Norm Varieties and the Chain Lemma (after Markus Rost)

NORM V ARIETIES AND THE CHAIN LEMMA (AFTER MARKUS ROST) Notes by Christian Haesemeyer and Chuck Weib el. The go al of this pap er is to pre sent proofs of tw o r esults of Mar k us Rost, the Chain Lemma 0.1 and the Norm Pr inciple 0.3. These are the steps needed to complete t he published veriﬁcation of the Blo ch -Kato c onje ctur e , that the norm residue m aps are isomorphisms K M n ( k ) /p ≃ → H n et ( k , Z /p ) for every prime p , every n and every ﬁeld k containing 1 /p . Throughout this pap er , p is a ﬁxed o dd prime, and k is a ﬁeld of characteristic 0, containing the p -th ro o ts of unity . W e ﬁx a n integer n ≥ 2 and an n -tuple ( a 1 , ..., a n ) o f units in k , such that the symbol { a } is nontrivial in the Milnor K - group K M n ( k ) /p . Asso ciated to this data ar e several no tions. A ﬁeld F over k is a splitting ﬁeld for { a } if { a } F = 0 in K M n ( F ) /p . A v ariety X ov er k is called a splitting variety if its function ﬁeld is a splitting ﬁeld; X is p -generic if a ny splitting ﬁeld F has a ﬁnite extension E / F of degree prime to p with X ( E ) 6 = ∅ . A Norm variety for { a } is a smo oth pr o jectiv e p -gener ic splitting v ariety for { a } of dimension p n − 1 − 1. The follo wing seq uenc e of theorems reduces the Bloch-Kato conjecture to the Chain Lemma 0 .1 and the Norm P rinciple 0.3; the notion of a R ost variety is deﬁned in 0 .5 b elow; the deﬁnitio n of a R ost m otive is g iven in [14] a nd [15], and will not b e needed in this pap er . (0) The Chain Lemma 0.1 and the Norm Principle 0.3 hold; this is proven here. (1) Given (0), Rost v arieties exist; this is Theorem 0.7 below, and is proven in [10, p. 25 3]. (2) If Rost v arieties exist then Rost motives exist; this is prov en in [15]. (3) If Rost motives exist then Blo ch-Kato is true; this is proven in [13] and [14]. Here is the statemen t of the Chain Lemma, whic h w e quote from [10, 5.1 ] and pr ov e in § 5. A ﬁeld is p - sp e cial if p divides the or der of every ﬁnite ﬁeld extension. thm:chainl emma Theorem 0. 1 (Rost’s Cha in Lemma) . L et { a } ∈ K M n ( k ) /p b e a nont rivial symb ol, wher e k is a p -sp e cial ﬁeld . Then ther e exists a smo oth p r oje ctive c el lular v ariety S/k and a c ol le ction of invertible she aves J = J 1 , J ′ 1 , . . . , J n − 1 , J ′ n − 1 e quipp e d with nonzer o p -forms γ = γ 1 , γ ′ 1 . . . , γ n − 1 , γ ′ n − 1 satisfying the fol lowing c onditions. (1) dim S = p ( p n − 1 − 1) = p n − p ; (2) { a 1 , . . . , a n } = { a 1 , . . . , a n − 2 , γ n − 1 , γ ′ n − 1 } ∈ K M n ( k ( S )) /p, { a 1 , . . . , a i − 1 , γ i } = { a 1 , . . . , a i − 2 , γ i − 1 , γ ′ i − 1 } ∈ K M i ( k ( S )) /p for 2 ≤ i < n . In p articular, { a 1 , . . . , a n } = { γ , γ ′ 1 , . . . , γ ′ n − 1 } ∈ K M n ( k ( S )) /p ; (3) γ / ∈ Γ( S, J ) ⊗ ( − p ) , as is evident fr om (2); (4) for any s ∈ V ( γ i ) ∪ V ( γ ′ i ) , the ﬁ eld k ( s ) splits { a 1 , . . . , a n } ; (5) I ( V ( γ i )) + I ( V ( γ ′ i )) ⊆ p Z for al l i , as fol lows fr om (4); (6) deg ( c 1 ( J ) dim S ) is r elatively prime to p . Rost’s Norm P rinciple concerns the g roup A 0 ( X, K 1 ), which we now deﬁne. Date : Octob er 26, 2018. 1 2 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) def:A0K1 Deﬁnition 0.2. (Ros t, [5]) F o r any regular scheme X , the g r oup A 0 ( X, K 1 ) is deﬁned to b e the group generated b y symbols [ x, α ], where x is a close d p oint o f X and α ∈ k ( x ) × , mo dulo the relations (i) [ x, α ][ x, α ′ ] = [ x, αα ′ ] and (ii) for every po int y o f dimension 1 the ima g e of the tame s ymbol K 2 ( k ( y )) → ⊕ k ( x ) × is zero. The functor A 0 ( X, K 1 ) is cov ariant in X for prop er ma ps , b ecause it is isomo r phic to the motivic homolog y group H − 1 , − 1 ( X ) = Hom DM ( Z , M ( X )(1)[1]) (s e e [10, 1.1]). It is also the K -c o homology g roup H d ( X, K d +1 ), where d = dim( X ). The reduced group A 0 ( X, K 1 ) is deﬁned to b e the q uotient of A 0 ( X, K 1 ) b y the diﬀerence of the tw o pro jections from A 0 ( X × X , K 1 ). As observed in [10, 1.2], there is a well deﬁned map N : A 0 ( X, K 1 ) → k × sending [ x, α ] to the norm of α . normprin T heorem 0 .3 (Nor m Principle) . Supp ose t hat k is a p -sp e cial ﬁeld and that X is a N orm variety for some nontrivial symb ol { a } . L et [ z , β ] ∈ A 0 ( X, K 1 ) b e such that [ k ( z ) : k ] = p ν for ν > 1 . Then ther e exists a p oint x ∈ X with [ k ( x ) : k ] = p and α ∈ k ( x ) × such that [ z , β ] = [ x, α ] in A 0 ( X, K 1 ) . W e will prov e the Norm Pr inciple 0.3 in se ction 9 b elow. Our pro ofs of these tw o r e sults a re based on 19 9 8 Ros t’s pr eprint [7], his web site [6] and Rost’s lectures [Rost] in 1 9 99-2 0 00 and 2005 . The idea for wr iting up these notes in publishable form origina ted during his 2 005 c o urse, a nd was reinvigorated by conv ers ations with Mar kus Rost at the Ab el Sy mp o s ium 2007 in O slo. As usual, all mistakes in this paper are the res po nsibility of the author s. Rost v arieties . In the rest of this in tro duction, we e x plain how 0.1 and 0.3 imply the pr oblematic Theo rem 0 .7, and hence complete the pr o of of the Blo ch-Kato conjecture. W e ﬁrst r ecall the notions o f a ν i -v ariety a nd a Rost v ar iety . Let X b e a smo oth pro jective v a riety o f dimension d > 0. Recall fr om [4, § 1 6] that there is a characteristic class s d : K 0 ( X ) → Z corresp o nding to the symmetric po lynomial P t d j in the Chern ro ots t j of a bundle; w e write s d ( X ) for s d of the tangent bundle T X . When d = p ν − 1, we know that s d ( X ) ≡ 0 (mo d p ); see [4, 16.6 and 16-E ] and [9, pp. 128–9] or [1, I I.7]. def:nu-var Deﬁnition 0. 4. (see [10, 1 .20]) A ν n − 1 -variety ov er a ﬁeld k is a smo oth pr o jective v ariety X o f dimension d = p n − 1 − 1, with s d ( X ) 6≡ 0 (mo d p 2 ). F or example, s d ( P d ) = d + 1 by [4, 16.6]. Th us the pro jective space P p − 1 is a ν 1 -v ariety , and so is any Bra uer-Severi v ariety of dimension p − 1. In Section 8, we will show that the bundle P ( A ) ov er S is a ν n -v ariety . def:Rostva r Deﬁn i tion 0.5 . A R ost variety for a sequence { a } = ( a 1 , ..., a n ) o f units in k is a ν n − 1 -v ariety s uch that: { a 1 , ..., a n } v anishes in K M n ( k ( X )) /p ; for ea ch i < n there is a ν i -v ariety mapping to X ; and the motivic ho mology sequence 6.3 (0.6) H − 1 , − 1 ( X × X ) π ∗ 0 − π ∗ 1 − − − − → H − 1 , − 1 ( X ) → H − 1 , − 1 ( k ) (= k × ) . is exact. Part o f Theo r em 0.7 states that Rost v arieties exist for every { a } . R emark 0 .6.1 . Rost orig inally deﬁned a Norm V ariety for { a } to be a pro jective splitting v ariety of dimension p n − 1 which is a ν n − 1 -v ariety . (S ee [Ro s t, 10 /20/ 9 9].) Theorem 0.7(2) says that our deﬁnition ag rees with Rost’s when k is p -sp ecial. Here is the s ta tement of Theore m 0.7, quoted fro m [10, 1.21]. It assumes that the Blo ch-Kato conjecture ho lds for n − 1. NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 3 thm:normva r Theorem 0.7. L et n ≥ 2 and 0 6 = { a } = { a 1 , . . . , a n } ∈ K M n ( k ) /p. Then: 0) Ther e exists a ge ometric al ly irr e ducible Norm variety for { a } . Assume further t hat k is p -sp e cial. If X is a Norm variety for { a } , then: 1) X is ge ometric al ly irr e ducible. 2) X is a ν n − 1 -variety. 3) e ach element of A 0 ( X, K 1 ) is of the form [ x, α ] , wher e x ∈ X is a close d p oint of de gr e e p and α ∈ k ( x ) × . The co nstruction of geometrically irr educible Nor m v arieties was carried out in [10, pp. 25 4–25 6]; this pr ov es part (0) o f Theorem 0.7. Part (1) was pr ov en in [10, 5.4]. P art (2 ) was pr ov en in [10, 5.2 ], assuming Rost’s Cha in Lemma (see 0.1), a nd part (3) was prov en in [10, p. 2 71], assuming not only the Chain Lemma but also the Norm Pr inciple (see 0.3 b elow). As stated in the intro duction of [10], the construc tio n of No rm v arieties and the pro of o f Theorem 0.7 ar e par t of a n inductive pro of of the Blo ch-Kato conjecture. W e point o ut that in the present pap er, the inductive a ssumption (that the Blo ch- Kato co njecture for n − 1 holds) is never used. It only a pp ea rs in [10] to prove that the ca ndida tes fo r norm v ar ie ties construc ted there ar e p -ge ne r ic splitting v arieties. (How ever, the Norm Pr inciple 0.3 is itself a statement ab o ut norm v arieties.) In particular, the Cha in Lemma 0.1 holds in all degr ees independently of the B lo ch- Kato conjecture. 1. F orms on vector bundles W e b egin with a presentation of some well known facts ab out p -fo rms. If V is a vector space over a ﬁeld k , a p -form on V is a symmetric p -linear function on V , i.e., a linea r map φ : Sym p ( V ) → k . It deter mines a p - ary form , i.e., a function ϕ : V → k s atisfying ϕ ( λv ) = λ p ϕ ( v ), by ϕ ( v ) = φ ( v , v , . . . , v ). If p ! is inv e r tible in k , p -linea r forms are in 1 –1 corres p o ndence with p -ary forms. If V = k then every p -form may b e written a s ϕ ( λ ) = aλ p or φ ( λ 1 , . . . ) = a Q λ i for so me a ∈ k . Up to isometry , non-zer o 1-dimensiona l p -for ms are in 1–1 corresp ondence w ith elements o f k × /k × p . Therefore an n -tuple of forms ϕ i determine a well-deﬁned element of K M n ( k ) /p which w e write as { ϕ 1 , . . . , ϕ n } . Of cours e the notion of a p -form on a pr o jective mo dule ov er a commutativ e ring makes s e nse, but it is a sp ecial case of p -for ms on lo cally fr ee mo dules (algebr aic vector bundles), which we now deﬁne. def:pform Deﬁnition 1. 