The slice filtration and Grothendieck-Witt groups

THE SLICE FIL TRA TION AND GROTHENDIECK-WITT GR OUPS MARC LEVINE Abstract. Let k be a p erfect ﬁeld of characteristic diﬀerent from t wo . W e sho w that the ﬁltration on the Grothendiec k-Witt group GW( k ) induced b y the slice ﬁl tration f or the sphere sp ectrum in the motivic stable homotop y category is the I -adic ﬁltration, where I is the augmenta tion ideal in GW( k ). Contents Int ro duction 1 1. Background and no tation 4 2. The homotopy coniveau to wer 5 3. Connectedness and genera tors for π 0 8 4. The Pon try agin-Tho m collapse map 12 5. ( P n /H , 1) and Σ n s G ∧ n m 15 6. The suspe nsion of a symbol 20 7. Computing the collapse map 25 8. T rans fers and P 1 -susp ension 26 9. Conclusion 28 10. Epilog: Conv ergence questio ns 39 References 40 Introduction Let k be a per fect ﬁeld of c haracter istic diﬀerent fro m tw o. A fundamen tal theorem of Morel [8, 11] states that the endomor phism ring of the motivic sphere sp ectrum S k ∈ S H ( k ) is na turally isomorphic to the Grothendieck-Witt ring of quadratic forms ov er k , GW ( k ). This result follows from Morel’s ca lculation [8, corolla r y 3.43] of the cor resp onding bi-g raded homotopy sheav es of S n ∧ G ∧ q m in the unstable motivic homotopy ca tegory H • ( k ) as the Milnor -Witt sheav es π m + p,p ( S n ∧ G ∧ q m ) ∼ = ( K M W q − p for n = m ≥ 2 , q ≥ 1 , p ≥ 0 , 0 for m < n, p, q ≥ 0 . Date : September 5, 2018. 2000 Mathematics Subje ct Classiﬁc ation. Primary 14C25, 19E15; Secondary 19E08 14F42 , 55P42. Key wor ds and phr ases. Algebraic cycles, M orel-V oevodsky stable homotopy categ ory , slice ﬁltration. Researc h supp orted by the Alexander v on Hum b oldt F oundation. 1 2 MARC LE VINE Ev aluating at k and taking m = n , p = q gives End H • ( k ) ( S m ∧ G ∧ q m ) = K M W 0 ( k ) for m ≥ 2 , q ≥ 1 . Combining this with Mor el’s iso morphism K M W 0 ( k ) ∼ = GW( k ) and stabilizing gives Morel’s theorem End S H ( k ) ( S k ) = GW( k ) . This a lso leads to the computation of the homoto py s hea f π p,p Σ ∞ s G ∧ q m (in the S 1 - stable homotopy catego ry S H S 1 ( k )) as K M W q − p , for all q ≥ 1, p ≥ 0. In another direction, V o evodsky [15] has deﬁned natural tow ers in S H ( k ) and S H S 1 ( k ), which ar e a nalogs of the cla ssical Postniko v tow er in S H ; we call each of these tow ers the T ate Postnikov tower (in S H ( k ) or S H S 1 ( k ), a s the case may b e). Just a s the classica l Postnik ov tow er measures the S n -connectivity of a sp ectrum, the T ate Postniko v tow er measures the S ∗ ,n connectivity of a motivic spectr um. In particular , the tower for S k . . . → f n +1 S k → f n S k → . . . → f 0 S k = S k gives a ﬁltration on the sheaf π 0 , 0 S k by F n T ate π 0 , 0 S k := im( π 0 , 0 f n S k → π 0 , 0 S k ) . W e hav e a similarly deﬁned ﬁltration o n π p,p Σ ∞ s G ∧ q m , whic h determines F n T ate π 0 , 0 S k : by F n T ate π 0 , 0 S k := lim − → q F n + q T ate π q,q Σ ∞ s G ∧ q m ( k ) . Our main result is the computatio n of F n T ate π p,p Σ ∞ s G ∧ q m , and ther eby F n T ate π 0 , 0 S k (on per fect ﬁelds) Theorem 1. L et k b e a p erfe ct ﬁeld of char acteristic 6 = 2 and let F b e a p erfe ct ﬁ eld extension of k . L et n, p ≥ 0 , q ≥ 1 b e inte gers and let N ( a, b ) = max(0 , min( a, b )) . Then via the identiﬁc ation given by Mor el’s isomorphism π p,p Σ ∞ s G ∧ q m ∼ = K M W q − p , we have F n T ate π p,p Σ ∞ s G ∧ q m ( F ) = K M W q − p ( F ) · I ( F ) N ( n − p,n − q ) , wher e I ( F ) ⊂ K M W 0 ( F ) is the augmentation ide al. After stabilizing, this gives F n T ate π p,p Σ q G m S k ( F ) = K M W q − p ( F ) I ( F ) N ( n − p,n − q ) , n, p, q ∈ Z , in p articular, F n T ate π 0 , 0 S k ( F ) = I ( F ) max( n, 0) . See theorem 9.14, cor ollary 9.15 and corolla ry 9.1 6 for details. R emark 1 . In ca se k is a ﬁeld of characteristic 0, we have a ﬁner result, na mely the ident ities stated in theor em 1 extend to identities on the corresp onding sheav es , for example F n T ate π p,p Σ ∞ s G ∧ q m = K M W q − p · I N ( n − p,n − q ) . Of course, one can mor e ge ne r ally consider the ﬁltration F ∗ T ate π a,b E o n the homo- topy shea ves π a,b E induced by the T ate Postnik ov tower f or an arbitrar y T -sp ectrum E ∈ S H ( k ). In g eneral, we ca nnot say anything ab out this ﬁltra tio n, but assum- ing a ce r tain connectedness condition, we can compute the ﬁltration on the ﬁrst non-v anishing homotopy sheav es, ev aluated on p erfect ﬁelds. THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 3 Theorem 2. L et k b e a p erfe ct ﬁeld of char acteristic 6 = 2 and let F b e a p erfe ct ﬁeld extension of k . T ake E ∈ S H ( k ) and supp ose that π a + b,b E = 0 for a < 0 , b ∈ Z . Then for n > p , F n T ate π p,p E ( F ) = [ π n,n E · K M W n − p ] b T r ( F ) . F or n ≤ p , we have t he identity of she aves F n T ate π p,p E = π p,p E . T o e xplain the nota tion: The ca nonical action o f π ∗ , ∗ S k on π ∗ , ∗ E , gives, for each ﬁnitely generated ﬁeld extensio n L of k , a r ight K M W −∗ ( L )-mo dule structure on π ∗ , ∗ E ( L ), giving us the subgro up π n,n E ( L ) · K M W n − p ( L ) of π p,p E ( L ). This extends to arbitrar y ﬁeld extensions of k by tak ing the evident co limit. Also, for each closed po int w ∈ A n F , w e hav e a ca nonically deﬁned tr ansfer map T r F ( w ) ∗ : π a,b E ( F ( w )) → π a,b E ( F ) (see § 8 for details). [ π n,n E · K M W n − p ] b T r ( F ) is the subgroup of π p,p E ( F ) generated by the subgroups T r F ( w ) ∗ ( π n,n E ( F ( w )) · K M W n − p ( F ( w ))), a s w runs ov er closed p oints of A n F . See theorem 9.1 2 for details. Theorem 1 is an easy consequence of theorem 2; one uses Mo rel’s unstable com- putations of the maps S a,b ∧ Sp ec F + → S m,n to r educe theo rem 1 to its T -stable version and then one uses the explicit presentation o f K M W ∗ to compute [ K M W q − n · K M W n − p ]] b T r ( F ) = K M W q − p ( F ) I N ( n − p,n − q ) ( F ) . Morel’s results on strictly A 1 -inv ar iant sheav es allow us to go from the statement on functions ﬁelds to the one for the sheav es. The restrictio n to per fect ﬁelds arises from a sepa rability assumption needed to compute the action o f transfer s on our selected gener a tors for F n T ate π p,p E . W e av oid characteristic t wo so a s to hav e a description of the homotopy sheav es o f the sphere sp ectrum in terms of Milnor- Witt K -theory . The pap er is org anized as follows. After se tting up our notation and going ov e r some ba ckground ma terial on motivic homo to py theory in sectio n 1, we recall some basic fa cts ab out the T ate Postniko v tow er in section 2. In s ection 3 we prov e some connectedness results for the terms f n E , s n E in the T ate P ostniko v tow er for an S 1 - sp ectrum E and give a des cription of gener ators for the subgroup F n T ate π 0 E ( F ), all under a certain connectedness assumption on E . The generators are then factored int o a pro duct o f t w o terms, one dep ending on E , the other only on the choice of a closed p oint of ∆ n F \ ∂ ∆ n F . W e analyze the se cond ter m in sections 4-8, our main result b eing a descr iption of this term as the n th susp ension of a “ symbol map” asso ciated to units u 1 , . . . , u n ∈ F × . This is the ma in c omputation achiev e d in this pap er. It is then rela tively simple to feed this result in to our description of the generator s for F n T ate π 0 E ( F ) to prov e theorems 1 and 2 in se c tion 9; we conclude in section 10 with some rema rks o n the conv ergence of the T ate Postniko v tow e r . I thank the r eferee for making s e veral helpful suggestio ns and for po inting out a num b er o f erro rs, including a n inco rrect formulation of theorem 2, in an earlier version of this pap er. Finally , I wis h to thank the editors for giving me the opp or- tunit y o f contributing to this v olume. As a small token of my gra titude to Eck art for all of his aid and supp ort over man y years, I dedicate this article to his memory . 4 MARC LE VINE 1. Back ground and n ot a tion Unless we sp ecify other wise, k will be a ﬁxed p erfect ba se ﬁeld, without restric- tion on the c haracter istic. F o r details on the following constructions, we refer the reader to [3, 4, 5, 8, 9, 1 1, 12]. W e write [ n ] := { 0 , . . . , n } (including [ − 1] = ∅ ) and let ∆ be the categ ory with ob jects [ n ], n = 0 , 1 , . . . , and morphisms [ n ] → [ m ] the or der-prese rving maps of sets. Giv e n a category C , the category o f simplicial ob jects in C is as usua l the category of functors ∆ op → C . Sp c will denote the category of simplicial sets, Sp c • the category of p ointed simplicial sets, H := Sp c [ W E − 1 ] the classical unstable homo to py categ ory and H • := Sp c • [ W E − 1 ] the p o inted version. W e denote the susp ension op er ator − ∧ S 1 by Σ s . Spt is the categ ory of susp ension sp ectra and S H := Spt [ W E − 1 ] the classical stable homotopy category . The motivic v ersions are as follows: Sm /k is the categor y of smo oth ﬁnite t yp e k -schemes. Sp c ( k ) is the ca tegory of Sp c -v a lued presheav es on Sm /k , Spc • ( k ) the Sp c • -v alued presheaves, a nd Spt S 1 ( k ) the Spt -v alued pres heav es . These a ll come with “ motivic” mo del structures (see for example [5]); we denote the co rresp onding homotopy categories by H ( k ), H • ( k ) and S H S 1 ( k ), resp ectively . Sending X ∈ Sm /k to the sheaf of sets on Sm /k r epresented b y X (also denoted X ) gives an embedding of Sm / k to Sp c ( k ); we ha ve the similarly deﬁned em bedding of the category o f smo o th p ointed s chemes over k in to Sp c • ( k ). All these categ ories ar e equipp e d with an internal Ho m, denoted H om . Let G m be the p ointed k -sc heme ( A 1 \ 0 , 1 ). In H • ( k ) we hav e the ob jects S a + b,b := Σ a s G ∧ b m , fo r b ≥ 1, S n, 0 := S n = Σ n s Spec k + . If X is a scheme with a k -p oint x , we write ( X , x ) for the cor resp onding o b ject in Sp c • ( k ) or H • ( k ). F o r a coﬁbration Y → X in Sp c ( k ), we us ua lly give the quo tien t X / Y the canonical base-p oint Y / Y , but on occa sion, we will give X / Y a base-p oint coming from a po int x ∈ X ( k ); w e write this as ( X / Y , x ). W e let T := A 1 / ( A 1 \ { 0 } ) and let Spt T ( k ) denote the categor y of T -sp ectra , i.e., sp ectra in Sp c • ( k ) with respec t to the T -susp ensio n functor Σ T := − ∧ T . Spt T ( k ) ha s a motivic mode l structure (see [5 ]) and S H ( k ) is the homotopy cat- egory . W e can also form the category of sp ectra in Spt S 1 ( k ) with resp ect to Σ T ; with an appr opriate mo del str ucture the resulting homotopy ca tegory is equiv alent to S H ( k ). W e will ignore the s ubtleties of this dis tinction and simply identify the t wo homotopy categories. Both S H S 1 ( k ) and S H ( k ) a re triangula ted ca tegories with sus p ensio n functor Σ s . W e hav e the triangle of inﬁn ite susp ension functors Σ ∞ and their rig ht a djoints Ω ∞ H • ( k ) Σ ∞ s / / Σ ∞ T % % J J J J J J J J J S H S 1 ( k ) Σ ∞ T   S H ( k ) H • ( k ) S H S 1 ( k ) Ω ∞ s o o S H ( k ) Ω ∞ T O O Ω ∞ T e e J J J J J J J J J bo th comm utative up to natura l iso morphism. Thes e are a ll left, resp. r ight der ived versions of Quille n adjoint pair s of functors on the under lying mo del categ ories. W e note that the susp ension functor Σ G m is in vertible o n S H ( k ). THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 5 F or X ∈ H • ( k ), w e hav e the bi-g raded homotopy sheaf π a,b X , deﬁned for b ≥ 0, a − b ≥ 0, as the Nisnevic h sheaf asso ciated to the pr esheaf on Sm /k U 7→ Hom H • ( k ) (Σ a − b s Σ b G m U + , X ) . These extend in the us ua l wa y to bi-g r aded homotopy s heaves π a,b E for E ∈ S H S 1 ( k ), b ≥ 0, a ∈ Z , and π a,b E fo r E ∈ S H ( k ), a, b ∈ Z , b y ta king the Nis- nevich sheaf ass o ciated to U 7→ Hom S H S 1 ( k ) (Σ a − b s Σ b G m Σ ∞ s U + , E ) o r U 7→ Hom S H ( k ) (Σ a − b s Σ b G m Σ ∞ T U + , E ) , as the case may b e. W e write π n for π n, 0 ; for e.g. E ∈ Spt S 1 ( k ) ﬁbr ant, π n E is the Nisnevich shea f asso ciated to the presheaf U 7→ π n ( E ( U )). F or F a ﬁnitely generated ﬁeld extension o f k , we may view Spec F as the g eneric po int of some X ∈ Sm /k . Thus, for a Nisnevich sheaf S on Sm /k , we may deﬁne S ( F ) a s the stalk o f S a t Spec F ∈ X . F or an arbitra ry ﬁeld extension F o f k (not necessarily ﬁnitely generated over k ), we deﬁne S ( F ) as the colimit o ver S ( F α ), as F α runs ov er subﬁelds o f F c ontaining k and ﬁnitely ge ne r ated ov er k . 