A counterexample to generalizations of the Milnor-Bloch-Kato conjecture

A coun terexample to generalizat ions of the Milnor-Blo c h-Kato conjecture ∗ By Mic hael Spieß and T ak ao Y amazaki Abstract W e construct an example of a torus T ov er a field K for which the Galois symbol K ( K ; T , T ) /nK ( K ; T , T ) → H 2 ( K, T [ n ] ⊗ T [ n ]) is no t injectiv e for s ome n . Here K ( K ; T , T ) is the Milnor K -gr oup attached to T intro duced by Somek aw a . W e show also that the motive M ( T × T ) gives a counterexample to ano ther g e neralization of the Milnor- Blo ch- Kato conjecture (pr o p osed by Beilinson). 1 In tro d uction Let K b e a field, m a p ositiv e int eger and n an inte ger pr im e to the c h ar- acteristic of K . The Milnor-Blo c h-Kato conjecture asserts that the Galois sym b ol (1) K M m ( K ) /nK M m ( K ) − → H m ( K, Z /n Z ( m )) from Milnor K -groups to Galois cohomology is b ij ective . Recently , Rost and V o evodsky hav e announ ced a pro of (sp ecial cases ha ve b een obtained earlier b y Merkurjev-Su slin, Ro st and V o ev o dsky). In [So], Somek aw a has in tro du ced certain gener alize d Milnor K -gr oups K ( K ; A 1 , . . . , A m ) attac hed to semi-ab elian v arieties A 1 , . . . , A m . If A 1 = . . . = A m = G m is the one-dimens ional split torus they agree with the u sual K M m ( K ). If m = 2, A 1 = Jac X and A 2 = Jac Y are the Jac ob ians of sm o oth, pro jectiv e and connected cur ves X and Y o v er K ha ving a K -rational p oint , then K ( K ; A 1 , A 2 ) is the kernel of the Albanese map CH 0 ( X × Y ) deg=0 → Alb X × Y ( K ). Somek a w a has defined a Galois symb ol (2) K ( K ; A 1 , . . . , A m ) /nK ( K ; A 1 , . . . , A m ) − → H m ( K, A 1 [ n ] ⊗ . . . ⊗ A m [ n ]) ∗ This w ork w as done d uring the second author staye d at Universit¨ at Bielefeld supported by SFB 701. He is grateful to the members there. 1 and conjectured that it is alwa ys injectiv e. In this n ote w e p resen t a coun- terexample (see section 2). Let us describ e it briefly . Let L/K b e a cyclic extension of degree n and σ a generator of the Galoi s group Gal( L/K ). Let T b e the k er n el of the norm map Res L/K G m → G m . W e sh o w th at the n orm K ( L ; T , T ) → K ( K ; T , T ) induces an isomorphism K 2 ( L ; T , T ) / (1 − σ ) → K 2 ( K ; T , T ). On the other han d , the corresp ond ing map of Galois coho- mology groups H 2 ( L, T [ n ] ⊗ T [ n ]) / (1 − σ ) → H 2 ( K, T [ n ] ⊗ T [ n ]) is neither injectiv e nor surjectiv e (for a s u itable choice of L/K ). No te that, since T is sp lit o v er L , the Galois symb ol K 2 ( L ; T , T ) /n → H 2 ( L, T [ n ] ⊗ T [ n ]) is bijectiv e. Consequent ly , K 2 ( K ; T , T ) /n → H 2 ( K, T [ n ] ⊗ T [ n ]) is in general not injectiv e. In the section 3 w e s ho w th at the motiv e M ( T × T ) giv es a coun terexam- ple to another generalizatio n of the Milnor-Blo c h-Kato conjecture (prop osed b y Beilinson). W e w ould lik e to thank Bruno Kahn for his comments on the first v ersion of this note. In p articular he pointed out to us th at our coun terexample to Somek a w a’s conjecture should also pro vide a co unt er examp le to Beilinson’s conjecture. 