Slices and Transfers

SLICES AND TRANSFERS MARC LEVINE Abstract. W e study the slice ﬁltration for S 1 -sp ectra, and rais e a num ber of questions regardings its prop erties. Contents Int ro ductio n 1 1. Inﬁnite P 1 -lo op sp ectra 4 2. An example 4 3. Co-transfer 6 4. Co-gro up structure o n P 1 10 5. Slice lo calizations and co-trans fer 13 6. Higher lo ops 24 7. Suppo rts and co-trans fers 25 8. Slices of lo op sp e c tra 29 9. T rans fers on the g eneralized cycle complex 32 10. The F riedlander -Suslin to wer 37 References 38 Introduction V o evodsky [21] has deﬁned an analog of the classical Postnik ov to wer in the setting of motivic stable homotopy theory b y r eplacing the simplicial suspension Σ s := − ∧ S 1 with P 1 -susp ension Σ P 1 := − ∧ P 1 ; we call this construction the motivic Postnikov tower . Let S H ( k ) deno te the mo tivic stable homotopy catego ry of P 1 -sp ectra. One of the main results on motivic Postnik ov tow er in this setting is Theorem 1. F or E ∈ S H ( k ) , the slic es s n E have the natur al structu r e of an H Z -mo dule, and henc e determine obje cts in t he c ate gory of motives D M ( k ) . The statement is a bit impr e cise, as the following expansion will make clear: Ostv ar-R¨ ondigs [17, 18] hav e shown that the homo topy categor y of strict H Z - mo dules is equiv alent to the category of motiv es DM ( k ). Additionally , V o evodsky Date : Octob er 30, 2018. 2000 Mathema tic s Subje ct Classiﬁc ation. Primary 14C25, 19E15; Second ary 19E08 14F42, 55P42. Key wor ds and phr ases. Algebraic cycles, M orel-V oev o dsky stable homotopy category , slice ﬁltration. Researc h s upp orted by the N SF grant D MS-0801220 and the A l exander von Humboldt F oundation. 1 2 MARC LEVINE [21] and the author [11] have shown that the 0th slice o f the sphere sp ectrum S in S H ( k ) is isomorphic to H Z . Thus, for E ∈ S H ( k ), the cano nical S -mo dule struc- ture on E induces a n H Z -mo dule s tr ucture on the slices s n E , in S H S 1 ( k ). This has bee n reﬁned to the mo del c a tegory level by Pelaez [19], sho wing that the slices of a P 1 -sp ectrum E hav e a natural structure of a strict H Z -mo dule, hence are motiv es. Let Spt S 1 ( k ) denote the ca tegory of S 1 -sp ectra. The mo tivic analo g is the category of eﬀective motives over k , D M ef f ( k ). W e consider the motivic Postnik ov tow er in the homotopy categor y o f S 1 -sp ectra, S H S 1 ( k ), and ask the question: (1) is there a ring ob ject in Spt S 1 ( k ), H Z ef f , such tha t the homotopy cate- gory of H Z ef f mo dules is equiv alent to the ca tegory of eﬀective motives D M ef f ( k )? (2) What pro per ties (if any) need an S 1 -sp ectrum E have so that the s lices s n E hav e a natural structure ofhav e a na tural s tr ucture as the Eilenberg- Maclane s pectr um of a ho motopy inv ariant complex of presheaves with transfer? Naturally , if H Z ef f exists as in (1), we are asking the slices in (2) to b e (strict) H Z ef f mo dules. Of co urse, a natural candidate for H Z ef f would be the 0- S 1 - sp ectrum o f H Z , Ω ∞ P 1 H Z , but as far a s I know, this prop erty has not yet b een inv es tigated. As we shall see, the 0- S 1 -sp ectrum o f a P 1 -sp ectrum do es hav e the prop erty that its ( S 1 ) slice s are motiv es, while one can give examples of S 1 -sp ectra fo r which the 0th slice do es not hav e this pro per t y . This suggests a rela tion of the question of the structure of the slices o f an S 1 -sp ectrum with a motivic version of the recognitio n problem: (3) How can one tell if a given S 1 -sp ectrum is an n -fold P 1 -lo op sp ectrum? In this pap er, we prove tw o main res ults ab out the “motivic” s tructure on the slices of S 1 -sp ectra: Theorem 2. Supp ose char k = 0 . L et E b e an S 1 -sp e ctrum. Then for e ach n ≥ 1 , ther e is a tower . . . → ρ ≥ p +1 s n E → ρ ≥ p s n E → . . . → s n E in S H S 1 ( k ) with the fol lowing pr op erties: (1) t he tower is natur al in E . (2) L et ¯ s p,n E b e t he c oﬁb er of ρ ≥ p +1 s n E → ρ ≥ p s n E . Then ther e is a ho motopy invariant c omplex of pr eshe aves with tr ansfers ˆ π p (( s n E ) ( n ) ) ∗ ∈ D M ef f − ( k ) and a natur al isomorphism in S H S 1 ( k ) , E M A 1 ( ˆ π p (( s n G ) ( n ) ) ∗ ) ∼ = ¯ s p,n E , wher e E M A 1 : D M ef f − ( k ) → S H S 1 ( k ) is the Eilenb er g-Maclane sp e ct ru m functor. This result is proven in section 9. One ca n say a bit more ab out the tower appea ring in theorem 2. F or instance, holim p ﬁb( ρ ≥ p s n E → s n E ) is weakly e q uiv alent to z ero, so the s pectr al sequence asso ciated to this tow er is w eakly con vergent. If s n E is globally N -co nnec ted (i.e., there is an N such tha t s n E ( X ) is N -c o nnected for all X ∈ Sm /k ) then the spectr al sequence is strongly conv ergent. SLICES AND TRANSFERS 3 In other words, the higher slices of an a rbitrary S 1 -sp ectrum hav e some so rt of trans fers “ up to ﬁltr ation”. The situation for the 0th slice app ears to b e mor e complicated, but for a P 1 -lo op sp ectrum we have at least the following result: Theorem 3. Supp ose char k = 0 . T ake E ∈ S H P 1 ( k ) . Then for al l m , the homo- topy she af π m ( s 0 Ω P 1 E ) has a natur al s t ructur e of a homotopy invariant she af with tr ansfers. W e actually prov e a more precise r esult (coro llary 8.5) which sta tes that the 0th slice s 0 Ω P 1 E is itself a presheaf with transfers , with v alues in the stable homo topy category S H , i.e., s 0 Ω P 1 E has “transfers up to homo topy”. This raises the questio n: (4) Is there an op erad acting on s 0 Ω n P 1 E which shows tha t s 0 Ω n P 1 E admits transfers up to homotopy and higher homotopies up to some lev el? Part of the motiv ation for this pap er came out of discussions with H ´ el` ene Esna ult concerning the (admittedly v a gue) question: Given a smo oth pro jective v ariety X ov er s o me ﬁeld k , that admits a 0-cycle of degr ee 1 , are there “ motivic” pr op erties of X that lea d to the e x istence o f a k -p oint? The fact that the exis tence of 0- cycles o f degree 1 has something to do with the transfer maps fro m 0-cycle s on X L to zer o-cycles on X , as L runs ov er ﬁnite ﬁeld extensio ns of k , while the lack of a transfer map in gener al appea rs to b e closely related to the subtlet y o f the existence of k -p oints led to our inquiry into the “motivic” nature of the spaces Ω n P 1 Σ n P 1 X + , or rather, their asso ciated S 1 -sp ectra. Notation and con v en tio ns . In this pap er, we will b e pa s sing from the unsta ble motivic (p ointed) homotopy categor y ov er k , H • ( k ), to the ho motopy category of motivic S 1 -sp ectra ov er k , S H S 1 ( k ), via the inﬁnite (simplicial) suspensio n functor Σ ∞ s : H • ( k ) → S H S 1 ( k ) F or a smo oth k -scheme X ∈ Sm /k and a subscheme Y of X (sometimes closed, sometimes op en), we let ( X , Y ) denote the homotopy pus h-out in the diagr am Y / /   X Spec k and as usual wr ite X + for ( X ∐ Sp ec k , Sp ec k ). W e often deno te Sp ec k by ∗ . F or an ob ject S o f H • ( k ), we often use S to denote Σ ∞ s S ∈ S H S 1 ( k ) when the context makes the meaning clear; we a lso use this co n ven tion w hen passing to v ario us lo calizations of S H S 1 ( k ). Regarding the catego ries Spt S 1 ( k ), S H S 1 ( k ) and S H ( k ), we will use the no- tation sp elled out in [11]. In addition to this source, we refer the reader to [8, 14, 15, 18, 18, 21]. F o r details on the categor y D M ef f ( k ), we refer the rea der to [3, 5]. De dic ation . This pa p er is warmly dedicated to Andrei Suslin, who has provided me more inspiration than I can hop e to tell. 4 MARC LEVINE 1. Infinite P 1 -loop spectra W e ﬁrst co nsider the case of the 0- S 1 -sp ectrum of a P 1 -sp ectrum. W e let Ω ∞ P 1 : S H ( k ) → S H S 1 ( k ) Ω ∞ P 1 ,mot : D M ( k ) → D M ef f ( k ) be the (derived) 0- sp e ctrum (resp. 0- complex) functor, let E M A 1 : D M ( k ) → S H ( k ) E M A 1 ef f : D M ef f ( k ) → S H S 1 ( k ) the resp ective Eilenberg -Maclane spe ctrum functors. Theorem 1. 1. Fix an inte ger n ≥ 0 . Then t her e is a functor M ot ef f ( s n ) : S H ( k ) → D M ef f ( k ) and a natu r al isomorphi sm ϕ n : E M A 1 ef f ◦ M ot ef f ( s n ) → s ef f n ◦ Ω ∞ P 1 of functors fr om S H ( k ) to S H S 1 ( k ) .In other wor ds, for E ∈ S H ( k ) , ther e is a c anonic al lifting of the slic e s ef f n (Ω ∞ P 1 E ) to a motive M ot ef f ( s n )( E ) . Pr o of. By Pelaez, there is a functor M ot ( s n ) : S H ( k ) → D M ( k ) and a natural isomorphism Φ n : E M A 1 ◦ M ot ( s n ) → s n i.e., the s lice s n E lifts canonically to a motive M ot ( s n )( E ). Now apply the 0-c omplex functor to deﬁne M ot ef f ( s n ) := Ω ∞ P 1 ,mot ◦ M ot ( s n ) . W e hav e canonical isomorphisms E M A 1 ef f ◦ Ω ∞ P 1 ,mot ◦ M ot ( s n ) ∼ = Ω ∞ P 1 ◦ E M A 1 ◦ M ot ( s n ) ∼ = Ω ∞ P 1 ◦ s n ∼ = s ef f n ◦ Ω ∞ P 1 as desired.  In other words, the slice s of an inﬁnite P 1 -lo op spectrum a re eﬀectiv e motives. 2. An example W e now show that the 0th slice o f an S 1 -sp ectrum is not alwa ys a mo tive. In fact, we will give a n example of an Eilenberg-Ma clane sp ectrum whose 0th slice do es not ha ve transfers. F or this, note the fo llowing: Lemma 2.1 . L et p : Y → X b e a ﬁnite Galois c over in Sm /k , with Galois gr oup G . L et F b e a pr eshe af with tr ansfers on Sm /k . Then the c omp osition p ∗ ◦ p ∗ : F ( Y ) → F ( Y ) SLICES AND TRANSFERS 5 is given by p ∗ ◦ p ∗ ( x ) = X g ∈ G g ∗ ( x ) Pr o of. Letting Γ p ⊂ Y × X b e the g raph of p , and Γ g ⊂ Y × Y the gr a ph of g : Y → Y for g ∈ G , one computes that Γ t p ◦ Γ p = X g ∈ G Γ g , whence the result.  Now let C b e a smooth pro jectiv e curve over k , ha ving no k -r ational p o int s. W e assume that C has genus g > 0, so every map A 1 F → C F ov er a ﬁeld F ⊃ k is constant ( C is A 1 -rigid ). Let Z C be the representable pr esheaf: Z C ( Y ) := Z [Hom Sm /k ( Y , C )] Z C is automatica lly a Nisnevich shea f; since C is A 1 -rigid, Z C is also homotopy inv ar ia nt . F urthermor e Z C is a bir ational sheaf, that is , for each dens e o pen im- mersion U → Y in Sm /k , the restriction map Z C ( Y ) → Z C ( U ) is an isomo rphism. Indeed, it is the same to say that Hom Sm /k ( Y , C ) → Hom Sm /k ( U, C ) is an isomor- phism. If now f : U → C is a morphism, then the closure ¯ Γ in Y × C of the gr a ph of f maps bir ationally to Y via the pr o jection. But since Y is reg ular, each ﬁb er of ¯ Γ → Y is ra tionally connected, hence maps to a p oint of C , and thus ¯ Γ → Y is birational and 1-1. By Zar iski’s main theorem, ¯ Γ → Y is an isomorphism, hence f extends to ¯ f : Y → C , as cla imed. Next, Z C satisﬁes Nisnevic h excision. This is just a general prop er ty of birational sheav es. In fact, let V j V / / f | V   Y f   U j U / / X be an elementary Nisnevich square, i.e., the square is ca rtesian, f is ´ etale, j U and j V are op en immersions, a nd f induces an iso morphism Y \ V → X \ U . W e may assume that U a nd V ar e dense in X and Y . L et F b e a bir a tional s heaf on Sm /k , and apply F to this diag ram. This gives us the squa re F ( X ) j ∗ U / / f ∗   F ( U ) f ∗ | V   F ( Y ) j ∗ V / / F ( V ) As the horizo n tal ar rows are is omorphisms, this square is ho mo topy cartesia n, as desired. In particular, the (simplicia l) E ilenberg-Mac la ne spectr um E M s ( Z C ) is weakly equiv alent as a presheaf o n Sm /k to its ﬁbrant mo del in S H S 1 ( k ) ( E M s ( Z C ) is quasi-ﬁbr ant ). In addition, the canonical map E M s ( Z C ) → s 0 ( E M s ( Z C )) 6 MARC LEVINE is an isomorphis m in S H S 1 ( k ). Indeed, sinc e E M s ( Z C ) is quasi-ﬁbr ant, a qua si- ﬁbrant mo del for s 0 ( E M s ( Z C )) may be computed by using the metho d of [11, § 5 ] as fo llows: T ake Y ∈ Sm / k and let F = k ( Y ). Let ∆ n F, 0 be the semi-lo c al algebr aic n -simplex, that is, ∆ n F, 0 = Sp ec( O ∆ n F ,v ), where v = { v 0 , . . . , v n } is the set of vertices in ∆ n F , a nd O ∆ n F ,v is the se mi- lo cal ring of v in ∆ n F . The as signment n 7→ ∆ n F, 0 forms a co s implicial subscheme of n 7→ ∆ n F and for a q uasi-ﬁbrant S 1 -sp ectrum E , there is a natural isomorphism in S H s 0 ( E )( Y ) ∼ = E (∆ ∗ F, 0 ) , where E (∆ ∗ F, 0 ) denotes the to ta l spectrum of the simplicial sp ectrum n 7→ E (∆ n F, 0 ). If now E happ ens to b e a biratio na l S 1 -sp ectrum, meaning tha t j ∗ : E ( Y ) → E ( U ) is a weak e q uiv alence for each dense open immersion j : U → Y in Sm /k , then the restriction map j ∗ : E (∆ ∗ Y ) → E (∆ ∗ F, 0 ) ∼ = s 0 ( E )( Y ) is a weak equiv alence. Thus, as E is qua si-ﬁbrant and hence homotopy inv ariant, we hav e the sequence of isomor phis ms in S H E ( Y ) → E (∆ ∗ Y ) → E (∆ ∗ F, 0 ) ∼ = s 0 ( E )( Y ) , and hence E → s 0 ( E ) is an iso morphism in S H S 1 ( k ). T a king E = E M s ( Z C ) veriﬁes our claim. Finally , Z C do es not a dmit tra ns fers. Indeed, supp ose Z C has tra nsfers. Let k → L be a Galois extension such that C ( L ) 6 = ∅ ; let G b e the Galois group. Since Z C ( k ) = { 0 } (as we ha ve assumed that C ( k ) = ∅ ), the push-forward map p ∗ : Z C ( L ) → Z C ( k ) is the zer o map, hence p ∗ ◦ p ∗ = 0 . But for each L -po int x of C , le mma 2 .1 tells us that p ∗ ◦ p ∗ ( x ) = X g ∈ G x g 6 = 0 , a contradiction. Thu s the homotopy sheaf π 0 ( s 0 E M s ( Z C )) = π 0 ( E M s ( Z C )) = Z C do es not admit transfers, giving us the example we were seeking. 3. Co-transfer W e recall how o ne uses the deformation to the normal bundle to deﬁne the “co-tra nsfer” ( P 1 F , 1) → ( P 1 F ( ¯ x ) , 1) for closed p oint ¯ x ∈ A 1 F ⊂ P 1 F , with chosen genera tor ¯ f ∈ m ¯ x /m 2 ¯ x . F or later use, we work in a somewhat mor e genera l se tting: Let R b e a semi-lo cal k - algebra, smo oth and ess en tially of ﬁnite type ov er k , and ¯ x a reg ula r closed subscheme of P 1 R \ { 1 } ⊂ P 1 R , such that the pr o jection ¯ x → Spec R is ﬁnite. W e cho ose a generator ¯ f ∈ m ¯ x /m 2 ¯ x , whic h we lift to a generator f for the idea l m ¯ x ⊂ O P 1 , ¯ x . Let µ : W ¯ x → P 1 × A 1 R be the blow-up of P 1 × A 1 R along ( ¯ x, 0) with exceptiona l divisor E . Let s ¯ x , C 0 be the pr o pe r tra nsforms s ¯ x = µ − 1 [ ¯ x × A 1 ], C 0 = µ − 1 [ P 1 × 0]. Let t be the s tandard parameter on A 1 ; the rationa l function f / t r estricts to a well-deﬁned rationa l pa rameter on E . W e identif y E with P 1 ¯ x by sending s ¯ x ∩ E to 0, C 0 ∩ E to 1 a nd the section on E deﬁned by f /t = 1 to ∞ . Note that this SLICES AND TRANSFERS 7 ident iﬁcatio n dep ends only on ¯ f ∈ m ¯ x /m 2 ¯ x . W e denote these s ubschemes of E by 0 , ∞ , 1, resp e ctiv ely . W e let W ( s ¯ x ) ¯ x , E (0) , ( P 1 F ) (0) be following ho motopy push-outs W ( s ¯ x ) ¯ x := ( W ¯ x , W ¯ x \ s ¯ x ) , E (0) := ( E , E \ 0) , ( P 1 R ) (0) := ( P 1 R , P 1 R \ 0) . Since ( A 1 ¯ x , 0) ∼ = ∗ in H • ( k ), the resp ective ide ntit y maps induce iso morphisms ( E , 1 ) → E (0) , ( P 1 R , 1) → ( P 1 R ) (0) . Combining with the iso morphism ( P 1 ¯ x , 1) ∼ = ( E , 1 ), the inclusion E → W ¯ x induces the map i 0 : ( P 1 ¯ x , 1) → W ( s ¯ x ) ¯ x . The homotopy purit y theore m of Morel-V o evodsky [15, theorem 2.2 3] implies as a sp ecial ca s e that i 0 is an iso morphism in H • ( k ). W e prove a mo diﬁcation of this result. Let s 1 := A 1 R × 1 . W e write W for W ¯ x , e tc., when the context ma kes the meaning clear. Lemma 3. 1. The identity on W induc es an isomorphism ( W , C 0 ∪ s 1 ) → W ( s ¯ x ) . in H • ( k ) . Pr o of. As s 1 ∼ = A 1 R , with C 0 ∩ s 1 = 0, the inclusio n ( C 0 , 0) → ( C 0 ∪ s 1 , 0) is an isomorphism in H • ( k ). Thu s, we need to show that ( W, C 0 ) → W ( s ¯ x ) is an isomorphism in H • ( k ). As W ( s ¯ x ) = ( W, W \ s ¯ x ), we need to show that C 0 → W \ s ¯ x is an isomorphism in H ( k ). Let U = W \ s ¯ x , E U = E ∩ U , V = U \ E U and C V 0 := V ∩ C 0 . W e ﬁr st show that the diagram C V 0 / /   V   C 0 / / U is co-ca rtesian in H ( k ). F or this, we know from Morel-V o evo dsky [15] that the coﬁb er of V → U is isomorphic to the Tho m space of the no r mal bundle N o f E U in U . As E ∼ = P 1 ¯ x and E ∩ s ¯ x → ¯ x is an isomorphis m, E U is isomorphic to A 1 ¯ x and, as ¯ x is semi-lo ca l, N is the tr ivial bundle. Thus we hav e the co-ca r tesian diag ram V / /   U   ∗ / / Σ P 1 A 1 ¯ x + . 8 MARC LEVINE Similarly , we ha ve the co-c a rtesian diagram C V 0 / /   C 0   ∗ / / Σ P 1 0 ¯ x + . As the inclusion 0 ¯ x → A 1 ¯ x is a n isomorphism in H ( k ), o ur ﬁr st diagram is c o- cartesian, as desired. Next, w e no te that the blo w-down ma p µ : W → P 1 × A 1 R induces isomorphisms V → P 1 × A 1 R \ ¯ x × A 1 R C 0 → P 1 R × 0 C V 0 → P 1 × 0 \ { ( ¯ x, 0 ) } Thu s U is iso morphic in H ( k ) to the homotopy pus h-out in the diagram P 1 × 0 \ { ( ¯ x, 0 ) } / /   P 1 × A 1 R \ ¯ x × A 1 R P 1 R × 0 But by the contradictibilit y of A 1 R , the upp er hor izontal arrow is a n isomor phism in H ( k ), and thus C 0 → U is an isomorphism in H ( k ), completing the pro of.  Lemma 3. 2. The inclusion E → W induc es an isomorphism ( P 1 ¯ x , 1) → ( W ¯ x , C 0 ∪ s 1 ) in H • ( k ) . Pr o of. W e have the commutativ e diagram ( P 1 ¯ x 1) / / & & L L L L L L L L L L ( W , C 0 ∪ s 1 )   W ( s ¯ x ) . The diag onal arr ow is an isomorphism in H • ( k ) b y Morel- V o evodsky; the vertical arrow is an isomo rphism b y lemma 3.1.  One usually deﬁnes the “co- tr ansfer” ( P 1 R , 1) → ( P 1 ¯ x , 1) as the comp osition ( P 1 R , 1) i 1 − → W ( s ¯ x ) ¯ x i − 1 0 − − → ( P 1 ¯ x , 1) . Instead, w e will use the comp osition co - tr ¯ x, ¯ f in H • ( k ), ( P 1 R , 1) i 1 − → ( W ¯ x , C 0 ∪ s 1 ) i − 1 0 − − → ( P 1 ¯ x , 1) . which is well-deﬁned b y lemma 3 .1. Comparing with the usual co-tr ansfer via the isomorphism of lemma 3.2, ( W ¯ x , C 0 ∪ s 1 ) id W − − → W ( s ¯ x ) ¯ x , SLICES AND TRANSFERS 9 shows that the tw o co-tra nsfer maps a gree. W e examine some prop erties of co - tr ¯ x, ¯ f . Let s b e the standar d para meter on P 1 . Lemma 3.3. T ake ¯ x = 0 , f = s . The map co - tr 0 , ¯ s : ( P 1 F , 1) → ( P 1 F , 1) is the identity in H • ( k ) . Similarly, t he map co - tr ∞ , ¯ s − 1 : ( P 1 F , 1) → ( P 1 F , 1) is the identity in H • ( k ) Pr o of. The as sertion for co - tr ∞ , ¯ s − 1 follows from the statement for co - tr 0 , ¯ s by ap- plying the automo rphism of ( P 1 , 1) exchanging 0 and ∞ . Identify A 1 with P 1 \ { 1 } , sending 0 to 0 and 1 to ∞ . This embeds the blow-up W := W 0 in the blow-up ¯ W of P 1 × P 1 F at (0 , 0), with i 1 being the inclus io n i ∞ : P 1 = P 1 × ∞ → P 1 × P 1 F . The cur ve C 0 on ¯ W has self-intersection -1 , and can thus b e blown down via a morphism ρ : W → W ′ . Let ∆ ⊂ P 1 F × P 1 \ { 1 } b e the r estriction o f the diagona l in P 1 F × P 1 , giving us the prop er trans fo rm µ − 1 [∆] o n W and the image ∆ ′ = ρ ( µ − 1 [∆]) o n W ′ . Similarly , let s ′ 0 = ρ ( s 0 ), s ′ 1 = ρ ( s 1 ); note that ρ ( C 0 ) ⊂ s ′ 1 . It is easy to check that s ′ 0 , ∆ ′ and s ′ 1 give disjoint sections of W ′ → P 1 \ { 1 } , hence there is a unique is omorphism (ov er P 1 \ { 1 } ) of W ′ with P 1 F × P 1 \ { 1 } sending ( s ′ 0 , s ′ 1 , ∆ ′ ) to (0 , 1 , ∞ ) × P 1 \ { 1 } . W e hav e in addition the commut ative diagram ( P 1 F , 1) i 0 / / i ′ 0 & & M M M M M M M M M M ( W , C 0 ∪ s 1 ) ρ   ( P 1 F , 1) i 1 o o i ′ 1 x x q q q q q q q q q q ( W ′ , s ′ 1 ) where i ′ 0 is the canonical identiﬁcation of P 1 with the ﬁb er of W ′ ov er 0, sending (0 , 1 , ∞ ) into ( s ′ 0 , s ′ 1 , ∆ ′ ), and i ′ 1 is deﬁned similarly . Finally , the intersection ∆ ∩ E is the point s/t = 1 used to deﬁne the isomorphis m E ∼ = P 1 F in the deﬁnitiio n of co - tr 0 , ¯ s . It follows fr om lemma 3.2 that a ll the morphisms in this diag ram a re isomorphisms in H • ( k ); as i ′− 1 0 ◦ i ′ 1 is clearly the identit y , the lemma is prov ed.  Lemma 3.4. L et R → R ′ b e a ﬂat extension of s mo oth semi-lo c al k -algebr as, essential ly of ﬁn it e typ e over k . L et ¯ x b e a close d subscheme of P 1 R \ { 1 } , ﬁnite and ´ etale over R . L et ¯ x ′ = ¯ x × R R ′ ⊂ P 1 R ′ . L et ¯ f b e a gener ator for m ¯ x /m 2 ¯ x , and let ¯ f ′ b e the extension to m ¯ x ′ /m 2 ¯ x ′ . Then the diagr am ( P 1 R ′ , 1) co - tr ¯ x ′ , ¯ f ′ / /   ( P 1 ¯ x ′ , 1)   ( P 1 R , 1) co - tr ¯ x, ¯ f / / ( P 1 ¯ x , 1) c ommut es. The pro of is easy and is left to the rea der. 10 MARC LEVINE 4. Co-gr oup structure on P 1 Let G m = A 1 \ { 0 } , which we consider as a p ointed scheme with bas e-p oint 1. W e reca ll the Mayer-Vietoris squa re for the standard cov er of P 1 G m j ∞ / / j 0   A 1 i ∞   A 1 i 0 / / P 1 Here i 0 ( A 1 ) = P 1 \ { ∞} , i ∞ ( A 1 ) = P 1 \ { 0 } and the inclusions are normalized by sending 1 to 1 = (1 : 1). T his g ives us the isomor phism in H • ( k ) of P 1 with the homotopy push-out in the diag r am G m j ∞ / / j 0   A 1 A 1 combining with the co n tra ctibilit y of A 1 gives us the ca nonical isomorphism ( P 1 , 1) ∼ = S 1 ∧ G m . This together with the standa r d co -group structure on S 1 , σ : S 1 → S 1 ∨ S 1 makes ( P 1 , 1) a co-gr oup ob ject in H • ( k ); let σ P 1 := σ ∧ id G m : ( P 1 , 1) → ( P 1 , 1) ∨ ( P 1 , 1) be the co-multiplication. In this section, w e discuss a mor e algebraic descr iption of this structure. The function f := s/ ( s − 1) 2 on P 1 \ { 1 } a nd the deforma tio n to the norma l bundle used in the previous section gives us the collapse map co - tr { 0 , ∞} , ¯ f : ( P 1 , 1) → ( P 1 , 1) ∨ ( P 1 , 1) . Lemma 4. 