2. The homotopy conivea u to wer Our computations rely hea vily on our model for the T ate Postnik ov to wer in S H S 1 ( k ), which w e br ieﬂy r ecall (for details , we refer the reader to [6 ]). W e star t by reca lling the T ate Postnik ov tower in S H S 1 ( k ) and introducing some nota tio n. Fix a p erfect base- ﬁeld k . Let Σ T : S H S 1 ( k ) → S H S 1 ( k ) be the T -susp ension functor . F or n ≥ 0, w e let Σ n T S H S 1 ( k ) be the lo ca lizing subcat- egory of S H S 1 ( k ) generated by inﬁnite susp ension sp e ctra of the for m Σ n T Σ ∞ s X + , with X ∈ Sm /k . W e no te that Σ 0 T S H S 1 ( k ) = S H S 1 ( k ). The inclusion functor i n : Σ n T S H S 1 ( k ) → S H S 1 ( k ) a dmits, by results of Neeman [13], a r ight a djoint r n ; deﬁne the functor f n : S H S 1 ( k ) → S H S 1 ( k ) b y f n := i n ◦ r n . The unit for the adjunction gives us the natur al morphism ρ n : f n E → E for E ∈ S H S 1 ( k ); similarly , the inclusion Σ m T S H S 1 ( k ) ⊂ Σ n T S H S 1 ( k ) for n < m gives the natura l transformation f m E → f n E , forming the T ate Postnikov tower . . . → f n +1 E → f n E → . . . → f 0 E = E . W e complete f n +1 E → f n E to a distinguished triangle f n +1 E → f n E → s n E → f n +1 E [1]; this turns out to b e functoria l in E . The ob ject s n E is the n th slic e of E . There is an analogous constr uction in S H ( k ): F or n ∈ Z , let Σ n T S H ef f ( k ) ⊂ S H ( k ) be the lo ca lizing category gener ated b y the T -susp ension sp ectra Σ n T Σ ∞ T X + , for X ∈ Sm /k . As ab ov e, the inclusion i n : Σ n T S H ef f ( k ) → S H ( k ) admits a left adjoint r n , giving us the truncatio n functor f n and the Postnik ov tower . . . → f n +1 E → f n E → . . . → E . Note that this tow er is in genera l inﬁnite in b oth directions. W e deﬁne the lay er s n E as a bove. By [6, theorem 7.4.1], the 0-space functor Ω ∞ T sends Σ n T S H ef f ( k ) to Σ n T S H S 1 ( k ). This fa ct, to g ether with the universal pro p er ties of the tr uncation functors f n in 6 MARC LE VINE S H S 1 ( k ) and S H ( k ), plus the fact that Ω ∞ T is a r ight adjoint, gives the cano nic a l isomorphism for n ≥ 0 (2.1) f n Ω ∞ T E ∼ = Ω ∞ T f n E . F urthermor e, for E ∈ S H S 1 ( k ), we hav e (by [6, theor e m 7.4.2 ]) the cano nical isomorphism (2.2) Ω G m f n E = f n − 1 Ω G m E . As Ω G m : S H ( k ) → S H ( k ) is a n auto-equiv alence, and res tricts to an equiv alence Ω G m : Σ n T S H ef f ( k ) → Σ n − 1 T S H ef f ( k ) , the analog ous identit y in S H ( k ) ho lds a s w ell. Deﬁnition 2.1. F or a ∈ Z , b ≥ 0, E ∈ S H S 1 ( k ), deﬁne the ﬁltr ation F n T ate π a,b E , n ≥ 0 , of π a,b E b y F n T ate π a,b E := im ( π a,b f n E → π a,b E ) . Similarly , for E ∈ S H ( k ), a , b, n ∈ Z , deﬁne F n T ate π a,b E := im( π a,b f n E → π a,b E ) . The main ob ject of this paper is to unders tand F n T ate π 0 E for suitable E . F or later use, we note the following Lemma 2. 2 . 1. F or E ∈ S H S 1 ( k ) , n, p, a, b ∈ Z with n, p, b, n − p , b − p ≥ 0 , t he adjunction isomorphism π a,b E ∼ = π a − p.b − p Ω p G m E induc es an isomorph ism F n T ate π a,b E ∼ = F n − p T ate π a − p,b − p Ω p G m E . Similarly, for E ∈ S H ( k ) , n, p, a, b ∈ Z , the adjunction isomorphism π a,b E ∼ = π a − p.b − p Ω p G m E induc es an isomorph ism F n T ate π a,b E ∼ = F n − p T ate π a − p,b − p Ω p G m E . 2. F or E ∈ S H ( k ) , a, b, n ∈ Z , with b, n ≥ 0 , we have a c anonic al isomorphism ϕ E ,a,b,n : π a,b f n E → π a,b Ω ∞ T f n E , inducing an isomorphism F n T ate π a,b E ∼ = F n T ate π a,b Ω ∞ T E . Pr o of. (1) By (2.2), adjunction induces isomorphisms F n T ate π a,b E := im( π a,b f n E → π a,b E ) ∼ = im( π a − p,b − p Ω p G m f n E → π a − p,b − p Ω p G m E ) = im( π a − p,b − p f n − p Ω p G m E → π a − p,b − p Ω p G m E ) = F n − p T ate π a − p,b − p Ω p G m E . The pro of for E ∈ S H ( k ) is the same. F or (2), the iso morphism ϕ E ,a,b,n arises from (2 .1) a nd the a djunction is o mor- phism Hom S H S 1 ( k ) (Σ ∞ s Σ a − b s Σ b G m U + , f n Ω ∞ T E ) ∼ = Hom S H S 1 ( k ) (Σ ∞ s Σ a − b s Σ b G m U + , Ω ∞ T f n E ) ∼ = Hom S H ( k ) (Σ ∞ T Σ a − b s Σ b G m U + , E ) .  THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 7 W e now turn to a discussion of o ur model for f n E ( X ), X ∈ Sm /k . W e start with the cosimplicial scheme n 7→ ∆ n , with ∆ n the algebr aic n -simplex Spec k [ t 0 , . . . , t n ] / P i t i − 1. The cosimplicial structure is given b y sending a map g : [ n ] → [ m ] to the map g : ∆ n → ∆ m determined by g ∗ ( t i ) = ( P j,g ( j )= i t j if g − 1 ( i ) 6 = ∅ 0 else. A fac e of ∆ m is a clo sed subscheme F deﬁned by equa tions t i 1 = . . . = t i r = 0; we let ∂ ∆ n ⊂ ∆ n be the close d subsch eme deﬁned by Q n i =0 t i = 0 , i.e., ∂ ∆ n is the union of all the prop er faces . T ake X ∈ Sm / k . W e let S ( q ) X ( m ) denote the set of closed subse ts W ⊂ X × ∆ m such that co dim X × F W ∩ X × F ≥ q for all faces F ⊂ ∆ m (including F = ∆ m ). W e make S ( q ) X ( m ) int o a par tially ordered set via inclusions of clo sed subsets. Sending m to S ( q ) X ( m ) and g : [ n ] → [ m ] to g − 1 : S ( q ) X ( m ) → S ( q ) X ( n ) gives us the simplicial po set S ( q ) X . Now take E ∈ Spt S 1 ( k ). F or X ∈ Sm / k and c losed subset W ⊂ X , we have the sp ectrum with s uppo rts E W ( X ) deﬁned as the homo topy ﬁb er of the restrictio n map E ( X ) → E ( X \ W ). This construction is functorial in the pa ir ( X , W ), w he r e we deﬁne a map f : ( Y , T ) → ( X , W ) a s a morphism f : Y → X in Sm /k with f − 1 ( W ) ⊂ T . Deﬁne E ( q ) ( X, m ) := ho c o lim W ∈S ( q ) X ( m ) E W ( X × ∆ m ) . The fact that m 7→ S ( q ) X ( m ) is a simplicial p oset, and ( Y , T ) 7→ E T ( Y ) is a functor from the category o f pairs to sp ectra shows that m 7→ E ( q ) ( X, m ) deﬁnes a simplicial sp ectrum. W e deno te the a sso ciated total sp ectrum by E ( q ) ( X ). F or q ≥ q ′ , the inclus ions S ( q ) X ( m ) ⊂ S ( q ′ ) X ( m ) induces a map of simplicia l pos ets S ( q ) X ⊂ S ( q ′ ) X and thus a mor phism of sp ectra i q ′ ,q : E ( q ) ( X ) → E ( q ′ ) ( X ). W e ha ve as well the natur a l map ǫ X : E ( X ) → T o t( E ( X × ∆ ∗ )) = E (0) ( X ) , which is a weak equiv a lence if E is homotopy inv ariant. T o gether, this forms the augmente d homotopy c onive au tower tow er E ( ∗ ) ( X ) := . . . → E ( q +1) ( X ) i q − → E ( q ) ( X ) i q − 1 − − − → . . . E (1) ( X ) i 0 − → E (0) ( X ) ǫ X ← − − E ( X ) with i q := i q,q +1 . Th us , for homotopy inv ariant E , we hav e the homo topy coniveau tow er in S H E ( ∗ ) ( X ) := . . . → E ( q +1) ( X ) i q − → E ( q ) ( X ) i q − 1 − − − → . . . E (1) ( X ) i 0 − → E (0) ( X ) ∼ = E ( X ) . Letting Sm / /k denote the subcateg ory of Sm /k with the same ob jects and with morphisms the smo o th morphisms, it is not hard to see that sending X to E ( ∗ ) ( X ) deﬁnes a functor from Sm / /k op to augmented towers of sp ectra. On the other ha nd, for E ∈ Spt S 1 ( k ), w e hav e the (augmented) T ate Postnik ov tow er f ∗ E := . . . → f q +1 E → f q E → . . . → f 0 E ∼ = E in S H S 1 ( k ), which we ma y ev a luate at X ∈ Sm /k , giving the to wer f ∗ E ( X ) in S H , aug men ted ov er E ( X ). 8 MARC LE VINE As a direct co nsequence of our main result (theorem 7.1 .1) from [6] we hav e Theorem 2 .3. L et E b e a qu asi-ﬁbr ant obje ct in Spt S 1 ( k ) for the mo del structur e describ e d in [4] , and take X ∈ Sm /k . Then ther e is an isomorphism of augmente d towers in S H ( f ∗ E )( X ) ∼ = E ( ∗ ) ( X ) over the identity on E ( X ) , whic h is natur al with r esp e ct to smo oth morphisms in Sm /k . In particular , we may use the ex plicit mo del E ( q ) ( X ) to understa nd ( f q E )( X ). R emark 2.4 . F or X, Y ∈ Sm /k with given k -p oints x ∈ X ( k ), y ∈ Y ( k ), we hav e a natural isomorphism in S H S 1 ( k ) Σ ∞ s ( X ∧ Y ) ⊕ Σ ∞ s ( X ∨ Y ) ∼ = Σ ∞ s ( X × Y ) i.e. Σ ∞ s ( X ∧ Y ) is a ca nonically deﬁned summand of Σ ∞ s ( X × Y ). In par ticular for E a quasi-ﬁbrant ob ject of Spt S 1 ( k ), we ha ve a natural isomo rphism in S H H om (Σ ∞ s ( X ∧ Y ) , E ) ∼ = hoﬁb ( E ( X × Y ) → hoﬁb( E ( X ) ⊕ E ( Y ) → E ( k ))) where the maps a re induced by the evident res triction maps. In particula r, we may deﬁne E ( X ∧ Y ) via the ab ove isomorphism, and our comparison results for T a te Postnik ov tow er and homotopy co niveau tow e r extend to v alues at smash pro ducts of smo oth p ointed schemes ov er k . 3. Connectedness and genera tors for π 0 As in sectio n 2, o ur base - ﬁeld k is p erfect. W e ﬁx a qua si-ﬁbrant S 1 -sp ectrum E ∈ Spt S 1 ( k ). Lemma 3.1. L et F b e a ﬁnitely gener ate d ﬁeld extension of k , x ∈ A n F a close d p oint. Then for every m > 0 , the map i 0 ∗ : E ( x, 0) ( A n × A m F ) → E ( x × F A m F ) ( A n × A m F ) induc e d by the map of p airs id A n × A m : ( A n × A m F , x × A m F ) → ( A n × A m F , ( x, 0)) is the zer o-map in S H . In p articular, t he induc e d map on homotopy gr oups is the zer o map. Pr o of. W e use the Mor el-V o evo dsky pur ity isomo rphisms in H • ( k ) [12, Theorem 3.2.23], with the isomorphisms deﬁned via a ﬁx e d c hoice of gener ators for the max- imal ideal m x ⊂ O A n F ,x and m 0 ⊂ O A m , 0 A n F × A m / ( A n F × A m \ { ( x, 0) } ) ∼ = Σ n + m T ( x, 0 ) + ∼ = Σ n T x × A m / ( x × A m \ { ( x, 0 ) } ) A n F × A m / ( A n F × A m \ x × A m ) ∼ = Σ n T x × A m + . Via these isomor phisms, the quotien t map q : A n F × A m / ( A n F × A m \ x × A m ) → A n F × A m / ( A n F × A m \ { ( x, 0 } ) is isomorphic to the n th T -susp ension of the quotient map q ′ : x × A m + → x × A m / ( x × A m \ { ( x, 0 ) } ) THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 9 As i 0 ∗ is the map induced by applying H om ( − , E ) to Σ ∞ s q , we need only show that q ′ factors thro ug h the ma p x × A m + → ∗ (in H • ( k )). This follows from the commutativ e diagram x × 1 + i ∼ / /   x × A m + q ′   ∗ / / x × A m / ( x × A m \ { ( x, 0 ) } ) , where 1 = (1 , . . . , 1) ∈ A m , since i is an isomor phism in H • ( k ) by homoto py inv ar ia nce.  W e have the re-indexed ho motopy sheav es Π n,m ( E ) := π n + m,m ( E ). W e hav e as well the sheaf π n E := π n, 0 E ; w e call E m -connected if π n ( E ) = 0 for all n ≤ m . Since E ( n ) ( X ) = T ot[ m 7→ E ( n ) ( X, m )], we hav e the strongly convergen t sp ectra l sequence (3.1) E 1 p,q ( X ) = π q E ( n ) ( X, p ) = ⇒ π p + q E ( n ) ( X ) , Now take X = Spe c F , F a ﬁnitely gener ated ﬁeld o ver k . F or dimensional reasons , we hav e S ( n ) F ( p ) = ∅ for p < n , and we therefore hav e an edge homomorphis m ǫ − n : π q − n E ( n ) ( X, n ) → π q E ( n ) ( X ) . F urthermor e, S ( n ) F ( n ) is the set of closed points w ∈ ∆ n F \ ∂ ∆ n F , s o ǫ − n can be written as ǫ − n : ⊕ w ∈ (∆ n F \ ∂ ∆ n F ) ( n ) π q − n E w (∆ n F ) → π q E ( n ) ( F ); here Y ( a ) denotes the set of co dimension a points on a scheme Y . Via the weak equiv alence E ( n ) ( F ) ∼ = f n E ( F ), we hav e the cano nical map ǫ − n : ⊕ w ∈ (∆ n F \ ∂ ∆ n F ) ( n ) π q − n E w (∆ n F ) → π q f n E ( F ) . Similarly , comp os ing with f n E → s n E , w e hav e the canonical map ǫ − n : ⊕ w ∈ (∆ n F \ ∂ ∆ n F ) ( n ) π q − n E w (∆ n F ) → π q s n E ( F ) . Prop ositi o n 3.2. L et E ∈ Spt S 1 ( k ) b e quasi-ﬁbr ant. Supp ose Π a, ∗ E ( F ) = 0 for al l a < 0 and for al l ﬁnitely gener ate d ﬁeld ext ensions F of k . Then for n ≥ 0 : 1. Π a, ∗ f n E and Π a, ∗ s n E ar e zer o for al l a < 0 . In p articular, f n E and s n E ar e − 1 -c onne cte d. 2. F or e ach ﬁn itely gener ate d ﬁeld F over k , the e dge homomorphisms ǫ − n : ⊕ w ∈ (∆ n F \ ∂ ∆ n F ) ( n ) π − n E w (∆ n F ) → π 0 ( f n E )( F ) ǫ − n : ⊕ w ∈ (∆ n F \ ∂ ∆ n F ) ( n ) π − n E w (∆ n F ) → π 0 ( s n E )( F ) ar e surje ctions. Pr o of. Using the dis tinguished triang le f n +1 E → f n E → s n E → Σ s f n +1 E we see that it suﬃces to prov e the statements fo r f n E . 