2 Coun terexample to Somek a w a’s conjecture Algebraic groups as Mack ey-functors Let K b e a fi eld. F or a finite field extension L/K and comm utativ e algebraic groups G o v er K and H o ve r L w e denote b y G L the base c hange of G to L and by Res L/K H the W eil restriction of H . The fun ctor G 7→ G L is left and right adjoin t to H 7→ Res L/K H . In p articular there are adjunction h omomorphisms ι L/K : G → Res L/K G L and N L/K : Res L/K G L → G . When L/K is a Ga lois extension, the Galois group G al ( L/K ) acts ca n onically on Res L/K G L . The follo wing s imple r esult, whose pro of will b e left to the reader, will b e used later. Lemma 1 L et L/K b e a cyclic Galois extension of de gr e e n , σ a gener ator of Gal( L/K ) and let G b e a c ommutative algebr aic gr oup over K . L et G ′ b e the kernel of N L/K : Res L/K G L → G so that G ′ L ∼ = G n − 1 L . Then the map Res L/K ( G L ) n − 1 ∼ = Res L/K G ′ L N L/K − → G ′ ֒ → Res L/K G L is given on the i -th summand by 1 − σ i . W e d en ote by C K the category of fin ite reduced K -sc hemes. Th us eac h ob ject of C K is isomorphic to Sp ec( E 1 × . . . × E r ) where E 1 , . . . , E r /K are finite field ext ensions. A comm u tativ e algebraic group G o v er K defin es 2 a Mackey-functor , i.e. a co - and con tra v arian t fu nctor G : C K → Mod Z satisfying (i), (ii) b elo w. If f : X → Y is a morphism we denote by f ∗ : G ( X ) → G ( Y ) and f ∗ : G ( Y ) → G ( X ) the homomorph isms induced b y co- and con tra v arian t fun ctorialit y resp ecti vely . (i) If X = X 1 ` X 2 ∈ Ob j( C K ) then G ( X ) = G ( X 1 ) ⊕ G ( X 2 ). (ii) If X ′ − − − − → f ′ Y ′   y g ′   y g X − − − − → f Y is a c artesian square in C K then g ∗ f ∗ = ( f ′ ) ∗ ( g ′ ) ∗ . If K ⊆ E 1 ⊆ E 2 are t wo finite field extensions and f : Sp ec E 2 → Sp ec E 1 the corresp onding map in C K then f ∗ (resp. f ∗ ) is giv en by ι E 2 /E 1 : G ( E 1 ) → G ( E 2 ) (resp. N E 2 /E 1 : G ( E 2 ) → G ( E 1 )). Lo cal sym b ols W e recall also the notion of a lo c al symb ol ([Se] a nd [So]) for G . Let X → Sp ec K is a pr op er non-singular algebraic curv e (n ote that w e do not assu m e that X is connected). L et K ( X ) denote the rin g of rational functions on X and | X | the set of closed p oin ts of X . F or P ∈ | X | w e den ote b y K P the quotien t field of the completio n b O X,P of O X,P , b y v P : K P → Z ∪ {∞} the normalized v aluation and by K ( P ) the residu e field of K P . The local symb ol at P is a homomorphism ∂ P : ( K P ) ∗ ⊗ G ( K P ) → G ( K ( P )). It is charact erized by th e foll o wing p rop erties: (i) If f ∈ ( K P ) ∗ and g ∈ G ( b O X,P ) then ∂ P ( f ⊗ g ) = v P ( f ) g ( P ). Here g ( P ) is the image of g under the canonical map G ( b O X,P ) → G ( K ( P )). (ii) F or f ∈ K ( X ) ∗ and g ∈ G ( K ( X )) w e h a ve P P ∈| X | N K ( P ) /K ( ∂ P ( f ⊗ g )) = 0. Milnor K -groups att ac hed to commutativ e algebraic groups Let G 1 , . . . , G m b e co m mutativ e algebraic group s ov er K . I n [So] Somek a wa has in tro du ced the Milnor K -group K ( K ; G 1 , . . . , G m ) (actuall y Somek a wa considered only the ca s e of semia b