1. co - tr { 0 , ∞} , ¯ f = σ P 1 in H • ( k ) . Pr o of. W e ﬁrst unwind the deﬁnition of σ P 1 in so me detail. The is omorphism α : S 1 ∧ G m → ( P 1 , 1) in H • ( k ) arises via a sequence of co mparison maps b et ween SLICES AND TRANSFERS 11 push-out diagrams: G m j ∞ / / j 0   A 1 A 1 ← I × G m 1 × G m i 1 o o j ∞ / / 1 × A 1 0 × G m i 0 O O j 0   0 × A 1 ↓ (4.1) I × G m 1 × G m i 1 o o / / ∗ ; 0 × G m i 0 O O   ∗ the ﬁrst map is induced by the ev ide nt pro jections and the second b y cont ra cting A 1 to ∗ . Thus, the open immersio n G m → P 1 go es ov er to the map I × G m → S 1 ∧ G m given by the b ottom push-out diag r am; as the inclusion { 1 / 2 } × G m → I × G m admits a deformation retract, we have the isomor phis m ρ : ( P 1 , G m ) → S 1 ∧ G m ∨ S 1 ∧ G m in H • ( k ), giving the commutativ e diagram ( P 1 , 1) ∼ / /   S 1 ∧ G m σ ∧ id   ( P 1 , G m ) ∼ / / S 1 ∧ G m ∨ S 1 ∧ G m . If we cons ider the middle push-out diagr am, we ﬁnd the is omorphism of ( P 1 , G m ) with (0 × A 1 , 0 × G m ) ∨ (1 × A 1 , 1 × G m ) in H • , with the ﬁrs t inclusio n the standar d one A 1 \ { 0 } → A 1 , and the second the inclusio n ( P 1 \ { 0 , ∞} , 1) → ( P 1 \ { 0 } , 1). The map fr o m the middle diagr am to the last diagr am furnishes the commutativ e diagram of isomorphisms (0 × A 1 , 0 × G m ) ∨ (1 × A 1 , 1 × G m ) β / / ϑ + + V V V V V V V V V V V V V V V V V V V S 1 ∧ G m ∨ S 1 ∧ G m α ∨ α   ( P 1 , 1) ∨ ( P 1 , 1) 12 MARC LEVINE in H • ( k ). Putting this all toge ther gives us the commutativ e diagram in H • ( k ): (4.2) ( P 1 , 1) ρ ∼ / /   S 1 ∧ G m   ( P 1 , G m ) ∼ / / δ & & ǫ * * U U U U U U U U U U U U U U U U U ( S 1 ∧ G m , I ∧ G m ) ∼   (0 × A 1 , 0 × G m ) ∨ (1 × A 1 , 1 × G m ) β ∼   S 1 ∧ G m ∨ S 1 ∧ G m α ∨ α ∼   ( P 1 , 1) ∨ ( P 1 , 1) , W e th us need to show tha t the resulting map ( P 1 , 1) → ( P 1 , 1) ∨ ( P 1 , 1) is g iven b y co - tr { 0 , 1 } , ¯ f . Let j 0 : A 1 → P 1 be the sta ndard aﬃne neighbor ho o d of 0, and j ∞ : A 1 → P 1 the standard aﬃne neig hborho o d of ∞ . The maps j 0 , j ∞ induce the isomorphisms in H • ( k ) j 0 : ( A 1 , G m ) → ( P 1 , j ∞ ( A 1 )) j ∞ : ( A 1 , G m ) → ( P 1 , j 0 ( A 1 )) giving together the isomorphism τ : ( P 1 , 1) ∨ ( P 1 , 1) → ( A 1 , G m ) ∨ ( A 1 , G m ): ( P 1 , 1) ∨ ( P 1 , 1) → ( P 1 , j ∞ ( A 1 )) ∨ ( P 1 , j 0 ( A 1 )) j − 1 0 ∨ j − 1 ∞ − − − − − − → ( A 1 , G m ) ∨ ( A 1 , G m ) . By co mpa ring the maps in this co mpo sition with the push-o ut dia grams in (4.1), we see that τ is the in verse to ϑ Let W → P 1 × A 1 be the blo w-up at ( { 0 , ∞} , 0) with exceptiona l divisor E . W e hav e the compositio n of isomorphisms in H • ( k ) (4.3) ( P 1 , G m ) i 1 − → ( W, W \ s { 0 , ∞} ) i 0 ← − ( E , C 0 ∩ E ) ϕ ← − ( P 1 , 1) ∨ ( P 1 , 1) . where ϕ is given by the isomorphism P 1 ∐ P 1 → E deﬁned v ia the function f . The op en cover ( j 0 , j ∞ ) : A 1 ∐ A 1 → P 1 of P 1 gives r ise to an o pe n cov er of W : Let µ ′ : W ′ → A 1 × A 1 be the blow-up at (0 , 0 ), then w e ha ve the lifting of ( j 0 , j ∞ ) to the ope n cov er ( ˜ j 0 , ˜ j ∞ ) : W ′ ∐ W ′ → W. Letting s ′ ⊂ W ′ be the prop er transform of 0 × A 1 to W ′ , we hav e the excis ion isomorphism in H • ( k ) ( ˜ j 0 , ˜ j ∞ ) : ( W ′ , W ′ \ s ′ ) ∨ ( W ′ , W ′ \ s ′ ) → ( W , W \ s { 0 , ∞} ) . SLICES AND TRANSFERS 13 This extends to a commutativ e diagram of isomorphisms in H • ( k ) (4.4) ( A 1 , G m ) ∨ ( A 1 , G m ) i 1 ∨ i 1   ( j 0 ,j ∞ ) / / ( P 1 , G m ) i 1   ( W ′ , W ′ \ s ′ ) ∨ ( W ′ , W ′ \ s ′ ) ( j 0 ,j ∞ ) / / ( W , W \ s { 0 , ∞} ) ( E ′ , E ′ ∩ C ′ 0 ) ∨ ( E ′ , E ′ ∩ C ′ 0 ) i 0 ∨ i 0 O O ( j 0 ,j ∞ ) / / ( E , E ∩ C 0 ) i 0 O O ( P 1 , 1) ∨ ( P 1 , 1) ϕ ′ ∨ ϕ ′ O O ( P 1 , 1) ∨ ( P 1 , 1) . ϕ O O Here the map ϕ ′ is deﬁned using the standar d co ordina te on A 1 as genera tor for m 0 /m 2 0 ; this is where we use the sp ecia l prop erty of the function f . Exa mining the push-out diagram (4.1), w e se e that the map ( j 0 , j ∞ ) : ( A 1 , G m ) ∨ ( A 1 , G m ) → ( P 1 , G m ) is in verse to the map ǫ in diagram (4.2). Let W 0 → P 1 × A 1 be the blow-up along (0 , 0), E 0 the exceptiona l div isor, C 0 0 the prop er transfor m of P 1 × 0. The inclusion j 0 gives us the commutativ e diag r am ( P 1 , 1) ∨ ( P 1 , 1)   ( A 1 , G m ) ∨ ( A 1 , G m ) ϑ 3 3 f f f f f f f f f f f f f f f f f f f f f f i 1 ∨ i 1   ( j 0 ∨ j 0 ) / / ( P 1 , A 1 ) ∨ ( P 1 , A 1 ) i 1 ∨ i 1   ( W ′ , W ′ \ s ′ ) ∨ ( W ′ , W ′ \ s ′ ) ( j 0 ∨ j ∞ ) / / ( W 0 , W 0 \ s 0 ) ∨ ( W 0 , W 0 \ s 0 ) ( E ′ , E ′ ∩ C ′ 0 ) ∨ ( E ′ , E ′ ∩ C ′ 0 ) i 0 ∨ i 0 O O ( j 0 ,j ∞ ) / / ( E 0 , E 0 ∩ C 0 0 ) ∨ ( E 0 , E 0 ∩ C 0 0 ) i 0 ∨ i 0 O O ( P 1 , 1) ∨ ( P 1 , 1) ϕ ′ ∨ ϕ ′ O O ( P 1 , 1) ∨ ( P 1 , 1) . ϕ ′ ∨ ϕ ′ O O By lemma 3.3 the comp osition a long the right-hand side of this diag ram is the ident ity on ( P 1 , 1) ∨ ( P 1 , 1), and th us the co mpo s ition along the left-ha nd side is ϑ : ( A 1 , G m ) ∨ ( A 1 , G m ) → ( P 1 , 1) ∨ ( P 1 , 1). Referring to diagra m (4.2 ), as ǫ = ( j 0 , j ∞ ) − 1 , the comp osition a long the righ t-hand side of (4.4) is the map δ . As the r ight-hand side of (4.4) is the deformation diag ram use d to deﬁne co - tr { 0 , ∞} , ¯ f , the lemma is prov ed.  5. Slice localiz a tions and co-transfer In g e neral, the co-tr ansfer maps do not have the prop erties necessar y to g ive a lo op-s pectr um Ω P 1 E an a ction by corr esp ondences. Howev e r, if we pass to a certain loca lization o f S H S 1 ( k ) deﬁned by the slice ﬁltration, the co-tra ns fer maps 14 MARC LEVINE do res p ect cor resp ondences, which will lead to the action of cor r esp ondences on s 0 Ω P 1 E . W e ha ve the lo c a lizing sub categor y Σ n P 1 S H S 1 ( k ), generated (as a lo calizing sub- category ) by ob jects of the fo r m Σ n P 1 E , for E ∈ S H S 1 ( k ). W e let S H S 1 ( k ) /f n denote the lo calization of S H S 1 ( k ) with resp ect to Σ n P 1 S H S 1 ( k ): S H S 1 ( k ) /f n = S H S 1 ( k ) / Σ n P 1 S H S 1 ( k ) . R emark 5.1 . Pelaez has s hown that there is a mo del s tructure on Spt S 1 ( k ) with homotopy ca tegory equiv alent to S H S 1 ( k ) /f n ; in particular , this lo calization o f S H S 1 ( k ) do es exist. Lemma 5.2. L et V → U b e a dense op en immersion in Sm /k , n ≥ 1 an inte ger. Then the induc e d m ap Σ n P 1 V + → Σ n P 1 U + is an isomorphism in S H S 1 ( k ) /f n +1 . Pr o of. W e can ﬁlter U by open s ubschemes V = U N +1 ⊂ U N ⊂ . . . ⊂ U 0 = U such that U i +1 = U i \ C i , with C i ⊂ U i smo oth and having trivial nor mal bundle in U i , of ra nk say r i ,, for i = 0 , . . . , N . By the Morel-V o evodsky purity theorem [1 5, theorem 2.23 ], the coﬁb er of U i +1 → U i is iso morphic in H • ( k ) to Σ r i P 1 C i + , and thus the coﬁber of Σ n P 1 U i +1+ → Σ n P 1 U i + is isomor phic to Σ r i + n P 1 C i + . Since V is dense in U , we have r i ≥ 1 for all i , proving the lemma.  W e reca ll the blo w-up W := W ¯ x → P 1 R × A 1 . Lemma 5.3 . L et m R ⊂ R b e the J ac obson ra dic al of R and let S b e the s emi- lo c alization of R [ t ] with r esp e ct t o the ide al t ( t − 1 ) + m . L et Sp ec S → Spec R [ t ] = A 1 R b e evident op en immersion. L et W S , s 1 ,S b e the r esp e ctive ﬁb er pr o ducts of W , s 1 with Sp ec S over A 1 R . Then the inclusion W S → W induc es an isomorphism ( W S , C 0 ∪ s 1 ,S ) → ( W, C 0 ∪ s 1 ) in S H S 1 ( k ) /f 2 . Pr o of. W e note that W × A 1 R A 1 R \ { 0 } ∼ = P 1 × A 1 R \ { 0 } and th us the co ﬁb er of W S → W is isomo rphic to the coﬁber o f ( P 1 × Spec S [ t − 1 ] , 1) → ( P 1 × A 1 R \ { 0 } , 1) W e can form Sp ec S [ t − 1 ] from A 1 R \ { 0 } a s a ﬁltered pr o jectiv e limit of op en subschemes U α of A 1 R \ { 0 } , by succes sively r emoving s moo th clo sed s ubschem es C α ⊂ U α from U α to fo r m U α +1 ; we may also assume that each C α has trivia l normal bundle in U α . The coﬁb er of ( P 1 × U α +1 , 1) → ( P 1 × U α , 1) is thus isomor phic in H • ( k ) to the pair of Thom spaces ( T h ( O r P 1 × C α ) , T h ( O r 1 × C α )) for some r ≥ 1. As this is isomor phic to Σ r P 1 ( P 1 C α , 1 C α ) ∼ = Σ r +1 P 1 C α + this coﬁb er is in Σ 2 P 1 S H S 1 ( k ) for a ll α . Since Σ 2 P 1 S H S 1 ( k ) is lo calizing, the coﬁb er of W S → W is in Σ 2 P 1 S H S 1 ( k ), as desired.  SLICES AND TRANSFERS 15 Lemma 5.4 . L et W ⊂ U b e a c o dimension ≥ r close d su bscheme of U ∈ Sm /k , let w 1 , . . . , w m b e the generic p oints of W of c o dimension = r in U . T hen in S H S 1 /f r +1 ther e is a c anonic al isomorphism ( U, U \ W ) ∼ = ⊕ m i =1 Σ r P 1 w i + . Sp e ciﬁc al ly, letting m i ⊂ O U,w i b e the maximal ide al, this isomorphism is indep en- dent of any choic e of isomorphism m i /m 2 i ∼ = k ( w i ) r . Pr o of. Let w = { w 1 , . . . .w m } and let O U,w denote the s e mi-lo cal ring of w in U . Let V ⊂ U b e the pro jective limit o f op en subschemes of U of the fo rm U \ C , wher e C is a clo s ed subset of U co ntained in W . Since V ∩ W = w , w e s ee that V ∩ W has trivial norma l bundle in V and thus by the Morel-V o evodsky purity iso morphism [15, lo c. cit] ( V , V \ V ∩ W ) ∼ = Σ r P 1 w + ∼ = ⊕ m r =1 Σ P 1 w i + . On the other hand, we can write V as a ﬁltered pro jective limit of op en subschemes U α of U , with U α +1 = U α \ C α , for some closed s ubset C α of W which is smo oth, contains no w i , and has trivial normal bundle in U α . In particular , C α has co di- menison ≥ r + 1 in U α , and th us ( U α , U α +1 ) ∼ = T h ( O n C α ) ∼ = Σ n P 1 C α + is in Σ r +1 P 1 S H S 1 ( k ). Therefore the map ( V , V \ V ∩ W ) → ( U, U \ W ) is an isomorphism in S H S 1 ( k ) /f r +1 . W e need o nly verify that the resulting isomo rphism ( U, U \ W ) ∼ = ⊕ m r =1 Σ P 1 w i + is indep endent of any choices. Letting O deno te the Henselization of w in V , we hav e the canonical excision isomorphism ( V , V \ V ∩ W ) ∼ = (Spec O , Spe c O \ w ) A choice of isomo rphism m w /m 2 w ∼ = k ( w ) r then gives the isomo rphism in H • ( k ) Σ r P 1 w + ∼ = (Spec O , Sp ec O \ w ); this choice of isomorphism is thus the only choice inv olved in constructing our isomorphism ( U, U \ W ) ∼ = ⊕ m i =1 Σ r P 1 w i + . Explicitly , the choice of isomor phism m w /m 2 w ∼ = k ( w ) r is reﬂected in the isomor phis m (Sp ec O , Sp ec O \ w ) ∼ = Σ r P 1 w + through the identiﬁcation of the ex c eptional divisor of the blo w-up of V × A 1 along w × 0 with P r w . Let Iso denote the k ( w ) scheme of isomor phisms m w /m 2 w ∼ = k ( w ) r . W e thus hav e a c a nonical morphism in H • ( k ) (5.1) Σ r P 1 Iso + → (Sp ec O , Spec O \ w ) such that, for each k ( w )-p oint α of Iso, the comp osition Σ r P 1 w + Σ r P 1 α − − − → Σ r P 1 Iso + → (Sp ec O , Spec O \ w ) is the isomorphism in H • ( k ) describ ed ab ov e. 16 MARC LEVINE W e note that Iso is a trivia l principal homogeneo us space for GL r /k ( w ) , hence isomorphic to an op en subscheme of A r 2 k ( w ) , and thus ( A r 2 , Iso) is in Σ 1 P 1 S H S 1 ( k ). W e thus have the iso morphism in S H S 1 ( k ) /f r +1 Σ r P 1 Iso + ∼ = Σ r P 1 A r 2 w + ∼ = Σ r P 1 w + , from whic h it eas ily follows that the map Σ r P 1 α in S H S 1 ( k ) /f r +1 is independent of the choice of α , co mpleting the pro o f.  