1. If E is a lo ca lly free O X -mo dule ov er a s cheme X then a p - form on E is a s ymmetric p -linear function on E , i.e., a linear map φ : Sym p ( E ) → O X . If E is in vertible, w e will sometimes id ent ify the p -form with the diagona l p -a ry form ϕ = φ ◦ ∆ : E → O X ; lo cally , if v is a section gener ating E then the form is determined by a = ϕ ( v ): ϕ ( tv ) = a t p . R emark 1.1 .1 . The geometric vector bundle ov er a sc heme X whose sheaf of sections is E is V = Sp ec ( S ∗ ( E ˇ )), where E ˇ is the dual O X -mo dule of E . W e will sometimes describ e p -forms in ter ms o f V . The pro jective space bundle as so ciated to E is π : P ( E ) = Pro j ( S ∗ ) → X , S ∗ = S ∗ ( E ˇ ). The tautologic al line bundle on P ( E ) is L = Sp ec (Sym O (1)), and its sheaf of sections is O ( − 1). The multiplication S ∗ ⊗ E ˇ → S ∗ (1) in the symmetric algebra induces a surjection of lo cally f ree s heav es π ∗ ( E ˇ ) → O (1) a nd hence an 4 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) injection O ( − 1) → π ∗ ( E ); this yields a ca nonical morphism L → π ∗ ( V ) of the asso ciated geometric vector bundles. def:tautfo rm Deﬁnition 1 . 2. Any p -form ψ : Sym p ( E ) → O X on E induces a canonical p -form ǫ on the tautolog ical line bundle L : ǫ : O ( − p ) = Sym p ( O ( − 1 )) → Sym p ( π ∗ E ) = π ∗ Sym p ( E ) ψ − → π ∗ O X = O P ( E ) . W e will use the following notational s ho rthand. F o r a scheme Z , a p oint q on some Z - scheme and a v ector bundle V on Z w e write V | q for the ﬁb er o f V at q , i.e., the k ( q ) v ector s pace q ∗ ( V ) for q → Z . If ϕ is a p -form o n a line bundle L , 0 6 = u ∈ L | q and a = ϕ | q ( u p ), then ϕ | q : ( L | q ) p → k ( q ) is the p -form ϕ | q ( tu p ) = at p . P(O+K) Example 1.3 . Given an inv er tible shea f L on X , and a p -form ϕ on L , the bundle V = O ⊕ L ha s the p -form ψ ( t, u ) = t p − ϕ ( u ). Then P ( V ) → X is a P 1 -bundle, and its tautolog ical line bundle L ha s the p -for m ǫ des crib ed in 1.2. Over a p oint in P ( V ) of the form ∞ = (0 : u ), the p -form on L | ∞ is ǫ (0 , λu ) = − λ p ϕ ( u ). If q = (1 : u ) is any other p o int on P ( V ) then the 1 -dimensional subspa c e L | q of the vector space V | q is generated by v = (1 , u ) and the p - form ǫ | q on L | q is determined by ǫ ( v ) = ψ (1 , u ) = 1 − ϕ ( u ) in the sense that ǫ ( λ v ) = λ p (1 − ϕ ( u )). One a pplication of these ide a s is the for ma tion of the s heaf of Kummer a lgebras asso ciated to a p -form. Recall that if L is a line bundle then the ( p − 1)st symmetric power of P ( O ⊕ L ) is Sym p − 1 P ( O ⊕ L ) = P ( A ( L )), where A ( L ) = L p − 1 i =0 L ⊗ i . Kummeralge bra Deﬁnition 1. 4. If L is a line bundle on X , equipp ed with a p -form φ , the Kum- mer a lgebr a A φ ( L ) is the v ector bundle A ( L ) = L p − 1 i =0 L ⊗ i regar ded as a bundle of algebra s as in [10, 3.11]; lo ca lly , if u is a section ge ner ating L then A ( L ) ∼ = O [ u ] / ( u p − φ ( u )). If x ∈ X and a = φ | x ( u ) then the k ( x )-algebra A| x is the Kummer algebr a k ( x )( p √ a ), whic h is a ﬁeld if a 6∈ k ( x ) p and Q k ( x ) otherwise. Since the nor m on A φ ( L ) is given b y a homogeneous p olyno mial of degree p , we may reg ard the norm as a map from Sym p A φ ( L ) to O . The canonical p -form ǫ on the tautolog ic al line bundle L on the pro jectiv e bundle P = P ( A ( L )), given in 1.2, agrees with the natura l p -form: L ⊗ p → Sym p π ∗ A ( L ) N − → O P , where π : P → X is the structure map and the cano nical inclusion of L in to π ∗ ( A ( L )) = ⊕ p − 1 0 π ∗ L ⊗ i induces the ﬁrst map. Recall from 1.2 and 1 .4 that φ is a p -form on L , ψ = (1 , − ϕ ) is a p - fo rm on O ⊕ L and ǫ is the canonical p -form on L induced from ψ . lem:pth-po wer Lemma 1. 5. Su pp ose that x ∈ X has φ | x 6 = 0 and t hat 0 6 = u ∈ L | x . Th en ǫ | (0: u ) 6 = 0 . Mor e over, φ ( u ) ∈ k ( x ) × p iﬀ ther e is a p oint ℓ ∈ P ( O ⊕ L ) over x so t hat ǫ | ℓ = 0 . Pr o of. Let w = ( t, su ) b e a point of L | x ov er ℓ = ( t : su ) ∈ P ( O ⊕ L ) | x . If t = 0 then ℓ = (0 : u ) and ǫ ( w ) = − s p φ ( u ), which is nonzero for s 6 = 0. If t 6 = 0 then ǫ | ℓ is determined b y the scalar ǫ ( w ) = ψ ( t, su ) = t p − s p φ ( u ). Thus ǫ | ℓ = 0 iﬀ φ ( u ) = ( t/s ) p .  R emark 1.5 .1 . Here is an alternative pro of, using the Kummer algebra K = k ( x )( a ), a = p p φ ( u ). Since ǫ ( w ) = ψ ( t, su ) is the no r m of the nonzero element t − sa in K , the norm ǫ ( w ) is zero iﬀ the Kummer algebra is split, i.e., φ ( u ) = a p ∈ k ( x ) × p . NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 5 Finally , the notation { γ , . . . , γ ′ n − 1 } in the Chain Lemma 0 .1 is a sp ecial ca se of the notation in the following deﬁnition. def:symbol Deﬁnition 1. 6. Given line bundles H 1 , . . . , H n on X , p -forms α i on H i , a nd a po int x ∈ X at which each form α i | x is nonzero, we write { α 1 , . . . , α n }| x for the element { α 1 | x , . . . , α n | x } of K M n ( k ( x )) /p descr ib ed befo re 1.1: if u i is a g enerator of H i | x and α i | x ( u i ) = a i then { α 1 , . . . , α n }| x = { a 1 , . . . , a n } . W e re cord the following useful consequence of this construction. lem:specia lize Lemma 1. 7. Su pp ose tha t the p -forms α i ar e a l l nonzer o at the generic p oint η of a sm o oth X . On the op en subset U of X of p oints x on which e ach α i | x 6 = 0 , the symb ol { α 1 | x , . . . , α n | x } in K M n ( k ( x )) /p is obtai ne d by sp e cializa tion fr om the symb ol in K M n ( k ( X )) /p . 2. The Chain Lemma when n = 2 . sec:n=2bis The goal of this section is to construct cer tain itera ted pro jective bundles to- gether with line bundles and p -forms o n them a s needed in the ca se n = 2 of the Chain Lemma 0.1. Our prese ntation is based up on Ro st’s lectures [Rost]. W e beg in with a generic cons tr uction, which star ts with a pair K 0 , K − 1 of line bundles on a v arie t y X 0 = X − 1 and produces a tow er of v arieties X r , equipp e d with distinguished lines bundles K r . E ach X r is a pro duct o f p − 1 pro jective line bundles ov er X r − 1 , so X r has relative dimension r ( p − 1) over X 0 . def:tower Deﬁnition 2. 1. Given a mor phism f r − 1 : X r − 1 → X r − 2 and line bundles K r − 1 on X r − 1 , K r − 2 on X r − 2 , we form the pro jective line bundle P ( O ⊕ K r − 1 ) over X r − 1 and its tautological line bundle L . By deﬁnition, X r is the pro duct Q p − 1 1 P ( O ⊕ K r − 1 ) ov er X r − 1 . W riting f r for the pro jection X r → X r − 1 , a nd L r for the exterior pro duct L ⊠ · · · ⊠ L on X r , we deﬁne the line bundle K r on X r to b e K r = ( f r ◦ f r − 1 ) ∗ ( K r − 2 ) ⊗ L r . X r f r − → X r − 1 f r − 1 − → X r − 2 · · · X 1 f 1 − → X 0 = X − 1 . ex:ktower Example 2.2 ( k -tower) . The k -tower is the tow er obtaine d when we start with X 0 = Spec ( k ), us ing the trivial line bundles K − 1 , K 0 . Note that X 1 = Q P 1 and K 1 = L 1 , while X 2 is a pro duct of pro jectiv e line bundles ov e r Q P 1 , and K 2 = L 2 . In the Chain Lemma (Theorem 0.1) for n = 2 we have S = X p in the k -tow er , and the line bundles are J = J 1 = K p , J ′ 1 = f ∗ p ( K p − 1 ). Before deﬁning the p -forms γ 1 and γ ′ 1 in 2.7, we quickly establish 2.6; this veriﬁes part (6) of Theorem 0.1, that the degree of c 1 ( K p ) p 2 − p is prime to p . If L is a line bundle ov er X , and λ = c 1 ( L ), the Chow r ing of P = P ( O ⊕ L ) is C H ( P ) = C H ( X )[ z ] / ( z 2 − λz ), wher e z = c 1 ( L ). If π : P → X then π ∗ ( z ) = − 1 in C H ( X ). Applying this observ atio n to the constr uctio n of X r out o f X = X r − 1 with λ r − 1 = c 1 ( K r − 1 ), we hav e C H ( X r ) = C H ( X r − 1 )[ z r, 1 , . . . , z r,p − 1 ] / ( { z 2 r,j − λ r − 1 z r,j | j = 1 , . . . , p − 1 } ) , where z r,j is the ﬁr st Chern class of the j th tautolo gical line bundle L . (F o rmally , C H ( X r − 1 ) is identiﬁed with a subring of C H ( X r ) via the pullback of cycles.) By induction on r , this yields the following r esult: 6 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) lem:CH(Xr) Lemma 2. 3. C H ∗ ( X r ) is a fr e e C H ∗ ( X 0 ) -mo dule. A b asis c onsists of the mono- mials Q z e i,j i,j for e i,j ∈ { 0 , 1 } , 0 < i ≤ r and 0 < j < p . As a gr ade d algebr a, C H ∗ ( X r ) /p ∼ = C H ∗ ( X 0 ) /p ⊗ R 0 R r , wher e R 0 = F p [ λ 0 , λ − 1 ] and R r = F p [ λ − 1 , λ 0 , . . . , λ r , z 1 , 1 , . . . , z r,p − 1 ] /I r , I r = ( { z 2 i,j − λ i − 1 z i,j | 1 ≤ i ≤ r , 0 < j < p } , { λ i − λ i − 2 − P p − 1 j =1 z i,j | 1 ≤ i ≤ r } ) . def:z-zeta Deﬁnition 2. 