10 MARC LE VINE Using the isomorphis m (2.2), we see that for (1), it suﬃces to show that f n E is − 1-connec ted. By a theorem of Mor el [11, lemma 3.3 .6], it suﬃces to s how that f n E ( F ) is − 1-connected for all ﬁnitely g enerated ﬁeld extensions F of k . W e ﬁrst show that, for each p ≥ n , a. π q E ( n ) ( F, p ) = 0 for q < − p b. The natural map ⊕ W ∈S ( n ) F ( p ) ,w ∈ W ∩ (∆ p F ) ( p ) π − p E w (∆ p F ) → π − p E ( n ) ( F, p ) is surjective. F or (a), le t W ⊂ ∆ p F be a closed subset. W e hav e the Gersten sp ectr al sequence E a,b 1 = ⊕ w ∈ W ∩ (∆ p F ) ( a ) π − a − b E w (Spec O ∆ p F ,w ) = ⇒ π − a − b E W (∆ p F ) . Since E is quasi- ﬁbrant, and ∆ p F is s mo oth over k , we hav e an isomorphism (via Morel-V oevo dsky purity [12, Theorem 3.2.2 3 ]) π m ( E w (Spec O ∆ p F ,w )) ∼ = π m ( E ( w + ∧ S 2 a,a )) , where a = co dim ∆ p F w . But π m ( E ( w + ∧ S 2 a,a )) = ( π m +2 a,a E )( F ( w )) which is zer o for m + a < 0. Since 0 ≤ a ≤ p , w e see that, for m < − p , π m E W (∆ p F ) = 0 . As E ( n ) ( F, p ) is a colimit ov er E W (∆ p F ) with W ∈ S ( n ) F ( p ), it follows that π m E ( n ) ( F, p ) = 0 for m < − p , proving (a). The same computation shows that π − p ( E w (Spec O ∆ p F ,w )) = 0 if co dim ∆ p F w < p , so (b) follows from the Gersten spe c tral sequence. Using the strongly conv ergent sp ectra l sequence (3 .1) , w e see that (a) implies that π q E ( n ) ( F ) = 0 for q < 0. Next, we show that c. π − p E ( n ) ( F, p ) = 0 for p > n. F or this, it suﬃces by (b) to show that for w ∈ W ∩ (∆ p F ) ( p ) with W ∈ S ( n ) F ( p ) and with p > n , the map (3.2) π − p E w (∆ p F ) → π − p E ( n ) ( F, p ) is the zero map. T o see this, no te that W do es not intersect any fa c e T of ∆ p F having dim F T < n . Thus, ther e is a linear W ′ ∼ = A p − n F ′ ⊂ ∆ p F containing w (for F ′ some extension ﬁeld of F co nt ained in F ( w )) with W ′ ∈ S ( n ) F ( p ): for a suitable degeneracy map σ : ∆ p → ∆ n one takes W ′ = σ − 1 ( σ ( w )). By lemma 3.1, the map E w (∆ p F ) → E W ′ (∆ p F ) is the ze ro ma p in S H ; pa s sing to the limit ov er all W ′′ ∈ S ( n ) F ( p ), we see that (3 .2) is the zero ma p, as claimed. In the sp ectr al sequence (3.1 ), we have E 1 p, − p = 0 for p > n ; we als o hav e E 1 p, − p = 0 for p < n since S ( n ) F ( p ) = ∅ if p < n for dimensiona l rea s ons. Th us, the only term co ntributing to π 0 E ( n ) ( F ) is E 1 n, − n . As the sp ectral s e quence is THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 11 strongly conv ergent, the edg e homomorphism in the sp ectral s e quence (3.1) induces a surjection ⊕ w ∈ S ( n ) F ( n ) π − n E w (∆ n F ) → π 0 E ( n ) ( F ) . Combining this with theo rem 2.3 gives us the sur jection ⊕ w ∈ S ( n ) F ( n ) π − n E w (∆ n F ) → π 0 ( f n E ( F )) . Similarly , the v anis hing π p E ( n ) ( F ) = 0 for p < 0 shows that f n E ( F ) is -1 connected.  W e th us hav e generators ⊕ w ∈ (∆ n F \ ∂ ∆ n F ) ( n ) π − n E w (∆ n F ) for π 0 f n E ( F ), and hence for our main ob ject of study , F n T ate π 0 E ( F ). W e examine the comp osition (3.3) π − n E w (∆ n F ) ǫ − n − − → π 0 f n E ( F ) ρ n − → π 0 E ( F ) more closely . Fix a closed p oint w in ∆ n F \ ∂ ∆ n F . W e have the quotient ma p c w : ∆ n F /∂ ∆ n F → ∆ n F / (∆ n F \ w ) and the canonical identiﬁcation E w (∆ n F ) = H om (Σ ∞ s ∆ n F / (∆ n F \ w ) , E ) . Thu s, given a n elemen t τ ∈ π − n ( E w (∆ n F )), we have the corr esp onding morphism τ : Σ ∞ s ∆ n F / (∆ n F \ w ) → Σ n s E and w e may comp os e with c w to give the map τ ◦ Σ ∞ s c w : Σ ∞ s ∆ n F /∂ ∆ n F → Σ n s E . As ea ch of the faces of ∆ n F are a ﬃne spaces o ver F , we hav e a canonical iso mo r- phism σ F : Σ n s Spec F + → ∆ n F /∂ ∆ n F in H • ( k ) (see the b eginning o f § 4 for details), giving us the elemen t π ( τ ) := τ ◦ Σ ∞ s ( c w ◦ σ F ) ∈ π n (Σ n s E ( F )) = π 0 ( E ( F )) . The following res ult is a direct conse q uence o f the deﬁnitions: Lemma 3.3 . F or τ ∈ π − n ( E w (∆ n F )) , π ( τ ) = ρ n ( ǫ − n ( τ )) . On the other hand, we hav e the Mor el-V o evo dsky purity isomorphism ( lo c. cit. ) (3.4) M V w : ∆ n F / (∆ n F \ w ) → w + ∧ S 2 n,n . The deﬁnition of M V w requires s ome a dditio nal choices; we complete our deﬁnition of M V w in § 5, where it is written as M V w = (id w + ∧ α ) ◦ mv w (see deﬁnition 4.3 and (5.3)). In any ca se, via M V w , we may factor π ( τ ) as π ( τ ) := τ ◦ Σ ∞ s ( c w ◦ σ F ) = ( τ ◦ Σ ∞ s M V − 1 w ) ◦ Σ ∞ s ( M V w ◦ c w ◦ σ F ) The term τ ◦ Σ ∞ s M V − 1 w is the morphism τ ◦ Σ ∞ s M V − 1 w : Σ ∞ s w + ∧ S 2 n,n → Σ n s E 12 MARC LE VINE which we may interpret as an element of π − n (Ω n T E ( w )), while the mo r phism Σ ∞ s ( M V w ◦ c w ◦ σ F ) is the inﬁnite sus pens ion of the map (3.5) Q F ( w ) := M V w ◦ c w ◦ σ F : Σ n s Spec F + → w + ∧ S 2 n,n . Conv er sely , giv en any element ξ ∈ π − n (Ω n T E ( w )), which we write a s a morphism ξ : w + ∧ S 2 n,n → Σ n s E we re cov er an element τ ∈ π − n ( E w (∆ n F )) as τ := ξ ◦ Σ ∞ s M V w , and th us the ele ment ξ ◦ Σ ∞ s Q F ( w ) ∈ π n (Σ n s E ( F )) = π 0 E ( F ) is in F n T ate π 0 E ( F ). Putting this all tog ether, we hav e Prop ositi o n 3.4. L et F b e a ﬁnitely gener ate d ﬁeld extension of k and let E ∈ Spt S 1 ( k ) b e quasi-ﬁbr ant. 1. L et w b e a close d p oint of ∆ n F \ ∂ ∆ n F , and take ξ w ∈ π − n (Ω n T E ( w )) . Then ξ w ◦ Σ ∞ s Q F ( w ) is in F n T ate π 0 E ( F ) . 2. Supp ose that Π a, ∗ E = 0 for al l a < 0 . Then F n T ate π 0 E ( F ) is gener ate d by elements of the form ξ w ◦ Σ ∞ s Q F ( w ) , ξ w ∈ π − n (Ω n T E ( w )) , as w runs over cl ose d p oints of ∆ n F \ ∂ ∆ n F . R emark 3 .5 . The prop ositio n extends without change to arbitra ry ﬁeld ex tensions F o f k , by a s imple limit arg ument. The next few sections will be dev oted to giving explicit f ormulas for the map Q F ( w ). In case w is a n F - p o in t of ∆ n \ ∂ ∆ n , we are able to do so directly; in general, we will need to pass to a n n -fold P 1 -susp ension b efore we can give an explicit formula. W e will then conclude with the pro of of our main res ult in § 9. 4. The P ontr y agin-Thom collapse map W e rec a ll a spe c ial case of Pont ryagin-Thom construction in H • ( k ). Let V n be the open subsch eme ∆ n \ ∂ ∆ n of ∆ n ; we use bar y centric co ordinates u 0 , . . . , u n on V n , giving us the identiﬁcation V n = Sp ec k [ u 0 , . . . , u n , ( u 0 · . . . · u n ) − 1 ] / X i u i − 1 . W e let H ⊂ P n be the hyperplane P n i =1 X i = X 0 and let 1 := (1 : 1 : . . . : 1) ∈ P n ( k ). Deﬁnition 4.1. Le t F b e ﬁnitely genera ted ﬁeld extension o f k a nd let w b e a closed p oint of V nF . The Pontryagin-Thom c ol lapse map asso ciated to w : P T F ( w ) : Σ n s Spec F + → ( P n F ( w ) /H F ( w ) , 1) . is the comp osition in H • ( k ) Σ n s Spec F + σ F ∼ / / ∆ n F /∂ ∆ n F c w − − → ∆ n F / (∆ n F \ { w } ) mv w ∼ / / ( P n F ( w ) /H F ( w ) , 1) for sp eciﬁc choices o f the isomorphisms in this comp ositio n, to be ﬁlle d in be low. THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 13 The ma p σ F is the standar d one g iven by the contractibilit y of ∆ n and a ll its faces, which gives an isomorphism in H • ( k ) of ∆ n /∂ ∆ n with the constant preshea f on the simplicial space ∆ n /∂ ∆ n : ∆ n ([ m ]) := Hom ∆ ([ m ] , [ n ]) and ∂ ∆ n ([ m ]) ⊂ ∆ n ([ m ]) the set of non-s urjective maps f : [ m ] → [ n ]. The isomorphism Σ n S 0 ∼ = ∆ n /∂ ∆ n in H • th us gives the isomorphism σ : Σ n S 0 → ∆ n /∂ ∆ n in H • ( k ) and thereby g ives ris e to the isomorphism in H • ( k ) (4.1) σ F : Σ n s Spec F + = Sp ec F + ∧ Σ n S 0 id ∧ σ − − − → Sp ec F + ∧ ∆ n /∂ ∆ n = ∆ n F /∂ ∆ n F . The map c w is the quotient map. The is omorphism mv w : ∆ n F / ∆ n F \ { w } → ( P n F ( w ) /H F ( w ) , 1) is the Morel-V o evo dsky pur ity isomorphism. This map dep ends in gener a l o n the choice of an isomorphis m ψ w : m w /m 2 w → F ( w ) n , where m w ⊂ O ∆ n F ,w is the maximal idea l; in additio n, w e need to make ex plicit the role of the chosen ba s e- po int 1. F or this, w e g o thro ugh the construction o f the purity iso morphism, g iving the explicit choices which lead to a well-deﬁned c hoice of isomorphism mv w . W e give V n × A 1 × ∆ n co ordinates u 0 , . . . , u n , x, t 0 , . . . , t n , with the u i the barycen- tric coo r dinates on V n , x the standar d co or dinate on A 1 and the t i the barycentric co ordinates on ∆ n . Let ( X 0 , X 1 , . . . , X n ) := ( x, t 1 − u 1 u 0 , . . . , t n − u n u 0 ) . The constructio n of mv w uses the blow-up o f A 1 × ∆ n F along 0 × w µ w : Bl 0 × w A 1 × ∆ n F → A 1 × ∆ n F . Let E w ⊂ Bl 0 × w A 1 × ∆ n F be the exceptiona l divisor. Then E w is a n F ( w )-scheme. Suppo se ﬁrst that w is separa ble ov er F . The clo sed point 0 × w o f A 1 × ∆ n F has the canonica l lifting to the clos ed po int 0 × w of A 1 × ∆ n F ( w ) ; let m 0 × w ⊂ O A 1 × ∆ n F , 0 × w and m ′ 0 × w ⊂ O A 1 × ∆ n F ( w ) , 0 × w denote the resp ective maximal ideals. As w is separable ov er F , the pro jection p : A 1 × ∆ n F ( w ) → A 1 × ∆ n F induces an isomorphism of gra ded F (0 × w )-algebra s p ∗ : ⊕ m ≥ 0 m m 0 × w /m m +1 0 × w → ⊕ m ≥ 0 m ′ m 0 × w /m ′ m +1 0 × w . The functions ( X 0 , X 1 ( w ) , . . . , X n ( w )) give g enerator s for the maximal ideal m ′ 0 × w ; as E w = P ro j F (0 × w ) ⊕ m ≥ 0 m m 0 × w /m m +1 0 × w ∼ = Pro j F (0 × w ) ⊕ m ≥ 0 m ′ m 0 × w /m ′ m +1 0 × w the image ( x 0 , x 1 ( w ) , . . . , x n ( w )) o f ( X 0 , X 1 ( w ) , . . . , X n ( w )) in m ′ 0 × w /m 2 0 × w give homogeneous co ordinates for E w , deﬁning an isomorphism q w := ( x 0 : x 1 ( w ) : . . . : x n ( w )) : E w → P n F ( w ) . Let H ( w ) ⊂ E w be the pull-back of H F ( w ) via q w , and let 1 w = q − 1 w (1). The prope r transform µ − 1 w [ A 1 × w ] ⊂ Bl 0 × w A 1 × ∆ n F maps isomorphically to A 1 × w via µ w , and intersects E w in a closed p oint ¯ w lying ov er 0 × w . Lemma 4.2 . 1. F or al l w ∈ V nF , we have 1 w 6 = ¯ w and ¯ w 6∈ H ( w ) . 2. q w ( ¯ w ) = (1 : 0 : . . . : 0) 14 MARC LE VINE Pr o of. Clea rly (2) implies (1). F o r (2), q w ( ¯ w ) is the image of 1 × w under ( X 0 : X 1 ( w ) : . . . : X n ( w )) : A 1 × ∆ n F ( w ) \ { 0 × w } → P n F ( w ) , which is (1 : 0 : . . . : 0).  Additionally , the quotient map r w : ( P n F ( w ) /H F ( w ) , 1) → P n F ( w ) / ( P n F ( w ) \ { (1 : 0 : . . . : 0) } ) is an is omorphism in H • ( k ), sinc e pro jection from (1 : 0 : . . . : 0 ) realize s P n F ( w ) \ { (1 : 0 : . . . : 0) } a s a n A 1 -bundle ov er P n − 1 F ( w ) with section H F ( w ) . This gives us the sequence of isomorphisms in H • ( k ): E w / ( E w \ { ¯ w } ) q w − − → P n F ( w ) / ( P n F ( w ) \ { (1 : 0 : . . . : 0) } ) r w ← − − ( P n F ( w ) /H F ( w ) , 1) . In case w is not separable ov er F , we ch o ose an y set of para meters X 1 ( w ), . . . , X n ( w ) for m w such that, taking X 0 = x , the iso mo rphism E w → P n w deﬁned b y the sequence x 0 , x 1 ( w ) , . . . , x n ( w ) sa tisﬁes the condition of lemma 4.2 ( F is inﬁnite, so (1) is satisﬁed for a genera l choice; the condition (2) is satisﬁed for a ll c hoices). W e then pro ceed as ab ove. Morel-V oevo dsky show that the inclus ions i w : E w → Bl 0 × w A 1 × ∆ n F and i 1 : ∆ n F = 1 × ∆ n F → A 1 × ∆ n F induce isomor phisms ¯ i w : E w / ( E w \ { ¯ w } ) → Bl 0 × w A 1 × ∆ n F / (Bl 0 × w A 1 × ∆ n F \ µ − 1 w [ A 1 × w ]) ¯ i 1 : ∆ n F / (∆ n F \ { w } ) → Bl 0 × w A 1 × ∆ n F / (Bl 0 × w A 1 × ∆ n F \ µ − 1 w [ A 1 × w ]) in H • ( k ) (see the pro of of [1 2, Theorem 3.2.23]). Deﬁnition 4.3 . The pur ity isomor phism mv w : ∆ n F / (∆ n F \ { w } ) ∼ − → ( P n F ( w ) /H F ( w ) , 1) . is deﬁned as the comp ositio n ∆ n F / (∆ n F \ { w } ) ¯ i 1 ∼ / / Bl 0 × w A 1 × ∆ n F / (Bl 0 × w A 1 × ∆ n F \ µ − 1 w [ A 1 × w ]) ¯ i w ∼ o o E w / ( E w \ { ¯ w } ) q w ∼ / / P n F ( w ) / P n F ( w ) \ { (1 : 0 : . . . : 0) } r w ∼ o o ( P n F ( w ) /H 1 F ( w ) , 1) . In ca se w is an F -rational p oint of ∆ n F , we have another des cription of mv w . The map q − 1 w ◦ ( X 0 : X 1 ( w ) : . . . : X n ( w )) : A 1 × ∆ n F ( w ) \ { 0 × w } → E w extends to a morphism p w : Bl 0 × w A 1 × ∆ n F → E w making Bl 0 × w A 1 × ∆ n F an A 1 -bundle ov er E w with section i w , and thus p w induces an isomorphism in H • ( k ) ¯ p w : Bl 0 × w A 1 × ∆ n F / (Bl 0 × w A 1 × ∆ n F \ µ − 1 w [ A 1 × w ]) → E w / ( E w \ { ¯ w } ) inv erse to ¯ i w . Thus THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 15 Lemma 4.4 . Su pp ose w is in ∆ n ( F ) . Then mv w = r − 1 w ◦ q w ◦ ¯ p w ◦ ¯ i 1 . W e can further simplify the ab ov e desc ription of mv w by no ting: Lemma 4.5 . Su pp ose w is in ∆ n ( F ) . L et (4.2) ϕ w : ∆ n F / (∆ n F \ { w } ) → P n F / P n F \ { (1 : 0 : . . . : 0) } b e t he map induc e d by (1 : X 1 ( w ) : . . . : X n ( w )) : ∆ n F → P n F . Then ϕ w = q w ◦ ¯ p w ◦ ◦ ¯ i 1 , henc e mv w = r − 1 w ◦ ϕ w . Pr o of. The identit y mv w = r − 1 w ◦ ϕ w follows directly from our descr iption ab ove of the maps q w and ¯ p w and lemma 4.4.  Altogether, this gives us the formula, for w ∈ ∆ n ( F ), (4.3) P T F ( w ) = r − 1 w ◦ ϕ w ◦ c w ◦ σ F . 