R emark 5.5 . As a particular case, we hav e a canonical isomorphism ψ 0 , ∞ : ( P 1 , G m ) → ( P 1 , 1) ∨ ( P 1 , 1) in S H S 1 ( k ) /f 2 , indep enden t of any c hoice of ge ne r ator for m 0 , ∞ /m 2 0 , ∞ . This sim- pliﬁes our des c ription of the co-multiplication on ( P 1 , 1) at least if w e work in S H S 1 ( k ) /f 2 , as b eing given by the comp osition ( P 1 , 1) → ( P 1 , G m ) ψ 0 , ∞ − − − → ( P 1 , 1) ∨ ( P 1 , 1) . Similarly , for each a ∈ P 1 ( F ), we hav e a ca nonical isomor phism ψ a : ( P 1 F , P 1 F \ { a } ) → ( P 1 F , 1) in S H S 1 ( k ) /f 2 ; by lemma 3 .3, the compositio n ( P 1 F , 1) → ( P 1 F , P 1 F \ { a } ) ψ 0 − − → ( P 1 F , 1) in S H S 1 ( k ) /f 2 is the iden tity . Lemma 5.6. L et µ n : ( P 1 F , ∞ ) → ( P 1 F , ∞ ) b e t he µ n (1 : t ) = (1 , t n ) . Assume the char acteristic of k is prime to n ! . Then in S H S 1 ( k ) /f 2 , µ n is multiplic ation by n . Pr o of. The proo f go es by induction on n , sta rting with n = 1 , 2 . The case n = 1 is trivial. F or n = 2, we hav e the commut ative diagram ( P 1 F , ∞ ) / / µ 2   ( P 1 F , P 1 F \ {± 1 } ) µ 2   ( P 1 F , ∞ ) / / ( P 1 F , P 1 F \ { 1 } ) The b ottom horiz ont al arrow is an isomo r phism in H • ( k ); using remark 5 .5 w e see that this diagram gives us the factor ization of µ 2 (in S H S 1 ( k ) /f 2 ) as ( P 1 F , ∞ ) σ − → ( P 1 F , ∞ ) ∨ ( P 1 F , ∞ ) id ∨ id − − − → ( P 1 F , ∞ ) . Here σ is the co -mult iplicatio n (using ∞ instead of 1 a s base-p oint). Since [(id ∨ id) ◦ σ ] ∗ is m ultiplication by 2, this tak es care of the case n = 2. In genera l, we consider the map ρ n : ( P 1 , ∞ ) → ( P 1 , ∞ ) sending (1 : t ) to (1 : w ) := (1 : t n − t n − 1 + 1 ). As ab ov e, we lo calize a round w = 1. Note that ρ − 1 n (1) = { 0 , 1 } ; we r eplace the targ e t P 1 with the Hense liz ation O at w = 1, and see that P 1 × ρ n O br eaks up into tw o comp onents. On the comp onent containing 0, the map ρ n is isomorphic to a Hensel lo ca l v ersio n of ρ n − 1 , and on the component containing 1, the map ρ n is isomorphic to the identit y . More explicitly , we have the following commutativ e diagra m (in S H S 1 ( k ) /f 2 ) ( P 1 F , ∞ ) / / µ n   ( P 1 F , P 1 F \ { 0 , 1 } ) µ n   ∼ / / ( P 1 F , ∞ ) ∨ ( P 1 F , ∞ ) µ n − 1 ∨ id   ( P 1 F , ∞ ) / / ( P 1 F , P 1 F \ { 1 } ) ∼ / / ( P 1 F , ∞ ) SLICES AND TRANSFERS 17 As the upp er row is the co-mult iplicatio n (in S H S 1 ( k ) /f 2 ), our induction hypo thesis shows that ρ n (in S H S 1 ( k ) /f 2 ) is multiplication b y n . On the other hand, w e may form the family of morphisms ρ n ( s ) : ( P 1 × A 1 , ∞ × A 1 ) → ( P 1 × A 1 , ∞ × A 1 ) sending ( t 0 : t 1 , s ) to ( t n 0 : t n 1 − st 0 t n − 1 1 + t n 0 ). By ho motopy inv ar iance, we hav e ρ n (0) = ρ n (1), and the induction go es through.  While we are on the sub ject, we might as well sho w Prop ositio n 5.7. The c o-gr oup (( P 1 , 1) , σ P 1 ) in S H S 1 ( k ) /f 2 is c o-c ommutative. Pr o of. In P 1 × P 1 , consider the dia gonal ∆ and a nt i-diag onal ∆ ′ , deﬁned b y x 0 y 1 − x 1 y 0 = 0 and x 1 y 1 − x 0 y 0 = 0, resp ectively . In aﬃne co ordinates , these a r e y = x and y = 1 / x , r esp ectively , hence ∆ ∩ ∆ ′ consists o f the tw o p oints (1 , 1) and ( − 1 , − 1). Th us, if we r estrict to U × P 1 , U = P 1 \ {± 1 } , the subscheme ∆ U ∪ ∆ ′ U of U × P 1 is ´ etale and ﬁnite ov er U a nd is disjoint from U × 1 . This gives us the map ˜ σ : ( U × P 1 , U × 1 ) → ( U × P 1 , U × P 1 \ ∆ U ∪ ∆ ′ U ) The comp osition U × P 1 → A 1 × P 1 → P 1 shows that the pro jection induces an isomorphism ( U × P 1 , U × 1 ) → ( P 1 , 1) in S H S 1 ( k ) /f 2 , while on the other side, we ha ve the Thom iso morphism ( U × P 1 , U × P 1 \ ∆ U ∪ ∆ ′ U ) ∼ = T h ( N ∆ U ) ∨ T h ( N ∆ ′ U ) ∼ = Σ P 1 U + ∨ Σ P 1 U + , which in turn is isomorphic to ( P 1 , 1) ∨ ( P 1 , 1) in S H S 1 ( k ) /f 2 . As the restriction of ˜ σ to 0 ∈ U is the map σ and the re striction to ∞ ∈ U is σ follow ed by the exchange isomorphism τ : ( P 1 , 1) ∨ ( P 1 , 1) → ( P 1 , 1) ∨ ( P 1 , 1) , (using r emark 5.5 to av oid sp ecifying the choice of triv ialization in the Thom iso- morphism) we hav e prov en the co-commutativit y .  W e now return to o ur study of proper ties of the co -transfer map in S H S 1 ( k ) /f 2 . W e alre ady kno w that, for a given closed p oint ¯ x ∈ P 1 F , the map co - tr ¯ x, ¯ f : ( P 1 F , 1) → ( P 1 F ( ¯ x ) , 1) in S H S 1 ( k ) /f 2 is indep endent of the choice of gener ator f ∈ m ¯ x /m 2 ¯ x ; we deno te this map b y co - tr ¯ x . Suppo se w e have a semi-lo cal smoo th k -alg ebra A , essentially of ﬁnite type, and a ﬁnite extension A → B , with B smooth ov er k . Supp ose fur ther that B is generated as an A -algebra by a single element x ∈ B : B = A [ x ] . W e say in this case that B is a simply gener ate d A -algebr a. Let f ∈ A [ T ] b e the minimal monic p olyno mial of x , giving us the p oint ¯ x ′ of A 1 A = Sp ec A [ T ] with ideal ( f ). W e identify A 1 A with P 1 A \ { 1 } as usual, giving us the subscheme ¯ x of P 1 A \ { 1 } , smo oth ov er k a nd ﬁnite ov e r Spe c A , in fact, canonically isomorphic to Spec B over Sp ec A via the choice of generator x . Let ϕ x : ¯ x → Sp ec B 18 MARC LEVINE be this isomor phism. W e let ¯ f b e the g enerator of m ¯ x /m 2 ¯ x determined by f . Via the comp osition ( P 1 A , 1) co - tr ¯ x, ¯ f − − − − − → ( P 1 ¯ x , 1) ϕ x × id − − − − → ( P 1 B , 1) we hav e the morphism co - tr x : ( P 1 A , 1) → ( P 1 B , 1) in H • ( k ). Lemma 5.8 . Supp ose t hat Sp ec B → Spe c A is ´ etale over e ach generic p oint of Spec A . Then the map co - tr x : ( P 1 A , 1) → ( P 1 B , 1) in S H S 1 ( k ) /f 2 is indep endent of the choic e of gener ator x for B over A . We write co - tr B / A for co - tr x . Pr o of. W e us e a defo rmation ar gument; we ﬁrst lo caliz e to r educe to the case of an ´ etale extensio n A → B . F or this, let a ∈ A b e a non-zero divisor , and let x be a g enerator for B as an A -algebra . Then x is a generato r for B [ a − 1 ] a s an A [ a − 1 ]-algebra and we have the commutativ e diagram P 1 A [ a − 1 ] / / co - tr x   P 1 A co - tr x   P 1 B [ a − 1 ] / / P 1 B , with ho rizontal arrows isomor phis ms in S H S 1 ( k ) /f 2 . Thus, we may assume that A → B is ´ eta le. Suppo se we hav e genera tors x 6 = x ′ for B ov e r A ; let d = [ B : A ]. Let s b e an indeterminate, let x ( s ) = sx + (1 − s ) x ′ ∈ B [ s ], a nd consider the extension ˜ B s := A [ s ][ x ( s )] of A [ s ], considere d as a subalgebra of B [ s ]. Clearly ˜ B s is ﬁnite ov er A [ s ]. Let m A ⊂ A b e the Jaco bson ra dical, and let A ( s ) b e the lo caliz a tion of A [ s ] at the ideal ( m A A [ s ] + s ( s − 1)). In other words, A ( s ) is the semi- lo cal ring of the set of closed p oints { (0 , a ) , (1 , a ) } in A 1 × Sp ec A , as a runs over the closed po in ts of Sp ec A . Deﬁne B ( s ) := B ⊗ A A ( s ) and B s := ˜ B s ⊗ A A ( s ) ⊂ B ( s ). Let y = (1 , a ) b e a clo sed p oint of A ( s ), with ma x imal ideal m y , a nd let x y be the image o f x in B ( s ) /m y B ( s ). Cle a rly x y is in the image o f B s → B ( s ) /m y B ( s ), hence B s → B ( s ) /m y B ( s ) is surjective. Similarly , B s → B ( s ) /m y B ( s ) is sur jectiv e for all y of the for m (0 , a ); b y Nak ayama’s lemma B s = B ( s ). Also, B ( s ) and A ( s ) are reg ular and B ( s ) is ﬁnite ov er A ( s ), hence B ( s ) is ﬂat over A ( s ) and th us B ( s ) is a free A ( s )-mo dule of rank d . Finally , B ( s ) is clear ly unramiﬁed ov er A ( s ), hence A ( s ) → B ( s ) is ´ e ta le. Using Nak ay ama’s lemma a g ain, w e see that B ( s ) is g enerated as a n A ( s ) mod- ule by 1 , x ( s ) , x ( s ) 2 , . . . , x ( s ) d − 1 . It follows that x ( s ) sa tisﬁes a monic p olynomial equation o f degree d ov er A ( s ), thus x ( s ) admits a monic minimal p olyno mial f s of degree d ov er A ( s ). Sending T to x ( s ) deﬁnes an isomor phism ϕ s : A ( s )[ T ] / ( f s ) → B ( s ) . W e let ¯ x s ⊂ A 1 A ( s ) = P 1 A ( s ) \ { 1 } b e the clo sed subscheme of P 1 A ( s ) corres p onding to f s ; the isomorphism ϕ s gives us the iso morphism ϕ s : ¯ x s → Sp ec B ( s ) . SLICES AND TRANSFERS 19 Thu s, we may deﬁne the map co - tr x ( s ) : ( P 1 A ( s ) , 1) → ( P 1 B ( s ) , 1) giving us the commutativ e diagram ( P 1 A , 1) co - tr x ′   i 0 / / ( P 1 A ( s ) , 1) co - tr x ( s )   ( P 1 A , 1) co - tr x   i 1 o o ( P 1 B , 1) i 0 / / ( P 1 B ( s ) , 1) ( P 1 B , 1) i 1 o o By lemma 5 .2 a nd a limit ar g umen t, the map ( P 1 A ( s ) , 1) → ( P 1 A [ s ] , 1) is an isomor- phism in S H S 1 ( k ) /f 2 . By homo topy inv ariance, it follows that the maps i 0 , i 1 are isomorphisms in S H S 1 ( k ) /f 2 , inv erse to the map ( P 1 A ( s ) , 1) → ( P 1 A , 1) induced by the pro jection Spec A ( s ) → Sp ec A . Therefore co - tr x ′ = co - tr x , as desired.  Lemma 5. 9. co - tr A/ A = id ( P 1 A , 1) . Pr o of. W e may cho o se 1 a s the g enerator for A ov er A , which gives us the po in t ¯ x = 0 ∈ P 1 A . The result now follows from lemma 3.3.  Lemma 5.10. L et A → C b e a ﬁ nite simply gener ate d extension and A ⊂ B ⊂ C a sub-ext ension, with B also simply gener ate d over A . We supp ose that A , B and C ar e smo oth over k , and that A → B and A → C ar e ´ etale over e ach generic p oint of Sp ec A , and B → C is ´ etale over e ach generic p oint of Sp ec B . Then co - tr C / A = co - tr C /B co - tr B / A . Pr o of. This is ano ther deforma tion a rgument. As in the pro o f of lemma 5.8, we may assume that A → B , B → C and A → C are ´ etale extensions. Le t y b e a gener ator for C ov er A , x a g enerator for B ov er A . These genera tors give us corres p onding closed subschemes ¯ y , ¯ x ⊂ P 1 A and ¯ y B ⊂ P 1 B . Let y ( s ) = sy + (1 − s ) x , giving ¯ y ( s ) ⊂ P 1 A ( s ) . Note that ¯ y (1 ) = ¯ y , ¯ y (0) red = ¯ x As in the pro of of lemma 5.8, the element y ( s ) of C ( s ) is a genera tor o ver A ( s ) after lo calizing at the p oints of Sp ec A ( s ) ly ing over s = 1. T he subscheme ¯ y ( s ) in a neighborho o d o f s = 0 is no t in general regular, hence y ( s ) is not a g enerator of C ( s ) ov er A ( s ). How ever, let µ : W := W ¯ x → P 1 × A 1 be the blow-up alo ng { ( ¯ x, 0 ) } , a nd let ˜ y ⊂ W A ( s ) be the prop er transform µ − 1 [ ¯ y ]. An elementary lo cal computation shows that this blow-up reso lves the singular ities of ¯ y ( s ), and that ˜ y is ´ etale ov er A ( s ); the argument use d in the pro of of lemma 5.8 go es through to show that A ( s )( ˜ y ) ∼ = C ( s ). In addition, let C 0 be the pr o pe r transform to W A ( s ) of P 1 × 0 and E the e x ceptional divis or, then ˜ y (0) is disjoint from C 0 . Finally , after identif ying E with P 1 A [ ¯ x ] (using a the monic minimal p olyno mial of x as a gene r ator for m ¯ x ), we may consider ˜ y (0) as a closed subs cheme of P 1 B ; the iso morphism A ( s )( ˜ y ) ∼ = C ( s ) leads us to conclude that A ( ˜ y (0)) = B ( ˜ y ) = C . By lemma 5.8, we may use ˜ y to deﬁne co - tr C /B . The map co - tr C / A in S H S 1 ( k ) /f 2 is deﬁned via the diagr am ( P 1 A , 1) → ( P 1 A , P 1 A \ ¯ y ) ∼ = ( P 1 C , 1) 20 MARC LEVINE where the v arious choices in volved lea d to equal maps. The inclusions i 1 : P 1 A → W A ( s ) , i 0 : P 1 A [ ¯ x ] → W A ( s ) induce isomorphis ms (in S H S 1 ( k ) /f 2 ) ( P 1 A , P 1 A \ ¯ x ) ∼ = ( W A ( s ) , W A ( s ) \ ˜ y ( s )) ∼ = ( P 1 A [ ¯ x ] , P 1 A [ ¯ x ] \ ˜ y (0 )) . As in the pro of of le mma 5 .8, we can use homotopy inv ariance to see tha t co - tr C / A is also equal to the comp osition ( P 1 A , 1) → ( P 1 A , P 1 A \ ¯ y ) i 1 − → ( W A ( s ) , W A ( s ) \ ˜ y ( s )) i − 1 0 − − → ( P 1 A [ ¯ x ] , P 1 A [ ¯ x ] \ ˜ y (0)) ∼ = ( P 1 C , 1) . Now let s 1 A ( s ) be the transform to W A ( s ) of the 1-sectio n. The ab ov e factoriza- tion of co - tr C / A shows tha t this map is also equal to the comp osition ( P 1 A , 1) i 1 − → ( W A ( s ) , C 0 ∪ s 1 A ( s ) ) i − 1 0 − − → ( P 1 A [ ¯ x ] , 1) → ( P 1 A [ ¯ x ] \ ˜ y (0 )) ∼ = ( P 1 C , 1) . Using lemma 5.3, this latter comp osition is co - tr C /B ◦ ( co - tr B / A ), as desired.  R emark 5.11 . 1. Supp ose we have simply generated ﬁnite ge ne r ically ´ etale exten- sions A 1 → B 1 , A 2 → B 2 , with A i smo oth, semi-lo cal and esse n tially of ﬁnite type ov er k . Then co - tr B 1 × B 2 / A 1 × A 2 = co - tr B 1 / A 1 ∨ co - tr B 2 / A 2 where we make the ev iden t identiﬁcation ( P 1 B 1 × B 2 , 1) = ( P 1 B 1 , 1) ∨ ( P 1 B 2 , 1) and sim- ilarly for A 1 , A 2 . 