4. F o r r = 1 , . . . , p , s e t z r = P p − 1 j =1 z r,j and ζ r = Q z r,j . It follows from Lemma 2.3 that λ i = λ i − 2 + z i and z p i = P z p r,j = P z r,j λ p − 1 r − 1 = z i λ p − 1 i − 1 in R r and hence in C H ( X r ) /p . By Lemma 2.3, if 1 ≤ r ≤ p then multiplication by Q ζ i ∈ C H r ( p − 1) ( X r ) is an isomorphism C H 0 ( X 0 ) /p ∼ − → C H 0 ( X r ) /p . If X 0 = Spe c ( k ) then C H 0 ( X r ) /p ∼ = F p , and is gener ated by Q ζ i . lem:deg(yz ) Lem ma 2.5 . If y ∈ C H 0 ( X 0 ) , the de gr e e of y · ζ 1 · · · ζ r is ( − 1) r ( p − 1) deg( y ) . Pr o of. The deg ree on X r is the compo s ition of the ( f i ) ∗ . The pro jection for m ula implies that ( f r ) ∗ ( ζ r ) = ( − 1 ) p − 1 , and ( f r ) ∗ ( y · ζ 1 · · · ζ r ) = ( y · ζ 1 · · · ζ r − 1 ) · ( f r ) ∗ ( ζ r ) = ( − 1 ) p − 1 y · ζ 1 · · · ζ r − 1 . Hence the result fo llows by induction on r .  part6/n=2 Prop ositi on 2.6. F or every 0-cycle y on X 0 and 1 ≤ r ≤ p , λ r = c 1 ( K r ) satisﬁes y λ r ( p − 1) r ≡ y ζ 1 · · · ζ r in C H 0 ( X r ) /p , and deg( y λ r ( p − 1) r ) ≡ deg ( y ) (mo d p ) . F or the k - t ower 2.2 (with y = 1 ), we have deg ( λ p 2 − p p ) ≡ 1 (mo d p ) . Pr o of. If r = 1 this follows from y λ − 1 = y λ 0 = 0 in C H ( X 0 ): λ 1 = z 1 + λ − 1 and y · ζ 1 ≡ y λ p − 1 1 . F or r ≥ 2, we have λ r = z r + λ r − 2 and z p r = z r λ p − 1 r − 1 by 2 .4. Because p − r ≥ 0, we hav e λ r ( p − 1) r =( z r + λ r − 2 ) p ( r − 1)+( p − r ) ≡ ( z p r + λ p r − 2 ) r − 1 · ( z r + λ r − 2 ) p − r mo d p =( z r λ p − 1 r − 1 + λ p r − 2 ) r − 1 ( z r + λ r − 2 ) p − r ≡ ζ r λ ( r − 1)( p − 1) r − 1 + T mo d p, where T ∈ C H ( X r − 1 )[ z r ] is a ho mogeneous p olynomial o f tota l degree < p − 1 in z r . By 2.3, the co eﬃcients o f y T are ele ment s o f C H ( X r − 1 ) of degr e e > dim( X r − 1 ), so y T must be zero. Then b y the inductive hypo thesis, y λ r ( p − 1) r − 1 ≡ y ζ r λ ( r − 1)( p − 1) r − 1 ≡ y ζ r · ( ζ 1 · · · ζ r − 1 ) in C H ∗ ( X r ) /p , a s c la imed. No w the deg r ee ass e rtion follows fro m Lemma 2 .5.  The p -forms. W e now turn to the p -forms in the Chain Lemma 0.1, using the k - tow er 2.2. W e will inductively equip the line bundles L r and K r of 2 .2 with p - forms Ψ r and ϕ r ; the γ 1 and γ ′ 1 of the Cha in Lemma 0 .1 will b e ϕ p and ϕ p − 1 . When r = 0 , w e equip the trivial line bundles K − 1 , K 0 on X 0 = Spec ( k ) with the p -forms ϕ − 1 ( t ) = a 1 t p and ϕ 0 ( t ) = a 2 t p . The p - form ϕ r − 1 on K r − 1 induces a p -form ψ ( t, u ) = t p − ϕ r − 1 ( u ) on O ⊕ K r − 1 and a p -form ǫ on the tautological line bundle L , as in Example 1.3. As obs e rved in E xample 1.3, at the point q = (1 : x ) of P ( O ⊕ K r − 1 ) we hav e ǫ ( y ) = ψ (1 , x ) = 1 − ϕ r − 1 ( x ). NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 7 def:gamma2 Deﬁnition 2. 7. The p -form Ψ r on L r is the pro duct form Q ψ : Ψ r ( y 1 ⊠ · · · ⊠ y p − 1 ) = Y ψ ( y i ) . The p -form ϕ r on K r = ( f r − 1 ◦ f r ) ∗ ( K r − 2 ) ⊗ L r is deﬁned to b e ϕ r = ( f r − 1 ◦ f r ) ∗ ( ϕ r − 2 ) ⊗ Ψ r . splitting/ n=2 Prop ositi on 2.8. L et x = ( x 1 , . . . , x p − 1 ) ∈ X r b e a p oint wi th r esidue ﬁ eld E = k ( x ) . F or − 1 ≤ i ≤ r , cho ose gener ators u i and v i for the one-dimensional E ve ctor sp ac es K i | x and L i | x r esp e ctively, in such a way that u i = u i − 2 ⊗ v i . (1) If ϕ i | x = 0 for some 1 ≤ i ≤ r then { a 1 , a 2 } E = 0 ∈ K 2 ( E ) /p . (2) If ϕ i | x 6 = 0 for al l i , 1 ≤ i ≤ r , then { a 1 , a 2 } E = ( − 1) r { ϕ r − 1 ( u r − 1 ) , ϕ r ( u r ) } ∈ K 2 ( E ) /p. Pr o of. By induction on r . Both parts are obvious if r = 0. T o prov e the ﬁrst part, w e m ay assume that ϕ i | x 6 = 0 for 1 ≤ i ≤ r − 1, but ϕ r | x = 0. W e ha ve u r = u r − 2 ⊗ v r and by the deﬁnition of ϕ r , we conc lude that 0 = ϕ r ( u r ) = ϕ r − 2 ( u r − 2 )Ψ r ( v r ) , whence Ψ r ( v r ) = 0. Now the elemen t v r 6 = 0 is a tensor pr o duct of s ections w j and Ψ r ( v r ) = Q ψ ( w j ) so ψ ( w j ) = 0 for a nonzer o sectio n w j of L | x j . By Lemma 1.5, ϕ r − 1 ( u r − 1 ) is a p th p ower in E . Consequently , { ϕ r − 2 ( u r − 2 ) , ϕ r − 1 ( u r − 1 ) } E = 0 in K 2 ( E ) /p . This symbo l equals ± { a 1 , a 2 } E in K 2 ( E ) /p , b y (2) a nd induction. This ﬁnishes the pro o f of the ﬁrst as sertion. F or the second claim, we c a n assume by induction that { a 1 , a 2 } E = ±{ ϕ r − 2 ( u r − 2 ) , ϕ r − 1 ( u r − 1 ) } E . Now ϕ r ( u r ) = ϕ r − 2 ( u r − 2 )Ψ r ( v r ) . But { ϕ r − 1 ( u r − 1 ) , N ϕ r − 1 ( v r − 1 ) } = 0 by Lemma 2.9 below. W e co nclude that { ϕ r − 2 ( u r − 2 ) , ϕ r − 1 ( u r − 1 ) } E ≡ −{ ϕ r − 1 ( u r − 1 ) , ϕ r ( u r ) } E mo d p ; this concludes the pro of of the second as sertion.  K2Tate Lemma 2. 9 . F or any ﬁeld k any a ∈ k × and any b in K a = k [ p √ a ] , the symb ol { a, N K a /k ( b ) } is t rivial in K 2 ( k ) /p . Pr o of. Because { a, b } = p { p √ a, b } v anishes in K 2 ( K a ]) /p , we hav e { a, N ( b ) } = N { a, b } = pN ( { p √ a, b } ) = 0.  Pro of of the Chain Lemma 0.1 fo r n = 2 . W e verify the conditions f or the v ariety S = X p in the k -tow er 2 .2; the line bundles J = J 1 = K p , J ′ 1 = f ∗ p ( K p − 1 ); the p - fo rms γ 1 and γ ′ 1 in 0.1 a re the for ms ϕ p and ϕ p − 1 of 2 .7. P a rt (1) of Theorem 0.1 is immediate from the construction of S = X p ; par ts (2 ) and (4) were pr ov en in Prop os itio n 2.8; par ts (3) and (5 ) follow fr o m (2 ) a nd (4); and part (6) is P rop osition 2.6 with y = 1.  8 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) Norm Principle f or n = 2 The N orm Principle for n = 2 was implicit in the Merkurjev- Sus lin paper [3, 4.3]. W e repro duce their short pro o f, which uses the the Severi-Bra uer v ar iety X of the cyclic division algebr a D = A ζ ( a, b ) attached to a nontrivial symbol { a, b } in K 2 ( k ) /p a nd a p th ro o t o f unit y ζ ; X is a Nor m v ariety for the symbol { a, b } . MSnorm Theorem 2. 10 (Nor m Principle for n = 2) . If x ∈ X and [ k ( x ) : k ] = p m for m > 1 then for al l λ ∈ k ( x ) ther e exists x ′ ∈ X and λ ′ ∈ k ( x ′ ) so that [ k ( x ′ ) : k ] ≤ p and [ x, λ ] = [ x ′ , λ ′ ] in A 0 ( X, K 1 ) . Pr o of. By Merkurjev-Suslin [3, 8.7.2], N : A 0 ( X, K 1 ) → k × is an injection with image Nrd( D ) ⊆ k × . Therefore the unit N ([ x, λ ]) of k can b e written as the reduced nor m of an element λ ′ ∈ D . The subﬁeld E = k ( λ ′ ) o f D ha s degr ee ≤ p , and cor resp onds to a p oint x ′ ∈ X . Since N ([ x ′ , λ ′ ]) = Nrd( λ ′ ) = N ([ x, λ ]), w e hav e [ x, λ ] = [ x ′ , λ ′ ] in A 0 ( X, K 1 ).  3. The Symbo l Chain sec:Symbol Chain Here is the pattern of the chain lemma in all weights. W e sta rt with a sequence a 1 , a 2 , . . . o f units o f k , and the function Φ 0 ( t ) = t p . F or r ≥ 1, w e inductively deﬁne functions Φ r in p r v ariables and Ψ r in p r − p r − 1 v ariables, taking v alues in k , a nd prov e (in 3.4) that { a 1 , ..., a r , Φ r ( x ) } ≡ 0 (mo d p ). Note that Φ r and Ψ r depe nd only up on the units a 1 , ..., a r . W e write x i for a sequence of p r v ariables x ij (where j = ( j 1 , . . . , j r ) and 0 ≤ j t < p ), and we inductively deﬁne Ψ r +1 ( x 1 , ..., x p − 1 ) = Y p − 1 i =1  1 − a r +1 Φ r ( x i )  , eq:Psi (3.1) Φ r +1 ( x 0 , ..., x p − 1 ) =Φ r ( x 0 )Ψ r +1 ( x 1 , ..., x p − 1 ) . eq:Phi (3.2) W e say that tw o ra tional functions a r e bir ational ly e quivalent if they can b e transformed into one a no ther by an auto morphism (ov er the base ﬁe ld k ) of the ﬁeld of ratio na l functions. ex:r=1 Example 3.3 . Ψ 1 ( x 1 , ..., x p − 1 ) = Q (1 − a 1 x p i ) and Φ 1 ( x 0 , ..., x p − 1 ) is x p 0 Q (1 − a 1 x p i ), the norm of the elemen t x 0 Q (1 − x i α 1 ) in the Kummer ex tension k ( x )( α 1 ), α 1 = p √ a 1 . Th us Φ 1 is bir ationally equiv alent to symmetrizing in the x i , follow ed by the norm from k [ p √ a 1 ] to k . More genera lly , Ψ r ( x 1 , ..., x p − 1 ) is the norm o f an e lement in k ( x 1 , ..., x p − 1 )( p √ a r ). ex:Weilres trict Ex ample 3.3.1 . It is us e ful to interpret the map Φ 1 geometrically . Let R k ( α ) /k A 1 denote the v a r iety , isomo rphic to A p , which is the W eil res triction ([16]) o f the a ﬃne line over k ( α ), so that there is a mor phis m N : R k ( α ) /k A 1 → A 1 corres p o nding to the norm map. The function k p → k ( α ) deﬁned by ( x 0 , s 1 , . . . , s p − 1 ) 7→ x 0 (1 − s 1 α + s 2 α 2 − · · · ± s p − 1 α p − 1 ) NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 9 induces a birational map A p m − → R k ( α ) /k A 1 . Fina lly , let q : A p − 1 → A p − 1 / Σ p − 1 ∼ = A p − 1 be the symmetrizing map se nding ( x 1 , . . . ) to the elementary symmetric func- tions ( s 1 , . . . ). Then the following diag ram commut es: A p = A 1 × A p − 1 1 × q / / Φ 1 + + X X X X X X X X X X X X X X X X X X X X X X X X X X X A 1 × A p − 1 birat. m / / R k ( α ) /k A 1 = A p N   A 1 . rem:p=2 R emark 3.3.2 . If p = 2, Φ 1 ( x 0 , x 1 ) = x 2 0 (1 − a 1 x 2 1 ) is birationa lly equiv alent to the norm form u 2 − a 1 v 2 for k ( √ a 1 ) /k , a nd Φ 2 = Φ 1 ( x 0 )[1 − a 2 Φ 1 ( x 1 )] is birationally equiv alent to the nor m form hh a 1 , a 2 ii = ( u 2 − a 1 v 2 )[1 − a 2 ( w 2 − a 1 t 2 )] for the quaternionic algebra A − 1 ( a 1 , a 2 ). More gener a lly , Φ n is birationally eq uiv alent to the Pﬁster form hh a 1 , ..., a r ii = hh a 1 , ..., a r − 1 ii ⊥ a n hh a 1 , ..., a r − 1 ii and Ψ r is equiv alent to the restric tion o f the Pﬁster form to the subspace deﬁned by the equatio ns x 0 = (1 , . . . , 1). rem:p=3 R emark 3.3.3 (Rost) . Supp o se that p = 3. Then Φ 2 is birationally equiv ale n t to (symmetrizing, follow ed b y ) the re duced norm of the algebra A ζ ( a 1 , a 2 ) a nd Φ 3 is equiv alent to the norm form of the exceptional Jor dan algebra J ( a 1 , a 2 , a 3 ). When r = 4, Ros t show ed that the set of nonzero v alues o f Φ 4 is a subgr oup of k × . F or the next lemma, it is useful to in tro duce the function ﬁeld F r in the p r v ariables x j 1 ,...,j r , 0 ≤ j t < p . Note that F r is isomorphic to the tensor pro duct of p copies of F r − 1 . lem:basicf act Lemma 3. 