 1 \ 0)) Similarly, the spectrum with support E W 0 (Y P 1 ) is the iterate homotopy fiber over the iagram E(Y P 1 ) E(Y P 1 \ W 0) E(Y P 1 \ W 0) E(Y P 1 \ W 0) The evient restriction maps yiel a map of the secon iagram to the first, an hence a canonical map of spectra θ W p (Y ) : E W 0 (Y P 1 ) (Ω T E) W (Y ); if E satisfies Zariski excision, then θ W p (Y ) is a weak equivalence, since (Y \ W ) P 1 Y (P 1 \ 0) = Y P 1 \ W 0.

20 20 MARC LEVINE Definition For E Spt(k), let E (p) //k be the presheaf on Sm//k Y E (p) (Y, ), an let E (P1 )(p) (0) //k be the presheaf on Sm//k Y E (p) Y 0 (Y P1, ). The maps θ W p (Y n ) yiel the map in Spt(Sm//k) The sequence θ E : E (P1 )(p) (0) //k (Ω T E) (p 1) //k. E (p) Y 0 (Y P1, ) E (p) (Y P 1, ) E (p) (Y (P 1 \ 0), ) gives rise to the map in Spt(Sm//k) τ E : E (P1 )(p) (0) //k Ω T (E (p) //k) Theorem Suppose that E Spt(k) satisfies Zariski excision. Then the iagram E (P1 )(p) (0) //k θ E τ E (Ω T E) (p 1) //k Ω T (E (p) //k) efines an isomorphism in HSpt(Sm//k) ξ p : (Ω T E) (p 1) //k Ω T (E (p) //k) Proof. By the localization theorem, τ E is a pointwise weak equivalence. Since the map θ E is a pointwise weak equivalence if E satisfies Zariski excision, the result follows. 4. Functoriality an Chow s moving lemma Fix a fiel k. In this section, we iscuss the extension of the presheaf E (p) //k on Sm//k to a presheaf on Sm/k Functoriality. Take E Spt(k). Recall from the previous section the presheaf E (p) //k, X E (p) (X, ). Let ρ : Sm//k Sm/k be the inclusion. Theorem Suppose that E Spt(k) is homotopy invariant an satisfies Nisnevic excision; if k is finite, assume in aition that E satisfies the axiom A3. Then,

21 THE HOMOTOPY CONIVEAU TOWER 21 (1) For each p 0 there is a presheaf E (p) Spt(k), together with an isomorphism φ p : E (p) ρ E (p) //k in HSpt(Sm//k). (2) There are natural transformations ξ p : E (p) E (p 1), p 0, making the iagram E (p) ρ φ p E (p) //k E (p 1) ρ φp 1 E (p 1) //k commute in HSpt(Sm//k). (3) The presheaf E (P1 )(p) (0) //k an natural transformation E (P1 )(p) (0) //k E (P1 )(p 1) (0) //k extens as in (1) an (2) to a presheaf E (P1 )(p) (0) Spt(k) an natural transformation ξ p : E (P1 )(p) (0) E (P1 )(p 1) (0), an the iagram of Theorem extens as in (2) to a iagram of weak equivalences E (P1 )(p) (0) θ E τ E (Ω T E) (p 1) Ω T (E (p) ) intertwining the transformations ξ p 1, ξ (0) (P 1 )(p) an Ω T (ξ p ). Setting ψ p := τ E θ 1 E, we have isomorphisms ψ p : (Ω T E) (p 1) Ω T (E (p) ), p 0, intertwining the transformations ξ p 1 an Ω T (ξ p ) (here we set (Ω T E) ( 1) := (Ω T E) (0), ξ 1 = ξ 0 ). The same result hols after replacing Ω T with Ω P 1. Aitionally, E (p) is a bifibrant object in Spt Nis (Sm/k) an the operation E (E (p), φ p, ξ p, ψ p ) is natural in E an preserves weak homotopy fiber sequences in Spt Nis (k). Proof. The theorem follows essentially from the main result of [11], with some moifications an extensions.