2. Let B 1 , B 2 be simply ge ne r ated ﬁnite gener ically ´ etale A algebras and let B = B 1 × B 2 . As a sp ecial case of lemma 5.10, we ha ve co - tr B / A = ( co - tr B 1 / A ∨ co - tr B 2 / A ) ◦ σ P 1 A Indeed, we ma y factor the extension A → B as A δ − → A × A → B 1 × B 2 = B . W e then use (1) and note that σ P 1 A = co - tr A × A/ A by le mma 4.1. Next, we make a lo cal calcula tion. Let ( A, m ) b e a lo ca l ring of essentially o f ﬁnite type and smo oth o ver k . Let s ∈ m b e a parameter and let B = A [ T ] /T n − s and let t ∈ B b e the image of T . Set Y = Sp ec B , X = Sp ec A , Z = Sp ec A/ ( s ), W = Sp ec B / ( t ); the extensio n A → B induces an isomor phism α : W ∼ − → Z . W e write co - tr Y /X for co - tr B / A , etc. This gives us the diagram in S H S 1 ( k ) /f 2 P 1 Z i Z / / P 1 X co - tr Y /X   P 1 W i W / / α O O P 1 Y . Lemma 5. 12. Su pp ose that n is prime t o char k . In S H S 1 ( k ) /f 2 we have co - tr Y /X ◦ i Z ◦ α = n × i W . Pr o of. First, supp ose we hav e a Nisnevich neighbor ho o d f : X ′ → X of Z in X , giving us the Nisnevich neighbo r ho o d g : Y ′ := Y × X X ′ → Y of W in Y . As co - tr Y /X ◦ f = g ◦ co - tr Y ′ /X ′ SLICES AND TRANSFERS 21 we may r e place X with X ′ , Y with Y ′ . Similar ly , we r educe to the case of A a Hensel DVR, i.e., the Henselizatio n of 0 ∈ A 1 F for s ome ﬁeld F , Z = W = 0, with s the image in A of the canonical co ordinate on A 1 F . The map co - tr Y /X is deﬁned b y the clo sed immersion Y i Y − → A 1 X = P 1 X \ { 1 } ⊂ P 1 X where i Y is the closed subscheme of A 1 = Sp ec A [ T ] deﬁned by T n − s , together with the isomorphism ( P 1 X , P 1 X \ Y ) ∼ = P 1 Y furnished by the blow-up µ : W Y → A 1 × A 1 X of A 1 × A 1 X along ( Y , 0). The comp osition co - tr Y /X ◦ i Z ◦ α is given b y the comp o sition ( P 1 W , 1) ∼ = ( P 1 W , P 1 W \ { 0 } ) α − → ( P 1 Z , P 1 Z \ { 0 } ) i Z − → ( P 1 X , P 1 X \ { 0 } ) id ← − ( P 1 X , 1) → ( P 1 X , P 1 X \ Y ) ∼ = ( P 1 Y , 1) . In b oth cases, the isomor phisms (in S H S 1 ( k ) /f 2 ) are indep enden t of a choice of the resp ective deﬁning equa tion. Let U → P 1 X be the Hensel lo cal neig h b orho o d of (0 , 0) in P 1 X , Spec O h P 1 X , (0 , 0) , and let U Z ⊂ U b e the ﬁber o f U ov er Z , i.e., the subscheme s = 0. W e may use excis io n to r e w r ite the ab ov e descr iption of co - tr Y /X ◦ i Z ◦ α as a comp osition as ( P 1 W , 1) ∼ = ( U Z , U Z \ { (0 , 0) } ) i Z − → ( U, U \ Y ) ∼ = ( P 1 Y , 1) . Similarly , letting i 0 : X → X × P 1 be the 0 -section, the map i W may be g iven b y the comp osition ( P 1 W , 1) ∼ = ( X, X \ Z ) i 0 − → ( U, U \ Y ) ∼ = ( P 1 Y , 1); again, the isomor phisms in S H S 1 ( k ) /f 2 are independent of choice o f deﬁning equa - tions. W e change co or dinates in U by the iso morphism ( s, t ) 7→ ( s − t n , t ). This trans- forms Y to the subscheme s = 0 , is the identit y on the 0-section, and tra nsforms s = 0 to the gra ph of t n + s = 0. Replacing s with − s , we have just switched the roles of Y a nd U Z . Let ϕ : U Z → U be the map ϕ ( t ) = ( t n , t ). After making our change of co or dina tes, the map co - tr Y /X ◦ i Z ◦ α is identiﬁed with ( P 1 W , 1) ∼ = ( U Z , U Z \ { (0 , 0) } ) ϕ − → ( U, U \ U Z ) ∼ = ( P 1 Y , 1) while the description of i W bec omes ( P 1 W , 1) ∼ = ( X, X \ Z ) i 0 − → ( U, U \ U Z ) ∼ = ( P 1 Y , 1); W e now construct an A 1 -family of maps ( U Z , U Z \ { (0 , 0) } ) → ( U, U \ U Z ). Let Φ : U Z × A 1 → U be the map Φ( t, v ) = ( t n , v t ). Note that Φ deﬁnes a map o f pairs Φ : ( U Z , U Z \ { 0 } ) × A 1 → ( U, U \ U Z ) . Clearly Φ( − , 1) = ϕ while Φ( − , 0) factors as U Z µ n − − → U Z β − → X i 0 − → U 22 MARC LEVINE where µ n is the map t 7→ t n and β is the iso morphism β ( t ) = s . Thus, we can rewrite co - tr Y /X ◦ i Z ◦ α as ( P 1 W , 1) ∼ = ( X, X \ Z ) µ n − − → ( X , X \ Z ) i 0 − → ( U, U \ U Z ) ∼ = ( P 1 Y , 1) W e identify X with the Hensel neighborho o d of 0 in P 1 Z . Using excis ion again, we hav e the commut ative diagram in H • ( k ) ( X, X \ Z ) µ n / /   ( X, X \ Z )   ( P 1 Z , P 1 Z \ { 0 } ) µ n / / ( P 1 Z , P 1 Z \ { 0 } ) ( P 1 Z , ∞ ) µ Z n / / O O ( P 1 Z , ∞ ) O O where the vertical ar rows are all iso morphisms. By lemma 5.6 the b ottom map is m ultiplication by n , whic h completes the pro of.  Lemma 5.1 3. L et A → B b e a ﬁnite simple ´ etale ex tension. L et X = Spec A , Y = Spec B , let i x : x → X b e t he close d p oint of X and i y : y → Y t he inclusion of y := x × X Y . Then co - tr Y /X ◦ i x = i y ◦ co - tr y /x . Pr o of. T ake an em b edding of Y in A 1 X = P 1 X \ { 1 } ⊂ P 1 X ; the ﬁber of Y → A 1 X ov er x → X is thus an embedding y → A 1 x = P 1 x \ { 1 } ⊂ P 1 x . The result follows e a sily from the commutativit y of the diagr am P 1 x \ y / /   P 1 x   P 1 X \ Y / / P 1 X  Prop ositio n 5.14. L et A → B b e a ﬁnite generic al ly ´ etale ex tension, with A a DVR and B a semi-lo c al princip al ide al ring. L et X = Spec A , Y = Spe c B , let i x : x → X b e the close d p oint of X and i y : y → Y the inclusion of y := x × X Y . Write y = { y 1 , . . . , y r } , with e ach y i irr e ducible. L et n i denote the r amiﬁc ation index of y i ; su pp ose that e ach n i is prime to char k . Then co - tr Y /X ◦ i x = r X i =1 n i · i y i ◦ co - tr y i /x . Pr o of. By passing to the Henseliz ation A → A h , w e may assume A is Hensel. By remar k 5.11(2), we ma y a ssume that r = 1 . Let A → B 0 ⊂ B be the maximal unramiﬁed s ubextens ion. As co - tr B / A = co - tr B /B 0 ◦ c o - tr B 0 /B , we reduce to the t wo cases A = B 0 , B = B 0 . W e no te that a ﬁnite se pa rable extension of Hensel DVRs A → B with tr iv ial re sidue ﬁeld extension degree is isomo rphic to an extensio n of the for m t n = s for so me s ∈ m A \ m 2 A . Thus, the ﬁrst ca se is lemma 5 .12, the second is lemma 5.13.  SLICES AND TRANSFERS 23 Consider the functor ( P 1 ? , 1) : Sm /k → S H S 1 ( k ) /f 2 sending X to ( P 1 X , 1) ∈ S H S 1 ( k ) /f 2 , which we co nsider as a S H S 1 ( k ) /f 2 -v alued presheaf on Sm /k op (w e co uld also write this functor as X 7→ Σ P 1 X + ). W e pro cee d to extend ( P 1 ? , 1) to a presheaf on S mC or ( k ) op ; we will a s sume that c har k = 0, so we do not need to worry abo ut inseparability . W e ﬁrst deﬁne the action o n the gener ators of Hom S mC or ( X, Y ), i.e., on ir r e- ducible W ⊂ X × Y such that W → X is ﬁnite and surjective over some com- po nent of X . As S H S 1 ( k ) /f 2 is an additive categ ory , it suﬃces to consider the case o f irr educible X . Let U ⊂ X b e a dense o pen subscheme. Then the map ( P 1 U , 1) → ( P 1 X , 1) induced by the inclusion is an is omorphism in S H S 1 ( k ) /f 2 . W e may therefore deﬁne the mor phism ( P 1 ? , 1)( W ) : ( P 1 X , 1) → ( P 1 Y , 1) in S H S 1 ( k ) /f 2 as the comp osition ( P 1 X , 1) ∼ = ( P 1 k ( X ) , 1) co - tr k ( W ) /k ( X ) − − − − − − − − − → ( P 1 k ( W ) , 1) p 2 − → ( P 1 Y , 1) . W e extend to linear ity to deﬁne ( P 1 ? , 1) on Hom S mC or ( X, Y ). Suppo se that Γ f ⊂ X × Y is the gr a ph of a morphism f : X → Y . It follows fr o m lemma 5.9 that co - tr k (Γ f ) /k ( X ) is the in verse to the isomorphis m p 1 : ( P 1 k (Γ f ) , 1) → ( P 1 k ( X ) , 1). Thus, the c ompo sition ( P 1 k ( X ) , 1) co - tr k (Γ f ) /k ( X ) − − − − − − − − − − → ( P 1 k (Γ f ) , 1) p 2 − → ( P 1 Y , 1) is the map induced by the restrictio n of f to Sp ec k ( X ). Since ( P 1 k ( X ) , 1) → ( P 1 X , 1) is a n isomor phism in S H S 1 ( k ) /f 2 , it follows tha t ( P 1 ? , 1)(Γ f ) = f , i.e., our def- inition of ( P 1 ? , 1) on Ho m S mC or ( X, Y ) re a lly is an extension of its deﬁnition on Hom Sm /k ( X, Y ). The main po in t is to chec k functoriality . Lemma 5. 15. F or α ∈ Hom S mC or ( X, Y ) , β ∈ Hom S mC or ( Y , Z ) , we have ( P 1 ? , 1)( β ◦ α ) = ( P 1 ? , 1)( β ) ◦ ( P 1 ? , 1)( α ) Pr o of. It suﬃces to consider the case of irreducible ﬁnite cor resp ondences W ⊂ X × Y , W ′ ⊂ Y × Z . If W is the graph of a ﬂat morphism, the result follows from lemma 3.4. As the action of corres po ndences is deﬁned at the generic p oint, w e may repla ce X with η := Sp ec k ( X ). Then W b ecomes a closed p oint of Y η and the corresp ondence W η : η → Y factors a s p 2 ◦ i W η ◦ p t 1 , wher e p 1 : W η → η p 2 : Y η → Y ar e the pro jections. Let W ′ η ⊂ Y η × Z b e the pull-back o f W ′ . As we hav e alre ady established naturality with resp ect to pull-back b y ﬂat maps, we r educe to showing ( P 1 ? , 1)( W ′ η ◦ i W η ) = ( P 1 ? , 1)( W ′ η ) ◦ ( P 1 ? , 1)( i W η ) . Since Y is qua si-pro jective, we ca n ﬁnd a sequence of closed subschemes of Y η W η = W 0 ⊂ W 1 ⊂ . . . ⊂ W d − 1 ⊂ W d = Y η such that W i is smo oth of co dimension d − i on Y η . Using ag ain the fact the co - tr is deﬁned at the generic po int , and that we hav e alrea dy pr ov en functorialit y with 24 MARC LEVINE resp ect to comp osition of morphis ms, we reduce to the ca se of Y = Spec O for some D VR O , and i η the inclusion of the closed p oint η o f Y . Let W ′′ → W ′ be the nor malization o f W ′ . Using functoriality with res pect to morphisms in Sm /k once mo r e, we may r eplace Z with W ′′ and W ′ with the transp ose of the gr aph of the pr o jection W ′′ → Y . Chang ing notation, we may assume that W ′ is the transp os e of the g raph of a ﬁnite morphis m Z → Y . This reduces us to the case consider e d in prop osition 5.14; this latter result completes the pro of.  W e will colle ct the results of this se ction, gener a lized to higher lo ops, in theo- rem 6.1 of the next section. 6. Higher l oops The r esults of these last sections ca rry over immediately to statements ab out the n - fold smash pro duct ( P 1 , 1) ∧ n for n ≥ 1. F o r clar it y and completeness , we list these explicitly in an omnibus theorem. Let R b e a semi-lo cal k -a lgebra, smo o th and essentially of ﬁnite type over k , a nd let ¯ x ⊂ P 1 R and f b e as in s e c tion 3. F or n ≥ 1, deﬁne co - tr n ¯ x, ¯ f : Σ n P 1 Spec R + → Σ n P 1 ¯ x + . be the map id ( P 1 , 1) ∧ n − 1 ( co - tr ¯ x, ¯ f ). Similarly , let A b e a semi- lo c al k - algebra, s mo o th a nd e s sentially of ﬁnite type ov er k . Let B = A [ x ] b e a s imply generated ﬁnite g enerically ´ etale A -algebr a. F o r n ≥ 1 , deﬁne co - tr n x : Σ n P 1 Spec A + → Σ n P 1 Spec B + . be the map Σ n P 1 ( co - tr x ). Theorem 6. 1. 1. F or ¯ x = 0 , f = s , we have co - tr n ¯ x, ¯ f = id . 2. L et R → R ′ b e a ﬂat extens ion of sm o oth semi-lo c al k -algebr as, ess en tial ly of ﬁnite typ e over k . L et ¯ x b e a sm o oth close d subscheme of P 1 R \ { 1 } , ﬁnite and generic al ly ´ etale over R . L et ¯ x ′ = ¯ x × R R ′ ⊂ P 1 R ′ . L et ¯ f b e a gener ator for t he m ¯ x /m 2 ¯ x , and let ¯ f ′ b e the extens ion to m ¯ x ′ /m 2 ¯ x ′ . Then the diagr am Σ n P 1 Spec R ′ + co - tr n ¯ x ′ , ¯ f ′ / /   Σ n P 1 ¯ x ′ +   Σ n P 1 Spec R + co - tr n ¯ x, ¯ f / / Σ n P 1 ¯ x + c ommut es. 3. The c o-gr oup structure Σ n − 1 P 1 ( σ P 1 ) on ( P 1 , 1) ∧ n is given by t he map co - tr n { 0 , ∞} , s/ ( s − 1) 2 : ( P 1 , 1) ∧ n → ( P 1 , 1) ∧ n ∨ ( P 1 , 1) ∧ n . 4. The c o-gr oup (( P 1 , 1) ∧ n , Σ n − 1 P 1 ( σ P 1 )) in S H S 1 ( k ) /f n +1 is c o-c ommutative. 