elian v arieties though h is construction w orks for arbitrary comm utativ e algebraic groups). It is giv en as K ( K ; G 1 , . . . , G m ) = M X G 1 ( X ) ⊗ . . . ⊗ G m ( X ) ! /R 3 where X ru ns through all ob jects of C K and wh ere the su b group R is gen- erated b y the follo wing elemen ts: (R1) If f : X → Y is a morphism in C K and if x i 0 ∈ G i 0 ( Y ) for some i 0 and x i ∈ G i ( X ) for i 6 = i 0 , then x 1 ⊗ . . . ⊗ f ∗ ( x i 0 ) ⊗ . . . ⊗ x m − f ∗ ( x 1 ) ⊗ . . . ⊗ x i 0 ⊗ . . . ⊗ f ∗ ( x m ) ∈ R . (R2) Let X → Sp ec K be a prop er non-singular cur ve, f ∈ K ( X ) ∗ and g i ∈ G i ( K ( X )). Assum e that for eac h P ∈ | X | there exists i ( P ) such that g i ∈ G i ( b O X,P ) for all i 6 = i ( P ). T h en X P ∈| X | g 1 ( P ) ⊗ . . . ⊗ ∂ P ( f ⊗ g i ( P ) ) ⊗ . . . ⊗ g m ( P ) ∈ R. F or X ∈ C K and x i ∈ G i ( X ) for i = 1 , . . . , m we write { x 1 , . . . , x m } X/K for the image of x 1 ⊗ . . . ⊗ x m in K ( K ; G 1 , . . . , G m ) (elemen ts of this form will b e referred to as symb ols ). A sequence of a lgebraic groups G ′ → G → G ′′ o ve r K will be called Zariski exact if G ′ ( E ) → G ( E ) → G ′′ ( E ) is exa ct for ev ery extension E /K . The pro of o f the follo w ing result is straigh tforward; hence will b e omitted. Lemma 2 L et m b e a p ositive inte ger and let i ∈ { 1 , . . . , m } . L et G 1 , . . . , G m b e c ommutative algebr aic gr oups over K and let G ′ i → G i → G ′′ i → 1 b e a Zariski exact se que nc e of c ommutative algebr aic gr oups over K . Then the se quenc e K ( K ; G 1 , . . . , G ′ i , . . . ) → K ( K ; G 1 , . . . , G i , . . . ) → K ( K ; G 1 , . . . , G ′′ i , . . . ) → 0 is exact as wel l. The norm map L et G 1 , . . . , G m b e comm utativ e algebraic groups o ver K and let L/K b e a finite extension. Set K ( L ; G 1 , . . . , G m ) : = K ( L ; ( G 1 ) L , . . . , ( G m ) L ). Then we hav e the norm m ap [So] (3) N L/K : K ( L ; G 1 , . . . , G m ) − → K ( K ; G 1 , . . . , G m ) defined on symb ols b y N L/K ( { x 1 , . . . , x m } X/L ) = { x 1 , . . . , x m } X/K for an y X ∈ C L and x i ∈ G i ( X ) ( i = 1 , . . . , m ) . W e giv e another inte r pretation of (3) b elo w when L/K is separable. It is based o n the follo wing result. Lemma 3 L et L/K b e a finite sep ar able extension and let i, m b e p ositive inte gers with i ≤ m . L et G 1 , . . . , G i − 1 , G i +1 , . . . , G m b e c ommutative alge- br aic gr oups over K and let G i b e a c ommutative algebr aic gr oup over L . Then, we have an isomorphism K ( K ; G 1 , . . . , Res L/K G i , . . . , G m ) ∼ = K ( L ; ( G 1 ) L , . . . , G i , . . . , ( G m ) L ) . 4 Pr o of. T o simplify the notation w e assume that i = m . W e denote b y π − 1 : C K → C L and π : C L → C K the functors π − 1 ( X → Sp ec K ) : = ( X ⊗ K L → Sp ec L ) , π ( Y → Sp ec L ) : = ( Y → Sp ec L → Sp ec K ) . π is left adjoint to π − 1 . F or X ∈ C K and Y ∈ C L let p X : X ⊗ K L − → X , ι Y : Y − → Y ⊗ K L. b e the adjunction mo rphisms. W e define homomorphisms φ : K ( K ; G 1 , . . . , G m − 1 , Res L/K G m ) − → K ( K ; ( G 1 ) L , . . . , ( G m − 1 ) L , G m ) , ψ : K ( L ; ( G 1 ) L , . . . , ( G m − 1 ) L , G m ) − → K ( K ; G 1 , . . . , G m − 1 , Res L/K G m ) . as f ollo ws. F or X ∈ C K , x 1 ∈ G 1 ( X ) , . . . , x m − 1 ∈ G m − 1 ( X ) and x m ∈ G m ( X ⊗ K L ) w e put φ ( { x 1 , . . . , x m } X/K ) = { p ∗ ( x 1 ) , . . . , p ∗ ( x m − 1 ) , x m } ( X ⊗ K L ) /L . Con versely , for Y ∈ C L and y 1 ∈ G 1 ( Y ) , . . . , y m ∈ G m ( Y ) let ψ ( { y 1 , . . . , y m − 1 , y m } Y /L = { y 1 , . . . , y m − 1 , ι ∗ ( y m ) } Y /K . One can easily v erify that these maps are w ell-defined a nd mutually inv erse to eac h other.  Let G 1 , . . . , G m b e comm utativ e alg ebraic groups o v er K and let L/K b e a finite sep arable extension. T ak e an y i ∈ { 1 , . . . , m } . The map N L/K : Res L/K ( G i ) L → G i induces a map K ( K ; G 1 , . . . , Res L/K ( G i ) L , . . . , G m ) − → K ( K ; G 1 , . . . , G m ), and the comp osition of it with the isomorphism ψ abov e coincides with the norm map (3). Wh en L/K is a Galois extension, th e action of Ga l ( L/K ) on Res L/K ( G i ) L induces its actio n on K ( L ; G 1 , . . . , G m ) ∼ = K ( K ; G 1 , . . . , Res L/K ( G i ) L , . . . , G m ) and w e ha ve N L/K ◦ σ = N L/K for all σ ∈ Gal( L/K ). This action do es not dep end on the choic e of i . Lemma 4 L et L/K b e a cyclic Galois extension and let σ ∈ Gal( L/K ) b e a gener ator. Supp ose that for two differ ent i ∈ { 1 , . . . , m } the se quenc e (4) Res L/K ( G i ) K N L/K − → G i − → 1 is Zariski exact. Then the se quenc e of ab elian gr oups K ( L ; G 1 , . . . , G m ) 1 − σ − → K ( L ; G 1 , . . . , G m ) N L/K − → K ( K ; G 1 , . . . , G m ) − → 0 is exact. 5 Pr o of. Supp ose that (4 ) is exact for i = m − 1 , m . Let G ′ m : = Ker( N L/K : Res L/K ( G m ) L → G m ). By L emmas 2 and 3 there are exact sequences K ( K ; G 1 , . . . , G ′ m ) → K ( L ; G 1 , . . . , G m ) N L/K − → K ( K ; G 1 , . . . , G m ) → 0 (5) K ( L ; G 1 , . . . , G m − 1 , G ′ m ) N L/K − → K ( K ; G 1 , . . . , G m − 1 , G ′ m ) − → 0 . (6) Since ( G ′ m ) L ∼ = ( G m ) n − 1 L ( n : = [ L : K ]) we can replace th e first group of (6) b y K ( L ; G 1 , . . . , G m ) n − 1 . By Lemma 1 the co m p osite K ( L ; G 1 , . . . , G m ) n − 1 → K ( K ; G 1 , . . . , G m − 1 , G ′ m ) → K ( L ; G 1 , . . . , G m ) is giv en on the i -th su mmand b y 1 − σ i . The assertion follo ws.  Galois symbol Let G 1 , . . . , G m b e connected comm utativ e algebraic groups o ve r K , and let n b e an in teger prime to the charact eristic of K . F or any finite extension L/K , w e ha ve a h omomorphism [So] (7) h L : K ( L ; G 1 , . . . , G m ) /n − → H m ( L, G 1 [ n ] ⊗ . . . ⊗ G m [ n ]) called the Ga lois symb ol . This is characte r ized by t he follo wing prop erties. (i) If x i ∈ G i ( L ) for i = 1 , . . . , m , then h L ( { x 1 , . . . , x m } L/L ) = ( x 1 ) ∪ . . . ∪ ( x m ). Here we write b y ( x i ) for the image of x i in H 1 ( L, G i [ n ]) b y the connecting h omomorphism associated to the exact sequence 1 → G i [ n ] → G i n → G i → 1 . (ii) If M /L/K is a to wer of finite extensions and if M /L is separable (resp. purely inseparable), then the d iagram K ( M ; G 1 , . . . , G m ) /n − − − − → h M H m ( M , G 1 [ n ] ⊗ . . . ⊗ G m [ n ])   y