4. { a 1 , ..., a r , Φ r ( x ) } = { a 1 , ..., a r , Ψ r ( x ) } = 0 ∈ K M r +1 ( F r ) /p . If b ∈ k is a nonzer o value of Φ r , then { a 1 , ..., a r , b } = 0 ∈ K M r +1 ( k ) /p . Pr o of. By Lemma 2 .9, { a r , Ψ r ( x ) } = 0 b ecaus e Ψ r ( x ) is a no rm of an element of k ( x )( α r ) by 3 .3. If r = 1 then { a 1 , Φ 1 ( x ) } = { a 1 , x p 0 } ≡ 0 as well. The result for F r follows by inductio n: { a 1 , ..., a r +1 , Φ r +1 ( x ) } = { a 1 , ..., a r +1 , Φ r ( x 0 ) }{ a 1 , ..., a r +1 , Ψ r +1 ( x ) } = 0 . The result for b follows from the ﬁrst asser tion, and sp ecializa tion from F r to k .  exist:NV R emark 3 .5 . F or any v alue b ∈ k × of Φ n , any des ingularizatio n X o f the pr o jective closure of the aﬃne hyper s urface X b = { x : Φ n ( x ) = b } will b e a Norm v ar ie ty for the symbol { a 1 , ..., a n , b } in K M n +1 ( k ) /p . Indeed, since dim( X b ) = p n − 1, w e see from Lemma 3.4 that every aﬃne p oint of X b splits the sym bo l. In particular, the generic po int of X b is a s plitting ﬁeld for this symbo l. By sp ecializ ation, every p oint of X b and X splits the symbol. The symmetric group Σ p − 1 acts on { x 1 , . . . , x p − 1 } and ﬁxes Φ n , so it acts on X b . It is eas y to see that X b / Σ p − 1 is birationally is omorphic to the Nor m v ariety constructed in [10, § 2] using the hypersur fa ce W deﬁned by N = b in the vector bundle of lo c. cit. By [1 0, 1.19], X is also a Norm v arie ty . C-move Deﬁnition 3. 6 . A mov e o f type C n on a sequence a 1 , ..., a n in k × is a transfor ma- tion of the kind: Type C n : ( a 1 , ..., a n ) 7→ ( a 1 , ..., a n − 2 , a n Ψ n − 1 ( x ) , a − 1 n − 1 ) . 10 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) Here Ψ n − 1 is a function of p n − 1 − p n − 2 new v aria bles x i = { x 1 , 1 , ..., x 1 ,p − 1 } . By Lemma 3.4, { a 1 , ..., a n } = { a 1 , ..., a n − 2 , a n Ψ n − 1 ( x ) , a − 1 n − 1 } , so the move do es not change the symbol in K M n ( k ). If we do this mov e p times, alwa y s with a new set of v ariables x i , we o btain a mov e ( a 1 , ..., a n ) 7→ ( a 1 , ..., a n − 2 , γ n − 1 , γ ′ n − 1 ) in which γ n − 1 , γ ′ n − 1 are functions o f p n − p n − 1 v ariables x i,j , 1 ≤ i < p , 1 ≤ j ≤ p . Since these moves do not change the symbol, we hav e eq:moves (3.7) { a 1 , ..., a n } = { a 1 , ..., a n − 2 , γ n − 1 , γ ′ n − 1 } in K M n ( k ). The functions γ n − 1 and γ ′ n − 1 in (3.7 ) a re the ones appea ring in the Chain Lemma 0.1. F ormally , if k ( x 1 ) is the function ﬁeld o f the move of type C n , then the function ﬁeld F ′ n of the mov e (3.7) is the tensor pro duct k ( x 1 ) ⊗ · · · ⊗ k ( x p ). W e will deﬁne a v ariety S n − 1 with function ﬁeld F ′ n . Using p n − 1 − p n − 2 more v a riables x ′ i,j (1 ≤ i < p , 1 ≤ j ≤ p ) we do p mov es of t yp e C n − 1 on ( a 1 , ..., a n − 2 , γ n − 1 ) to get the sequence ( a 1 , ..., a n − 3 , γ n − 2 , γ ′ n − 2 , γ ′ n − 1 ). The function ﬁeld of this move is F ′ n − 1 ⊗ F ′ n , and we will deﬁne a v ariety S n − 2 with this function ﬁeld, together with a mor phism S n − 2 → S n − 1 . Next, apply p mov es of type C n − 2 , then p mov es o f type C n − 3 , and so o n, ending with p mov es of type C 2 . W e have the s equence ( γ 1 , γ ′ 1 , γ ′ 2 , ..., γ ′ n − 1 ) in p n − p v ar iables x 1 , ..., x p − 1 . Moreov er, we se e fr om Lemma 3.4 that eq:pmoves (3.8) { a 1 , . . . , a n } = { γ 1 , γ ′ 1 , γ ′ 2 , ..., γ ′ n − 1 } in K M n ( k ) . The net eﬀect will be to construct a tow er eq:Rtower (3.9) S = S 1 f r − → S 2 − → · · · → S n − 2 − → S n − 1 − → S n = Sp ec( k ) . Let S be a ny v ariety cont aining U = A p n − p as a n a ﬃne open, s o that k ( S ) = k ( x 1 , ..., x p − 1 ), each x i is p n − 1 v ariables x i,j and all line bundles on U a r e trivia l. Then pa rts (1) and (2) o f the Cha in Lemma 0.1 ar e immediate fr om (3 .7) and (3.8). Now the only thing to do is to constr uct S = S 1 , extend the line bundles (and forms) from U to S , a nd prove pa rts (4) and (6) of 0.1. 4. Model P n − 1 f or m oves of type C n sec:model In this section, we constr uct a tower of v arieties P r and Q r ov er S ′ , with p -forms on lines bundles over them, whic h will pro duce a model o f the forms Ψ r and Φ r in (3.1) and (3.2). This to w er, depic ted in (4.0), is deﬁned in 4.2 b elow. eq:PQtower (4.0) P n − 1 → · · · → P r − → Q r − 1 → P r − 1 − → · · · → Q 1 → P 1 − → Q 0 = S ′ The passag e from S ′ to the v ariety P n − 1 is a mo del for the mov es of t yp e C n deﬁned in 3.6. def:Q-bund le Deﬁnition 4 . 1. Let X b e a v ar ie t y ov er some ﬁxed base S ′ . Giv en line bundles K , L on X , we can for m the vector bundle V = O ⊕ L , the P 1 -bundle P ( V ) ov er X , and L . T aking pr o ducts ov er S ′ , set P = Y p − 1 1 P ( O ⊕ L ); Q = X × S ′ P On P a nd Q , we hav e the exterio r pro ducts of the tauto logical line bundles: L (1 , . . . , 1) = L ⊠ L ⊠ · · · ⊠ L on P, K ⊠ L (1 , . . . , 1 ) on Q. NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 11 Given p -forms ϕ a nd φ o n K and L , resp ectively , the line bundle L has the p -form ǫ , a s in E xample 1 .3, and the line bundles L (1 , . . . , 1) and K ⊠ L (1 , . . . , 1 ) are equipp e d with the pr o duct p -forms Ψ = Q ǫ and Φ = ϕ ⊗ Ψ. R emark 4.1.1 . Let x = ( x 1 , . . . , x p − 1 ) denote the g eneric p oint of X p − 1 . The function ﬁelds of P and Q a re k ( P ) = k ( x )( y 1 , . . . , y p − 1 ) and k ( Q ) = k ( x 0 ) ⊗ k ( P ). W e ma y r epresent their g eneric p o int s in co o rdinate fo r m a s a ( p − 1)-tuple { (1 : y i ) } , where the y i generate L ov er x i . Then y = { (1 , y i ) } is a generato r o f L (1 , . . . , 1) at the generic p oint, and Ψ( y ) = Q (1 − φ ( y i )), Φ( y ) = ϕ ( x 0 )Ψ( y ). ex:Q-bundl e Ex ample 4.1.2 . An imp or tant sp ecial ca se ar ises when we b egin with tw o line bun- dles H on S ′ , K on X , with p -forms α a nd ϕ . In this ca se, w e set L = H ⊗ K and equip it with the pro duct form φ ( u ⊗ v ) = α ( u ) ϕ ( v ). A t the generic po int q o f Q we can pick a gener ator u ∈ H | q and set y i = u ⊗ v i ; the forms res e mble the fo rms of (3.1) a nd (3.2 ): Ψ( y ) = Y  1 − α ( u ) ϕ ( v i )  , Φ( y ) = ϕ ( v 0 ) Ψ( y ) . R emark 4.1 .3 . Supp ose a g roup G acts on S ′ , X , K a nd L , and K 0 , L 0 are nontrivial 1-dimensional representations so that at every ﬁxed p oint x of X (a) k ( x ) = k , (b) L x ∼ = L 0 . Then G acts o n P (resp., Q ) with 2 p − 1 ﬁxed p oints y ov er each ﬁxed po int of X p − 1 (resp., of X p ), each with k ( y ) = k , and each ﬁb er of L = L (1 , . . . , 1) (resp., K ⊠ L ) is the representation L j 0 (resp., K 0 ⊗ L j 0 ) for some j (0 ≤ j < p ). Indeed, G a cts nontrivially on each term P 1 of t he ﬁber Q P 1 , so that the ﬁxed po ints in the ﬁb er are the p oints ( y 1 , ..., y p − 1 ) with each y i either (0 : 1 ) or (1 : 0). W e now deﬁne the tow er (4.0) of P r and Q r ov er a ﬁxed bas e S ′ , by induction on r . W e start with line bundles H 1 , . . . , H r , and K 0 = O S ′ on S ′ , and set Q 0 = S ′ . PQtower Deﬁnition 4 .2. Given a v ariety Q r − 1 and a line bundle K r − 1 on Q r − 1 , we fo rm the v arieties P r = P a nd Q r = Q using the co nstruction in Deﬁnition 4 .1, with X = Q r − 1 , K = K r − 1 and L = H r ⊗ K r − 1 as in 4.1 .2. T o emphasize that P r only depe nds up on S ′ and H 1 , . . . , H r , we will sometimes wr ite P r ( S ′ ; H 1 , . . . , H r ). As in 4.1, P r has the line bundle L (1 , . . . , 1), and Q r has the line bundle K r = K r − 1 ⊠ L (1 , . . . , 1). Suppo se that we are giv en p -for ms α i 6 = 0 on H i , a nd we set Φ 0 ( t ) = t p on K 0 . Inductiv ely , the line bundle K r − 1 on Q r − 1 is equipp ed with a p -form Φ r − 1 . As de s crib ed in 4.1 a nd 4 .1.2, the line bundle L (1 , . . . , 1) on P r obtains a p - fo rm Ψ r from the p -form α r ⊗ Φ r − 1 on L = H r ⊗ K r − 1 , and K r obtains a p -form Φ r = Φ r − 1 ⊗ Ψ r . Example 4.2 .1 . Q 1 = P 1 is Q p − 1 1 P 1 ( O ⊕ H 1 ) ov er S ′ , equipp ed with the line bundle K 1 = L (1 , . . . , 1 ). If H 1 is a trivial bundle with p - fo rm α 1 ( t ) = a 1 t p then Φ 1 is the p -form Φ 1 of Exa mple 3.3. P 2 is Q p − 1 1 P 1 ( O ⊕ H 2 ⊗ K 1 ) ov e r Q p − 1 1 , and K 2 = K 1 ⊠ L (1 , . . . , 1). dim/n=2 Lemma 4.3. If r > 0 then dim( P r /S ′ ) = ( p r − p r − 1 ) and dim( Q r /S ′ ) = p r − 1 . Pr o of. Set d r = dim( Q r /S ′ ). This follows easily by induction from the form ula s dim( P r +1 /S ′ ) = ( p − 1)( d r + 1), dim ( Q r +1 /S ′ ) = p ( d r + 1) − 1.  