5. F or an extens ion A → B as ab ove, the map co - tr n x : Σ n P 1 Spec A + → Σ n P 1 Spec B + is indep endent of the choic e of x , and is denote d co - tr n B / A . SLICES AND TRANSFERS 25 6. S u pp ose that char k = 0 . The S H S 1 ( k ) /f n +1 -value d pr eshe af on Sm /k op Σ n P 1 ? + : Sm / k → S H S 1 ( k ) /f n +1 extends to an S H S 1 ( k ) /f n +1 -value d pr eshe af on S mC or ( k ) op , by sending a gen- er ator W ⊂ X × Y of Ho m S mC or ( X, Y ) t o t he morphism Σ n P 1 X + → Σ n P 1 Y + in S H S 1 ( k ) /f n +1 determine d by the diagr am Σ n P 1 Spec k ( X ) + ∼ / / co - tr n k ( W ) /k ( X )   Σ n P 1 X + Σ n P 1 Spec k ( W ) + p 2   Σ n P 1 Y + in S H S 1 ( k ) /f n +1 . The assertion that Σ n P 1 Spec k ( X ) + → Σ n P 1 X + is an isomorphism in S H S 1 ( k ) /f n +1 is p art of the statement. We write the map in S H S 1 ( k ) /f n +1 asso ciate d to α ∈ Hom S mC or ( X, Y ) as co - tr n ( α ) : Σ n P 1 X + → Σ n P 1 Y + . 7. Suppor ts and co-transfers In this sectio n, w e ass ume that char k = 0. W e consider the fo llowing situatio n. Let i : Y → X b e a co dimension one close d immersion in Sm /k , and let Z ⊂ X be a pure co dimensio n n clo sed subset of X such that i − 1 ( Z ) ⊂ Y also has pure co dimension one. W e let T = i − 1 ( Z ), X ( Z ) := ( X , X \ Z ), Y T = ( Y , Y \ T ), so that i induces the map of p ointed spaces i : X ( Z ) → Y ( T ) Let z be the set of g eneric p oints of Z , O X,z the semi-lo ca l r ing of z in X , X z = Spec O X,z and X ( z ) z = ( X z , X z \ z ). W e let t be the set of generic points of T , and let O X,t be the semi-lo cal ring of t in X , X t = Sp ec O X,t . Set Y t := X t × X Y and let Y ( t ) t = ( Y t , Y t \ t ). Lemma 7. 1. Ther e ar e c anonic al isomorphisms in S H S 1 ( k ) /f n +1 X ( Z ) ∼ = X ( z ) z ∼ = Σ n P 1 z + ; Y ( T ) ∼ = Y ( t ) t ∼ = Σ n P 1 t + . Pr o of. This follows fro m lemma 5.4.  Thu s, the inclusion i gives us the map in S H S 1 ( k ) /f n +1 : i : Σ n P 1 t + → Σ n P 1 z + . On the other hand, we ca n deﬁne a map i co - tr : Σ n P 1 t + → Σ n P 1 z + as follows: Let Z t = Z ∩ X t ⊂ X t . Since Y has co dimension one in X and intersects Z prop erly , t is a co llection of codimeniso n one p oints of Z , and thus Z t is a semi- lo cal reduced scheme of dimensio n one. Let p : ˜ Z t → Z t be the normaliza tio n, and 26 MARC LEVINE let ˜ t ⊂ ˜ Z t be the set of points lying over t ⊂ Z t . W rite ˜ t = ∪ j ˜ t j . F or each j , w e let n j denote the multiplicit y at ˜ t j of the pull-back Cartier divisor Y t × X t ˜ Z t , and let t j = p ( ˜ t j ). This gives us the diagra m ˜ t ˜ i / / p   ˜ Z t p   z j o o t i / / Z. Note tha t j is an isomorphism in S H S 1 ( k ) /f 1 . W e deﬁne i co - tr to b e the comp osition Σ n P 1 t + P j n j co - tr n ˜ t j /t − − − − − − − − − → Σ n P 1 ˜ t + Σ n P 1 ˜ i − − − → Σ n P 1 ˜ Z + Σ n P 1 j − 1 − − − − − → Σ n P 1 z + in S H S 1 ( k ) /f n +1 . Lemma 7. 2. i = i co - tr in S H S 1 ( k ) /f n +1 . Pr o of. Using Nisnevich excision, we may replace X with the Hens e lization of X along t ; we may als o assume that t is a single point. Via a limit argument, we may then replace X with a s mo o th aﬃne scheme of dimension n + 1 ov er k ( t ). Thus we may take Z to b e a reduced closed subscheme of X of pure dimension one ov er k ( t ). W e may also assume that Y is the ﬁber ov er 0 of a mor phism X → A 1 k ( t ) for which the restr iction to Z is ﬁnite. As we ar e working in S H S 1 ( k ) /f n +1 , we may r eplace ( X , Z ) with ( X ′ , Z ′ ) if there is a morphism f : X → X ′ which ma kes ( X , t ) a Hensel neighbor ho o d of ( X ′ , f ( t )) and such that the restriction of f to Z ′ is birational. Using Gabb er’s presentation lemma [6, lemma 3 .1], we may assume that X = A n +1 k ( t ) , that t = 0 and that Y is the co ordina te hyperpla ne X n +1 = 0. W e write F for k ( t ) and changing notation write simply 0 instead of t . After a suitable linear change of co ordinates in A n +1 F , w e may a ssume that ea ch co ordinate pro jection q : A n +1 F → A r F q ( x 1 , . . . , x n +1 ) = ( x i 1 , . . . , x i r ) , r = 1 , . . . , n , restr icts to a ﬁnite morphism on Z , a nd that Z → q ( Z ) is bir ational if r ≥ 2 . W e now reduce to the cas e in which Z is co ntained in the co or dinate subspa c e X ′ = A 2 F deﬁned by X 1 = . . . = X n − 1 = 0 . F or this, consider the map m : A 1 × A n +1 F → A 1 × A n +1 F m ( t, x 1 , . . . , x n +1 ) = ( t, tx 1 , . . . , tx n − 1 , x n , x n +1 ) Let Z = m ( A 1 × Z ) ⊂ A 1 × A n +1 F . By our ﬁniteness a ssumptions, Z is a (re- duced) c lo sed subscheme of A 1 × A n +1 F , and each ﬁb er Z t ⊂ t × A n +1 F is birationally isomorphic to Z × F F ( t ). Consider the inclusion map ( A 1 × Y ) ( A 1 × 0) → ( A 1 × X ) ( Z ) The maps i 0 , i 1 : Y (0) → ( A 1 × Y ) ( A 1 × 0) SLICES AND TRANSFERS 27 are clearly isomorphisms in H • ( k ), and the maps i 1 : X ( Z ) → ( A 1 × X ) ( Z ) i 0 : X ( Z 0 ) → ( A 1 × X ) ( Z ) are easily seen to b e is omorphisms in S H S 1 ( k ) /f n +1 . Combining this with the commutativ e diagram Y (0) / / i 1   X ( Z ) i 1   ( A 1 × Y ) ( A 1 × 0) / / X ( Z ) Y (0) / / i 0 O O X ( Z 0 ) i 0 O O shows that we can r eplace Z with Z 0 ⊂ X ′ . Having do ne this, we see that the map Y (0) → X ( Z ) is just the n − 1-fo ld P 1 susp ension of the map ( Y ∩ X ′ ) (0) → ( X ′ ) ( Z ) This reduces us to the case n = 1. Since p 2 : Z → A 1 F is ﬁnite, we may replace A 1 × A 1 F with P 1 × A 1 F . Then the map Y (0) → X ( Z ) is isomorphic to ( P 1 × 0 , ∞ × 0 ) → X ( Z ) . W e extend this to the isomorphic map ( P 1 × A 1 F , ∞ × A 1 F ) → X ( Z ) = ( P 1 × A 1 F , P 1 × A 1 F \ Z ) . Let s be the generic p oint of A 1 F , Z s the ﬁbe r o f p 2 ov er s . Then the inclusions ( P 1 × 0 , ∞ × 0 ) j 0 − → ( P 1 × A 1 F , ∞ × A 1 F ) j s ← − ( P 1 × s, ∞ × s ) ( P 1 × A 1 F , P 1 × A 1 F \ Z ) j s ← − ( P 1 × s, P 1 s \ Z s ) are isomorphisms in S H S 1 ( k ) /f 2 , and th us the map i 0 : Y (0) ∼ = ( P 1 × 0 , ∞ × 0 ) → X ( Z ) = ( P 1 × A 1 F , P 1 × A 1 F \ Z ) is isomorphic in S H S 1 ( k ) /f 2 to the collapse map ( P 1 × s, ∞ × s ) → ( P 1 × s, P 1 s \ Z s ) . Therefore, the map i : Σ P 1 0 + → Σ P 1 z + we need to consider is equal to the c o -transfer map co - tr Z s /s : Σ P 1 s + → Σ P 1 z s + comp osed with the (canonical) isomor phisms Σ P 1 0 + i 0 − → Σ P 1 s + ; Σ P 1 z s + ∼ = Σ P 1 z + , the latter isomor phism arising by noting that z s is a generic p oint of Z ov er F . The result now follows directly from prop osition 5.14.  28 MARC LEVINE Deﬁnition 7.3. 1. T ake X , X ′ ∈ Sm /k , a nd let Z ⊂ X , Z ′ ⊂ X ′ be pure co dimension n closed subsets. T ake a g enerator A ∈ Hom S mC or ( X, X ′ ), A ⊂ X × X ′ . L et q : A N → A b e the no rmalization o f A . Let z be the set of g eneric po in ts of Z , let a b e the set of generic p oints o f A ∩ X × Z ′ and let a ′ = q − 1 ( a ). Suppo se that (1) A N → X is ´ etale on a neighborho o d of a ′ (2) p X ( a ) is contained in Z . Let O A N ,a be the semi-lo cal ring o f a ′ in A N , a nd let A N a ′ = Sp ec O A N ,a ′ ; deﬁne X z similarly . Deﬁne co - tr n ( W ) : X ( Z ) → X ′ ( Z ′ ) to be the ma p in S H S 1 ( k ) /f n +1 given by the follo wing comp osition: X ( Z ) ∼ = X ( z ) z ∼ = Σ n P 1 z + co - tr n a ′ /z − − − − − − → Σ n P 1 a ′ + ∼ = A N ( a ′ ) a ′ p X ′ − − → X ′ ( Z ′ ) . 2. Let Hom S mC or ( X, X ′ ) Z,Z ′ ⊂ Hom S mC or ( X, X ′ ) b e the subgr oup generated by A sa tisfying (a) and (b). W e extend the deﬁnition of the mo r phism c o - tr n ( A ) to Hom S mC or ( X, X ′ ) Z,Z ′ by line a rity . Note that we implicity inv oke lemma 7.1 to e nsure that the isomor phisms used in the deﬁnition of co - tr n ( A ) exist and are cano nical; condition (1) implies in par - ticular that A is smo oth in a neighborho o d of a , so we ma y use lemma 7.1 for the isomorphism Σ n P 1 a + ∼ = A ( a ) a . Lemma 7.4. T ake X , X ′ , X ′′ ∈ Sm /k , and let Z ⊂ X , Z ′ ⊂ X ′ and Z ′′ ⊂ X ′′ b e a pur e c o dimension n close d su bsets. T ake α ∈ Hom S mC or ( X, X ′ ) Z,Z ′ , α ′ ∈ Hom S mC or ( X ′ , X ′′ ) Z ′ ,Z ′′ . Then α ′ ◦ α is in Hom S mC or ( X, X ′′ ) Z,Z ′′ and co - tr n ( α ′ ) ◦ co - tr n ( α ) = co - tr n ( α ′ ◦ α ) . Pr o of. W e may as sume tha t α and α ′ are gener ators A a nd A ′ . W e may replace X , X ′ and X ′′ with the resp ective strict Henseliza tions along z , z ′ and z ′′ . W rite z = { z 1 , . . . , z r } , z ′ = { z ′ 1 , . . . , z ′ s } , z ′′ = { z ′′ 1 , . . . , z ′′ t } . Then A and A ′ break up as a disjoint union of g raphs of morphisms f j k : X z k → X ′ z ′ j ; g ij : X ′ z ′ j → X ′′ z ′′ i and A ′ ◦ A is thus the sum of the graphs o f the co mpo sitions g ij ◦ f j k . Therefor e, each irreducible comp onent o f the supp o rt of A ′ ◦ A is smo o th. This veriﬁes c o ndition (1) of deﬁnition 7.3; the condition (2) is easy and is left to the re a der. The co mpatibilit y of co - tr n with the co mp os itio n of corresp ondences follows directly from theorem 6.1(6).  Prop ositio n 7.5. L et i : ∆ 1 → ∆ b e a close d immersion of quasi-pr oje ctive schemes in Sm /k , take X , X ′ ∈ Sm /k and α ∈ Hom S mC or ( X, X ′ ) . L et Z ⊂ X × ∆ , Z ′ ⊂ X ′ × ∆ b e close d c o dimension n subsets. Supp ose t hat (1) Z 1 := Z ∩ X × ∆ 1 and Z ′ 1 := Z ′ ∩ X ′ × ∆ 1 have c o dimension n in X × ∆ 1 , X ′ × ∆ 1 , r esp e ct ively. (2) α × id ∆ is in Ho m S mC or ( X × ∆ , X ′ × ∆) Z,Z ′ (3) α × id ∆ 1 is in Ho m S mC or ( X × ∆ 1 , X ′ × ∆ 1 ) Z 1 ,Z ′ 1 SLICES AND TRANSFERS 29 Then the diagr am in S H S 1 ( k ) /f n +1 ( X × ∆ 1 ) ( Z 1 ) id × i   co - tr n ( α × id) / / ( X ′ × ∆ 1 ) ( Z ′ 1 ) id × i   ( X × ∆) ( Z ) co - tr n ( α × id) / / ( X ′ × ∆) ( Z ′ ) c ommut es. Pr o of. Since ∆ is b y a ssumption quasi- pro jective, we may factor ∆ 1 → ∆ as a sequence of closed co dimension 1 immersions ∆ 1 = ∆ d → ∆ d − 1 → . . . → ∆ 1 → ∆ 0 = ∆ such that ea ch closed immersio n ∆ i → ∆ satisﬁes the conditions of the prop ositio n. This reduces us to the case of a co dimension one closed immersion. W e may r eplace X × ∆, X ′ × ∆, etc., with the r esp e ctiv e semi-lo cal schemes ab out the generic p oints o f Z 1 and Z ′ 1 . As ∆ 1 has co dimension one on ∆, it follows that the nor malizations Z N , Z ′ N of Z and Z ′ are smo oth over k . Let ˜ i : ˜ z → Z N , ˜ i ′ : ˜ z ′ → Z ′ N be the p oints of Z N , Z ′ N lying ov er Z 1 , Z ′ 1 , r esp ectively , which we write as a disjoin t union of clo s ed points ˜ z = ∐ j ˜ z j ; ˜ z ′ = ∐ j ˜ z ′ j . By lemma 7 .1 and lemma 7.2, w e may rewrite the diagram in the statement of the prop osition as Σ n P 1 Z 1+ P j m j co - tr n ˜ z j / Z 1   co - tr n ( α × id Z N 1 ) / / Σ n P 1 Z ′ 1+ P j m ′ j co - tr n ˜ z ′ j / Z ′ 1   Σ n P 1 ˜ z + ˜ i   Σ n P 1 ˜ z ′ + ˜ i ′   Σ n P 1 Z N co - tr n ( α × id Z N ) / / Σ n P 1 Z ′ N where α × id Z N , α × id Z 1 denote the cor r esp ondences induced b y α × id ∆ and α × id ∆ 1 , and the m j , m ′ j are the r e le v ant in tersection multiplicities. The commutativitiy of this dia gram follows from the functoriality of the maps co - tr n − / − ( − ) w ith resp ect to the compo sition of corre s po ndences (theorem 6 .1).  8. Slices o f loop s pectra T ake E ∈ S H S 1 ( k ). F ollowing V o evodsky’s r emarks in [21], Neeman’s version of Brown repr esentabilit y [16] gives us the motivic Postnik ov tow er . . . f n +1 E → f n E → . . . → f 0 E = E , where f n E → E is universal for morphisms from an ob ject of Σ n P 1 S H S 1 ( k ) to E . The lay er s n E is the n slic e of E , and is characteriz e d up to unique isomorphism by the distinguished tr iangle (8.1) f n +1 E → f n E → s n E → Σ s f n +1 E . 