N M /L   y K ( L ; G 1 , . . . , G m ) /n − − − − → h L H m ( L, G 1 [ n ] ⊗ . . . ⊗ G m [ n ]) is comm utativ e, where the right vertic al map is the corestriction (resp . the m ultiplication b y [ M : L ] under t he id en tification H m ( M , G 1 [ n ] ⊗ . . . ⊗ G m [ n ]) ∼ = H m ( L, G 1 [ n ] ⊗ . . . ⊗ G m [ n ]) ). Prop erty (i) implies in particular that (7) coincides with the usu al Ga- lois symbol (1) in th e case G 1 = . . . = G m = G m . In [So] Rema rk 1.7, Somek a w a c onjectured that th e Galois symb ol asso ciated to s emiab elian v a- rieties should b e in jectiv e. 6 Galois cohomology of cyclic extensions Let L/K b e a cyclic Galois extension of degree n and let σ b e a generator of G : = Gal( L/K ). F or a discrete G K -mo dule M , tensoring the short exact sequence o f G -mo du les (8) 0 − → Z − → Z [ G ] 1 − σ − → Z [ G ] − → Z − → 0 with M yields a distinguished triangle (9) M [1] α − → C · ( M ) β − → M γ − → M [2] in the deriv ed categ ory D ( G K ). Here w e denote by C · ( M ) the complex Res L/K M 1 − σ − → Res L/K M concen trated in degree − 1 and 0. The spectral sequence E p,q 1 = H q ( K, C p ( M )) = ⇒ E p + q = H p + q ( K, C · ( M )) induces short exac t sequences (10) 0 → H q ( L, M ) G → H q ( K, C · ( M )) → H q +1 ( L, M ) G → 0 . It is ea s y to see that the comp osite H q +1 ( K, M ) α − → H q ( K, C · ( M )) → H q +1 ( L, M ) G is the restrictio n and H q ( L, M ) G → H q ( K, C · ( M )) β − → H q ( K, M ) is induced b y the corestriction. In particular we ha ve γ ( H q ( K, M )) ⊆ Ker(res : H q +2 ( K, M ) → H q +2 ( L, M )) hence (11) nγ ( H q ( K, M )) = 0 . F or an in teger m p rime t o c har K and r ∈ N w e write Z /m Z ( r ) : = µ ⊗ r m and H 3 ( L/K, Z /m Z (2)) : = Ker( H 3 ( K, Z /m Z (2)) res − → H 3 ( L, Z /m Z (2))) . By r estricting α : H 3 ( K, Z /m Z (2)) → H 2 ( K, C · ( Z /m Z (2))) to th e su b group H 3 ( L/K, Z /m Z (2)) and comp osing it with the in ve rse of the fir st map in (10) w e obtain a map (12) H 3 ( L/K, Z /m Z (2)) → Ker( H 2 ( L, Z /m Z (2)) G cor → H 2 ( K, Z /m Z (2))) . Lemma 5 Assume that n is prime to c har K and µ n 2 ( K ) ⊂ K . Then the homomo rphism (12) is inje ctive for m = n . 7 Pr o of. It is enough to sho w that γ : H 1 ( K, Z /n Z (2)) → H 3 ( K, Z /n Z (2)) is zero. Consider the comm utativ e d iagram H 1 ( K, Z /n Z (2)) − − − − → H 1 ( K, Z /n 2 Z (2))   y γ   y γ H 3 ( K, Z /n Z (2)) − − − − → H 3 ( K, Z /n 2 Z (2)) induced b y the canonical in jection Z /n Z (2) → Z /n 2 Z (2). Th e assum ption µ n 2 ( K ) ⊂ K implies that the upp er h orizon tal map can b e iden tified with K ∗ / ( K ∗ ) n − → K ∗ / ( K ∗ ) n 2 , x ( K ∗ ) n 7→ x n ( K ∗ ) n 2 . In particular the imag e is con tained in nH 1 ( K, Z /n 2 Z (2)). By (1 1) it is mapp ed under γ to nγ ( H 1 ( K, Z /n 2 Z (2))) = 0. On the other hand it is a simple consequence of the Merku r jev-Suslin theorem [MS] that the low er horizon tal map is injectiv e. Hence γ ( H 1 ( K, Z /n (2))) = 0.  