5. ( P n /H , 1) and Σ n s G ∧ n m Our main task in this sectio n is to constr uct an explicit isomorphism α : ( P n /H , 1) ∼ − → Σ n s G ∧ n m . W e ﬁrst r e c all some elemen tary constructio ns inv o lving homotopy colimits ov er sub c ategories of the n - cub e . Let C b e a small category and let F : C → Sp c ( k ) b e a functor. Let N : ∆ op → Sets b e the ner ve of C . F or σ = ( s 0 f 1 − → s 1 → . . . f n − → s n ) ∈ N n ( C ) deﬁne F ( σ ) := F ( s 0 ). Bousﬁeld-K an [2] deﬁne ho colim F to b e the simplicial ob ject of Sp c ( k ) with n -simplices ho colim F n := ∐ σ ∈N n ( C ) F ( σ ); for g : [ n ] → [ m ] in ∆, ho colim F ( g ) : ho co lim F m → ho colim F n is the map sending ( F ( s 0 ) , σ = ( s 0 , . . . , s m )) to ( F ( s ′ 0 ) , σ ′ = ( s ′ 0 , . . . , s ′ n )), with σ ′ = N ( g )( σ ), s ′ 0 = s g (0) and the map F ( s 0 ) → F ( s ′ 0 ) is F ( s 0 → s g (0) ). ho colim F is the geometric rea lization of hoco lim F . F or a functor F : C → Sp c • ( k ) w e use ess ent ially the same deﬁnition of ho colim F as a simplicial ob ject o f Sp c • ( k ), repla cing dis joint union ∐ with pointed union ∨ , and we use the p ointed version of geometric realizatio n to deﬁne ho colim F in Sp c • ( k ). Concretely , hoco lim F is the co- equalizer of ∨ g :[ n ] → [ m ] ho colim F m ∧ ∆ n + / / / / ∨ n ho colim F n ∧ ∆ n + The essential prope rty of ho colim we will need is the following: Prop ositi o n 5.1 ([2]) . L et C b e a ﬁnite c ate gory, F , G : C → Sp c • ( k ) f unctors, and ϑ : F → G a n atur al tr ansformation. Su pp ose that ϑ ( c ) : F ( c ) → G ( c ) is an isomorphi sm in H • ( k ) for e ach c ∈ C . Then ho colim ϑ : ho colim F → ho co lim G is an isomorphi sm in H • ( k ) . The analo gous r esult hold s after r eplac ing Sp c • ( k ) and H • ( k ) with Sp c ( k ) and H ( k ) . 16 MARC LE VINE This is of co ur se just a sp ecial ca se of the genera l re sult v a lid for functor s from a small (not just ﬁnite) ca teg ory to a prop er simplicial model category . See for example [4] for details. R emark 5.2 . Let F : C → Sp c ( k ) b e a functor. Supp ose our index categ ory C is a pro duct C 1 × C 2 . W e ma y form the bi-simplicial ob ject ho colim 2 F of Sp c ( k ), with ( n, m )-simplices ho colim 2 F n,m := ∐ ( σ 1 ,σ 2 ) ∈N ( C 1 ) n ×N ( C 2 ) m F ( σ 1 , σ 2 ) where F ( σ 1 , σ 2 ) = F ( s 0 × s ′ 0 ) if σ = ( s 0 → . . . ) a nd σ ′ = ( s ′ 0 → . . . ); the morphisms are deﬁned similarly . As N ( C 1 × C 2 ) is the diagonal simplicial set asso cia ted to the bi-s implicial set N ( C 1 ) × N ( C 2 ), it follows that ho c o lim F is the diag onal simplicial ob ject of Sp c ( k ) asso ciated to the bi-simplicial o b ject ho colim 2 F , and thus w e have the natural isomorphism of geometr ic r ealizations ho colim F = | ho colim F | ∼ = | ho colim 2 F | in Sp c ( k ). Similar rema rks hold in the p ointed ca se. Let  n +1 be the p oset of subsets of [ n ], or de r ed under inclus io n. F or J ⊂ J ′ ⊂ { 0 , . . . , n } , we let  n +1 J ≤∗≤ J ′ be the full sub categor y o f subs e ts I with J ⊂ I ⊂ J ′ ,  n +1 J < ∗≤ J ′ the full subcatego r y of subsets I w ith J ( I ⊂ J ′ , etc. W e s o metimes omit J if J = ∅ o r J ′ if J ′ = [ n ]. If | J | = r + 1, we let i ′ J : { 0 , . . . , r } → J be the unique or de r -preser ving bijection, and let i J :  r +1 →  n +1 ∗≤ J be the r e sulting isomo rphism of categ o ries. Clearly i J induces the isomorphism of sub catego ries i J :  r +1 ∗ < [ r ] →  n +1 ∗ r . Identifying  r +1 ∗ < [ r ] with  n +1 ∗ < [ r ] via the inclusion [ r ] ⊂ [ n ], extend F to a functor σ n − r F :  n +1 ∗ < [ n ] → Sp c • ( k ) by setting σ n − r F ( J ) = ∗ if J is not a prop er subset of [ r ]. Similarly , let G :  r +1 ∗ < [ r ] ×  s → Sp c • ( k ) b e a functor and take n > r . Iden tifying  r +1 ∗ < [ r ] ×  s with a full s ubca tegory of  r +1 ∗ < [ r ] ×  n − r + s via the inclusio n [ s ] ⊂ [ n − r + s ], extend G to a functor c n − r G :  r +1 ∗ < [ r ] ×  n − r + s → Sp c • ( k ) by setting c n − r G ( J , I ) = ∗ if I 6⊂ [ s ]. Example 5.3 . Let X b e in Sp c • ( k ). No ting that  1 ∗ < [0] is the one- po int category , we write X for the functor  1 ∗ < [0] → Sp c • ( k ) with v alue X . This giv es us the functors c n X :  n +1 → Sp c • ( k ) , σ n X :  n +1 ∗ < [ n ] → Sp c • ( k ) . Explicitly , c n X ( ∅ ) = σ n X ( ∅ ) = X and b o th functors hav e v alue ∗ a t J 6 = ∅ . THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 17 Lemma 5.4 . Ther e ar e natur al isomorph isms Π c : hoc olim c n − r G → ho co lim G ∧ ([0 , 1] , 1) ∧ n − r Π σ : ho colim σ n − r F → Σ n − r s ho colim F . in Sp c • ( k ) Pr o of. W e pro cee d b y induction on n − r ; it suﬃces to handle the c a se r = n − 1 . W e ﬁrst take care of the isomorphis m Π c . Via remar k 5 .2, it suﬃces to give a n isomorphism | ho colim 2 c 1 G | ∼ = ho colim G ∧ ([0 , 1] , 1) , where w e use the pro duct decomp ositio n  r +1 ∗ < [ r ] ×  s +1 = (  r +1 ∗ < [ r ] ×  s ) ×  1 . Fix an m simplex σ of N (  r +1 ∗ < [ r ] ×  s ) and let c G σ :  1 → Sp c • ( k ) be the functor c G σ ( ∅ ) = G ( σ ) ∧ ∆ m + , c G σ ([0]) = ∗ . Then ho colim c G σ ∼ = G ( σ ) ∧ ∆ m + ∧ ([0 , 1] , 1) with the isomor phism natur al in σ . The re sult follows directly fro m this. Next, g iven F :  n < [ n − 1] → Sp c • ( k ), let c ′ F :  n → Sp c • ( k ) b e the extension of F to  n deﬁned by setting c ′ F ([ n − 1 ]) = ∗ . W e claim ther e is a na tural isomorphism ho colim c ′ F ∼ = ho colim F ∧ ([0 , 1] , 1 ) in Sp c • ( k ). Indeed, we have the bijection (for m > 0) N (  n ) m = N (  n ∗ < [ n − 1] ) m ∐ N (  n ∗ < [ n − 1] ) m − 1 with the ﬁrst component c o ming from the inclus ion of  n ∗ < [ n − 1] in  n , a nd the second arising by sending σ = ( s 0 → . . . → s m − 1 ) to ( s 0 → . . . → s m − 1 → [ n − 1]). F or m = 0, the same cons truction gives N (  n ) 0 = N (  n ∗ < [ n − 1] ) 0 ∐ { [ n − 1] } . As, for a simplicial set C , the m -simplices of C × [0 , 1] / C × 1 have exactly the s ame description, our claim follows easily . Finally , we can write the categ o ry  n +1 ∗ < [ n ] as a (strict) pushout  n +1 ∗ < [ n ] =  n ∐  n ∗ < [ n ]  n ∗ < [ n ] ×  1 . This leads to an isomor phism o f ho colim σ 1 F as a pushout ho colim σ 1 F ∼ = ho colim c ′ F ∨ ho colim F ho colim c F ∼ = ho colim F ∧ ([0 , 1] , 1 ) ∨ ho colim F ∧ 0 + ho colim F ∧ ([0 , 1 ] , 1) = Σ 1 s ho colim F .  As in sectio n 4, let H ⊂ P n be the hyperplane P n i =1 X i = X 0 and let 1 := (1 : 1 : . . . : 1) ∈ P n ( k ). W e deﬁne an isomo rphism α : ( P n /H , 1) ∼ − → Σ n s G ∧ n m in H • ( k ) as follows: Let U i ⊂ P n be the standard a ﬃne o pen subset X i 6 = 0 . W e identif y U i with A n in the usual wa y via co or dinates ( X 0 /X i , . . . ˆ X i /X i , . . . , X n /X i ), which we wr ite 18 MARC LE VINE as x i 1 , . . . , x i n , or simply x 1 , . . . , x n . F o r each index set I ⊂ { 0 , . . . , n } , we hav e the int ersection U I := ∩ i ∈ I U i . F or I = { i 1 < . . . < i r } , we use co o rdinates in U i 1 to iden tify U I ∼ = Spec k [ x 1 , . . . , x n , x − 1 i 2 , . . . , x − 1 i r ] ∼ = A n −| I | +1 × G | I |− 1 m . The op en cov er U := { U 0 , . . . , U n } of P n ident iﬁes P n (in H ( k )) with the homo- topy colimit ov er  n +1 ∗ < [ n ] of the functor P n U :  n +1 ∗ < [ n ] → Sp c ( k ) P n U ( J ) := U J c . W e thus have the functor P n U , 1 :  n +1 ∗ < [ n ] → Sp c • ( k ) P n U , 1 ( J ) := ( U J c , 1) and the isomorphism in H • ( k ), ho colim P n U , 1 ∼ = ( P n , 1). Next, w e note that the h yp erplane H ⊂ P n is cov ered by the a ﬃne op en subsets U 1 , . . . , U n . The op en cov er U 1 := { H ∩ U 1 , . . . , H ∩ U n } o f H identiﬁes H (in H ( k )) with the homotopy co limit o ver  n +1 ∗ < [ n ] of the functor H U 1 :  n +1 ∗ < [ n ] → Sp c ( k ) H U 1 ( J ) := ( H ∩ U J c for 0 ∈ J ∅ f or 0 6∈ J. Let P n U , 1 / H U 1 :  n +1 ∗ < [ n ] → Sp c • ( k ) be the functor deﬁned by P n U , 1 / H U 1 ( J ) := ( ( U J c /H ∩ U J c , 1) for 0 ∈ J ( U J c , 1) for 0 6∈ J. By our discussio n, the maps P n U , 1 / H U 1 ( J ) → ( P n /H , 1) induced by the inclusions U J c ֒ → P n give rise to a n isomorphism in H • ( k ) ǫ 1 : ho colim P n U , 1 / H U 1 → ( P n /H , 1) . T o simplify the no tation, we denote P n U , 1 / H U 1 by F for the next few para graphs. W e claim that, for each J 6 = ∅ with 0 6∈ J , we hav e ( U J / ( H ∩ U J ) , 1 ) ∼ = ∗ in H • ( k ). Indeed, supp ose for exa mple that n ∈ J , and use co ordina tes ( x n 1 , . . . , x n n ) = ( X 0 /X n , . . . , X n − 1 /X n ) on U J . W e have the pro jection p : U J → U J ∩ ( X 0 = 0) p ( x n 1 , . . . , x n n ) = (0 , x n 2 , . . . , x n n ) . Since 0 6∈ J , x n 1 is not in verted on U J , and thus p makes U J an A 1 -bundle ov er U J ∩ ( X 0 = 0 ). p ha s the section s (0 , x n 2 , . . . , x n n ) := (1 + n X i =2 x n i , x n 2 , . . . , x n n ) , THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 19 ident ifying U J ∩ ( X 0 = 0) with H ∩ U J ; this together with homotopy in v aria nc e in H • ( k ) prov es our claim. Th us F ( J ) ∼ = ∗ in H • ( k ) for all J with 0 ∈ J . In addition F ( { 1 , 2 , . . . , n } ) = ( U 0 , ∗ ) ∼ = ( A n , ∗ ), which is a lso isomorphic to ∗ in H • ( k ). Let i 0 :  n ∗ < [ n − 1] →  n +1 ∗ < [ n ] inclusion functor induced b y the inclusion [ n − 1] → [ n ] se nding i ∈ [ n − 1] to i + 1 , and let ω :  n +1 ∗ < [ n ] →  n +1 ∗ < [ n ] be the automo rphism induced by the cyc lic p ermutation ω of [ n ], ω ( i ) := ( i + 1 for 0 ≤ i < n 0 for i = n. Let F | 0 :  n ∗ < [ n − 1] → Sp c • ( k ) be the functor F ◦ i 0 . W e hav e the evident quotient map q : F ◦ ω → σ 1 F | 0 , which by o ur discussion ab ov e is a term-wis e isomorphism in H • ( k ). B y lemma 5.4, q induces the isomorphisms in H • ( k ) (5.1) ho colim F → ho colim σ 1 F | 0 → Σ 1 s ho colim F | 0 . W e now turn to the functor F | 0 . This is just the punctured n -cub e corresp onding to the op en cover U ′ := { U 0 ∩ U 1 , . . . , U 0 ∩ U n } of U 0 \ (1 : 0 : . . . : 0) (with base-p oint 1), i.e. ( A n \ 0 , 1). W e thus ha ve the isomor phis m in H • ( k ) ho colim F | 0 ∼ = ( U 0 \ (1 : 0 : . . . : 0) , 1) ∼ = ( A n \ 0 , 1) . Let C ⊂ U 0 \ (1 : 0 : . . . : 0) be the union of the aﬃne hyperplanes x 0 i = 1 , i = 1 , . . . , n . As the inclusion 1 → C is an isomor phism in H ( k ), w e hav e the is omorphism in H • ( k ) ( U 0 \ (1 : 0 : . . . : 0) , 1) ∼ = U 0 \ (1 : 0 : . . . : 0) /C . Letting ¯ F | 0 be the quotient of F | 0 given by ¯ F | 0 ( J ) = U i 0 ( J ) c /C ∩ U i 0 ( J ) c , we thus have the isomo rphisms in H • ( k ) ho colim F | 0 ∼ = ho colim ¯ F | 0 ∼ = U 0 \ (1 : 0 : . . . : 0) /C . On the o ther ha nd, fo r ea ch J ( { 1 , . . . , n } , the inclus ion C ∩ U 0 ∩ U J → U 0 ∩ U J is an isomor phism in H ( k ), and thus ¯ F | 0 ( J ) ∼ = ∗ for all J 6 = ∅ . Since ¯ F | 0 ( ∅ ) ∼ = G ∧ n m we ha ve the q uotient map ¯ F | 0 → σ n − 1 G ∧ n m ; our discussio n tog e ther with lemma 5.4 th us gives us the is omorphism in H • ( k ) ho colim F | 0 ∼ = ho colim ¯ F | 0 ∼ = Σ n − 1 s G ∧ n m . T ogether with (5.1), this giv es us the sequence of iso morphisms in H • ( k ) ( P n /H , 1) ∼ = ho colim P n U , 1 / H U 1 ∼ = Σ s ho colim F | 0 ∼ = Σ n s G ∧ n m . W e denote the co mpo sition b y (5.2) α : ( P n /H , 1) ∼ − → Σ n s G ∧ n m . Now that we have deﬁned α , we can complete our deﬁnition of the purit y iso- morphism (3.4): (5.3) M V w := (id w + ∧ α ) ◦ m v v (see deﬁnition 4.3 for the deﬁnition of mv w ). 20 MARC LE VINE R emark 5 .5 . T a ke n > 1. Le t H ∞ ⊂ P n be the hype rplane X 0 = 0 and for let C 1 ⊂ U 1 be the union of the hyper planes x 1 i = 1, i = 1 , . . . , n . Let G ∞ n :  n +1 ∗ < [ n ] → Sp c • ( k ) b e the functor G ∞ n ( J ) := ( U J c /H ∞ ∩ U J c for 1 ∈ J U J c / [( H ∞ ∪ C 1 ) ∩ U J c ] for 1 6∈ J. W e note that the inclusion (0 : 1 : 0 : . . . : 0) → H ∞ ∩ C 1 is a n A 1 -weak equiv alence; using this it is ea sy to mo dify the arguments used in this section to s how that the ident ity map G ∞ ( ∅ ) → G ∧ n m extends to a map of functors G ∞ n → σ n G ∧ n m , whic h is a termwise iso morphism in H • ( k ), giving us the isomorphism ho colim G ∞ n ∼ = Σ n s G ∧ n m in H • ( k ). F urthermo r e, we hav e the sequence of isomor phisms in H • ( k ): P n /H ∞ → P n / [ H ∞ ∐ C 1 ∩ H ∞ ∩ U 1 C 1 ] → ho colim G ∞ n . Putting these together gives us the isomorphism (5.4) α ∞ : P n /H ∞ → Σ n s G ∧ n m in H • ( k ). F or n = 1, we note that H = 1, so ( P 1 /H , 1) = ( P 1 , H ). T o deﬁne α ∞ , w e just comp ose α : ( P 1 /H , 1) → Σ s G m with the iso morphism τ : ( P 1 , H ∞ ) → ( P 1 , H ) given by τ ( X 0 : X 1 ) = ( X 1 − X 0 : X 1 ) . W e will use these mo dels for Σ n s G ∧ n m to construct tra ns fer maps in § 8. 