22 22 MARC LEVINE Let f : Y X be a morphism in Sm/k. We have efine in [11, 7.4] a homotopy coniveau tower on X aapte to f:... E (p) (X, ) f E (p 1) (X, ) f... E (0) (X, ) f The simplicial spectrum E (p) (X, ) f is efine using the support conitions S (p) X (n) f aapte to f: S (p) X (n) f := {W X n W S (p) X (n) an (f i) 1 (W ) S (p) Y (n)}, with E (p) (X, n) f := hocolim W S (p) X (n) f E W (X n ). We have also efine a category L(Sm/k) with objects morphisms f : Y X in Sm/k; the operation (f : Y X) X efines a faithful functor L(Sm/k) Sm/k, making L(Sm/k) op a lax fibere category over Sm/k op. In aition, sening f : Y X to E (p) (X, ) f efines a functor E (p) ( )? on L(Sm/k) op. Sening X to holim π 1 (X) E (p) ( )? gives a lax functor from Sm/k op to spectra, which is then regularize to an honest presheaf by applying a type of homotopy colimit construction aapte from work of Dwyer-Kan (see [11, 7.3]). The bifibrant replacement of this presheaf (for r the Nisnevic-local moel structure on Spt(k) gives the esire presheaf E (p). To make the whole homotopy coniveau tower functorial, replace the presheaf category Spt(k) with Spt(Sm/k N), where N is the sequence category n.... Taking N to be iscrete, the Nisnevic topology on Sm/k inuces a topology on Sm/k N. We procee exactly as above, constructing a presheaf of spectra Ê( ) on Sm/k N, an then take E ( ) to be the functorial fibrant moel for the Nisneviclocal moel structure. This efines the natural transformations ξ p. The same approach, applie to the iagram in Theorem 3.3.4, gives the natural tranformations ψ p such that the whole package has the esire compatibilities. Remark If E Spt S 1(k) is fibrant, then E is homotopy invariant an satisfies Nisnevic excision. It follows from the naturality in E an the fact that E E (p) preserves homotopy cofiber sequences that the operations E (E (p), φ p, ξ p, ψ p ) escens to exact functors, resp. natural transformations, on SH S 1(k), at least if k is an infinite fiel. If k is a finite fiel, we can consier the full subcategory SH S 1(k) fin with objects those E which satisfy axiom A3. It is obvious that SH S 1(k) fin is a triangulate subcategory. In the case of a finite basefiel, we have the functors, resp. natural transformations, as above on SH S 1(k) fin.

23 THE HOMOTOPY CONIVEAU TOWER 23 A useful consequence of Theorem 4.1.1(3) is Corollary Take E Spt(k) satisfying the hypotheses of Theorem Then the canonical map E (p) E inuces an isomorphism on taking pth loop spaces Ω p T E(p) Ω p T E. Proof. The composition ψ 1... ψ p gives the isomorphism Ω p T E Ω p T E(p) ; it follows from the explicit construction of ψ p on Ω T (E (p 1) //k) that the composition is the ientity. Ω p T E Ωp T E(p) Ω p T E Remark To state the next result, we nee to escribe how one extens a presheaf E Spt(k) to Zariski localizations of X Sm/k. Let S = {x 1,..., x n } be a finite set of points in X, an let O be the semi-local ring O X,S. We set E(O) := colim S U X E(U) where U runs over open subschemes of X containing S. This efines E(F ) for F a finitely an separately generate fiel extension of k by choosing a smooth moel X with F = k(x). Corollary Uner the hypotheses of Theorem 4.1.1, for integers p, r 0, there is a presheaf E (p/p+r) Spt(k) whose restriction to Sm//k is isomorphic to E (p/p+r) (?, ) : Sm//k op Spt in HSpt(Sm//k). In aition (1) The functor E (0/1) is birational: The restriction map E (0/1) (X) E (0/1) (k(x)) is a weak equivalence. (2) The functor E (0/1) is rationally invariant: If F F (t) is a pure transcenental extension of fiels (finitely an separably generate over k), then E (0/1) (F ) E (0/1) (F (t)) is a weak equivalence. Proof. The main statement follows from Theorem For (1), fix an irreucible X Sm/k, an let Z X be a proper close subset. We have the localization fiber sequence E (0/1) Z (X, ) E (0/1) (X, ) E (0/1) (X \ Z, ).