The coun terexample Let L/K b e as in the last section and let T : = Ker( N L/K : Res L/K G m → G m ). W e mak e the follo wing assu mptions n is prime to c h ar K and µ n 2 ( K ) ⊂ K , (13) H 3 ( L/K, Z /n Z (2))) 6 = 0. (14) Prop osition 6 The Galois symb ol K ( K ; T , T ) /n → H 2 ( K, T [ n ] ⊗ T [ n ]) i s not inje ctive. Pr o of. Let σ b e a generator of G : = Gal( L/K ). The e xact sequence 1 − → G m − → Res L/K G m 1 − σ − → Res L/K G m N L/K − → G m − → 1 yields t w o s h ort exac t sequences 1 − → G m − → Res L/K G m − → T − → 1 , (15) 1 − → T − → Res L/K G m − → G m − → 1 . (16) Corresp ond ingly , (8) induces t w o short exact sequences (17) 0 → Z → Z [ G ] → X → 0 , 0 → X → Z [ G ] → Z → 0 where X denotes th e co charac ter group of T . Note that the sequence (15) is Zariski exact by Hilb ert 90. Since the map Res L/K G m → T factors through 8 Res L/K G m → Res L/K T → T the sequence Res L/K T → T → 1 is Zariski exact as well. By Lemma 4 the upp er horizonta l map in the diag ram ( K ( L ; T , T ) /n ) G N L/K − − − − → K ( K ; T , T ) /n   y   y H 2 ( L, T [ n ] ⊗ T [ n ]) G cor − − − − → H 2 ( K, T [ n ] ⊗ T [ n ]) is an isomorphism. The vertical maps are Galois sym b ols. Sin ce T L is a split torus the left v ertical map is an isomorp hism b y the Merkurjev-Sus lin theorem [MS]. Th u s to finish the pro of it remains to sho w that the lo wer v ertical arro w is not injectiv e. Note that T [ n ] ∼ = Z /n Z (1) ⊗ X . Hence the assertion follo ws from Lemma 5 and Lemma 7 b elo w . Lemma 7 Ther e exists homomorphisms of G -mo dules e : Z → X ⊗ Z X and f : X ⊗ Z X → Z such that f ◦ e : Z → Z is multiplic ation by n − 1 . Pr o of. F or a G -mo dule M w e w rite M ∨ for the G -mo du le Hom ( M , Z ). Let ( , ) : Z [ G ] ⊗ Z Z [ G ] − → Z b e the symmetric pairing giv en by (18) ( g , g ′ ) =  1 if g = g ′ , 0 if g 6 = g ′ . It yields an isomorph ism Z [ G ] → Z [ G ] ∨ . F or a subm o dule M ⊆ Z [ G ] let M ⊥ = { x ∈ Z [ G ] | ( x, m ) = 0 ∀ m ∈ M } . Then w e hav e X ⊥ = Z S and ( Z S ) ⊥ = X where S = P n − 1 i =0 σ i . Thus (18) yields an isomorphism X ∼ = ( Z [ G ] / Z S ) ∨ . By (17) we ha ve Z [ G ] / Z S ∼ = X , hence X ⊗ Z X ∼ = X ⊗ Z X ∨ ∼ = Hom( X , X ) Th us it suffices to pro v e the assertion for Hom( X, X ). Obviously , for th e t wo maps e : Z → Hom( X, X ) , m 7→ m id X and f : Hom( X , X ) → Z , τ 7→ T r( τ ) w e hav e f ◦ e = rank( X ) = n − 1.  Remark 8 It is easy to co n struct examples w here the assumptions (1 3 ) and (14) ab o v e are satisfied. F or instance if K is a 2-lo cal field satisfying prop erty (13) and L/K is an y cyclic extension of degree n th en (14) holds b y [Ka]. 3 Coun terexample to a c onjecture of Beilinson W e fi rst introduce some n otation and recall a f ew facts fr om [V o1 ] an d [MVW]. Let K b e a field of c haracteristic zero. Let C or K denote the a d di- tiv e category of finite corresp ondences ([MVW], 1.1). The ob jects of C or K 9 are smo oth separated K -schemes of finite type and for X , Y ∈ Ob j( C or K ) the g roup of morphism s C or K ( X, Y ) is the free ab elia n group generated b y in tegral closed sub sc hemes W of X × Y whic h are finite and surj ective o ve r X . Let D − ( S hv Nis ( C or K )) (resp . D − ( S hv et ( C or K ))) d en ote the d e- riv ed category of complexes of Nisnevic h (resp. ´ etale) shea ve s with