6. The suspension of a symbol Let ˜ ρ : V n → G n m be the map ρ ( u 0 , . . . , u n ) := ( − u 1 u 0 , . . . , − u n u 0 ) . Comp osing with the quotient map G n m → G ∧ n m gives us the map ρ : V n + → G ∧ n m . Our next main task is to g ive an explicit algebro -geometric description of Σ n s ρ . More genera lly , for f : T → V n a morphism in Sm /k , we will give a description of Σ n s ( ρ ◦ f ). W e b egin b y g iv ing a description o f Σ n s T + as a certain homoto py co limit. F or this, co nsider the scheme A 1 × ∆ n , with co or dinates x, t 0 , . . . , t n : A 1 × ∆ n = Sp ec k [ x, t 0 , . . . , t n ] / X i t i − 1 . F or i = 1 , . . . , n , let U ′ i ⊂ A 1 × ∆ n be the subscheme deﬁned by t i = 0, and let U ′ 0 ⊂ A 1 × ∆ n be the subsc heme deﬁned by x = 1. F or I ⊂ { 0 , . . . , n } , let U ′ I := ∩ i ∈ I U ′ i , the intersection taking place in A 1 × ∆ n . This gives us the punctured n + 1- cube ˆ G T n :  n +1 ∗ < [ n ] → Sp c ( k ) with ˆ G T n ( J ) := T × U ′ J c . As ab ove, use barycentric co or dinates u 0 , . . . , u n for V n . W e pull these back to T via f , and write u i for f ∗ ( u i ), letting the context mak e the meaning clear . Set ( X 0 , X 1 , . . . , X n ) := ( x, t 1 − u 1 u 0 , . . . , t n − u n u 0 ) THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 21 and set ( x i 1 , . . . , x i n ) := ( X 0 /X i , . . . , \ X i /X i , . . . , X n /X i ); i = 0 , . . . , n. Inside T × A 1 × ∆ n , we have the “hyperplane” H ( T ) deﬁned b y n X i =1 X i = X 0 . Fix an index I = ( i 0 , . . . , i r ) with 0 ≤ i 0 < . . . < i r ≤ n , and write the comple- men t of I in { 0 , . . . , n } as I c = ( j 1 , . . . , j n − r ) with j 1 < . . . < j n − r . W e have the isomorphism ϕ I := id × ( x i 0 j 1 , . . . , x i 0 j n − r ) : T × U ′ I → T × A n − r . In addition, let H I ⊂ A n − r be the hyperplane deﬁned by n − r X ℓ =1 x ℓ = 1 if i 0 = 0 , n − r X ℓ =2 x ℓ = x 1 if i 0 > 0 . Then ϕ I restricts to an isomorphism of H ( T ) ∩ T × U ′ I with T × H I , and thus the pro jection p 1 : H ( T ) ∩ T × U ′ I → T a nd inclusio n ι : H ( T ) ∩ T × U ′ I → T × U ′ I are isomorphisms in H ( k ). F or J ( [ n ], J 6 = ∅ , deﬁne G T ′ n ( J ) to b e the pushout in the diag ram H ( T ) ∩ T × U ′ J c   ι / / p 1   ˆ G T n ( J ) i ( J )      T × U ′ J c T s J / / _ _ _ _ _ _ G T ′ n ( J ) . Since ι is a coﬁbration and a weak equiv a lence in Sp c ( k ), so is s J . As p 1 is also a weak equiv alence in Sp c ( k ), i ( J ) is a weak equiv alence in Sp c ( k ) a s well. W e set G T ′ n ( ∅ ) := ˆ G T n ( ∅ ) = T × U ′ [ n ] ∼ = T . This deﬁnes for us the functor G T ′ n :  n +1 ∗ < [ n ] → Sp c ( k ) that ﬁts int o a diag ram ( T the constant functor) ˆ G T n i   T s / / G T ′ n with i and s term- wise iso morphisms in H ( k ) and s a term-wise coﬁbratio n in Sp c ( k ). F or n = 1 , deﬁne G T 1 ( J ) := ( G T ′ 1 ( J ) / s ( T ) for J 6 = ∅ G T ′ 1 ( ∅ ) + ∼ = T + for J = ∅ . giving us the functor G T 1 :  2 ∗ < [1] → Sp c • ( k ) 22 MARC LE VINE F or n > 1, take ∅ 6 = J ⊂ [ n ] and let Π ′ J ⊂ P n be the dimension n − | J | linear subspace de ﬁned by ∩ j ∈ J ( X j = 0). Let Π J ⊂ P n be the dimension n − | J | + 1 linear s pa ce spanned b y 1 and Π ′ J and le t A J ⊂ Π J be the aﬃne s pace Π J \ Π ′ J . Since Π ′ J is not contained in H , the intersection A J ∩ H is a co dimension one aﬃne space A J,H in A J . Clearly A J ⊃ A J ′ for J ⊂ J ′ , so we have the functor A / A H :  n +1 ∗ < [ n ] → Sp c ( k ) J 7→ A J c / A J c ,H . Let ∗ J be the base-p oint in A J / A J,H and let s ′ J : T → T × A J / A J,H be the morphism identifying T with T × ∗ J . Le t 1 J be the image of 1 ∈ A J in the quo tien t A J / A J,H . W e have the mo rphism s ′ J, 1 : T → T × A J / A J,H ident ifying T with T × 1 J . F or J 6 = ∅ , let G T n ( J ) b e the push-out in the dia gram T ∐ T s ′ J c × s ′ J c , 1 / / s J ∐ p   T × A J c / A J c ,H      G T ′ n ( J ) ∐ ∗ / / _ _ _ _ _ G T n ( J ) . where p : T → ∗ is the cano nic a l map; we give G T n ( J ) the base-p oint ∗ . W e set G T n ( ∅ ) = T + with its canonical base-p oint. Using the functoriality of G T ′ n and A / A H deﬁnes the functor (6.1) G T n :  n +1 ∗ < [ n ] → Sp c • ( k ) . Lemma 6.1 . F or e ach J 6 = ∅ , G T n ( J ) ∼ = ∗ in H • ( k ) . Pr o of. T ake J ⊂ [ n ], J 6 = ∅ . F or n = 1, s : T → G T ′ 1 ( J ) is a co ﬁbration and weak equiv alence in Sp c ( k ), and th us the quotient G T ′ 1 ( J ) / T is contractible. F or n > 1, the morphisms s J : T → G T ′ n ( J ), s ′ J : T → T × A J / A J,H and s ′ J, 1 : T → T × A J / A J,H are co ﬁbrations and weak equiv alences in Sp c ( k ); since 1 J 6∈ A J,H , the map s ′ J × s ′ J, 1 : T ∐ T → T × A J / A J,H is a coﬁbration. Let G T ′′ n ( J ) be the push-out in the diagra m T s ′ J c / / s J   T × A J c / A J c ,H ι      G T ′ n ( J ) / / _ _ _ _ _ G T ′′ n ( J ) . Then ι is a co ﬁbration and a weak equiv alence, hence the same is true for the comp osition T s ′ J, 1 − − → T × A J / A J,H ι − → G T ′′ n ( J ) . As G T n ( J ) = G T ′′ n ( J ) / T , it follows that G T n ( J ) is contractible.  Letting T :  1 ∗ < [0] → Sp c • ( k ) b e the functor T ( ∅ ) = T + , we have the evident quotient map G T n → σ n T , i.e., we se nd G n ( ∅ ) = T + to σ n T ( ∅ ) = T + by the identit y map, and the other maps ar e the canonical ones G T n ( I ) → ∗ . THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 23 By lemma 5.4 and lemma 6 .1, this map induces an isomo rphism (6.2) β T : ho colim G T n → Σ n s T + in H • ( k ). R emark 6.2 . The functors G T n , G T ′ n and ˆ G T n are all functors in T , where for e x ample g : T ′ → T g ives the morphism ˆ G n ( f ) : ˆ G T ′ n → ˆ G T n by the collectio n of maps f × id : T ′ × U ′ J c → T × U ′ J c . The map G T ′ n → G T n is natural in T , as is the map β T . Let ∆( V n ) ⊂ V n × ∆ n be the graph of the inclusion V n → ∆ n ; b y a s light abuse of notatio n, we write 0 × ∆( V n ) ⊂ V n × A 1 × ∆ n for the image of 0 × ∆( V n ) ⊂ A 1 × V n × ∆ n under the exchange of factors A 1 × V n × ∆ n → V n × A 1 × ∆ n . Deﬁne the morphism ϕ : V n × A 1 × ∆ n \ 0 × ∆( V n ) → P n by ϕ ( u 0 , . . . , u n , x, t 0 , . . . , t n ) := ( X 0 : X 1 : . . . : X n ) , where as ab ov e X 0 = x , X i = ( t i − u i ) /u 0 , i = 1 , . . . , n . Since V n × U ′ i ∩ 0 × ∆( V n ) = ∅ for each i = 0 , . . . , n , the restriction of ϕ to ∪ n i =0 V n × U ′ i is thu s a mor phis m, and ther efore gives a w ell-deﬁned mo rphism of functors  n +1 ∗ < [ n ] → Sp c ( k ), ˜ ϕ ∗ : ˆ G V n n → P n , where P n is the consta nt functor . Given a mo rphism f : T → V n , we comp ose ˜ ϕ J with f × id, giving the morphism of functor s ˜ ϕ T ∗ : ˆ G T n → P n . Adjoining the pro jections T × U ′ J c → T gives us the morphism of functors ( p 1 , ˜ ϕ T ∗ ) : ˆ G T n → T × P n . Passing to the quotients, ( p 1 , ˜ ϕ T ∗ ) induces the map of functors ( p 1 , ϕ T ′ ∗ ) : G ′ T n → T × ( P n /H ). W e extend ( p 1 , ϕ T ′ ∗ ) to a map of functors  n +1 ∗ < [ n ] → Sp c • ( k ) p 1 ∧ ϕ T ∗ : G T n → T + ∧ ( P n /H , 1) by using the inclusions A J c → P n , a nd sending the base-p oint in T + to the bas e- po int in T + ∧ ( P n /H , 1). This gives us the ma p in Sp c • ( k ) (6.3) Φ T : ho colim G T n → T + ∧ ( P n /H , 1) . Lemma 6.3 . L et f : T → V n b e a morphism in Sm /k . Then the diagr am Σ n s T + Σ n s (id T + ∧ ρ ◦ f ) / / T + ∧ Σ n s G ∧ n m ho colim G T n Φ T / / β T O O T + ∧ ( P n /H , 1) id ∧ α O O c ommutes in H • ( k ) . Pr o of. W e work thro ugh our descr iption of α and β T , a dding so me in termediate steps. W e introduce a n additional functor ( P n / H U , 1) :  n +1 ∗ < [ n ] → Sp c • ( k ) J 7→ ( U J c /H ∩ U J c , 1) By Mayer-Vietoris, the canonical map ho co lim( P n / H U , 1) ǫ − → ( P n /H , 1) induced by the cov er U is an isomor phism in H • ( k ). The collectio n of quo tien t ma ps U J c → U J c /H ∩ U J c or identit y ma ps g ive the map γ : P n U , 1 / H U 1 → ( P n / H U , 1). 24 MARC LE VINE W e als o hav e the functor σ n G ∧ n m . Iden tifying U 0 ...n with G n m via the co ordinates ( x 0 1 , . . . , x 0 n ), the quo tien t map U 0 ...n ∼ = G n m → G ∧ n m extends canonically to the quotient map δ : P n U , 1 / H U 1 → σ n G ∧ n m . F rom our discus sion on the isomor phis m α , we hav e the comm utative diag ram of isomorphisms in H • ( k ) (6.4) ( P n /H , 1) α + + X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ho colim( P n / H U , 1) ǫ 5 5 k k k k k k k k k k k k k k k ho colim P n U , 1 / H U 1 γ o o ǫ 1 O O δ / / ho colim σ n G ∧ n m ϑ / / Σ n n G ∧ n m . Note that, for each J 6 = ∅ , [ n ], we hav e A J ⊂ U J , since for j ∈ J , the intersection Π J ∩ ( X j = 0) is equa l to Π ′ J . Also, the map ˜ ϕ J : ˜ G n ( J ) → P n has image contained in U J c . W e deﬁne the map o f functors ψ T ∗ : G T n → T + ∧ ( P n / H U , 1) as follows: for J 6 = ∅ , [ n ], we use the map ( p 1 , ϕ T ′ J ) : G T ′ n ( J ) → T × U J c / ( U J c ∩ H ) on G T ′ n ( J ), and the map T × A J c id × i J − − − − → T × U J c induced by the inclusion i J : A J c ֒ → U J c . O ne chec ks that these descend to a well deﬁned map on the quotient ψ T J : G T n ( J ) → T + ∧ ( P n / H U , 1)( J ) . F or J = ∅ , we us e (id T , ϕ T ′ ∅ ) : T → T × U 0 ...n /H ∩ U 0 ...n This gives us the co mm utative diag ram of functors G T n ψ T ∗ / /   T + ∧ ( P n / H U , 1) T + ∧ P n U , 1 / H U 1 id ∧ γ O O id ∧ δ   σ n T id T ∧ σ n ( ρ ◦ f ) / / T + ∧ σ n G ∧ n m which induces the co mmu tative diag ram (in H • ( k )) on the ho mo topy colimits ho colim G T n Ψ T / / β T   T + ∧ ho colim( P n / H U , 1) id ∧ ϑ ◦ δ ◦ γ − 1   Σ n s T + Σ n s (id ∧ ρ ◦ f ) / / T + ∧ Σ n s G ∧ n m . THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 25 Combining this with our dia gram (6.4) and noting that Φ T = (id ∧ ǫ ) ◦ Ψ T yields the commutativ e diagra m in H • ( k ) ho colim G T n Ψ T / / Φ T + + β T   T + ∧ ho colim( P n / H U , 1) id ∧ ( ϑ ◦ δ ◦ γ − 1 )   id ∧ ǫ / / T + ∧ ( P n /H , 1) id ∧ α u u j j j j j j j j j j j j j j j Σ n s T + Σ n s (id ∧ ρ ◦ f ) / / T + ∧ Σ n s G ∧ n m , completing the pro of.  7. Computing the collapse ma p W e retain the notation fro m §§ 4, 5 and 6. O ur task in this section is to us e lemma 6.3 to give an explicit computation of Q F ( w ) as the n th suspensio n o f a map ρ w : Spec F + → w + ∧ G ∧ n m , at leas t for w an F -po int of ∆ n \ ∂ ∆ n . In g e neral, we will need to tak e a further P 1 -susp ension befo re desuspending , which we do in the next section. F or F a ﬁnitely gene r ated ﬁeld extensio n of k and w a closed p oint o f ∆ n F \ ∂ ∆ n F , we hav e the Pon tryagin-Thom colla pse map (deﬁnition 4.1) P T F ( w ) : Σ n s Spec F + → ( P n F ( w ) /H F ( w ) , 1) . W e have as well the map (3 .5) Q F ( w ) : Σ n s Spec F + → w + ∧ Σ n s G ∧ n m = w + ∧ S 2 n,n It follows fr om the deﬁnition of M V w (5.3), P T F ( w ) a nd mv w (deﬁnition 4.3) that (7.1) Q F ( w ) = (id w + ∧ α ) ◦ P T F ( w ) , where w e identify ( P n F ( w ) /H F ( w ) , 1) with w + ∧ ( P n /H , 1) and w he r e α : ( P n /H , 1) → Σ n s G ∧ n m is the isomorphism (5.2). Consider an F -p o in t w : Spec F → ∆ n of ∆ n . Given element s z 1 , . . . , z n of F × , we hav e the corresp onding map [ z 1 ] ∧ F . . . ∧ F [ z n ] : Spec F + → Sp ec F + ∧ G ∧ n m given as the comp ositio n Spec F + id ∧ ( z 1 ,...,z n ) − − − − − − − − → Sp ec F + ∧ ( G n m , 1) → Spec F + ∧ G ∧ n m . W e use the nota tion ∧ F to denote the smash pro duct for po ints F -s chemes ( X , x ), ( Y , y ): ( X, x ) ∧ F ( Y , y ) := X × F Y / ( X × F y ∨ x × F Y ) , and note that [ z 1 ] ∧ F . . . ∧ F [ z n ] really is the ∧ F -pro duct of the maps [ z i ]. Prop ositi o n 7.1. T ake w = ( w 0 , . . . , w n ) ∈ (∆ n \ ∂ ∆ n )( F ) . Then Q F ( w ) = Σ n s [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ] . Pr o of. W e hav e for ea ch V n -scheme T → V n the functor (6.1); applying this con- struction for the morphism w : Sp ec F → V n , gives us the functor G w n :  n +1 ∗ < [ n ] → Sp c • ( k ) . W e rec a ll the subschemes U ′ i , i = 0 , . . . , n a nd H of A 1 × ∆ n from § 6. 