30 MARC LEVINE The fact that this distinguished tria ngle determines s n E up to unique isomor phism rather than just up to isomorphism follows fro m (8.2) Hom S H S 1 ( k ) (Σ n +1 P 1 S H S 1 ( k ) , s n E ) = 0 T o see this, just use the universal pr o pe r t y of f n +1 E → E and the long exact sequence o f Homs asso ciated to the distinguished triangle (8.1 ). In particular, using the description of Hom S H S 1 ( k ) /f n +1 ( − , − ) via right fra c tions w e hav e Lemma 8. 1. F or al l F, E ∈ S H S 1 ( k ) and n ≥ 0 , the natu ra l map Hom S H S 1 ( k ) ( F, s n E ) → Ho m S H S 1 ( k ) /f n +1 ( F, s n E ) is an isomorphism. See also [20, prop osition 5-3 ] W e reca ll the de-lo oping formula [11, theorem 7.4.2 ] s n (Ω P 1 E ) ∼ = Ω P 1 ( s n +1 E ) for n ≥ 0 . T ake F ∈ Sp c • ( k ). F or E ∈ Spt S 1 ( k ), we ha ve H om int ( F, E ) ∈ S H , which for F = X + is just E ( X ), and in ge ne r al is formed as the homotopy limit as so ciated to the description of F as a homotopy colimit of repres en table ob jects, i.e., take the Kan extension to Sp c • of the functor E : Sm / k op → Spt S 1 ( k ). This gives us the “internal Hom” functor H om S H S 1 ( k ) ( F, − ) : S H S 1 ( k ) → S H S 1 ( k ) and more generally H om S H S 1 ( k ) /f n +1 ( F, − ) : S H S 1 ( k ) /f n +1 → S H S 1 ( k ) , with natural transfor mation H om S H S 1 ( k ) ( F, − ) → H om S H S 1 ( k ) /f n +1 ( F, − ) . These have v alue o n E ∈ Spt S 1 ( k ) deﬁned by ta king a ﬁbr a nt mo del ˜ E of E (in S H S 1 ( k ) or S H S 1 ( k ) /f n +1 , as the case may b e) a nd forming the pr esheaf o n Sm /k X 7→ H om int ( F ∧ X + , ˜ E ) . Putting the de-lo oping formula toge ther with lemma 8.1 gives us Prop ositio n 8.2. F or E ∈ S H S 1 ( k ) we have natu r al isomorphisms s 0 (Ω n P 1 E ) ∼ = Ω n P 1 s n E ∼ = H om S H S 1 ( k ) /f n +1 (( P 1 , 1) ∧ n , s n E ) Pr o of. Indeed, the ﬁr st isomor phism is just the de-lo oping isomo rphism re p ea ted n times. F or the seco nd, w e hav e Ω n P 1 s n E ∼ = H om S H S 1 ( k ) (( P 1 , 1) ∧ n , s n E ) ∼ = H om S H S 1 ( k ) /f n +1 (( P 1 , 1) ∧ n , s n E ) the second isomorphism following from le mma 8.1.  Deﬁnition 8.3. Supp ose tha t char k = 0. T ake E ∈ S H S 1 ( k ) and ta ke α ∈ Hom S mC or ( X, Y ). Deﬁne the tr ansfer T r Y /X ( α ) : (Ω n P 1 s n E )( Y ) → (Ω n P 1 s n E )( X ) SLICES AND TRANSFERS 31 as follows: (Ω n P 1 s n E )( Y ) ∼ = H om int S H S 1 ( k )( Y + , Ω n P 1 s n E ) ∼ = H om int S H S 1 ( k ) (Σ n P 1 Y + , s n E ) ∼ = H om int S H S 1 ( k ) /f n +1 (Σ n P 1 Y + , s n E ) co - tr n ( α ) ∗ − − − − − − − → H om int S H S 1 ( k ) /f n +1 (Σ n P 1 X + , s n E ) ∼ = H om int S H S 1 ( k ) ( X + , Ω n P 1 s n E ) (Ω n P 1 s n E )( X ) . Theorem 8.4. Supp ose that char k = 0 . F or E ∈ S H S 1 ( k ) , the maps T r( α ) ex tend the pr eshe af Ω n P 1 s n E : Sm / k op → S H to an S H -value d pr eshe af with t r ansfers Ω n P 1 s n E : S mC or ( k ) op → S H Pr o of. This fo llows from the deﬁnition of the maps T r( α ) and theorem 6.1, the main po in t be ing that the maps T r( α ) fac tor thro ugh an internal Hom in S H S 1 ( k ) /f n +1 .  Corollary 8.5 . Supp ose that char k = 0 . F or E ∈ S H S 1 ( k ) , ther e is an extension of the pr eshe af s 0 Ω P 1 E : Sm /k op → S H to an S H -value d pr eshe af with t r ansfers s 0 Ω P 1 E : S mC or ( k ) op → S H . Pr o of. This is just the c ase n = 1 of theorem 8.4, together with the de-lo o ping isomorphism s 0 Ω P 1 E ∼ = Ω P 1 s 1 E .  R emark 8.6 . The co rollary is actually the main result, in tha t one can deduce theorem 8.4 from corolla ry 8.5 (applied to Ω n − 1 P 1 E ) and the de-loo ping for m ula Ω n P 1 s n E ∼ = s 0 Ω n P 1 E = s 0 Ω P 1 (Ω n − 1 P 1 E ) . As the maps co - tr n ( α ) ar e deﬁned by smashing co - tr 1 ( α ) with an identit y map, this pro cedure do es indeed give back the maps T r( α ) : Ω n P 1 s n E ( Y ) → Ω n P 1 s n E ( X ) as deﬁned ab ov e. pr o of of t he or em 3. The weak transfers deﬁned above give rise to homotopy inv ari- ant sheav es with tra nsfers in the usua l sens e by tak ing the sheav es of homotopy groups of the motivic sp ectrum in q uestion. F or insta nce, co rollary 8.5 gives the sheaf π m ( s 0 Ω P 1 E ) the structure of a ho motopy inv aria nt sheaf with transfers, in particular, an eﬀective motive. In fact, these are bir ational motives in the sense of Kahn-Hub er- Sujatha [7, 1 0], as s 0 F is a bir ational S 1 -sp ectrum for each S 1 - sp ectrum F . The classical Postnik ov tow er th us gives us a spectra l sequence E 2 p,q := H − p ( X Nis , π q ( s 0 Ω P 1 E )) = ⇒ π p + q ( s 0 Ω P 1 E ( X )) 32 MARC LEVINE with E 2 term a “generalized mo tivic cohomology” of X . As the sheav es π q ( s 0 Ω P 1 E ) are mo tiv es, w e may repla ce Nisnevich cohomo logy with Zariski cohomolo gy; as the sheaves π q ( s 0 Ω P 1 E ) are bira tio nal, i.e., Za riski lo cally tr ivial, the higher Zarisk i cohomolog y v a nishes, giving us π n ( s 0 Ω P 1 E ( X )) ∼ = H 0 ( X Zar , π n ( s 0 Ω P 1 E )) = π n ( s 0 Ω P 1 E ( k ( X )) .  In shor t, we have shown tha t the 0th slice o f a P 1 -lo op sp ectrum ha s trans fer s in the weak sense. W e hav e a lr eady seen in section 2 that this do es no t hold for a n arbitrar y ob ject o f S H S 1 ( k ); in the next section we will see that the hig her slices of a n a rbitrary S 1 -sp ectrum do hav e transfers, albeit in a n even weaker sense than the one used above. 9. Transfers on the generaliz ed cycl e complex W e b egin b y recalling from [11, theorem 7 .1.1] mo dels for f n E and s n E ( X ) that are reminiscent o f Blo ch’s higher c ycle complex [1 ]. T o s implify the notation, we will always a ssume that w e hav e tak en a mo del E ∈ Spt S 1 ( k ) which is quasi-ﬁbra n t. F or a scheme X of ﬁnite t yp e and lo cally equi-dimensio nal ov er k , let S ( n ) X ( m ) be the set of clos ed subs ets W of X × ∆ m of co dimension ≥ n , such that, for ea ch face F of ∆ n , W ∩ X × F has co dimension ≥ n on X × F (or is empt y). W e order by S ( n ) X ( m ) inclusion. F or X ∈ Sm /k , we let E ( n ) ( X, m ) := lim − → W ∈S ( n ) X ( m ) E ( W ) ( X × ∆ m ) , where E ( W ) ( X ) is by deﬁnition the homotopy ﬁb er o f the restrictio n map E ( X × ∆ n ) → E ( X × ∆ n \ W ). Similarly , for 0 ≤ n ≤ n ′ , w e deﬁne E ( n/n ′ ) ( X, m ) := lim − → W ∈S ( n ) X ( m ) ,W ′ ∈S ( n ′ ) X ( m ) E ( W \ W ′ ) ( X × ∆ m \ W ′ ) The co nditions on the intersections of W with X × F for faces F mea ns that m 7→ S ( n ) X ( m ) for m a cosimplicial set, denoted S ( n ) X , for each n and tha t S ( n ′ ) X is a cos implicia l subs et of S ( n ) X for n ≤ n ′ . Thus the restric tio n maps for E make m 7→ E ( n ) ( X, m ) and m 7→ E ( n/n ′ ) ( X, m ) simplicial sp ectra, denoted E ( n ) ( X, − ) and E ( n/n ′ ) ( X, − ). W e denote the asso ciated total sp ectra by | E ( n ) ( X, − ) | and | E ( n/n ′ ) ( X, − ) | . The inclusion S ( n ′ ) X ( m ) → S ( n ) X ( m ) for n ≤ n ′ and the evident r e striction ma ps give the sequence | E ( n ′ ) ( X, − ) | → | E ( n ) ( X, − ) | → | E ( n/n ′ ) ( X, − ) | which is ea s ily s een to be a w eak homo to p y ﬁber sequence. W e note that | E (0) ( X, − ) | = E ( X × ∆ ∗ ); as E is homo topy inv ariant, the cano n- ical map E ( X ) → | E (0) ( X, − ) | is th us a weak eq uiv alence. W e therefore have the tow er in S H (9.1) . . . → | E ( n +1) ( X, − ) | → | E ( n ) ( X, − ) | → . . . → | E (0) ( X, − ) | ∼ = E ( X ) SLICES AND TRANSFERS 33 with n th layer isomor phic to | E ( n/n +1) ( X, − ) | . W e call this tow er the homotopy c onive au tower for E ( X ). In this reg ard, o ne of the main results from [11] states Theorem 9. 1 ([11 , theorem 7.1.1]) . Ther e is a c anonic al isomorphism of the tower (9.1) with the motivic Postnikov tower evaluate d at X : . . . → f n +1 E ( X ) → f n E ( X ) → . . . → f 0 E ( X ) = E ( X ) , giving a c anonic al isomorphism s n E ( X ) ∼ = | E ( n/n +1) ( X, − ) | . W e can further mo dify this des c ription o f s n E ( X ) as fo llows: Since s n is an idempo ten t functor, we ha ve s n E ( X ) ∼ = s n ( s n E )( X ) ∼ = | ( s n E ) ( n/n +1) ( X, − ) | Note that | ( s n E ) ( n/n +1) ( X, − ) | ﬁts in to a weak homotopy ﬁb er sequence | ( s n E ) ( n +1) ( X, − ) | → | ( s n E ) ( n ) ( X, − ) | → | ( s n E ) ( n/n +1) ( X, − ) | Using theorem 9.1 in reverse, we hav e the isomor phis m in S H | ( s n E ) ( n +1) ( X, − ) | ∼ = f n +1 ( s n E )( X ) But as f n +1 ◦ f n ∼ = f n +1 , we see that f n +1 ( s n E ) ∼ = 0 in S H S 1 ( k ) and th us | ( s n E ) ( n ) ( X, − ) | ∼ = | ( s n E ) ( n/n +1) ( X, − ) | ∼ = s n E ( X ) W e may therefo re use the simplicial mo del | ( s n E ) ( n ) ( X, − ) | for s n E ( X ). W e will need a reﬁnement of this constructio n, whic h takes into account the int era ction of the suppor t conditions with a given corr espo ndence. Deﬁnition 9.2 . L e t A ⊂ Y × X b e a g enerator in Hom S mC or ( Y , X ); for each m , we let A ( m ) ∈ Hom S mC or ( Y × ∆ m , X × ∆ m ) denote the corresp ondence A × id ∆ m . Let S ( n ) X,A ( m ) be the subset of S ( n ) X ( m ) consisting of those W ′ ∈ S ( n ) X ( m ) such that (1) W := p Y × ∆ m ( A × ∆ m ∩ Y × W ′ ) is in S ( n ) Y ( m ). (2) A ( m ) is in Hom S mC or ( Y × ∆ m , X × ∆ m ) W ,W ′ . F or an arbitra ry α ∈ Hom S mC or ( Y , X ), write α = r X i =1 n i A i with the A i generator s and the n i non-zero integers and de ﬁne S ( n ) X,α ( m ) := ∩ r i =1 S ( n ) X,A i ( m ) . If we hav e in a dditio n to α a ﬁnite corres po ndence β ∈ Hom S mC or ( Z, Y ), we le t S ( n ) X,α,β ( m ) ⊂ S ( n ) X,α ( m ) b e the set of W ⊂ X × ∆ m such that W is in S ( n ) X,α ( m ) and p Y × ∆ m ( Y × W ∩ | α | × ∆ m ) is in S ( n ) Y ,β ( m ). F or f : Y → X a ﬂat mor phism, one has S ( n ) X, Γ f ( m ) = S ( n ) X ( m ) and for g : Z → Y a ﬂat morphism, and α arbitra ry , one has S ( n ) X,α, Γ f ( m ) = S ( n ) X,α ( m ) 34 MARC LEVINE Note that m 7→ S ( n ) X,α ( m ) and m 7→ S ( n ) X,α,β ( m ) deﬁne cosimplicial subsets of m 7→ S ( n ) X ( m ). W e deﬁne the simplicial sp ectr a E ( n ) ( X, − ) α and E ( n ) ( X, − ) α,β using the suppo rt conditions S ( n ) X,α ( m ) and S ( n ) X,α,β ( m ) instead of S ( n ) X ( m ): E ( n ) ( X, m ) α := lim − → W ∈S ( n ) X,α ( m ) E ( W ) ( X × ∆ m ) E ( n ) ( X, m ) α,β := lim − → W ∈S ( n ) X,α,β ( m ) E ( W ) ( X × ∆ m ) giving us the sequence of simplicial sp ectra E ( n ) ( X, − ) α,β → E ( n ) ( X, − ) α → E ( n ) ( X, − ) . The main “moving lemma” [12, theor em 2.6.2(2)] yields Prop ositio n 9.3 . F or X ∈ Sm /k aﬃne, and E ∈ Spt S 1 ( k ) quasi-ﬁbr ant, the maps | E ( n ) ( X, − ) α,β | → | E ( n ) ( X, − ) α | → | E ( n ) ( X, − ) | ar e we ak e quivalenc es. W e pro ce e d to the main co nstruction of this section. Co nsider the simplicial mo del | ( s n E ) ( n ) ( X, − ) | for s n E ( X ). F o r e ach m , w e may consider the cla s sical Postnik ov tow er for the spectr um ( s n E ) ( n ) ( X, m ), whic h we wr ite as . . . → τ ≥ p +1 ( s n E ) ( n ) ( X, m ) → τ ≥ p ( s n E ) ( n ) ( X, m ) → . . . → ( s n E ) ( n ) ( X, m ) , where τ ≥ p +1 ( s n E ) ( n ) ( X, m ) → ( s n E ) ( n ) ( X, m ) is the p -connected cov er of ( s n E ) ( n ) ( X, m ). The p th lay er in this tow er is o f cour se the Eilenberg-Ma clane sp ectrum on π p (( s n E ) ( n ) ( X, m )), or ra ther its p th susp en- sion. T aking a functoria l mo del fo r the p -connected cov er, we hav e for each p the simpicial sp e ctrum m 7→ τ ≥ p +1 ( s n E ) ( n ) ( X, m ) giving us the tow e r of total sp ectra (9.2) . . . → | τ ≥ p +1 ( s n E ) ( n ) ( X, − ) | → | τ ≥ p ( s n E ) ( n ) ( X, − ) | → . . . → | ( s n E ) ( n ) ( X, − ) | . The layers in this tow er are then (up to susp ension) the E ilen b erg- Maclane sp ectrum on the chain complex π p ( s n E ) ( n ) ( X, ∗ ), with diﬀerential as usual the alterna ting sum of the face maps. The chain complex e s π p ( s n E ) ( n ) ( X, ∗ ) are eviden tly functorial for smoo th maps and inher it the homotopy inv aria nce prop erty from ( s n E ) ( n ) ( X, ∗ ) (see [12, theorem 3.3.5]). Somewhat more surpr ising is Lemma 9. 