Cho osing gener ators u i for H i at the g e ne r ic p oint o f S ′ , we g e t units a i = α i ( u i ). 12 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) lem:agreeg eneric Lemma 4 .4. At the generic p oints of P r and Q r , t he p - forms Ψ n and Φ n of 4.2 agr e e with the forms deﬁne d in (3.1) and (3.2) . Pr o of. This follows by induction on r , using the analysis of 4.1.2. Giv en a p oint q = ( q 1 , . . . , q p ) o f Q p − 1 r − 1 and a p oint { (1 : y i ) } on P r ov er it, y = { (1 , y i ) } is a nonzero point on L (1 , . . . , 1) and y i = 1 ⊗ v i for a section v i of K r − 1 . Since ǫ (1 , y i ) = 1 − a r Φ r − 1 ( v i ) and Ψ r ( y ) = Q ǫ (1 , y i ), the for ms Ψ r agree. Similarly , if v 0 is the generator o f K r − 1 ov er the generic p oint q 0 then y ′ = v 0 ⊗ y is a g enerator of K r and Φ r ( y ′ ) = Φ r − 1 ( v 0 )Ψ r ( y ) , which is als o in agreement with the form ula in (3.2).  Recall that K 0 is the trivial line bundle, and that Φ 0 is the standard p -fo rm Φ 0 ( v ) = v p on K 0 . Every p oint o f P r = Q P ( O⊕ L ) has the for m w = ( w 1 , . . . , w p − 1 ), and the pro jection P r → Q Q r − 1 sends w ∈ P r to a p oint x = ( x 1 , . . . , x p − 1 ). part4 Prop ositi on 4.5. L et s ∈ S ′ b e a p oint such that a 1 | s , . . . , a r | s 6 = 0 . 1. If Ψ r | w = 0 for some w ∈ P r , then { a 1 , . . . , a r } vanishes in K M r ( k ( w )) /p . 2. If Φ r | q = 0 for some q = ( x 0 , w ) ∈ Q r , { a 1 , . . . , a r } vanishes in K M r ( k ( q )) /p . Pr o of. Since Φ r = Φ r − 1 ⊗ Ψ r , the assumption that Ψ r | w = 0 implies that Φ r | q = 0 for any x 0 ∈ Q r − 1 ov er s . Conv ers ely , if Φ r | q = 0 then e ither Ψ r | w = 0 o r Φ r − 1 | x 0 = 0. Since Φ 0 6 = 0, w e may pro ceed by induction on r and as s ume that Φ r − 1 | x j 6 = 0 for e a ch j , so that Φ r | q = 0 is equiv a lent to Ψ r | w = 0. By co nstruction, the p -form o n L = H r ⊗ K r − 1 is φ ( u r ⊗ v ) = a r Φ r − 1 ( v ), where u r generates the vector space H r | s and v is a section of K r − 1 . Since Ψ r | w is the pro duct of the forms ǫ | w j , some ǫ | w j = 0 . Lemma 1.5 implies that a r Φ r − 1 ( v ) is a p th p ow er in k ( x j ), and hence in k ( w ), for a ny gener a tor v of K r − 1 | x j . By Lemma 3.4, { a 1 , . . . , a r − 1 , Φ r − 1 } = 0 and hence { a 1 , . . . , a r } = { a 1 , . . . , a r − 1 , a r Φ r − 1 } = 0 in K M r ( k ( w )) /p , as claimed.  W e conclude this section with some ident ities in C H ( P n ) /p C H ( P n ), given in 4.8. T o simplify the statements and pro ofs b elow, we write ch( X ) for C H ( X ) /p C H ( X ), and adopt the following notation. Deﬁnition 4.6. Set η = c 1 ( H n ) ∈ ch 1 ( S ′ ), and γ = c 1 ( L (1 , . . . , 1)) ∈ ch 1 ( P n ). W riting P for the bundle P ( O ⊕ H n ⊗ K n − 1 ) over Q n − 1 , let c ∈ ch( P ) deno te c 1 ( L ) and let κ ∈ ch( Q n − 1 ) denote c 1 ( K n − 1 ). W e write c j , κ j ∈ ch( P n ) for the images of c and κ under the j th co o rdinate pullbacks ch( Q n − 1 ) → ch( P ) → ch( P n ). Lemma12 Lemma 4.7. Su pp ose that H 1 , . . . , H n − 1 ar e trivial. Then (a) γ p n = γ p n − 1 η d in ch( P n ) , wher e d = p n − p n − 1 ; (b) If in additio n H n is trivial, t hen γ d = − Q c j κ e j , wher e e = p n − 1 − 1 . (c) If S ′ = Sp ec k then the zer o-cycles κ e ∈ ch 0 ( Q n − 1 ) and γ d ∈ ch 0 ( P n ) have deg( κ e ) ≡ ( − 1 ) n − 1 and deg( γ d ) ≡ − 1 mo dulo p. NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 13 Pr o of. First note that b eca use K n − 1 is deﬁned over the e -dimensional v ariety Q n − 1 (Spec k ; H 1 , ..., H n − 1 ), the element κ = c 1 ( K n − 1 ) s atisﬁes κ p n − 1 = 0. Thu s ( η + κ ) p n − 1 = η p n − 1 and hence ( η + κ ) d = η d . Now the element c = c 1 ( L ) satisﬁes the relatio n c 2 = c ( η + κ ) in ch ( P ) and hence c p n = c p n − 1 ( η + κ ) d = c p n − 1 η d in c h p n ( P ). Now re call tha t P n = Q P . Then γ = P c j and γ p n = X c p n j = X c p n − 1 j η d = γ p n − 1 η d . When H n is trivial we have η = 0 and hence c 2 = c κ . Setting b j = c p n − 1 j = c j κ e j , we have γ d = γ p n − 1 ( p − 1) = ( P b j ) p − 1 . T o ev aluate this, we use the algebra trick that since b 2 j = 0 for a ll j and p = 0 we hav e ( P b j ) p − 1 = ( p − 1)! Q b j = − Q b j . F or (c), note that if S ′ = Sp ec k then η = 0 and γ d is a zer o-cycle on P n . By the pro jection formula for π : P n → Q Q n − 1 , part (b) yields π ∗ γ d = ( − 1) p Q κ e j . Since each Q n − 1 is an itera ted pro jectiv e space bundle, C H ( Q Q n − 1 ) = ⊗ p − 1 1 C H ( Q n − 1 , and the degree of Q κ e j is the pro duct of the deg rees o f the κ e j . By induction on n , these degrees ar e a ll the same, and nonzero, so deg ( Q κ e j ) ≡ 1 (mo d p ). It remains to establis h the inductiv e for mula for deg( κ e ). Since it is c le a r for n = 0, and the Q i are pro jective spa c e bundles, it suﬃces to compute that c 1 ( K n ) p n − 1 = κ e γ d in ch( Q n ) = ch( Q n − 1 ) ⊗ ch( P n ). Since κ e +1 = 0 and c 1 ( K n ) = κ + γ we hav e c 1 ( K n ) p n − 1 = κ e +1 + γ p n − 1 = γ p n − 1 , and hence c 1 ( K n ) d = γ d . Since γ d +1 = 0, this yields the desired calcula tion: c 1 ( K n ) p n − 1 = c 1 ( K n ) e c 1 ( K n ) d = ( κ + γ ) e γ d = κ e γ d .  cor:RtoCH Corollary 4.8. Ther e is a ring homomorphi sm F p [ λ, z ] / ( z p − λ p − 1 z ) → ch( P n ) , sending λ t o η p n − 1 and z to γ p n − 1 . 5. Model f o r p mo ves sec:Modelf orMoves In this section we construct maps S n − 1 → S n which mo del the p mov es of type C n deﬁned in 3.6. Each such move intro duces p n − 1 − p n − 2 new v ariables, and will be mo delled by a map Y r → Y r − 1 of r elative dimension p n − 1 − p n − 2 , using the P n − 1 construction in 4.2. The r esult (Deﬁnition 5.1) will b e a tow er of the form: J n − 1 = L p L p − 1 L 2 L 1 L 0 = J n S n − 1 = Y p f p − → Y p − 1 − → · · · → Y 2 f 2 − → Y 1 f 1 − → Y 0 = S n . Fix n ≥ 2 , a v ariety S n , and line bundles H 1 , . . . , H n − 2 , H n and J n on S n . The ﬁrst step in the tower is to form Y 0 = S n and Y 1 = P n − 1 ( S n ; H 1 , . . . , H n − 2 , J n ), with line bundles L 0 = J n and L 1 = H n ⊗ L (1 , . . . , 1) as in 4 .2. In f orming the other Y r , the base in the P n − 1 construction 4.2 will b ecome Y r − 1 and o nly the ﬁnal line bundle will change (from J n to L r − 1 ). Here is the for mal deﬁnition. Ytower Deﬁnition 5. 1 . F o r r > 1, we deﬁne mor phisms f r : Y r → Y r − 1 and line bundles L ⊠ r and L r on Y r as follows. Inductively , we are given a morphism f r − 1 : Y r − 1 → Y r − 2 and line bundles L r − 1 on Y r − 1 , L r − 2 on Y r − 2 . Set L ⊠ r = L (1 , . . . , 1), Y r = P n − 1 ( Y r − 1 ; H 1 , . . . , H n − 2 , L r − 1 ) f r − → Y r − 1 , L r = f ∗ r f ∗ r − 1 ( L r − 2 ) ⊗ L ⊠ r . 14 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) Finally , w e write S n − 1 for Y p and set J n − 1 = L p , J ′ n − 1 = f ∗ p ( L p − 1 ). By Lemma 4.3, dim( Y r / Y r − 1 ) = p n − 1 − p n − 2 and hence dim( S n − 1 /S n ) = p n − p n − 1 . F or example, when n = 2 and and H 1 is trivial, this tow er is exactly the tow er of 2.1: we hav e Y r = P 1 ( Y r − 1 ; L r − 1 ) = Q P 1 ( O ⊕ L r − 1 ). R emark 5.1.1 . The line bundles J n − 1 and J ′ n − 1 will b e the line bundles of the Cha in Lemma 0 .1. The rest of tow er (3.9) will be obtained in Deﬁnition 5.8 by rep eating this construction and setting S = S 1 . The rest of this section, culminating in Theor e m 5.9, is devoted to pr oving pa rt (6) o f the Chain Lemma, that the degree of the zero-cy cle c 1 ( J 1 ) dim S is relatively prime to p . In prepar a tion, we need to compa r e the degre e s of the zer o-cycles c 1 ( J n − 1 ) dim S n − 1 on S n − 1 and c 1 ( J n ) dim S n on S n . In order to do so, w e in tro duce the following alg ebra. def:Ar Deﬁnition 5. 2 . W e deﬁne the graded F p -algebra A r and ¯ A r by ¯ A r = A r /λ − 1 A and: A r = F p [ λ − 1 , λ 0 , . . . , λ r , z 1 , . . . , z r ] / ( { z p i − λ p − 1 i − 1 z i , λ i − λ i − 2 − z i | i = 1 , . . . r } ) . RtoCHY R emark 5.2 .1 . By Corollary 4.8, there is a homomor phism A p ρ → ch( Y p ), s e nding λ r to c 1 ( L r ) p n − 2 and z r to c 1 ( L ⊠ r ) p n − 2 . When H n − 1 is trivial, ρ factors through ¯ A p . lem:19 Lemma 5. 