26 MARC LE VINE W e note that U ′ 0 = 1 × ∆ n , H ∩ U ′ 0 is the face t 0 = 0, and that U ′ 0 ∩ U ′ i is the face t i = 0, fo r i = 1 , . . . , n . Thus, collapsing the U ′ i , i = 1 , . . . , n , H ∩ U ′ 0 and all the A J to a p oint, and sending U ′ 0 to ∆ n by the pro jection map gives a w ell deﬁned morphism in Sp c • ( k ), a : ho colim G w n → Sp ec F + ∧ ∆ n /∂ ∆ n , which is an iso morphism in H • ( k ). In addition, we hav e the commutativ e diagram of isomorphisms in H • ( k ) (7.2) ho colim G w n a ∼ / / β w ∼   Spec F + ∧ ∆ n /∂ ∆ n Σ n s Spec F + , σ F ∼ 5 5 l l l l l l l l l l l l l where σ F is the isomorphism (4.1) and β w is the isomo rphism (6.2). Let ˜ r w : Sp ec F + ∧ ( P n /H , 1) → P n F / ( P n F \ { (1 : 0 : . . . : 0) } ) be the compos ition of the isomorphism Sp ec F + ∧ ( P n /H , 1) ∼ = ( P n F /H F , 1) follow ed by the quotient map r w : ( P n F /H F , 1) → P n F / ( P n F \ { (1 : 0 : . . . : 0 ) } ). It follows directly from the deﬁnition of the map Φ w (6.3) and the map ϕ w (4.2) that the diagram ho colim G w n Φ w / / a   Spec F + ∧ ( P n /H , 1) ˜ r w   Spec F + ∧ ∆ n /∂ ∆ n c w / / ∆ n F / ∆ n F \ { w } ϕ w / / P n F / ( P n F \ { (1 : 0 : . . . : 0) } ) . commutes. Com bining this with the diagr am (7.2) and our descr iption (4.3) of P T F ( w ) gives us the co mm utative diag ram Σ n s Spec F + P T F ( w ) ) ) R R R R R R R R R R R R R ho colim G w n Φ w / / β w ∼ O O Spec F + ∧ ( P n /H , 1) But b y lemma 6.3, (Σ n s [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ]) ◦ β w = (id Sp ec F + ∧ α ) ◦ Φ w ; since β w is an isomor phism, this gives us Σ n s [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ] = (id Sp ec F + ∧ α ) ◦ P T F ( w ) . Our formula (7.1) for Q F ( w ) completes the pro of.  8. Transfers and P 1 -suspension W e now consider the genera l ca se of a closed po int w ∈ V nF ⊂ ∆ n F . Consider the map j : ∆ n → P n j ( t 0 , . . . , t n ) := (1 : t 1 , . . . : t n ); j is an ope n immersion, identif ying ∆ n with U 0 and V n with U 0 ...n \ H ⊂ P n . THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 27 W e deﬁne the tr ansfer map T r F ( w ) : S 2 n,n ∧ Spec F + → S 2 n,n ∧ Spec F ( w ) + asso ciated to a close d p oint w ∈ A n F , separable ov er F , as the co mpo sition S 2 n,n ∧ Spec F + α ∞ ∧ id ∼ o o P n F /H ∞ F c jw − − → P n F / P n F \{ j ( w ) } ) ¯ p ◦ ¯ j ∼ o o ∆ n F ( w ) / (∆ n F ( w ) \{ w } ) mv ∞ w − − − → P n F ( w ) /H ∞ F ( w ) α ∞ ∧ id ∼ / / S 2 n,n ∧ Spec F ( w ) + . The map ¯ j is induced from j , ¯ p is induced fro m the pro jection p : ∆ n F ( w ) → ∆ n F , and w ∈ ∆ n F ( w ) is the cano nical lifting of w ∈ ∆ n F to ∆ n F ( w ) = w × F ∆ n F . The ma p ¯ p ◦ ¯ j is a n iso morphism by Nisnevich excis ion (whic h is where w e use the separability of w over F ). The map mv ∞ w is the Mor el-V o evo dsky purity isomorphism, where we use the generato rs ( t 1 − w 1 , . . . , t n − w n ) for m w , together with the is omorphism r w : P n F ( w ) /H ∞ F ( w ) → P n F ( w ) / ( P n F ( w ) \ (1 : 0 : . . . : 0)) induced by the identit y on P n F ( w ) . The map α ∞ is the isomorphism (5.4). Lemma 8.1 . Su pp ose that w is in V n ( F ) . Then T r F ( w ) = id . Pr o of. Let w 0 = (1 : 0 : . . . : 0) ∈ U 0 ⊂ P n ( k ), giving us the purity isomorphism mv ∞ w 0 : U 0 / ( U 0 \ w 0 ) → P n /H ∞ deﬁned via the choice of generators ( x 1 , . . . , x n ) for m w 0 . The morphism ( x : x 1 : . . . : x n ) : A 1 × U 0 \ 0 × w 0 → P n extends to an A 1 -bundle π := ( x : x 1 : . . . : x n ) : Bl 0 × w 0 A 1 × U 0 → P n F urthermor e, the restriction o f π to 1 × U 0 extends to the iden tit y map P n → P n . F rom this, it fo llows that morphism in H • ( k ), T r F ( w 0 ) : S 2 n,n → S 2 n,n is the identit y . On the other hand, let T w : P n F → P n F be the automorphism extending translatio n by w on U 0 . The n T w acts by the iden tity o n P n F /H ∞ F , a s we can extend T w to the A 1 family of automorphisms t 7→ T tw connecting T w with id. F urthermor e, T ∗ − w ( x 1 , . . . , x n ) = ( x 1 − w 1 , . . . , x n − w n ). F rom this it fo llows that T r F ( w ) = T w ◦ T r F ( w 0 ) ◦ T − w = id .  Prop ositi o n 8 .2. L et w = ( w 0 , . . . , w n ) b e a close d p oint of V nF , sep ar able over F . Then the S 2 n,n -susp ension of Q F ( w ) : id S 2 n,n ∧ Q F ( w ) : S 2 n,n ∧ Spec F + ∧ S n, 0 → S 2 n,n ∧ w + ∧ S 2 n,n is e qual to the map Σ n s  (id S 2 n,n ∧ [ − w 1 /w 0 ] ∧ F ( w ) . . . ∧ F ( w ) [ − w n /w 0 ]) ◦ T r F ( w )  . 28 MARC LE VINE Pr o of. W rite ∗ F for Spe c F . W e hav e the comm utative diag ram S 2 n,n ∧ ∗ F + ∧ S n, 0 α − 1 ∞ ∧ σ F   T r F ( w ) ∧ id / / S 2 n,n ∧ w + ∧ S n, 0 id ∧ σ   P n /H ∞ ∧ ∗ F + ∧ ∆ n /∂ ∆ n id ∧ c w   c j ( w ) ∧ id * * U U U U U U U U U U U U U U U U U S 2 n,n ∧ w + ∧ ∆ n /∂ ∆ n id ∧ c w   P n / ( P n \ j ( w )) ∧ ∆ n /∂ ∆ n id ∧ c w   ( α ∞ ◦ m ∞ j ( w ) ) ∧ id 4 4 j j j j j j j j j j j j j j j j P n /H ∞ ∧ ∆ n F / (∆ n F \ w ) c j ( w ) ∧ id / / id ∧ mv w   P n / ( P n \ j ( w )) ∧ ∆ n / (∆ n \ w ) ( α ∞ ◦ m ∞ j ( w ) ) ∧ id * * T T T T T T T T T T T T T T T T id ∧ mv w   P n /H ∞ ∧ w + ∧ ( P n /H , 1) c j ( w ) ∧ id * * U U U U U U U U U U U U U U U U U α ∞ ∧ id w + ∧ α   S 2 n,n ∧ w + ∧ ∆ n / (∆ n \ w ) id ∧ mv w   P n / ( P n \ j ( w )) ∧ w + ∧ ( P n /H , 1) ( α ∞ ◦ m ∞ j ( w ) ) ∧ id * * T T T T T T T T T T T T T T T T S 2 n,n ∧ w + ∧ ( P n /H , 1) id ∧ id w + ∧ α   S 2 n,n ∧ w + ∧ S 2 n,n S 2 n,n ∧ w + ∧ S 2 n,n ; the co mm utativity follows either by deﬁnitio n of T r F ( w ), or b y iden tities of the form ( a ∧ 1) ◦ (1 ∧ b ) = (1 ∧ b ) ◦ ( a ∧ 1), or (in the bo tto m pentagon) lemma 8.1. The comp osition along the left-ha nd side is id S 2 n,n ∧ [(id w + ∧ α ) ◦ P T F ( w )]; a long the right-hand side we hav e id S 2 n,n ∧ [(id w + ∧ α ) ◦ P T F ( w ) ( w )]. Since w is F ( w )- rational, we may apply prop o s ition 7.1 and our formula (7.1) for Q F ( w ) to complete the pro of.  9. Conclusion W e can now put all the pieces together. F o r E ∈ Spt S 1 ( k ) ﬁbra nt, we ha ve the asso ciated ﬁbr ant o b ject Ω n T E := H om Spt ( k ) ( S 2 n,n , E ), that is , Ω n T E is the pr esheaf (Ω n T E )( X ) := E ( X + ∧ S 2 n,n ). F o r e ach n ≥ 1, we have the cano nical map ι n : E → Ω n T Σ n T E . Replacing S 2 n,n with S n,n = G ∧ n m , w e hav e the ﬁbrant ob ject Ω n G m E := H om Spt ( k ) ( S n,n , E ) , deﬁned as the presheaf (Ω n G m E )( X ) := E ( X + ∧ G ∧ n m ). Given a clo sed po int w ∈ V nF , we deﬁne the map T r F ( w ) ∗ : π m (Ω n T E ( w )) → π m (Ω n T E ( F )) THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 29 as the comp osition π m (Ω n T E ( w )) = Hom S H S 1 ( k ) (Σ ∞ s ( S 2 n,n ∧ w + ) , Σ − m s E ) Σ ∞ s ( T r F ( w ) )) ∗ − − − − − − − − − − → Hom S H S 1 ( k ) (Σ ∞ s ( S 2 n,n ∧ Spec F + ) , Σ − m s E ) = π m (Ω n T E ( F )) . Deﬁnition 9.1. T a ke E ∈ S H S 1 ( k ) and let n ≥ 1 b e an integer. An n -fold T - delo oping of E is a n an ob ject ω − n T E o f S H S 1 ( k ) and an iso morphism ι n : E → Ω n T ω − n T E in S H S 1 ( k ). Given an n -fold T -delo o ping of E , ι n : E → Ω n T ω − n T E , the map T r F ( w ) ∗ for Ω n T ω − n T E induces the “transfer map” ι − 1 n ◦ T r F ( w ) ∗ ◦ ι n : π m ( E ( w )) → π m ( E ( F )) , which we wr ite simply as T r F ( w ) ∗ . R emarks 9 .2 . 1. The tra nsfer map T r F ( w ) ∗ : π m ( E ( w )) → π m ( E ( F )) may po s si- bly dep end on the choice o f n -fo ld T -delo oping, we do not have an example, how ever. 2. An n − b -fold T - delo oping of E g ives rise to a n n -fold T -delo o ping of Ω b G m E . Thu s, via the adjunction iso morphism Π a,b E ∼ = π a Ω b G m E we hav e a tr ansfer map T r F ( w ) ∗ : Π a,b E ( w ) → Π a,b E ( F ) for w a closed po int of V nF , separa ble over F . 3. If E = Ω ∞ T E for some E ∈ S H ( k ), then E admits canonical n - fold T -delo opings , namely ω − n T E := Ω ∞ T Σ n T E . Indeed, in S H ( k ), Σ T is the inv ers e to Ω T and Ω ∞ T commutes with Ω T . F or a morphism ϕ : Σ ∞ s w + → E , we hav e the susp ension Σ n T ϕ : Σ n T Σ ∞ s w + → Σ n T E , the comp o sition Σ n T ϕ ◦ Σ ∞ s T r F ( w ) ∗ : Σ n T Σ ∞ s Spec F + → Σ n T E and the adjoint morphis m (Σ n T ϕ ◦ Σ ∞ s T r F ( w ) ∗ ) ′ : Σ ∞ s Spec F + → Ω n T Σ n T E . Suppo se w e hav e an n -fold de-lo oping o f E , ι n : E → Ω n T ω − n T E . This gives us the adjoint ι ′ n : Σ n T E → ω − n T E and Ω n T ι ′ n : Ω n T Σ n T E → Ω n T ω − n T E . Let δ n : E → Ω n T Σ n T E be the unit for the adjunction. Lemma 9.3 . 1. ι n = Ω n T ι ′ n ◦ δ n 2. ι − 1 n ◦ Ω n T ι ′ n ◦ (Σ n T ϕ ◦ Σ ∞ s T r F ( w )) ′ = T r F ( w ) ∗ ( ϕ ) . Pr o of. The tw o assertions follow fr o m the universal pro p er ty of adjunction.  30 MARC LE VINE Before proving o ur main results, we show that the tr ansfer maps resp ect the Postnik ov ﬁltra tion F ∗ T ate π m E . Lemma 9. 4 . Su pp ose E admits an n -fold T -delo oping ι n : E → Ω n T ω − n T E . Then for e ach ﬁnitely gener ate d ﬁeld F over k and e ach close d p oint w ∈ A n F sep ar able over F , we have T r F ( w ) ∗ ( F q T ate π m E ( w )) ⊂ F q T ate π m E ( F ) . Pr o of. T ake q ≥ 0, and let τ q : f q E → E b e the ca nonical mor phism. As ab ov e, let ι ′ n : Σ n T E → ω − n T E be the adjo int of ι n and let δ n : E → Ω n T Σ n T E be the unit of the adjunction. By lemma 9.3, we ha ve the factoriza tion o f ι n as E δ n − → Ω n T Σ n T E Ω n T ι ′ n − − − → Ω n T ω − n T E . This gives us the co mm utative diag ram f q E δ n   τ q / / E ι n   Ω n T Σ n T f q E τ ′ q / / Ω n T ω − n T E , where τ ′ q := Ω n T ι ′ n ◦ Ω n T Σ n T τ q . Since ι n : E → Ω n T ω − n T E is an isomorphism, the comp osition ι n ◦ τ q : f q E → Ω n T ω − n T E satisﬁes the universal pro p erty of f q Ω n T ω − n T E → Ω n T ω − n T E . By [6, theorem 7.4.1], Ω n T Σ n T f q E is in Σ q T S H S 1 ( k ), hence there is a canonical morphism θ : Ω n T Σ n T f q E → f q E extending our ﬁrst diagra m to the commutativ e diag r am f q E ι n   τ q / / E ι n   Ω n T Σ n T f q E τ ′ q / / θ O O Ω n T ω − n T E . ι − 1 n O O Using the universal pro p e r ty of τ q , we see that θ ◦ ι n = id f q E , i.e., Ω n T Σ n T f q E = f q E ⊕ R and the restriction of τ ′ q to R is the zer o map. W e deﬁne the tra nsfer map T r F ( w ) ∗ : π m f q E ( w ) → π m f q E ( F ) by using the tra ns fer map for Ω n T Σ n T f q E and this splitting. The second diagr am th us gives rise to the commutativ e diagram π m f q E ( w ) T r F ( w ) ∗   τ q / / π m E ( w ) T r F ( w ) ∗   π m f q E ( F ) τ q / / π m E ( F ) , which yields the result.  THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 31 R emark 9.5 . One ca n deﬁne tra nsfer maps in a more general s e tting, that is, for a closed p oint w ∈ A n F and any c hoice of par ameters for m w ⊂ O A n ,w . The same pro of as used for lemma 9 .4 shows that these mo r e general tra nsfer maps r esp ect the ﬁltration F ∗ T ate π m E . Theorem 9.6 . L et E ∈ Spt ( k ) b e ﬁbr ant, and let F b e a ﬁeld extensions of k . 1. F or e ach w = ( w 0 , . . . , w n ) ∈ V n ( F ) , and e ach ρ ∈ π 0 Ω n G m E ( F ) , the element ρ ◦ Σ ∞ s ([ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ]) : Σ ∞ s Spec F + → E is in F n T ate π 0 E ( F ) . 2. Supp ose that E admits an n -fold T -delo oping ι n : E → Ω n T ω − n T E . The n for w = ( w 0 , . . . , w n ) a close d p oint of V nF , sep ar able over F , and ρ w ∈ π 0 Ω n G m E ( w ) (9.1) T r F ( w ) ∗ [ ρ w ◦ Σ ∞ s ([ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ])] is in F n T ate π 0 E ( F ) . 3. Supp ose that E admits an n -fold T -delo oping ι n : E → Ω n T ω − n T E . and that Π a, ∗ E = 0 for al l a < 0 . Su pp ose further that F is p erfe ct. Then F n T ate π 0 E ( F ) is gener ate d by elements of the form (9.1) , as w runs over close d p oint of V nF and ρ w over elements of π 0 Ω n G m E ( w ) . Pr o of. (1) fo llows directly fr om prop ositio n 3.4 and pro po sition 7.1, no ting that the isomorphism Ω n G m E ∼ = Σ n s Ω n T E gives the identiﬁcation π − n Ω n T E ( w ) ∼ = π 0 Ω n G m E ( w ) ∼ = Hom S H S 1 ( k ) (Σ ∞ s w + ∧ G ∧ n m , E ) . F or (2), the fact that this element is in F n T ate π 0 ( E ( F )) follows from (1) and lemma 9.4. F or (3), that is, to see that these ele ments ge nerate, take one o f the g enerator s γ := ξ w ◦ Σ ∞ s Q F ( w ) of F n T ate π 0 E ( F ), as given by pro p o sition 3.4, that is, w is a closed p oint of V nF and ξ w is in π − n (Ω n T E ( w )) = π 0 (Ω n G m E ( w )). Since F is p erfect, w is separable ov er F . T ake the n -fold T -susp ensio n o f γ Σ n T γ : Σ ∞ s (Σ n T Spec F + ) → Σ n T E , giving b y adjunction and co mpo sition with Ω n T ( ι ′ n ) the morphism Ω n T ( ι ′ n ) ◦ (Σ n T γ ) ′ : Σ ∞ s Spec F + → Ω n T ω − n E . It follows from the universal prop erties of adjunction tha t (Σ n T γ ) ′ = δ n ◦ γ , hence b y lemma 9.3 we ha ve (9.2) Ω n T ( ι ′ n ) ◦ (Σ n T γ ) ′ = Ω n T ( ι ′ n ) ◦ δ n ◦ γ = ι n ◦ γ . W rite Σ n T γ = (Σ n T ξ w ) ◦ (Σ ∞ s Σ n T Q F ( w )) . By prop osition 8.2 we have Σ n T Q F ( w ) = Σ n s (Σ n T [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ] ◦ T r F ( w )) , and th us Σ n T γ = Σ n T ( ξ w ◦ Σ n s [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ]) ◦ Σ n s T r F ( w ) . 