24 24 MARC LEVINE with E (0/1) Z (X, ) the cofiber of E (1) Z (X, ) E(0) Z (X, ). Since each close subset W Z n has coimension at least one on X n, the map E (1) Z (X, n) E(0) Z (X, n) is an isomorphism for each n. Thus E (0/1) Z (X, ) = 0 in SH an E (0/1) (X, ) E (0/1) (X \ Z, ) is a weak equivalence. (2) follows by taking limits. For (2), the homotopy property implies that E (0/1) (F, ) E (0/1) (A 1 F, ) is a weak equivalence. Since E (0/1) (A 1 F, ) E(0/1) (F (t), ) is a weak equivalence by (1), the result is prove The purity theorem. Using the functoriality of the E (p), we can exten the e-looping theorem to a version of the Thom isomorphism. Fix a scheme X in Sm/k an an E Spt(k). We may restrict E to Sm/X, giving the presheaf E X : Sm/X op Spt. If we have a close subset Z of X, we have the functor (f : U X) E f 1 (Z) (U), which we enote by EX Z. If f : Y X is a morphism in Sm/k, we have the pushforwar f : Spt(Y ) Spt(X), efine by f F (U X) := F (U X Y ). Clearly, f preserves weak equivalences, hence escens to f : HSpt(Y ) HSpt(X). Lemma Let i : Z X be a coimension close embeing, with X an Z in Sm/k. Suppose that E : Sm/k op Spt is homotopy invariant an satisfies Nisnevic excision, an that the normal bunle N Z/X is trivial. Then a choice of isomorphism φ : N Z/X = Z A etermines a natural isomorphism in HSpt(X), natural in (Z, X, φ). ω ψ : E Z X i (Ω T E Z ), Proof. Using Nisnevic excision, the inclusion A (P 1 ) inuces a natural isomorphism Ω T E Z (Y ) (E Z ) Y 0 (Y A ). for Y Z in Sm/Z. Letting E 0 Z ( A ) enote the presheaf we thus have the isomorphism Y (E Z ) Y 0 (Y A ), Ω T E Z (E Z ) 0 ( A ).

25 THE HOMOTOPY CONIVEAU TOWER 25 in Spt(Z). Let s : Z N Z/X be the zero-section, p : N Z/X Z the projection an enote the presheaf on Sm/Z. Taking a eformation to the normal bunle as in [14] gives a natural isomorphism of EX Z with i E s(z) (N Z/X ) in HSpt. The chosen isomorphism φ : N Z/X = Z A sens s(z) over to Z 0. As the eformation iagram is preserve by pullback with respect to a smooth U X, we actually have an isomorphism in HSpt(X), proving the result. This immeiately yiels Proposition Let E : Sm/k op Spt be a homotopy invariant presheaf satisfying Nisnevic excision. Let i : Z X be a coimension close embeing in Sm/k such that the normal bunle N Z/X is trivial. Then for all p 0 we have isomorphisms in SH: E (p) Z (X, ) = (Ω T E) (p ) (Z, ) E (p/p+1) Z (X, ) = (Ω T E) (p /p +1) (Z, ), where, for n < 0, we set (Ω T E)(n) = (Ω T E)(0) an E (n/n+1) =. The isomorphisms may epen on the choice of trivialization of N Z/X, but are natural in the category of close embeings i with trivialization of N i. We also have Corollary Suppose k is perfect. Let X be in Sm/k, an let E be as in Proposition For each N 0, there is a spectral sequence E 1 p,q(e) := x X (p)π p+q Ω p T E(N p/n p+s) (k(x), ) = π p+q E (N/N+s) (X, ). Proof. This follows from the localization property Corollary an Proposition by the usual limit process. Finally, the birationality an rational invariance of E (0/1) enable us to prove an extene form of the purity isomorphism for the layers of the homotopy coniveau tower. Let X be in Sm/k an let W X be a close subset with coim X W. We let W 0 W be the smooth locus of W an W 0 () W 0 the union of those components of W 0 having coimension exactly on X. Corollary Suppose k is perfect. LetW X be a close subset of X with coim X W, an let U be a ense open subset of W 0 (). Then there is a canonical isomorphism σ : (E (/+1) ) W (X) (Ω T E) (0/1) (U).