3 . In ¯ A r , every element u of de gr e e 1 satisﬁes u p 2 = u p λ p 2 − p 0 . Pr o of. W e will show that ¯ A r embeds into a product of graded r ing s o f the form Λ k = F p [ λ 0 ][ v 1 , . . . , v k ] / ( v p 1 , . . . , v p k ). In each entry , u = aλ 0 + v with v p = 0 and a ∈ F p , so u p = aλ p 0 and u p 2 = aλ p 2 0 , whence the result. Since ¯ A r +1 = ¯ A r [ z ] / ( z p − λ p − 1 r z ) is ﬂat ov er ¯ A r , it em beds b y induction into a pro duct of gra ded ring s of the for m Λ ′ = Λ k [ z ] / ( z p − u p − 1 z ), u ∈ Λ k . If u 6 = 0, there is a n embedding of Λ ′ int o Q p − 1 i =0 Λ k whose i th c omp onent sends z to iu . If u = 0, then Λ ′ ∼ = Λ k +1 .  rem:indep R emark 5 .3.1 . It follows that if m > 0 a nd ( p 2 − p ) | m then u kp + m = λ m 0 u kp . prop:cor22 Prop ositi o n 5.4. In ¯ A r , λ p N − p p = λ p N − p 2 0 ( Q z p − 1 i + T λ 0 ) , wher e deg ( T ) = p 2 − p − 1 . Pr o of. By Deﬁnition 5 .2, ¯ A p is free over F p [ λ 0 ], with the elements Q z m i i (0 ≤ m i < p ) for ming a basis. Thus any ter m of degree p N − p is a linear combination of F = λ p N − p 2 0 Q z p − 1 i and terms of the form λ m 0 0 Q z m i i where P m i = p N − p 2 and m 0 > p N − p 2 . It suﬃces to determine the co eﬃcient of F in λ p N − p p . Since λ p N − p p = λ p N − p 2 0 λ p 2 − p p by Rema rk 5 .3.1, it suﬃces to consider N = 2, when F = Q z p − 1 i . As in the pro of of Prop os ition 2 .6, if p ≥ r ≥ 2 we compute in the ring ¯ A r that λ r ( p − 1) r =( z r + λ r − 2 ) p ( r − 1)+( p − r ) = ( z p r + λ p r − 2 ) r − 1 · ( z r + λ r − 2 ) p − r =( z r λ p − 1 r − 1 + λ p r − 2 ) r − 1 ( z r + λ r − 2 ) p − r = z p − 1 r λ ( r − 1)( p − 1) r − 1 + T , where T ∈ ¯ A r − 1 [ z r ] is a homogeneo us p olyno mia l of total degree < p − 1 in z r . By induction on r , the c o eﬃcient o f ( z 1 · · · z r ) p − 1 in λ r ( p − 1) r is 1 for all r .  NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 15 thm16base Lemma 5. 5. If S n = Spe c ( k ) and c = c 1 ( J n − 1 ) ∈ C H 1 ( S n − 1 ) , then deg( c dim S n − 1 ) ≡ 1 (mod p ) . Pr o of. Set d = dim( S n − 1 ) = p n − p n − 1 ; under the map A p ρ → ch ( S n − 1 ) o f 5.2.1, the degree p 2 − p part of A p maps to C H d ( S n − 1 ). In pa rticular, the zero-cycle c d = ρ ( λ p ) p 2 − p equals the pro duct o f the ρ ( z i ) p − 1 = c 1 ( L ⊠ i ) d/p by Prop osition 5.4 (the T λ ∗ 0 term maps to zero for dimensional reasons). Because S n − 1 = Y p is a pro duct of iterated pro jective space bundles, C H 0 ( Y p ) is the tens o r pro duct o f their C H 0 groups, and the degree of c d is the pro duct o f the degrees of the c 1 ( L ⊠ i ) d/p , each of which is − 1 by Lemma 4.7. It fo llows that deg( c d ) ≡ 1 (mo d p ).  Theorem16 Theorem 5. 6. If S n has dimension p M − p n and H 1 , . . . , H n − 1 ar e trivial then the zer o-cycles c 1 ( J n − 1 ) dim S n − 1 ∈ C H 0 ( S n − 1 ) and c 1 ( J n ) dim S n ∈ C H 0 ( S n ) have the same de gr e e mo dulo p : deg( c 1 ( J n − 1 ) dim S n − 1 ) = deg ( c 1 ( J n ) dim S n ) (mod p ) . Pr o of. By 5.2.1, there is a homomorphism A p ρ → c h( S n − 1 ), sending λ r to c 1 ( L r ) p n − 2 and z r to c 1 ( L ⊠ r ) p n − 2 . Because H n − 1 is trivial, ρ fac tors through ¯ A p . Set N = M − n + 2 and y = λ p N − p 2 0 , so ρ ( y ) = c 1 ( J n ) dim S n ∈ ch 0 ( S n ). F r om Prop os itio n 5 .4 we hav e λ p N − p p ≡ y Q z p − 1 i mo dulo ker( ρ ). F rom Lemma 2.5, the degree of this e le men t equals the degr ee of y mo dulo p .  The p -forms . W e now deﬁne the p -for ms on the line bundles J n − 1 and J ′ n − 1 . using the tow er (5.1). Supp ose that the line bundles L − 1 = H n and L 0 = J n on S n are equipp ed with the p -forms β − 1 and β 0 . W e endo w the line bundle L 1 in Deﬁnition 5.1 with the p -form β 1 = f ∗ ( β − 1 ) ⊗ Ψ n − 1 ( β 0 ); inductively , we endow the line bundle L r with the p -form β r = f ∗ ( β r − 2 ) ⊗ Ψ n − 1 ( β r − 1 ) . Example. When n = 2 and H 1 is trivia l s aw that the tow e r 5.1 is e x actly the tow er of 2.1. In addition, the p -form β r = Ψ 1 ( β r − 1 ) agr ees with the p -form ϕ r = f ∗ ( ϕ r − 2 ) ⊗ Ψ r of 2.7. lem:agreeP si Lemma 5 . 7. If β 0 = α n − 1 and β − 1 = α n , then (at the generic p oint of Y 1 ) the p -form β p agr e es with the form α n Ψ n − 1 in (3.6) . Pr o of. By Lemma 4.4, the form agree s with the form of (3.1).  Rtower Deﬁnition 5. 8 . The tower (3 .9) o f v arieties S i is obtained by do wn ward induction, starting with S n = Sp ec( k ) and J n = H n − 1 . Construction 5.1 yields S n − 1 , J n − 1 and J ′ n − 1 . Inductively , w e rep ea t constr uctio n 5.1 for i , starting with the output S i +1 and J i +1 of the previous s tep, to pro duce S i , J i and J ′ i . By down ward induction in the tow er (3.9), each J i and J ′ i carries a p -for m, which we ca ll γ i and γ ′ i , r esp ectively . By 5.7, thes e fo r ms ag ree with the forms γ i and γ ′ i of (3.7) a nd (3.8 ). Since dim( S i /S i − 1 ) = p i +1 − p i we ha ve dim( S i /S n ) = p n − p i . Th us if we combine Lemma 5.5 a nd Theorem 5.6, we obtain the following result. thm:part6 Theorem 5. 9. F or e ach i < n , deg ( c 1 ( J i ) dim S i ) ≡ − 1 (mo d p ) . Theorem 5 .9 establishes par t (6) of the Cha in Lemma 0 .1, that deg ( c 1 ( J 1 ) dim S 1 ). 16 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) Pr o of of the Chain L emma 0.1. W e verify t he conditions for the v ariety S = S 1 in the to wer (3.9); the line bundles J i and J ′ i and their p -forms ar e obtained b y pulling back from the bundles and forms deﬁned in 5.8. Part (1) of Theorem 0.1 is immediate from the c o nstruction of S ; part (6) is Theorem 5.9, combined with Lemma 5.5. Part (2) was just e s tablished, and part (4) w as proven in P rop osition 4.5; parts (3) and (5) fo llow from (2) and (4). This completes the pr o of of the Chain Lemma.  6. Nice G -actions sec:trucki ng W e will ex tend the Chain Lemma to include an actio n by G = µ n p on S , J i , J ′ i leaving γ i and γ ′ i inv aria nt, such that the action is admissible in the following sense. def:Gnice Deﬁnition 6. 1. (Rost, cf. [7, p.2]) Let G b e a group acting on a k -v ariety X . W e say that the a ction is nic e if Fix G ( X ) is 0-dimensio nal, and consists o f k -p oints. When G als o acts on a line bundle L ov er X , the a ction on the geometric bundle L is nic e exa ctly when G ac ts nontrivially on L | x for every ﬁxed point x ∈ X , and in this case Fix G ( L ) is the zer o-section ov er Fix G ( X ). Suppo se that G acts nic e ly on each of several line bundles L i ov er X . W e say that G acts nic ely o n { L 1 , . . . , L r } if for each ﬁxed p oint x ∈ X the image of the ca nonical representation G → Q Aut( L i | x ) = Q k ( x ) × is Q G i , with ea ch G i nontrivial. rem:Gprodu ct R emark 6 .1.1 . If X i → S are equiv ariant maps and the X i are nice, then G also acts nicely on X 1 × S X 2 . How ever, even if G acts nicely on line bundles L i it may not a ct nicely on L 1 ⊠ L 2 , b ecause the r epresentation ov er ( x 1 , x 2 ) is the pro duct representation L 1 | x 1 ⊗ L 2 | x 2 . ex:Gprojec tive Example 6.2 . Supp ose that G acts nicely on a line bundle L o ver X . Then the induced G -action on P = P ( O ⊕ L ) and its ca nonical line bundle L is nice. Indeed, if x ∈ X is a ﬁxed p oint then the ﬁxed p oints of P | x consist of the t wo k -p o int s { [ O ] , [ L ] } , and if L | x is the repr esentation ρ then G a cts o n L at these ﬁxed p oints as ρ and ρ − 1 , resp ectively . By 6 .1.1, G also acts nicely on the pro ducts P = Q P ( O ⊕ L ) and Q = X × S ′ P of Deﬁnition 4.1, but it do e s not act nicely o n L (1 , . . . , 1). ex:GKummer Example 6.3 . The gr oup G a ls o acts nicely on the Kummer alg ebra bundle A = A ( L ) o f 1.4, and on its pro jective space P ( A ). Indeed, an elemen tary calcula tio n shows that Fix G P ( A ) consists o f the p sections [ L i ], 0 ≤ i < p ov er Fix G ( X ). In each ﬁb er , the (vertical) tangent s pace at eac h ﬁxed point is the representation ρ ⊕ · · · ⊕ ρ p − 1 . If G = µ p , this is the reduced regular repr esentation. Over a ny ﬁxed p oint x ∈ X , L | x is trivia l, and the symmetric group Σ p acts on the bundle A| x , permuting the ﬁxed points. This induces isomorphisms betw een the tangent spa ces at these p oints. ex:GB Example 6.3.1 . The action o f G on Y = P ( O ⊕ A ) is not nice. In this case, an elementary calcula tion shows tha t Fix G ( Y ) consists of the p oints [ L i ] of P ( A ), 0 < i < p , together with the pro jective line P ( O ⊕ O ) ov er ev ery ﬁxed p oint x o f X . F or each x , the (vertical) tangent spa ce a t [ L i ] is 1 ⊕ ρ ⊕ · · · ⊕ ρ p − 1 ; if G = µ p , this is the r e gular representation. When G = µ n p , the following lemma allows us to assume tha t the action on L | x is induced by the standard repres e ntation µ p ⊂ k × , via a pr o jection G → µ p . NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 17 lem:Glines Lemma 6. 4. Any nontrivial 1-dimensional r epr esentation ρ of G = µ n p factors as the c omp osition of a pr oje ction G → µ p with the standar d r epr esentation of µ p . Pr o of. The representation ρ is a nonzer o element of ( Z /p ) n = G ∗ = Hom( µ n p , G m ), and π is the Pontry a gin dual o f the induced map Z /p → G ∗ sending 1 to ρ .  