32 MARC LE VINE Using (9 .2) and lemma 9.3, we have ι n ◦ γ = Ω n T ( ι ′ n ) ◦ (Σ n T γ ) ′ = Ω n T ( ι ′ n ) ◦ [Σ n T ( ξ w ◦ Σ n s [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ]) ◦ Σ n s T r F ( w )] ′ = ι n ◦ T r F ( w ) ∗ ( ξ w ◦ Σ n s [ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ]) , or γ = T r F ( w ) ∗ [ ρ w ◦ Σ ∞ s ([ − w 1 /w 0 ] ∧ F . . . ∧ F [ − w n /w 0 ])] .  W e now assume that E = Ω ∞ T E for so me ﬁbrant T -sp ectrum E ∈ Spt T ( k ). Let S k denote the motivic sphere sp ectrum in Spt T ( k ), that is, S k is a ﬁbrant mo del of the suspe ns ion sp ectrum Σ ∞ T S 0 k . W e pro ceed to re-interpret theo rem 9.6 in terms of the canonical actio n of π 0 Ω ∞ T S k ( F ) on π 0 E ( F ), which w e no w recall, along with some of the fundamen tal c omputations of Morel relating the Grothendieck-Witt group with endomorphisms of the motivic sphere sp ectrum. W e reca ll the Milnor- Witt sheav es of Morel, K M W n (see [8, section 2] for de- tails). The graded shea f K M W ∗ := ⊕ n ∈ Z K M W n has struc tur e o f a Nisnevich sheaf of ass o ciative g raded ring s. F or a ﬁnitely ge ne r ated ﬁeld F over k , the grade d ring K M W ∗ ( F ) := K M W ∗ ( F ) has generato rs [ u ] in degr ee 1, for u ∈ F × , and an additional genera tor η in degree − 1, with relations • η [ u ] = [ u ] η • [ u ][1 − u ] = 0 (Stein b erg r elation) • [ uv ] = [ u ] + [ v ] + η [ u ][ v ] • η (2 + η [ − 1]) = 0. F or later use , w e note the following r esult: Lemma 9.7. L et F b e a ﬁeld, u 1 , . . . , u n ∈ F × with P i u i = 1 . The n [ u 1 ] · . . . · [ u n ] = 0 in K M W 0 ( F ) . Pr o of. W e use a num b er of relations in K M W ∗ ( F ), pro ved in [8, lemma 2.5, 2 .7]. F or u ∈ F × we let denote the elemen t 1 + η [ u ] ∈ K M W 0 ( F ). W e hav e the following relatio ns, for a, b ∈ F × , i) K M W 0 ( F ) is central in K M W ∗ ( F ) ii) [ a ][1 − a ] = 0 for a 6 = 1 iii) [ ab ] = [ a ] + [ b ] iv) [ a − 1 ] = − [ a ] v) [ a ][ − a ] = 0 vi) [1 ] = 0. These yield the additiona l relation vii) [ a ][ − a − 1 ] = 0. This follows by noting that [ a ][ − a − 1 ] = [ a ]( − < − a − 1 > [ − a ]) (iv) = ( − < − a − 1 > )[ a ][ − a ] (i) = 0 (v) W e prov e the lemma by induction on n , the case n = 1 b eing the relation (vi), the case n = 2 the Steinberg rela tion (ii). Induction reduces to showing [ u ][ v ] = [ u + v ][ − v /u ] for u + v 6 = 0 THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 33 (in case u + v = 0 w e use (v) to contin ue the induction). F or this, we hav e [ u ][ v ] = [ u ][ v ] + [ u ][ − u − 1 ] (vii) = [ u ][ v ] + [ u ] [ − u − 1 ] (i) = [ u ][ − v /u ] (iii) = [ u ][ − v /u ] + [1 + v /u ][ − v /u ] (ii) = [ u + v ][ − v / u ] (iii)  F or u ∈ F × , let denote the quadratic form uy 2 in the Grothendieck-Witt group GW ( F ). Sending [ u ] η to − 1 extends to an is omorphism [8, lemma 2.10] ϑ 0 : K M W 0 ( F ) → GW( F ) . In addition, for n ≥ 1 , the imag e of × η n : K M W n ( F ) → K M W 0 ( F ) is an ideal η n K M W n ( F ) in K M W 0 ( F ) and ϑ 0 maps η n K M W n ( F ) isomorphica lly onto the idea l I ( F ) n , where I ( F ) ⊂ GW ( F ) is the a ug ment ation ideal of quadra tic forms of virtual rank zero. F or each u ∈ F × , we have the corres po nding mo rphism [ u ] : Sp e c F + → G m W e have a s well the canonica l pro jection η ′ : A 2 \ { 0 } → P 1 . Using a construction similar to the o ne w e used to show that P 2 /H ∼ = Σ 2 s G ∧ 2 m , one constructs a canonica l isomorphism in H • ( k ), ( A 2 \ { 0 } , 1) ∼ = Σ 1 s G ∧ 2 m , and thus η ′ yields the morphism η : Σ 1 s G ∧ 2 m → Σ 1 s G m in H • ( k ). F or E , F ∈ Spt S 1 ( k ), let Hom ( E , F ) deno te the Nisnevich s he a f asso ciated to the presheaf U 7→ Hom S H S 1 ( k ) ( U + ∧ E , F ) . W e have the fundamental theo r em of Morel: Theorem 9. 8 ([8, cor ollary 3.43]) . Supp ose char k 6 = 2 . L et m, p, q ≥ 0 , n ≥ 2 b e inte gers. Then sendi ng [ u ] ∈ K M W 1 ( F ) to the morphism [ u ] and sending η ∈ K M W − 1 ( F ) t o the morphism η yields isomorphisms Hom H • ( k ) (Spec F + ∧ S m ∧ G ∧ p m , S n ∧ G ∧ q m ) ∼ = ( 0 if m < n K M W q − p ( F ) if m = n and q > 0 . As we will be relying o n Mo r el’s theorem, we assume for the res t of the pap er that the characteristic o f k is diﬀerent from t wo. Passing to the S 1 -stabilization, theore m 9.8 gives Π 0 ,p Σ ∞ s G ∧ q m = K M W q − p for p ≥ 0 , q ≥ 1 , (9.3) Π a,p Σ ∞ s G ∧ q m = 0 for p ≥ 0 , q ≥ 1 , a < 0 . Passing to the T -stable setting, Morel’s theor em gives π p,p Σ q G m S k ∼ = K M W q − p for p, q ∈ Z (9.4) π a + p,p Σ q G m S k = 0 for p, q ∈ Z , a < 0 . Comp osition of mor phisms gives us the (rig ht ) action of the bi- graded sheaf of rings π ∗ , ∗ S k on π ∗ , ∗ E for each T -sp ectrum E , and thus, the action o f K M W −∗ on 34 MARC LE VINE π ∗ , ∗ E . If we let E be the S 1 -sp ectrum Ω ∞ T E , then Π a,b E = π a + b,b E for all b ≥ 0. Thu s, via lemma 2.2(2) we th us hav e the rig ht multip lication Π a,b − m E ⊗ K M W − m → Π a,b E . Let I ⊂ K M W 0 be the sheaf of a ugmentation ideals. The K M W −∗ -mo dule structure on Π a, ∗ E gives us the ﬁltra tion F M W n Π a,b E of Π a,b E , deﬁned b y F M W n Π a,b E := im[Π a,n E ⊗ K M W n − b → Π a,b E ]; n ≥ 0 . Lemma 9. 9. Supp ose E = Ω ∞ T E for some E ∈ S H ( k ) . F or inte gers n, b, p ≥ 0 , with n − p, b − p ≥ 0 , the adjunction isomo rphism Π a,b E ∼ = Π a,b − p Ω p G m E induc es an isomorphism F M W n Π a,b E ∼ = F M W n − p Π a,b − p Ω p G m E . Pr o of. This follows easily from the fact that the adjunction isomorphism Π a, ∗ E ∼ = Π a, ∗− p Ω p G m E is a K M W ∗ -mo dule isomorphism.  Deﬁnition 9.10. Let E = Ω ∞ T E for so me E ∈ S H ( k ), F a ﬁeld e xtension of k . T ake in tegers a, b , n with n, b ≥ 0. F ollowing rema rk 9.2(2), we hav e the transfer maps T r F ( w ) : Π a,b E ( F ( w )) → Π a,b E ( F ) for each closed p oint w ∈ V nF , separa ble over F . 1. Let F M W b T r n Π a,b E ( F ) denote the subgr oup of Π a,b E ( F ) generated by elemen ts of the form T r F ( w ) ∗ ( x ); x ∈ F M W n Π a,b E ( F ( w )) as w runs ov er closed p oints o f V nF , separable ov er F . 2. Let [Π a,b E · I n ] b T r ( F ) denote the subgro up of Π a,b E ( F ) generated by elemen ts of the form T r F ( w ) ∗ ( x · y ); x ∈ Π a,b E ( F ( w )) , y ∈ I ( F ( w )) n , as w runs ov er closed p oints o f V nF , separable ov er F . R emark 9.11 . It fo llows directly from the deﬁnitions that, for w a close d p oint o f V nF , x ∈ K M W n − b ( F ), y ∈ Π a,n E ( F ( w )), w e hav e T r F ( w ) ∗ ( y · p ∗ x ) = T r F ( w ) ∗ ( y ) · x, where p ∗ x ∈ K M W n − b ( F ( w )) is the extension of s c alars of of x . In particular, [Π a,b E · I n ] b T r ( F ) is a K M W 0 ( F )-submo dule of Π a,b E ( F ) c o ntaining Π a,b E ( F ) I ( F ) n . Theorem 9.12. L et k b e a p erfe ct ﬁeld of char acteristic 6 = 2 . L et E = Ω ∞ T E for some E ∈ S H ( k ) with Π a,b E = 0 for al l a < 0 , b ≥ 0 . L et n > p ≥ 0 b e inte gers and let F b e a p erfe ct ﬁeld ex t ension of k . Then F n T ate Π 0 ,p E ( F ) = F M W b T r n Π 0 ,p E ( F ) . F or p ≥ n ≥ 0 , we have the identity of she aves F n T ate Π 0 ,p E = Π 0 ,p E . THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 35 Pr o of. Fir s t suppo s e n > p . By lemma 2.2 and lemma 9.9, w e reduce to the case p = 0 . The fact that we ha ve an inclusion o f K M W 0 ( F )-submo dules of Π 0 , 0 E ( F ), F n T ate Π 0 , 0 E ( F ) ⊂ F M W b T r n Π 0 , 0 E ( F ) , follows fro m theorem 9.6. Indeed, as F is p er fect, ea ch element of the form (9.1) is of the fo rm T r F ( w )( ρ w · z ), with ρ w ∈ Π 0 ,n E ( w ), z ∈ K M W n ( F ( w )), hence in F M W b T r n Π 0 , 0 E ( F ). T o show the other inclusion, it suﬃces b y lemma 9.4 and theorem 9.6 to show that, for each ﬁeld K ﬁnitely genera ted ov er k , the elements [ − u 1 /u 0 ] · . . . · [ − u n /u 0 ], with ( u 0 , . . . , u n ) ∈ V n ( K ), genera te K M W n ( K ) as a mo dule over K M W 0 ( K ). W e no te that the ma p s ending ( u 0 , . . . , u n ) to (1 /u 0 , − u 1 /u 0 , . . . , − u n /u 0 ) is an inv olution of V n , so it suﬃces to show that the elements [ u 1 ] · . . . · [ u n ], with ( u 0 , . . . , u n ) ∈ V n ( K ), genera te. Sending ( u 0 , . . . , u n ) to ( u 1 , . . . , u n ) identiﬁes V n with ( A 1 \ { 0 } ) n \ H . B ut by deﬁnition K M W n ( K ) is ge nerated by elements [ u 1 ] · . . . · [ u n ] with u i ∈ K × ; it thus suﬃces to show that [ u 1 ] · . . . · [ u n ] = 0 in K M W n ( K ) if P i u i = 1; this is lemma 9.7. If p ≥ n ≥ 0 , the universal prop erty of f n E → E g ives us the is o morphism for U ∈ Sm /k Hom S H S 1 ( k ) (Σ ∞ s Σ p G m U + , E ) ∼ = Hom S H S 1 ( k ) (Σ ∞ s Σ p G m U + , f n E ) , since Σ ∞ s Σ p G m U + is in Σ p T S H S 1 ( k ) for U ∈ Sm /k . As these groups of morphisms deﬁne the presheaves who se resp ective sheaves are Π 0 ,p E ( F ) and Π 0 ,p f n E , the map Π 0 ,p f n E → Π 0 ,p E is an isomorphism, hence F n T ate Π 0 ,p E = Π 0 ,p E .  R emark 9 .13 . The rea der may ob ject that the collection of transfer maps used to deﬁne F M W b T r n Π 0 ,p E ( F ) is rather artiﬁcial. Howev er , the fa ct that the g eneral transfer maps mentioned in r emark 9.5 resp ect the ﬁltration F ∗ T ate π m E , together with theorem 9.12, shows that, if we were to a llow arbitra r y trans fer maps in our deﬁnition of F M W b T r n Π 0 ,p E ( F ), we would arrive at the same subgroup of Π 0 ,m E ( F ). Our main res ult for a T -sp ectrum, theorem 2, follows easily fr om theorem 9.1 2: Pr o of of t he or em 2. Using lemma 2.2, we reduce to the case p = 0. Essentially the same ar gument a s used at the end of the pro of of theorem 9.12 pr ov es the part of theorem 2 for n ≤ 0. If n > 0, then for b ≥ 0 , w e hav e π a,b E ∼ = π a,b Ω ∞ T E (lemma 2.2) π a,b f n E ∼ = π a,b Ω ∞ T f n E ∼ = π a,b f n Ω ∞ T E (lemma 2.2) and (2.1) Thu s, in case n > 0, theorem 2 for E is equiv alent to theor e m 9.12 for Ω ∞ T E , completing the pro of.  Finally , we can pr ov e our main r esult for the motivic sphere sp ectr um, theo rem 1. Let E = Σ q G m S k . Then Morel’s isomor phism (9.4 ) and lemma 2.2 give Π a,b Ω ∞ T E = ( K M W q − b for a = 0 , b ≥ 0 0 for a < 0 , b ≥ 0 . 36 MARC LE VINE Theorem 9.1 4. L et k b e a p erfe ct ﬁeld of char acteristic 6 = 2 . 1. F or al l n > p ≥ 0 , q ∈ Z , and al l p erfe ct ﬁeld extensions F of k , we have F n T ate Π 0 ,p Ω ∞ T Σ q G m S k ( F ) = K M W q − p ( F ) I ( F ) N ⊂ K M W q − p ( F ) , wher e N = N ( n − p, n − q ) := max(0 , min( n − p, n − q )) . In p articular, F n T ate π 0 , 0 S k ( F ) = I ( F ) n ⊂ GW ( F ) . 2. F or n ≤ p , we have the identity of she aves F n T ate Π 0 ,p Ω ∞ T Σ q G m S k = K M W q − p . 