26 26 MARC LEVINE in SH. Proof. Let X 0 = X \(W \U). By the localization property for E (/+1), the restriction (E (/+1) ) W (X) (E (/+1) ) U (X 0 ) is a weak equivalence, so we reuce to the case X = X 0, W = U. By consiering the eformation to the normal bunle, we have a canonical isomorphism (E (/+1) ) W (X) = (E (/+1) ) W (N) in SH, where N is the normal bunle of W in X an W is inclue in N by the zero section i 0 : W N. Let N 0 := N \ {i 0 (W )} with projection q : N 0 W. Using Corollary an the localization property for E (/+1) again, the pull-back by q inuces weak equivalences (E (/+1) ) W (N) q (E (/+1) ) N 0 (q N) (Ω T E) (0/1) (W ) q (Ω T E) (0/1) (N 0 ). Using the iagonal section δ : N 0 q N 0 q N as 1, we have a canonical isomorphism φ : q N = N 0 A 1. This in turn gives a canonical trivialization of the normal bunle of i 0 (N 0 ) in q N, hence a canonical isomorphism in SH This completes the construction. (E (/+1) ) N 0 (q N) = (Ω T E) (0/1) (N 0 ). 5. Generalize cycles We use the results of the previous sections to give an interpretation of the layers in the homotopy coniveau tower The semi-local. We recall that n has the vertices v 0,..., v n, where v i is the close subscheme efine by t j = 0 j i. For a scheme X, we let n 0(X) be the intersection of all open subschemes U X n with X v i U for all i. Remark If X is a semi-local scheme with close points x 1,..., x m, then n 0(X) is just the semi-local scheme Spec O X n,s, where S is the close subset {x i v j i = 1,..., m, j = 0,..., n}. In particular n 0(X) is an affine scheme if X is semi-local.

27 THE HOMOTOPY CONIVEAU TOWER 27 For a scheme T, we let 0(T ) enote the cosimplicial in-scheme n n 0(T ); if T is semi-local, then 0(T ) is a cosimplicial semi-local scheme. For F a fiel, we write 0,F for 0(Spec F ) 5.2. Some vanishing theorems. We fix a presheaf E : Sm/k op Spt. For this section, we assume that E is homotopy invariant an satisfies Nisnevic excision; if k is finite, we also suppose that E satisfies the axiom A3. We note that these hypotheses pass to E (p) an E (p/p+r) for all p, r 0. Finally, we assume that k is perfect. Lemma Let F = E (p) : Sm/k op Spt with p > 0. Then for X in Sm/k, F (0/1) (X, ) is weakly contractible. Proof. Noting that F (0/1) (X, ) = F (0/1) (Spec k(x), ) in SH (Corollary 4.1.5), we reuce to the case of a fiel K. In this case, we have F (0/1) (K, ) = E (p) ( 0,K ). Since each n 0,K is semi-local, an hence affine, it follows from the construction of the functor E (p) in [11, 7] that we have the natural weak equivalences of presheaves on Or (i.e. simplicial spectra) [n E (p) ( n 0,K] = [n E (p) ( n K,0, ) fn ], where f n : m 0,K n 0,K is the isjoint union of the inclusions of faces m 0,K n 0,K. Thus, F (0/1) (K, ) is weakly equivalent to the total space of the bisimplicial spectrum (n, m) E (p) (n, m), where E (p) (n, m) is the limit of the spectra with support E W ( n 0,K K m K), as W runs over all close subsets of n 0,K K m K satisfying coim F F (W F F ) p for all faces F m K, F n 0,K. For each m, we have the restriction to a face (say the face t m+1 = 0) δ : E (p) (, m + 1) E (p) (, m), with right inverse given by pull-back by the corresponing coegeneracy map σ : E (p) (, m) E (p) (, m + 1). By the same argument as for the homotopy property for E (p) (X, ) (see [11, Theorem 3.3.5]), one shows that σ δ is homotopic to the ientity, hence δ is a homotopy equivalence.