4. The c omplexes π p ( s n E ) ( n ) ( X, ∗ ) satisfy Nisnevich excision. Pr o of. Let W ⊂ X × ∆ m be a close d subset in S ( n ) X ( m ), and let w b e the set of generic points of W . Then s n E ( W ) ( X × ∆ m ) ∼ = s n E (Σ n P 1 w + ) ∼ = Ω n P 1 ( s n E )( w ) ∼ = s 0 (Ω n P 1 E )( w ) . This gives us the following description of π p (( s n E ) ( n ) ( X, m )): ( s n E ) ( n ) ( X, m ) ∼ = ⊕ w π p ( s 0 (Ω n P 1 E )( w )) SLICES AND TRANSFERS 35 where the dir ect sum is over the set T ( n ) X ( m ) o f g eneric p oints of irr educible W ∈ S ( n ) X ( m ). Now let i : Z → X b e a clo sed subset with op en complement j : U → X . F or each m , we thus hav e the exact sequence 0 → ⊕ w ∈ Z × ∆ m ∩T ( n ) X ( m ) π p ( s 0 (Ω n P 1 E )( w )) → ⊕ w ∈ T ( n ) X ( m ) π p ( s 0 (Ω n P 1 E )( w )) → ⊕ w ∈ T ( n ) X ( m ) ∩ U × ∆ m π p ( s 0 (Ω n P 1 E )( w )) → 0 Deﬁne the sub complex π p ( s n E ) ( n ) ( X, ∗ ) Z of π p ( s n E ) ( n ) ( X, ∗ ) and quo tien t c o m- plex π p ( s n E ) ( n ) ( U X , ∗ ) of π p ( s n E ) ( n ) ( X, ∗ ) by taking supp orts in W ∈ S ( n ) X ( m ) ∩ Z × ∆ ∗ , W ∈ S ( n ) X ( m ) ∩ U × ∆ ∗ . W e thus have the term- wise exact sequence of complexes 0 → π p ( s n E ) ( n ) ( X, ∗ ) Z → π p ( s n E ) ( n ) ( X, ∗ ) → π p ( s n E ) ( n ) ( U X , ∗ ) → 0 The lo ca lization technique of [13, theorem 8.10] (for details, see [11, theo rem 3.2.1]) implies that the inclusion π p ( s n E ) ( n ) ( U X , ∗ ) → π p ( s n E ) ( n ) ( U, ∗ ) is a quasi-isomor phism, a nd we therefore ha ve the qua si-isomorphis m π p ( s n E ) ( n ) ( X, ∗ ) Z → cone( π p ( s n E ) ( n ) ( X, ∗ ) j ∗ − → π p ( s n E ) ( n ) ( U, ∗ )[ − 1] . But the left-hand side o nly dep ends on the the Nisnevich neig h b orho o d of Z in X , which yields the des ired Nisnevich excisio n proper t y .  W e will use the results o f section 7 to give X 7→ π p ( s n E ) ( n ) ( X, ∗ ) the structure of a complex of homotopy inv aria n t presheav es with transfer on Sm /k , i.e. a motive. F or this, w e consider the co mplex e s π p ( s n E ) ( n ) ( X, ∗ ) α , π p ( s n E ) ( n ) ( X, ∗ ) α,β con- structed ab ov e. The r eﬁned supp ort condition a re cons tr ucted so that, for each W ∈ S ( n ) X,α ( m ), α is in Hom S mC or ( Y , X ) W ′ ,W , where W ′ = p 1 ( Y × ∆ m × W ∩ | α | × ∆ m ) . W e may therefo re use the morphism co - tr n ( α × id ∆ m ) to deﬁne the map T r Y /X ( α )( m ) : π p (( s n E ) ( n ) ( X, m )) α → π p ( s n E ) ( n ) ( Y , m ) . By prop osition 7.5, the maps T r Y /X ( m ) deﬁne a map of complexes T r Y /X ( α ) : π p ( s n E ) ( n ) ( X, ∗ ) α → π p ( s n E ) ( n ) ( Y , ∗ ) . Similarly , given β ∈ Hom S mC or ( Z, Y ), we ha ve the map of complexes T r Y /X ( α ) β : π p ( s n E ) ( n ) ( X, ∗ ) α,β → π p ( s n E ) ( n ) ( Y , ∗ ) β . Note that, due to pos s ible “ca ncellations” o ccurring when one takes the comp osition α ◦ β , we ha ve only an inclusio n S ( n ) X,α,β ( m ) ⊂ S ( n ) X,α ◦ β ( m ) giving us a natural compariso n map ι α,β : π p ( s n E ) ( n ) ( X, ∗ ) α,β → : π p ( s n E ) ( n ) ( X, ∗ ) α ◦ β . Using our moving lemma again, w e se e that ι α,β is a quasi-iso morphism in case X is aﬃne. 36 MARC LEVINE Lemma 9. 5. Su pp ose char k = 0 . F or α ∈ Hom S mC or ( Z, Y ) , β ∈ Hom S mC or ( Z, Y ) , we have T r Z/ Y ( β ) ◦ T r Y /X ( α ) β = T r Z/X ( α ◦ β ) ◦ ι α,β . Pr o of. This follows fro m lemma 7.4.  W e have alrea dy noted that complexes π p ( s n E ) ( n ) ( X, ∗ ) are functorial in X fo r ﬂat mo rphisms in Sm /k , in particular for smo oth morphisms in Sm /k . Let g Sm /k denote the s ub categ ory of Sm /k with the sa me o b jects and with mor phisms the smo oth mor phisms. The transfer maps we hav e deﬁned on the reﬁned complexes, together with the moving lemma 7.4 yield the following result: Theorem 9. 6. Su pp ose char k = 0 . Consider t he pr eshe af π p (( s n E ) ( n ) ( − , ∗ )) : g Sm /k op → C − ( Ab ) on g Sm /k op . L et ι : g Sm /k → S mC or ( k ) b e the evident inclusion and let Q : C − ( Ab ) → D − ( Ab ) b e the evident additive functor. Ther e is a c omplex of pr eshe aves with tr ansfers ˆ π p (( s n E ) ( n ) ) ∗ : S mC or ( k ) op → C − ( Ab ) and an isomorphism of functors fr om g Sm /k op to D − ( Ab ) Q ◦ π p (( s n E ) ( n ) ( − , ∗ )) ∼ = Q ◦ ˆ π p (( s n E ) ( n ) ) ∗ ◦ ι. Pr o of. W e g ive a r ough sketch o f the constructio n here ; for details we r efer the reader to [9 , pr op osition 2.2.3], which in turn is an elab oration of [12, theor em 7.4.1]. The construction of ˆ π p (( s n E ) ( n ) ) ∗ is accomplished by ﬁrst taking a homotopy limit ov er the complex e s π p ( s n E ) ( n ) ( X, ∗ ) α . These are then functorial on S mC or ( k ) op , up to homo topy equiv ale nces a rising from the replac e men t o f the index catego ry for the homo topy limit with a certain coﬁna l s ubca tegory . One then forms a reg- ularizing homotopy c o limit that is strictly functorial on S mC or ( k ) op , a nd ﬁnally , one replaces this preshea f with a ﬁbrant mo del. The mo ving lemma for aﬃne schemes (prop ositio n 9.3) implies that the homotopy limit co nstruction yie lds for each aﬃne X ∈ Sm /k a complex cano nically quasi-isomo rphic to π p ( s n E ) ( n ) ( X, ∗ ); this prop erty is inherited by the regula rized ho motopy colimit. As the complexes π p ( s n E ) ( n ) ( X, ∗ ) satisfy Nisnev ic h excision (lemma 9.4) and are homotopy inv ar i- ant for al l X , this implies that the ﬁbrant model ˆ π p (( s n E ) ( n ) ) ∗ is canonically quas i- isomorphic to π p ( s n E ) ( n ) ( X, ∗ ) for all X ∈ Sm /k .  Corollary 9.7 . Su pp ose char k = 0 . ˆ π p (( s n E ) ( n ) ) ∗ is a homotopy invariant c om- plex of pr eshe aves with tr ansfer. Pr o of. By theorem 9.6, we have the iso mo rphism in D − ( Ab ) ˆ π p (( s n E ) ( n ) ) ∗ ∼ = π p (( s n E ) ( n ) ( − , ∗ )) . for all X ∈ Sm /k . As the preshea f π p (( s n E ) ( n ) ( − , ∗ )) is homoto p y in v a riant, so is ˆ π p (( s n E ) ( n ) ) ∗ .  SLICES AND TRANSFERS 37 pr o of of t he or em 2. As in the pro o f of theorem 9.6, the metho d of [1 2, theorem 7.4.1], shows that the tow er (9.2) extends to a tow er (9.3) . . . → ρ ≥ p +1 s n E → ρ ≥ p s n E → . . . → s n E in S H S 1 ( k ) with v alue (9 .2) at X ∈ Sm /k , and with the coﬁb er of ρ ≥ p +1 s n E → ρ ≥ p s n E naturally isomor phic to E M A 1 ef f ( ˆ π p (( s n E ) ( n ) ) ∗ ). By cor ollary 9.7, the presheav es ˆ π p (( s n E ) ( n ) ) ∗ deﬁne ob jects in D M ef f − ( k ). Thus, we have shown that the lay ers in the to wer (9.3 ) hav e a “ motivic” structure, proving theo rem 2.  10. The Friedlan der-Suslin tower As the r e ader ha s surely noticed, the la c k of functoria lit y for the simplicia l sp ec- tra E ( n ) ( X, − ) cr eates annoying technical problems when we wish to extend the construction of the homotopy co niveau tow er to a tow er in S H S 1 ( k ). In their work on the sp ectral sequence from motivic cohomolo g y to K -theory , F riedlander and Suslin [4] hav e construc ted a c ompletely functorial version of the homotopy coniveau tow e r , us ing “quasi-ﬁnite supp orts”. Unfortunately , the comparis on be tw een the F riedlander -Suslin version and E ( n ) ( X, − ) is prov en in [4] only for K -theory and mo- tivic cohomo logy . In this last section, we recall the F riedlander- Suslin construction and for m the co njecture that the F riedlander-Suslin tow er is na turally isomorphic to the homotopy coniveau to wer. Let Q ( n ) X ( m ) be the s et o f closed s ubs e ts W of A n × X × ∆ m such that, for each irreducible compo nent W ′ of W , the pro jection W ′ → X × ∆ m is quasi-ﬁnite. F or E ∈ Spt S 1 ( k ), we let E ( n ) F S ( X, m ) := lim − → W ∈Q ( n ) X ( m ) E ( W ) ( A n × X × ∆ m ) As the condition deﬁning Q ( n ) X ( m ) are preserved under ma ps id A n × f × g : A n × X ′ × ∆ m ′ → A n × X × ∆ m , where f : X ′ → X is an arbitra r y map in Sm /k , and g : ∆ m ′ → ∆ m is a structure map in ∆ ∗ , the sp ectra E ( n ) F S ( X, m ) deﬁne a simplicial sp ectrum E ( n ) F S ( X, − ) and these simplicia l s pectr a, for X ∈ Sm /k , e xtend to a preshea f of simplicial sp ectra on Sm /k : E ( n ) F S (? , − ) : Sm /k op → ∆ op Spt . Similarly , if we take the linear embedding i n : A n → A n +1 = A n × A 1 , x 7→ ( x, 0), the pull-ba ck by i n × id pres erves the supp ort conditions, and thus gives a well- deﬁned map of simplicial sp ectra i ∗ n : E ( n +1) F S ( X, − ) → E ( n ) F S ( X, − ) , forming the tow er of presheav e s on Sm /k (10.1) . . . → E ( n +1) F S (? , − ) → E ( n ) F S (? , − ) → . . . W e ma y c o mpare E ( n ) F S ( X, − ) a nd E ( n ) ( X, − ) using the metho d of [4] as fo llows: The simplicial sp ectra E ( n ) ( X, − ) are functorial for ﬂat ma ps in Sm /k , in the evident manner. They satisfy homotopy inv ariance, in that the pull-back map p ∗ : E ( n ) ( X, − ) → E ( n ) ( A 1 × X , − ) 38 MARC LEVINE induces a weak equiv alence o n the to tal sp ectra . W e hav e the e vident inclusion of simplicial sets F S ( n ) X ( − ) ֒ → S ( n ) A n × X ( − ) inducing the map ϕ X,n : E ( n ) F S ( X, − ) → E ( n ) ( A n × X , − ) T ogether with the weak equiv a lence p ∗ : | E ( n ) ( X, − ) | → | E ( n ) ( A n × X , − ) | , the maps ϕ X,n induce a map of tow e rs of total spectra in S H (10.2) ϕ X, ∗ : | E ( ∗ ) F S ( X, − ) | → | E ( ∗ ) ( X, − ) | . Conjecture 1 0.1. F or e ach X ∈ Sm /k and e ach quasi-ﬁbr ant E ∈ Spt S 1 ( k ) , the map (10.2) induc es an isomorphism in S H of t he towers of total sp e ct r a. Combined with the w eak equiv ale nc e given by homotopy inv a r iance and the results of [11], this would give us an isomorphism in S H S 1 ( k ): f n E ∼ = | E ( n ) F S (? , − ) | References [1] Blo ch, S., Al gebraic cycles and higher K -theory , Adv. i n M ath. 61 (1986), no. 3, 267–304. [2] Blo ch, S. and Lich tenbaum, S., A spectral sequence for motivic cohomology , preprint (1995). [3] Cisi nski, D. -C. and D´ eglise, F. 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Math. 219 (2008), 689–727. [19] Pelaez-Menaldo, J.P . Multiplicative structure on the m otivic Postn iko v tow er . preprint , 2007. [20] V erdier, J. L., Categorie derivees . [21] V o ev o dsky , V . A p ossible new a pproach to the motivic sp ectral s equence for algebraic K - theory , R e c ent pr o gr ess in homotopy the ory (Baltimor e, MD, 2000) 371–379, Con temp. Math., 29 3 (Amer. Math. So c., Providence, RI, 2002). SLICES AND TRANSFERS 39 Nor theastern University, Dep ar tment of Ma themati cs, Boston, MA 02115, U.S.A. Universit ¨ at Duisburg-Essen, F akul t ¨ at Ma thema tik, Cam pus Essen, 45117 Essen, Ger- many E-mail addr e ss : marc.levi ne@uni-due.de

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