The construction of the P r and Q r in 4 .2 is natural in the giv en line bund les H 1 , . . . , H n ov er S ′ , and so is the construction of the Y r , S r and S in 5.1 and 5 .8. Since Q n i =1 Aut( H i ) acts on the H i , this gro up (and a ny subg roup) will act on the v ariety S of the Chain Lemma. W e will show that it ac ts nice ly o n S . Recall from Deﬁnition 4.2 that P r and Q r are deﬁned by the cons truction 4.1 using the line bundle L r = H r ⊗ K r − 1 ov er Q r − 1 . lem:nicePQ Lemma 6. 5. If S ′ = Sp ec( k ) , t hen G = µ r p acts n ic ely on L r , P r and Q r . This implies that a ny subgro up of Q r i =1 Aut( H i ) containing µ r p also acts nicely . Pr o of. W e pro ceed by inductio n on r , the case r = 1 b eing 6.2, so we may ass ume that µ r − 1 p acts nicely o n Q r − 1 . By 6.1.1, it suﬃces to show that G = µ r p acts nicely on P ( O ⊕ L r ), where L r = H r ⊗ K r − 1 . Since th e ﬁnal component µ p of G acts trivially on K r − 1 and Q r − 1 and non trivially on H r , G = µ r − 1 p × µ p acts nicely o n L r . By Example 6.2, G acts nicely on P ( O ⊕ L r ).  The pro o f of Lemma 6.5 go es thr ough in slig ht ly greater gener a lity . cor:nicePQ Corollary 6.6. Supp ose that G = µ n p acts nic ely on S ′ and on the line bund les { H 1 , . . . , H r } over it. Then G acts nic ely on L r , P r and Q r . Pr o of. Without loss of gener a lity , we may replace S ′ by a ﬁxed point s ∈ S ′ , in which case G a cts nicely on { H 1 , . . . , H r } through the sur jection µ n p → µ r p . No w we are in the s ituation of Lemma 6.5.  ex:niceY Example 6.6.1 . Since µ n − 1 p acts nicely on Y = P n − 1 ( S ′ ; H 1 , . . . , H n − 1 ) a nd on the bundle K n − 1 , while µ p of G = µ n p acts solely on H n , it follows that the group µ n p = µ n − 1 p × µ p acts nicely o n { H 1 , . . . , H n − 1 , H n ⊗ L (1 , . . . , 1) } ov er Y . W e can now pro c e s s the tow er of v arieties Y r deﬁned in 5.1. F or notational conv enience, we wr ite H n − 1 for J n . The case r = 0 of the following assertion uses the conv en tion that L 0 = H n − 1 and L − 1 = H n . Gtwisting Prop ositi on 6.7. Supp ose that G = G 0 × µ n p acts n ic ely on S n and (via G → µ n p ) on { H 1 , . . . , H n } . Then G acts n ic ely on e ach Y r , and on its line bun d les { H 1 , . . . , H n − 2 , L r , L r − 1 } . Pr o of. The question b eing lo ca l, we may replace S ′ by a ﬁxed p oint s ∈ S ′ , and G b y µ n p . W e pro ceed b y induction on r , the cas e r = 1 being Example 6.6.1, since L 1 = H n ⊗ L (1 , . . . , 1 ). Inductively , suppose th at G a cts nicely on Y r and on { H 1 , . . . , H n − 2 , L r , L r − 1 } . Thus ther e is a factor of G iso mo rphic to µ p which acts nontrivially on L r but acts trivially on { H 1 , . . . , H n − 2 , L r } . Hence this factor acts tr iv ially o n Y r +1 = P n − 1 ( Y r ; H 1 , . . . , H n − 2 , L r ) a nd it s line bundle L ⊠ , and nontrivially on L r +1 = L r − 1 ⊗ L ⊠ . The asser tion fo llows.  GonSJ Corollary 6.8. G = µ n p acts n ic ely on ( S, J ) . 18 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) Pr o of. By Deﬁnition 5.1, S n − 1 = Y p , J n − 1 = L p and J ′ n − 1 = L p − 1 . By 6.7 with r = p , G acts nice ly on S n − 1 and on { H 1 , . . . , H n − 2 , J n − 1 , J ′ n − 1 } . By down ward induction, G = µ n − i p × µ i p acts nicely on S i and { H 1 , . . . , H i − 1 , J i , J ′ i } for all i ≤ n . The case i = 1 is the conclus ion, since ( S, J ) = ( S 1 , J 1 ).  R emark 6.8.1 . If G = µ n p acts nicely on S ′ , Ro s t [7, p.2 ] would s ay that a ﬁxed p oint s ∈ S ′ is twisting for { H 1 , . . . , H r } if the map G → µ r p ⊂ Q k ( s ) × = Q Aut( H i | s ) is a surjection. 7. G -fixed poin t equiv alences Let A = A ( J ) b e the Kummer algebra ov er the v ar iety S of the Cha in Lemma 0.1, a s in 1.4. The gr oup G = µ n p acts nicely on S and J by 6 .8, and o n A and P ( A ) by 6.3. In this section, we intro duce tw o G -v arieties ¯ Y and Q , pa rametrized by nor m conditions, and show that they are G -ﬁxed po int equiv alent to P ( A ) and P ( A ) p , resp ectively . This will b e used in the next section to show tha t ¯ Y is G -ﬁxed po int equiv alent to the W eil r estriction of Q E for any K ummer extension E of k . W e b egin b y deﬁning ﬁxed p oint equiv alence and the v ariety Q . def:fpe Deﬁnition 7 .1. L et G b e an algebraic group. W e say that tw o G -v arieties X and Y are G -ﬁx e d p oint e quivalent if Fix G X and Fix G Y are 0- dimens io nal, lie in the smo oth lo cus of X and Y , and there is a sepa rable extension K of k and a bijection Fix G ( X K ) → Fix G ( Y K ) under whic h the families of tangen t spaces a t the ﬁxed po ints are iso morphic as G -repres e nt ations ov er K . def:Q Deﬁnition 7. 2. Reca ll fro m 1 .4 that the nor m A N − → O S is equiv ariant, and ho- mogeneous of deg ree p . W e deﬁne the G -v ar iety Q over S × A 1 , a nd its ﬁb er Q w ov er w ∈ k , by the eq ua tion N ( β ) = w : Q = { [ β , t ] ∈ P ( A ⊕ O ) × A 1 : N ( β ) = t p w } , Q w = { [ β , t ] ∈ P ( A ⊕ O ) : N ( β ) = t p w } , for w ∈ k . Since dim( S ) = p n − p we have dim( Q w ) = p n − 1. If w 6 = 0, then it is prov ed in [10, § 2 ] that Q w is geometr ic ally irreducible and that the op en subscheme where t 6 = 0 is smoo th. If w 6 = 0, Q w is disjoint from the sec tion σ : S ∼ = P ( O ) → P ( A ⊕ O ); ov er each po int of S , the p oint (0 : 1) is disjoint from Q w . Hence the pro jection P ( A ⊕ O ) − σ ( S ) → P ( A ) from these p o ints induces an equiv ariant mor phism π : Q w → Y = P ( A ), π ( β , t ) = β . This is a cov er of degree p ov er its ima ge, since π ( β , t ) = π ( β , ζ t ) for all ζ ∈ µ p . thm:Xb Theorem 7. 3. If w 6 = 0 , G acts nic ely on Q w and Fix G Q w ∩ ( Q w ) sing = ∅ . Mor e- over, Q w and Y = P ( A ) ar e G -ﬁxe d p oint e quivalent over the ﬁeld ℓ = k ( p √ b ) . Pr o of. Since the ma ps Q w π − → Y → S ar e equiv aria nt , π maps Fix G Q w to Fix G Y , and b oth lie ov er the ﬁnite set Fix G S of k -ra tio nal p oints. Since the ta ngent s pace T y is the pro duct o f T s S and the tangent space of the ﬁbe r Y s , and similarly for Q w , it suﬃces to co nsider a G -ﬁxed point s ∈ S . By 6.7 and Lemma 6.4, G acts nont rivially o n L = J | s via a pr o jection G → µ p . By Example 6.3, G acts nicely o n P ( A ). Thu s there is no harm in as suming that G = µ p and that L is the standard 1-dimensiona l r e pr esentation. NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 19 Let y ∈ Y b e a G -ﬁx e d p oint lying ov er s . By 6 .2, the tangent space of Y | s at y is the reduced reg ular representation, and y is one o f [1], [ L ], . . . [ L p − 1 ]. W e saw in E x ample 6.3 .1 tha t a ﬁxed p oint [ a 0 : a 1 : · · · : a p − 1 : t ] of G in P ( A ⊕ O ) | s is either one of the p oints e i = [ · · · 0 : a i : 0 · · · : 0 ], which do no t lie o n Q w , or a point on the pr o jective line { [ a 0 : 0 : t ] } . By insp ection, Q w ⊗ k ℓ meets the pro jective line in the ℓ -p oints [ ζ p √ b : 0 : · · · : 0 : 1 ], ζ ∈ µ p . Ea ch of these p points is smo oth on Q w , and the tangent spa ce (ov er s ) is the reduced regula r representation of G .  R emark 7 .3 .1 . Since π ([ ζ p √ b : 0 : · · · : 0 : 1 ]) = [1 ] fo r all ζ , Fix G ( Q w ) π − → Fix G ( Y ) is not a scheme isomorphism ov er ℓ . cor:NV R emark 7.4 . F o r any w ∈ k × of N , any desingular ization Q ′ of Q w is a s mo oth, ge- ometrically irreducible splitting v a riety for the symbol { a 1 , ..., a n , w } in K M n +1 ( k ) /p . Assuming the Blo ch-Kato conjecture for n , Suslin and Joukhovitski show it is a norm v ar iety in [10, § 2]. Note that the v ariety X w of 3 .5 is biratio na lly a cov er of Q w . T o construct ¯ Y , we ﬁx a K ummer extensio n E = k ( ǫ ) of k . Let B be the O S - subbundle ( A ⊗ 1 ) ⊕ ( O S ⊗ ǫ ) of A E = A ⊗ k E and let N B : B → O S ⊗ k E b e the map induced by the norm on A E . Deﬁnition 7. 5. Let U b e the v ariety P ( A ) × P ( B ) × ( p − 1) ov er S × p , and let L be the line bundle L ( A ) ⊠ L ( B ) ⊠ ( p − 1) ov er U , given as the exterior pro duct of the ta utological bundles. The pro duct of the v ar ious norms deﬁnes an algebraic morphism N : L → O S ⊗ E . lem:nospli talgebras Lemma 7.6 . L et u ∈ U b e a p oint over ( s 0 , s 1 , . . . , s p − 1 ) , and write A i for the k ( s i ) -algebr a A| s i . Then the fol lowing hold. (1) If { a } do esn ’t spl it at any of the p oints s 0 , . . . , s p − 1 , then the norm map N : L u → k ( u ) ⊗ E is non- zer o. (2) If { a }| s 0 6 = 0 in K M n ( k ( s 0 )) /p , then A 0 is a ﬁeld. (3) F or i ≥ 1 , if { a }| E ( s i ) 6 = 0 in K M n ( E ( s i )) /p then A i ⊗ E is a ﬁeld. Pr o of. The ﬁrst asser tion follows from part (4) of the Chain Lemma 0.1, since by 1.4 the norm on L is induced fro m the p -form γ 1 on J . A ssertions (2–3) follow from part (2) of the C ha in Lemma, since { a } 6 = 0 implies that γ is nontrivial.  def:barT Deﬁnition 7 . 