28 28 MARC LEVINE Thus, the inclusion E (p) (, 0) E (p) (, ) is a weak equivalence. However, if W is an irreucible close subset of n 0,K which intersects all faces in coimension p > 0, then in particular, W contains no vertex of n 0,K. Since n 0,K is semi-local with close points the vertices, this implies that W is empty, that is, E (p) (, 0) is weakly contractible, proving the result. Proposition Let F = E (p) : Sm/k op Spt with p > 0. Then F (q/q+1) is weakly contractible for all 0 q < p. Similarly, (E (p/p+1) ) (q/q+1) is weakly contractible for 0 q < p. Proof. Since the operation F F (q/q+1) is compatible with taking homotopy cofibers, the secon assertion follows from the first. For the first assertion, the case p = 1 is hanle by Lemma 5.2.1; we prove the general case by inuction on p. Note that Theorem 4.1.1(3) gives us the weak equivalence ψ : Ω T F (Ω T E) (p ) in Spt(Sm//k). Thus, by our inuctive assumption, (Ω T F )(q/q+1) is weakly contractible for 0 q < p. By the Gersten spectral sequence (Corollary 4.2.3), this implies that the restriction map F (q/q+1) ( n K) F (q/q+1) ( n 0,K) is a weak equivalence for 0 q < p, for all fiels K finitely generate over k an for all n. Thus, we have the isomorphisms in SH F (q/q+1) (K) = (F (q/q+1) ) (0) (K, ) = (F (q/q+1) ) (0/1) (K, ) for 0 q < p. Applying Lemma to F, (F (q/q+1) ) (0/1) (K, ) is weakly contractible for q > 0, which completes the proof. Proposition Let F = E (p/p+1) : Sm/k op Spt with p > 0. Then F (p+r) is weakly contractible for all r > 0. Proof. F (p+r) (X) is isomorphic in SH to the total space of the simplicial spectrum F (p+r) (X, ). F (p+r) (X, n) in turn is the limit of the spectra with support F W (X n ), where W is a close subset of X n which, among other properties, has coimension p + r > p. By Corollary 4.2.4, it follows that F (p+r) (X, n) is weakly contractible, whence the result The main result. Theorem Let k be a perfect fiel. Let E Spt(k) be a homotopy invariant presheaf satisfying Nisnevic excision; if k is finite, we also assume that E satisfies the axiom A3. Take integers 0 p q. Then

29 THE HOMOTOPY CONIVEAU TOWER 29 (1) Applying the functor (q) to the canonical map E (p) E inuces a weak equivalence (E (p) ) (q) E (q). in Spt(k). (2) Applying the natural transformation (p) i to E (q) inuces a weak equivalence (E (q) ) (p) E (q) in Spt(k). (3) We have a natural isomorphism in HSpt(k) (E (q/q+1) ) (p/p+1) = { 0 for q p, E (p/p+1) for q = p. Proof. For (1), we apply (q) to the tower giving the tower E (p) E (p 1)... E (E (p) ) (q) (E (p 1) ) q)... E (q) with layers (E (r/r+1) ) (q), r = 0,..., p 1, which all vanish by Proposition For (2), we use the same argument, applying the tower of functors (p) (p 1)... i to E (q) an using Proposition For (3), the case p > q follows from Proposition The same argument as for (2), replacing E (q) with E (q/q+1) hanles the case p < q an shows that the map (E (q/q+1) ) (q) E (q/q+1) inuce by applying the natural transformation (q) i to E (q/q+1) is an isomorphism. Since (E (q/q+1) ) (q+1) is weakly contractible, the natural map (E (q/q+1) ) (q) (E (q/q+1) ) (q/q+1) is also an isomorphism, completing the proof. Corollary Let E : Sm/k op Spt be a presheaf satisfying the same hypotheses as in Theorem Let X be in Sm/k. Then E (p/p+1) (X) is naturally isomorphic in SH to the total spectrum of a simplicial spectrum E (p/p+1) s.l. (X, ), with E (p/p+1) s.l. (X, n) = (Ω p T E)(0/1) (k(x)) in SH. x X (p) (n)