3. In c ase k has char acteristic zer o, we have the identity of she aves F n T ate Π 0 ,p Ω ∞ T Σ q G m S k = K M W q − p I N ⊂ K M W q − p . with N as ab ove. Pr o of. Let N b e as deﬁned in the statement of the theorem. W e ﬁrs t note (3) follows from (1), in fact, from (1) for all ﬁelds extensions F ﬁnitely generated ov er k . Indeed, F n T ate Π 0 ,p Ω ∞ T Σ q G m S k is the image o f the map Π 0 ,p f n Ω ∞ T Σ q G m S k → Π 0 ,p Ω ∞ T Σ q G m S k induced by the ca nonical mor phis m f n Ω ∞ T Σ q G m S k → Ω ∞ T Σ q G m S k . By results of Morel [9, theor em 3 and lemma 5], b oth homotopy sheav es a re strictly A 1 -inv ar iant sheav e s of ab elia n g roups. But the categor y of stric tly A 1 -inv ar iant sheav es of ab elian groups is a belia n [9 , lemma 6 .2.13], hence F n T ate Π 0 ,p Ω ∞ T Σ q G m S k is a lso strictly A 1 -inv ar iant. It follows, e.g., from Morel’s isomorphis m π 0 Ω ∞ T Σ m G m S ∼ = π − m, − m S ∼ = K M W m that the sheaves K M W m are s trictly A 1 -inv ar iant; as K M W q − p I N is the ima ge of the map × η M : K M W q − p + M → K M W q − p , where M = N if q − p ≥ 0, M = p − q + N if q − p < 0, it follows that K M W q − p I N is strictly A 1 -inv ar iant as well. Our ass ertion follows from the fact that a strictly A 1 -inv ar iant s heaf F is zero if and only F ( k ( X )) = 0 for all X ∈ Sm /k , which in turn is an easy conseq uence o f [11, lemma 3.3.6 ]. Next, supp ose n − p ≤ 0. Then N = 0 and F n T ate Π 0 ,p Ω ∞ T Σ q G m S k = F n − p T ate Π 0 , 0 Ω p G m Ω ∞ T Σ q G m S k (lemma 2.2) = Π 0 , 0 Ω p G m Ω ∞ T Σ q G m S k ( n − p < 0) = Π 0 ,p Ω ∞ T Σ q G m S k (adjunction) = K M W q − p (Morel’s theorem) proving (2); we ma y thu s assume n − p > 0. By (9.4), w e may apply theorem 9.1 2, which tells us that F n T ate Π 0 ,p Ω ∞ T Σ q G m S k ( F ) is the subgroup of Π 0 ,p Ω ∞ T Σ q G m S k ( F ) = K M W q − p ( F ) g enerated by elements of the form T r F ( w ) ∗ ( y · x ) with y ∈ Π 0 ,n Ω ∞ T Σ q G m S k ( F ( w )) = K M W q − n ( F ( w )) x ∈ K M W n − p ( F ( w )) . THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 37 Suppo se that n − q < 0, so N = 0. Then q − n ≥ 0 and n − p > 0, and thus the pro duct map µ n − p,q − n : K M W n − p ( F ( w )) ⊗ K M W q − n ( F ( w )) → K M W q − p ( F ( w )) = Π 0 ,p Ω ∞ T Σ q G m S k ( F ( w )) is surjective. Since the map T r F ( w ) is an iso morphism for w ∈ V n ( F ), we see that F n T ate Π 0 ,p Ω ∞ T Σ q G m S k ( F ) = Π 0 ,p Ω ∞ T Σ q G m S k ( F ) . Suppo se n − q ≥ 0. Then × η n − q : K M W 0 ( F ( w )) → K M W q − n ( F ( w )) is surjective. If n − p ≥ n − q , then the ima ge of µ n − p,q − n is the same a s the imag e of the triple pro duct K M W q − p ( F ( w )) ⊗ K M W n − q ( F ( w )) ⊗ K M W q − n ( F ( w )) → K M W q − p ( F ( w )); as the image of µ n − q,q − n : K M W n − q ( F ( w )) ⊗ K M W q − n ( F ( w )) → K M W 0 ( F ( w )) is I ( F ( w )) n − q , we see that the image of µ n − p,q − n is K M W q − p ( F ( w )) I ( F ( w )) n − q and th us F n T ate Π 0 ,p Ω ∞ T Σ q G m S k ( F ) = [Π 0 ,p Ω ∞ T Σ q G m S k I N ] b T r ( F ) . Similarly , if n − q ≥ n − p , then the image of µ n − p,q − n is the same as the image of the triple pro duct K M W q − p ( F ( w )) ⊗ K M W n − p ( F ( w )) ⊗ K M W p − n ( F ( w )) → K M W q − p ( F ( w )) which is K M W q − p ( F ( w )) I ( F ( w )) n − p . Thus F n T ate Π 0 ,p Ω ∞ T Σ q G m S k ( F ) = [Π 0 ,p Ω ∞ T Σ q G m S k I N ] b T r ( F ) in this case as well. Thu s, to complete the pro of, it s uﬃces to show that, for w a closed po int of V nF , and N ≥ 0 an integer, w e hav e (9.5) T r F ( w ) ∗  K M W q − p ( F ( w )) I ( F ( w )) N  ⊂ K M W q − p ( F ) I ( F ) N . First s uppo se that q − p ≥ 0 . T a ke a c losed po int w ∈ V nF and elements x 1 , . . . , x N ∈ F ( w ) × , y ∈ K M W q − p ( F ( w )). W e have T r F ( w ) ∗ ( y · [ x 1 ] η · . . . · [ x N ] η ) = T r F ( w ) ∗ ( y · [ x 1 ] · . . . · [ x N ] η N ) = T r F ( w ) ∗ ( y · [ x 1 ] · . . . · [ x N ]) · η N . where we use r emark 9.11 in the la st line. Since q − p ≥ 0, K M W q − p ( F ) I ( F ) N is the image in K M W q − p ( F ) of the map − × η N : K M W q − p + N ( F ) → K M W q − p ( F ) , which veriﬁes (9.5). In case q − p < 0, write y = y 0 η p − q , with y 0 ∈ K M W 0 ( F ( w )). As ab ove, we ha ve T r F ( w ) ∗ ( y · [ x 1 ] η · . . . · [ x N ] η ) = T r F ( w ) ∗ ( y 0 · [ x 1 ] · . . . · [ x N ]) · η p − q + N , which is in η p − q · [ K M N W N ( F ) η N ] = K M W q − p ( F ) I ( F ) N , as desired.  Theorem 9.1 4 yields the main res ult for the S 1 -sp ectra Σ ∞ s G ∧ q m by using the S 1 -stable consequences of Morel’s unstable computations, theorem 9.8. 38 MARC LE VINE Corollary 9.15. L et k b e a p erfe ct ﬁeld of char acteristic 6 = 2 . 1. F or al l n > p ≥ 0 , q ≥ 1 , and al l p erfe ct ﬁ eld ex tensions F of k , we have F n T ate Π 0 ,p Σ ∞ s G ∧ q m ( F ) = K M W q − p ( F ) I ( F ) N ( n − p,n − q ) ⊂ K M W q − p ( F ) , with N ( n − p, n − q ) as in the or em 9.14. 2. F or n ≤ p , we have F n T ate Π 0 ,p Σ ∞ s G ∧ q m = Π 0 ,p Σ ∞ s G ∧ q m . 3. If char k = 0 , we have t he identity of she aves F n T ate Π 0 ,p Σ ∞ s G ∧ q m = K M W q − p I N ( n − p,n − q ) ⊂ K M W q − p . Pr o of. As in the pr o of of theorem 9.1 4, it suﬃces to prov e (1). The main po int is that Mor el’s uns table co mputations show that the G m -stabilization map Hom S H S 1 ( k ) (Σ m s Σ ∞ s G ∧ p m ∧ Spec F + , Σ ∞ s G ∧ q m ) → Hom S H S 1 ( k ) (Σ m s Σ ∞ s G ∧ p +1 m ∧ Spec F + , Σ ∞ s G ∧ q +1 m ) is an isomorphism for all m ≤ 0, p ≥ 0 a nd q ≥ 1. Let E ( p, q ) = Ω p G m Σ ∞ s G ∧ q m , and let E ( q − p ) = Ω ∞ T Σ − p G m Σ ∞ T G ∧ q m = Ω ∞ T Σ q − p G m S k . Then π a E ( p, q ) = Π a,p Σ ∞ s G ∧ q m . Thu s Π a, ∗ E ( p, q ) = 0 for m < 0 and so we may apply pro p osition 3.4 to give generator s of the form ξ w ◦ Σ ∞ s Q F ( w ) for F n − p T ate Π 0 , 0 Ω p G m Σ ∞ s G ∧ q m ( F ) = F n T ate Π 0 ,p Σ ∞ s G ∧ q m ( F ) . But ξ w is in π − n + p Ω n − p T E ( p, q )( w ) = π 0 ,n − p E ( p, q )( w ) . Similarly , we ha ve generato rs ξ ′ w ◦ Σ ∞ s Q F ( w ) for F n − p T ate π 0 E ( p − q )( F ), with ξ ′ w ∈ π 0 ,n − p E ( p − q )( w ) . But the stabilization map π 0 ,n − p E ( p, q )( w ) → π 0 ,n − p E ( p + 1 , q + 1)( w ) is an isomor phism, and hence we hav e an is o morphism from the ge ne r ators for F n − p T ate π 0 E ( p, q )( F ) to the generator s for F n − p T ate π 0 E ( q − p )( F ) = lim − → m F n − p T ate π 0 E ( p + m, q + m )( F ) . As the map π 0 E ( p, q )( F ) → π 0 E ( q − p )( F ) = K M W q − p ( F ) is an isomorphism, it follows that the surjection F n − p T ate π 0 E ( q − p )( F ) → F n − p T ate π 0 E ( q − p ) . is an isomorphism as well. By theorem 9.14, w e hav e F n − p T ate π 0 E ( q − p ) = K M W q − p ( F ) I ( F ) N ⊂ K M W q − p ( F ) , completing the pro of.  THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 39 Theorem 9.14 also gives us the T -stable version Corollary 9.16. L et k b e a p erfe ct ﬁeld of char acteristic 6 = 2 . F or n, p, q ∈ Z , and F a p erfe ct ﬁeld ex tensions of k , we have F n T ate π p,p Σ q G m S k ( F ) = K M W q − p ( F ) I ( F ) N ( n − p,n − q ) ⊂ K M W q − p ( F ) F or n ≤ p , we have F n T ate π p,p Σ q G m S k = K M W q − p . If char k = 0 , we have F n T ate π p,p Σ q G m S k = K M W q − p I N ( n − p,n − q ) ⊂ K M W q − p . Pr o of. Using lemma 2.2 and lemma 9.9 as in the pro of of theorem 9.12 w e hav e F n T ate π p,p Σ q G m S k = F n − p + r T ate π r,r Σ q − p + r G m S k for all integers r . As our a ssertion is also stable under this shift op era tion, we may assume that p , q ≥ 0. W e note that S k is in S H ef f ( k ), hence so a re all Σ q G m S k for q ≥ 0, and thus F n T ate π p,p Σ q G m S k = π p,p Σ q G m S k for n < 0, p, q ≥ 0. The truncation functors f n , n ≥ 0, on S H ( k ) and S H S 1 ( k ) commute with Ω ∞ T , and π a,p Ω ∞ T E = π a,p E for E ∈ S H ( k ), p ≥ 0. This reduces us to computing c o mputing F n T ate π p,p Ω ∞ T Σ q G m S k for n, p, q ≥ 0, whic h is theorem 9.14.  10. Epilog: Convergence questions V o evodsk y has stated a conjecture [14, conjecture 13 ] that would imply that for E = Σ ∞ T X + , X ∈ Sm /k , the T a te Postniko v tower is conv ergent in the fo llowing sense: for all a, b, n ∈ Z , one has ∩ m F m T ate π a,b f n E = 0 . Our computation of F n T ate π p,p Σ ∞ T G ∧ q m gives some evidence for this conv ergence con- jecture. Prop ositi o n 10.1. L et k b e a p erfe ct ﬁeld with char k 6 = 2 . F or al l p, q ≥ 0 , and al l p erfe ct ﬁeld ext ensions F of k , we have ∩ n F n T ate π p,p Σ ∞ T G ∧ q m ( F ) = 0 . Pr o of. In lig h t of theorem 9.14, the a ssertion is that the I ( F )-adic ﬁltration on K M W q − p ( F ) is separated. B y [10, th´ eor` eme 5.3 ], for m ≥ 0, K M W m ( F ) ﬁts in to a cartesian square of GW( F )-mo dules K M W m ( F ) / /   K M m ( F ) P f   I ( F ) m q / / I ( F ) m /I ( F ) m +1 , where K M m ( F ) is the Milnor K -group, q is the quotient map and P f is the map sending a sym b ol { u 1 , . . . , u m } to the class of the Pﬁster form <> mo d I ( F ) m +1 . F or m < 0 , K M W m ( F ) is isomorphic to the Witt group of F , W ( F ), that is, the q uo tient of GW( k ) b y the idea l ge ner ated by the h yp erb olic form x 2 − y 2 . Also, the map GW ( F ) → W ( F ) gives an isomorphism of I ( F ) r with its imag e in W ( F ) for all r ≥ 1. Thu s K M W m ( F ) I ( F ) n = ( I ( F ) n ⊂ W ( F ) for m < 0 , n ≥ 0 I ( F ) n + m ⊂ GW ( F ) for m ≥ 0 , n ≥ 1 . 40 MARC LE VINE The f act that ∩ n I ( F ) n = 0 in W ( F ) or equiv a lently in GW( F ) is a theorem of Arason and Pﬁster [1].  R emarks 1 0.2 . 1. The pro of in [10] that K M W m ( F ) ﬁts int o a car tesian square a s ab ov e relies the Milnor co njecture. 2. V oevo dsky’s conjecture [ lo c. cit. ] asserts the convergence for a wider class of ob jects in S H ( k ) than just the T - susp ension spectr a of smo oth k -schemes. The selected class is the triangulated catego ry generated by Σ n T Σ ∞ T X + , X ∈ Sm /k , n ∈ Z and the taking of direct summands. Howev er , as p ointed out to me by Igor Kriz, the conv er gence fails for this lar ger class of ob jects. In fact, ta ke E to b e the Mo ore spectr um S k /ℓ for some prime ℓ 6 = 2 . Since Π a,q S k = 0 for a < 0, pr op o- sition 3.2 sho ws that Π a,q f n S k = 0 for a < 0, and thus we have the right e x act sequence for all n ≥ 0 π 0 , 0 f n S k × ℓ − − → π 0 , 0 f n S k → π 0 , 0 f n E → 0 . In particular, we have F n T ate π 0 , 0 E ( k ) = im ( F n T ate π 0 , 0 S k ( k ) → π 0 , 0 S k ( k ) /ℓ ) = im ( I ( k ) n → GW( k ) /ℓ ) . T ake k = R . Then GW ( R ) = Z ⊕ Z , with virtua l r a nk and v irtual index giving the t wo facto rs. The a ugmentation idea l I ( R ) is th us isomo rphic to Z via the index and it is not hard to see that I ( R ) n = (2 n − 1 ) ⊂ Z = I ( R ). Th us π 0 , 0 E = Z /ℓ ⊕ Z /ℓ and the ﬁltration F n T ate π 0 , 0 E is constant, equal to Z /ℓ = I ( R ) /ℓ , and is therefore not separated. The conv er g ence prop erty is thus not a “tr ia ngulated” o ne in general, and there- fore seems to be quite subtle. How ever, if the I -a dic ﬁltration on GW ( F ) is ﬁnite (po ssibly of v arying length depending o n F ) for all ﬁnitely generated F over k , then o ur computatio ns (at least in characteristic zero) show that the ﬁltr ation F ∗ T ate π p,p Σ ∞ T G ∧ q m is at least lo c ally ﬁnite, and th us has b etter triangulated prop er- ties; in particular, for ℓ 6 = 2, π 0 , 0 ( S k /ℓ ) = Z /ℓ, F n T ate π 0 , 0 ( S k /ℓ ) = 0 for n > 0 , as the aug ment ation ideal in GW( F ) is purely tw o-primar y to r sion, and I π 0 , 0 S k /ℓ = 0. One can therefore ask if V o evodsky’s conv er gence conjecture is true if o ne as- sumes the ﬁniteness of the I ( F )-adic ﬁltration on GW( F ) for all ﬁnitely gener ated ﬁelds F ov e r k . References [1] Arason, J. K. and Pﬁster, A. , Beweis des Kr ullschen Dur chsc hnittsatze s f ¨ ur den Wittring . In v ent . Math. 12 (1971), 173–176. [2] Bousﬁe ld, A. K. and Kan, D. M. Homot opy limits, completions and lo ca lizatio ns . Lecture N otes in Mathematics, 304 . Spri nger-V erlag, Berlin-New Y ork, 1972. [3] Goerss, P . G.; Jardine, J. F., Lo calization theories for sim pl icial pr eshea ves . Canad. J. Math. 50 (1998), no. 5, 1048–1089. [4] Jardine, J. F., Simplicial presheav es , J. Pur e Appl. Algebra 47 (1987), no. 1, 35–87 [5] Jardine, J.F., M otivic symmetric sp ectra , Do c. Math. 5 (2000) , 445–55 3. [6] Levine , M., The homotop y conive au tow er . J T opology 1 (2008) 217–267 . [7] Levine , M., Chow’s moving l emma in A 1 -homotop y theo ry . K -theory 37 (1-2) (2006) 129-209. [8] Morel, F., A 1 -Algebraic top ology ov er a ﬁeld , preprint 19.11.2006. h ttp://www.mathematik.uni-m uenc hen.de/ ∼ morel/A1homoto py .p df [9] Morel, F., The stable A 1 -connectivit y theorems , K -theory , 2005, 35 , pp 1-68. THE SLICE FIL TRA TION AND GR OTHENDIECK-WITT GROUPS 41 [10] Morel, F. Sur les puissances de l’ id´ eal fondamenta l de l’ anneau de Witt . Comment. Math. Helv. 79 (2004), no. 4, 689–703. [11] Morel, F., An i nt ro duction to A 1 -homotop y theory . Contemp or ary developments in algebr aic K -the ory 357–441, ICTP Lect. Notes, XV , Ab dus Salam In t. Cent . Theoret. Phys., T rieste, 2004. [12] Morel, F. and V oevodsky , V., A 1 -homotop y theory of sc hemes , Inst. Hautes ´ Etudes Sci. Publ. Math. 90 (1999), 45–143. [13] Nee man, A, T riang ulated categor ies . Annals of Mathematics Stud ies, 148 . Pri nceton Unive rsity Press, Princeton, NJ, 2001. [14] V o ev odsky , V., Op en problems in the motivic stable homotopy theory . I . Motives, p olylo g - arithms and Ho dge the ory, Part I (Irvine, CA, 1998) , 3–34, In t. Press Lect. Ser., 3 , I, Int . Press, Somerville, MA, 2002. [15] V o ev odsky , V. A p ossible new approach to the motivic spectral sequence for algebraic K - theory , R e c ent pr o gr ess in homotopy the ory (Baltimor e, MD, 2000) 371–379, Conte mp. Math., 29 3 (Amer. Math. So c., Providence, RI, 2002). Universit ¨ at Duisburg-Essen, F akult ¨ at Ma thema tik, Campus Essen, 4 5117 Essen, Ger- many E-mail addr ess : marc.levin e@uni-due. de

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