transfer b ound ed from ab ov e. The catego r y of effectiv e motivic complexes DM eff , – Nis ( K ) (resp. ´ etale ef- fectiv e m otivic complexes DM eff , – et ( K )) is t he full sub catego r y of D − ( S hv Nis ( C or K )) (resp. D − ( S hv et ( C or K ))) which consists of complexes C ⋆ with ho- motop y inv ariant cohomology sh ea v es H i ( C ⋆ ) for all i (see [V o1], 3.1 or [MVW], 14.1, resp. 9.2). DM eff , – Nis ( K ) and DM eff , – et ( K ) are triangulated tensor catego r ies. Th ey are equipp ed with th e t-structur e induced from the stan- dard t-structure on D − ( S hv Nis ( C or K )) (resp . D − ( S hv et ( C or K ))). T h ere is a co v arian t functor M : C or K → DM eff , – Nis ( K ) , X 7→ M ( X ) and w e ha ve M ( X × Y ) = M ( X ) ⊗ M ( Y ). T here is also the ”c hange of top ology” f u nctor α ∗ : DM eff , – Nis ( K ) → DM eff , – et ( K ). It is a tensor functor which admits a righ t adjoin t R α ∗ : DM eff , – et ( K ) → DM eff , – Nis ( K ). Beilinson [Be ] has p rop osed th e follo win g generalization of the Milnor- Bloch-Kato conjecture: F or an y smo oth affine K -sc heme X the adjun ction morphism M ( X ) → Rα ∗ α ∗ M ( X ) induces an isomorph ism on cohomology in degrees ≤ 0, i.e. the map (19) a X : M ( X ) − → t ≤ 0 Rα ∗ α ∗ M ( X ) is an isomorphism in DM eff , – Nis ( K ). If X = ( G m ) d = G m × . . . × G m ( d -fold pro d uct of G m ) we h av e M ( X ) ∼ = ( Z ⊕ Z (1)[1]) d . Thus a X is an isomorphism if and only if (20) Z ( n ) − → t ≤ n Rα ∗ α ∗ Z ( n ) is an isomorp hism for all n ≤ d . It is kno w n (compare [SV]) that the Milnor-Bloch-Kato conjecture is equiv alen t to the assertion that (20 ) is an isomorphism for all n ≥ 0. Let L/K b e a separable quadratic extension and let T : = Ker( N L/K : Res L/K G m → G m ). W e s h all s ho w that (19) is in general not an isomor- phism for X = T n for n ≥ 2. By ([HK], 7.3) there exists a canonical decomp osition M ( T ) = Z ⊕ Z ( L/K, 1)[1] where Z ( L/K, 1) i s the cone of the morphism Z (1) → Res L/K Z (1). Remarks 9 (a) Here is a more explicit description of the m otiv e Z ( L/K, 1). The torus T defines a homotop y inv ariant ´ etale (hence Nisnevic h) sh eaf with transfer and therefore an element of DM eff , – Nis ( K ). W e ha ve Z ( L/K, 1) ∼ = T [ − 1] . 10 This can b e dedu ced from the corresp ondin g s tatemen t for G m ([MVW], 4.1) and th e exactness of ( 15) (as a sequence in S hv Nis ( C or K )). (b) 1 Let A 1 , . . . , A n b e semi-ab elian v arieties o ver K . It should b e p ossible to identify the generalized Milnor K -group K ( K ; A 1 , . . . , A n ) with a Hom- group in DM eff , – Nis ( K ). F or that w e vie w A 1 , . . . , A n again as elemen ts in S hv Nis ( C or K ). Then w e exp ect that K ( K ; A 1 , . . . , A n ) ∼ = Hom DM eff , – Nis ( K ) ( Z , A 1 ⊗ . . . ⊗ A n ) . If A 1 = . . . = A n = G m this is prov ed in ([MVW], lecture 5) an d it is lik ely that the pro of giv en there can b e adapted to the case of arbitrary semi-ab elian v arieties. F or p, q ≥ 0 and n = p + q we defi ne Z ( L/K, p, q ) : = Z ( L/K, 1) ⊗ p ⊗ Z ( q ) and den ote by C ( p, q ) the cone of Z ( L/K, p, q ) − → t ≤ n Rα ∗ α ∗ Z ( L/K, p, q ). Note that Z ( L/K, p, q )[ n ] is a direct s u mmand of M ( T p × ( G m ) q ). W e also put C ( n ) : = C (0 , n ). W e hav e (21) C ( n ) ∼ = ( t ≥ n +1 Rα ∗ Q / Z ( n ))[ − 1] This f ollo ws from the Milnor-Blo ch-Kat o co n jecture (in fact for our p urp ose w e need (21) only after localizatio n at the prime 2 wh ere it foll o ws from the Milnor conjecture [V o2]). T ensoring Z (1) → Res L/K Z (1) → Z ( L/K, 1) → Z (1)[1] w ith Z ( L/K, p − 1 , q ) (for p ≥ 1 , q ≥ 0) yields a distinguished triangle Z ( L/K, p − 1 , q + 1) → Res L/K Z ( n ) → Z ( L/K, p, q ) → Z ( L/K, p − 1 , q + 1)[1] hence also a t riangle (22) C ( p − 1 , q + 1) → Res L/K C ( n ) → C ( p, q ) → C ( p − 1 , q + 1)[1] . The follo wing Lemma follo ws easily b y indu ction o n q using (21) and (22). Lemma 10 L et p ≥ 1 , q ≥ 0 and n = p + q . Then we have H k ( C ( p, q )) = 0 for k < q + 2 and H q +2 ( C ( p, q ))( K ) ∼ = H n +1 ( L/K, Q / Z ( n )) wher e H n +1 ( L/K, Q / Z ( n )) : = K er ( H n +1 ( K, Q / Z ( n )) res − → H n +1 ( L, Q / Z ( n ))) . 1 This remark has b een communicated to us by B. Kahn. 11 Since [ L : K ] = 2 w e ha ve H n +1 ( L/K, Q / Z ( n )) ∼ = H n +1 ( L/K, Q 2 / Z 2 ( n )) ∼ = H n +1 ( L/K, Z / 2 Z ( n )) ∼ = H n +1 ( L/K, Z / 2 Z ) (the second isomorphism is a consequence of the Milnor conjecture). No w the follo win g Prop osition follo ws by app lying Lemma 10 for ( p, q ) = (2 , 0) and ( n, 0). Prop osition 11 (a) Ther e exists a short exact se quenc e 0 − → H 0 ( M ( T × T ))( K ) − → R 0 α ∗ α ∗ M ( T × T )( K ) − → H 3 ( L/K, Z / 2 Z ) − → 0 In p articular if H 3 ( L/K, Z / 2 Z ) 6 = 0 then (19) is not an isomorp hism for X = T × T . (b) Mor e ge ne r al ly let n b e an inte ger ≥ 2 and assume that H n +1 ( L/K, Z / 2 Z ) 6 = 0 . Then the map (19) is not an isomorphism for X = T n . Mor e pr e cisely either the map H 2 − n ( M ( X )) − → R 2 − n α ∗ α ∗ M ( X ) is not surje ctive or H 3 − n ( M ( X )) − → R 3 − n α ∗ α ∗ M ( X ) is not inje ctive. An n -lo cal field K of c h aracteristic 0 provides an example where the ab o ve assumption holds. In fact by [Ka] we ha ve H n +1 ( L/K, Z / 2 Z ) ∼ = Z / 2 Z for suc h fields. References [Be] A.A. Beilinson, T alk at Fields Institute at T oronto on 21 March 2007. [HK] A. Hub er, B. 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In: Th e arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 117–189, NA TO Sci. S er. C Math. Ph ys. Sci., 548, Klu we r Acad. Publ., Dord r ec ht, 20 00. [V o1] V. V o evodsk y , T riangulate d c ate gories of motives over a field . In: V. V o evodsk y , A. Su slin, E.M. F riedlander: C ycles, T ransfers, and Motivic Homology Theories. Ann als of Math. Studies 143 , Prince- ton Univ ersit y Press, 200 0, 188–2 38. [V o2] V. V o ev o dsk y , Motivic c ohomolo gy with Z / 2 -c o efficients. Pub l. Math., Inst. Ha u tes ´ Etud. Sci. 98, 59-104 (2003). Mic hael S pieß T ak ao Y amazaki F akult¨ at f ¨ ur Mat hematik Mathematica l Ins titute Univ ersit¨ at Bielefeld T ohoku Univ ersit y P ostfac h 100131 Aoba D-3350 1 Bielefeld, Germany Sendai 980- 8578, Japan mspiess@math.un i-bielefeld.de ytak ao@math.tohoku.ac.jp 13