7. Let A E denote the W eil restr iction Res E /k A 1 , characterized by A E ( F ) = F ⊗ k E ([16 ]). Let ¯ Y denote the s ubv ariety of P ( L ⊕ O ) × A E consisting of all p o ints ([ α : t ] , w ) such that N ( α ) = t p w in E . W e write ¯ Y w for the ﬁbe r ov er a p oint w ∈ A E . Note that dim( ¯ Y w ) = p n +1 − p = p dim( Q w ). rem:points onT Notation 7.8 . Let ([ α : t ] , w ) b e a k -r ational p oint on ¯ Y , so that w ∈ A E ( k ) = E . W e may regard [ α : t ] ∈ P ( L ⊕ O )( k ) as b eing giv en by a p oint u ∈ U ( k ), ly ing ov er a point ( s 0 , . . . , s p − 1 ) ∈ S ( k ) × p , and a nonzero pair ( α, t ) ∈ L u × k (up to scalars ). F r om the deﬁnition of L , w e see that (up to scalars) α determines a p - tuple ( b 0 , b 1 + t 1 ǫ, . . . , b p − 1 + t p − 1 ǫ ), where b i ∈ A| s i and t i ∈ k . When α 6 = 0, b 0 6 = 0 a nd for all i > 0, b i 6 = 0 o r t i 6 = 0. Finally , writing A i for A| s i , the no rm condition says that in E : N A 0 /k ( b 0 ) Y p − 1 i =1 N A i ⊗ E /E ( b i + t i ǫ ) = t p w. 20 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) If k ⊆ F is a ﬁeld extensio n, then an F -p oint of ¯ Y is de s crib ed a s ab ov e , replac ing k b y F and E by E ⊗ k F everywhere. rem:morepo intsonT R emark 7 .8.1 . If w 6 = 0 , then α 6 = 0, b ecaus e N ( α ) = t p w and ( α, t ) 6 = (0 , 0). lem:tnotze ro Lemma 7 . 9. If ¯ Y has a k -p oint with t = 0 then { a }| E = 0 in K M n ( E ) /p . Pr o of. W e use the description of a k -p oint of ¯ Y f rom 7.8. I f t = 0, then α 6 = 0, therefore b 0 6 = 0 ∈ A 0 and b i + t i ǫ 6 = 0 ∈ A i ⊗ E . By Lemma 7.6, if { a }| E 6 = 0 in K M n ( E ) /p then A 0 and a ll the alg ebras A i ⊗ E are ﬁelds, so that N ( α ) = N A 0 /k ( b 0 ) Q p − 1 i =1 N A i ⊗ E /E ( b i + t i ǫ ) 6 = 0 , a contradiction to t p w = 0.  Consider the pro jection ¯ Y → A E onto the second factor, and write ¯ Y w for the (scheme-theoretic) ﬁber ov er w ∈ A E . Com bining 7 .6 with 7.9 we o bta in the following consequence (in the notation of 7 .8): cor:Theore m5 Corollary 7.10 . If { a } 6 = 0 in K M n ( E ) /p and w 6 = 0 is such that ¯ Y w has a k -p oint, then A 0 and the A i ⊗ E ar e ﬁelds and w is a pr o duct of norms of an element of A 0 and elements in the subset s A i + ǫ of A i ⊗ k E . rem:kpoint givesNP R emark 7 .10.1 . In Theorem 7.13 we will see that if w is a gener ic element o f E then such a k -p oint exists. The group G = µ n p acts nicely on S a nd J b y 6.8, and on A and P ( A ) by 6.3. It acts trivially o n A E , so G a cts on B , U and ¯ Y (but not nicely; s e e 6 .1.1). In the notation o f 7.8, if ([ α : t ] , w ) is a ﬁx e d p o int of the G -action on ¯ Y then the p oints u 0 ∈ P ( A ) and s i ∈ S are ﬁxed, and therefore a re k -r ational (see 6 .1). If u is deﬁned over F , each p oint ( b i : t i ) is ﬁxed in B | s i . Since S acts nicely on J , Example 6.3.1 shows that if t = 0 then either t i 6 = 0 (and b i ∈ F ⊂ A i ⊗ F ) or els e t i = 0 and 0 6 = b i ∈ J | ⊗ r i s i ⊗ F ⊆ A i ⊗ F is for some r i , 0 ≤ r i < p . lem:Tfix1 Lemma 7. 11. F or al l w , Fix G ¯ Y w is disjoi nt fr om the lo cus wher e t = 0 . Pr o of. Supp ose ([ α : 0 ] , w ) is a ﬁxed po int deﬁned over a ﬁeld F containing k . As explained ab ov e, b 0 6 = 0 and (for each i > 0) b i + t i ǫ 6 = 0 and either t i 6 = 0 or there is an r i so that b i ∈ J r i | s i ⊗ F . Let I b e the set of indices such that t i 6 = 0. By Ex ample 6.3, b 0 ∈ J | ⊗ r 0 s 0 for so me r 0 , a nd hence N A 0 ( b 0 ) is a unit in k , bec ause the p - fo rm γ is nontrivial on J | s 0 . Likewise, if i / ∈ I , then N A i ⊗ F /F ( b i ) is a unit in F . Now supp ose i ∈ I , i.e., t i 6 = 0, and r ecall that in this cas e b i ∈ F ⊂ A i ⊗ F . If we write E F for the alg e bra E ⊗ F ∼ = F [ ǫ ] / ( ǫ p − e ), then the norm fro m A i ⊗ E F to E F is simply the p -th p ow er on element s in E F , so N A i ⊗ E F /E F ( b i + t i ǫ ) = ( b i + t i ǫ ) p as an ele ment in the alg e br a E F . T aking the pro duct, and keeping in mind t = 0 , we get the equa tion Y i ∈ I N A i ⊗ E F /E F ( b i + t i ǫ ) = Y i ∈ I ( b i + t i ǫ ) p = 0 . Because E F is a se pa rable F -algebra, it has no nilp otent e lement s. W e co nclude that Y i ∈ I ( b i + t i ǫ ) = 0 . The left hand side o f this equation is a p olynomial o f degree at mo st p − 1 in ǫ ; since { 1 , ǫ, . . . , ǫ p − 1 } is a basis of F ⊗ E ov er F , that p olyno mial mu st b e zero . This implies that b i = t i = 0 for s ome i , a contradiction.  NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 21 prop:thm6( 1) Prop osi tion 7.12. If w ∈ A E is generic then Fix G ¯ Y w lies in the op en subvariety wher e t Q p i =1 t i 6 = 0 . rem:w-fibe r Remark 7.12.1 . The o pen subv ar iety in 7.12 is G -is omorphic (by s e tting t and all t i to 1 ) to a closed subv ariety of A ( A ) p , namely the ﬁbe r over w of the map N A⊗ E /E : A ( A ) p → A E deﬁned by N ( b 0 , . . . , b p − 1 ) = N A 0 /k ( b 0 ) Y p − 1 1 N A i ⊗ E /E ( b i + ǫ ) . Indeed, A ( A ) p is G -isomorphic to an o p en subv a riety of ¯ Y and N A i ⊗ E /E is the restriction of α 7→ N ( α ). Pr o of. By Lemma 7.1 1, Fix G ¯ Y w is disjoin t from the lo cus where t = 0, so we may assume that t = 1. Since w is generic, we may also take w 6 = 0. So let ([ α : 1] , w ) be a ﬁxed p oint deﬁned over F ⊇ k for which t j = 0. As in the pr o of of the pr evious lemma, w e collect those indices i such that t i 6 = 0 into a se t I , a nd write E F for E ⊗ k F . Recall that for i ∈ I , we hav e b i ∈ F . Since j / ∈ I , w e hav e that | I | ≤ p − 2 . F or i / ∈ I , N A i ⊗ E F /E F ( b i + t i ǫ ) = N A i ⊗ F /F ( b i ) ∈ F × (the norm cannot b e 0 as t p w = w 6 = 0 by assumption). So we g et that Y i ∈ I ( b i + t i ǫ ) p = ξ w for some ξ ∈ F × . If we view ξ w as a p oint in P ( E )( F ) = ( E F − { 0 } ) /F × , then we get an equation of the form h Y i ∈ I ( b i + t i ǫ ) p i = [ w ] . But the left-hand side lies in the image of the morphism Q i ∈ I P 1 → P ( E ) whic h sends [ b i : t i ] ∈ P 1 ( F ) to [ Q ( b i + t i ǫ ) p ] ∈ P ( E )( F ) . Since | I | ≤ p − 2, this imag e is a prop er closed subv ariety , pr oving the ass ertion for generic w .  Theorem6 Theorem 7 .13. F or a generic close d p oint w ∈ A E , ¯ Y w is G -ﬁxe d p oint e quivalent to t he disjoi nt un ion of ( p − 1 )! c opies of P ( A ) p Pr o of. Since b oth lie over S , it suﬃces to consider a G -ﬁxed p o int s = ( s 0 , . . . , s p − 1 ) in S ( k ) p and prov e the a s sertion for the ﬁxed po ints ov er s . Be cause G acts nicely on S and J , k ( s ) = k and (by Lemma 6.4) G acts on J s via a pr o jection G → µ p as the standa r d r epresentation of µ p . Note that J s = J s i for all i . By Example 6 .3, ther e are precis ely p ﬁxed p oints on P ( A ) lying over a given ﬁxed p o int s i ∈ S ( k ), and at ea ch of these p oints the (vertical) tangent space is the reduced regular r epresentation of µ p . Thu s each ﬁxed p oint in P ( A ) p is k -rationa l, the num b er of ﬁxed p o int s over s is p p , and each o f their tangent spaces is the sum of p copies o f the reduced regula r representation. Since w is generic, we saw in 7.12 that all the ﬁxed p oints of ¯ Y w satisfy t 6 = 0 and t i 6 = 0 for 1 ≤ i ≤ p − 1. By Remark 7.12.1, they lie in the aﬃne op en A ( A ) p of P ( L ⊕ O ). Because µ p acts nicely o n J s , an F -p oint b = ( b 0 , . . . , b p − 1 ) of A ( A ) p is ﬁxed if and only if each b i ∈ F . That is, Fix G ( A ( A ) p ) = A p . No w the norm map restricted to the ﬁxed- p o int set is just the map A p → A E sending b to b p 0 Q p − 1 i =1 ( b i + ǫ ) p . This map is ﬁnite of degree p p ( p − 1)!, and ´ etale for gener ic w , so Fix G ( ¯ Y w ) has p p ( p − 1 )! geometric p oints for gene r ic w . This is the sa me num ber as the ﬁxed po ints in ( p − 1)! copies of P ( A ) o ver s , so it s uﬃces to c heck their tangent space r epresentations. 22 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) A t ea ch ﬁxed p o in t b , the tangent s pace of A ( A ) p (or ¯ Y ) is the sum of p co pie s of the regula r r epresentation of µ p . Since this tang ent space is also the sum of the tangent space of A p (a trivial representation of G ) and the nor mal bundle of A p in ¯ Y , the normal bundle must then b e p copies of the r educed reg ular representation of µ p . Since the tang e nt space of A p maps isomor phically onto the tangent space of A E at w , the tangent space of ¯ Y w is the sa me as the normal bundle of A p in ¯ Y , as require d.  R emark 7.13.1 . The ﬁx ed p oints in ¯ Y w are not nec essarily r ational po ints, and we only know that the isomo rphism of the tangent spa ces at the ﬁxed p oints holds on a separ able extensio n of k . This is pa r allel to the s itua tion with the ﬁxed p oints in Q w describ ed in Theor em 7.3. 8. A ν n -v ariety. sec:bpathe orem The following re s ult will b e needed in the pro of of the norm principle. thm:toddPA Theorem 8. 1. L et S b e t he variety of the chain lemma for some symb ol { a } ∈ K M n ( k ) /p and A = L p − 1 i =0 J ⊗ i the she af of Ku mmer algebr as over S . Then the pr oje ctive bund le P ( A ) has dimension d = p n − 1 and p 2 ∤ s d ( P ( A )) . Pr o of. Let π : P ( A ) → S be the pr o jection. The sta tement a b out the dimension is trivial. In the Gro thendieck g roup K 0 ( P ( A )), we have tha t [ T P ( A ) ] = π ∗ ([ T S ]) + [ T P ( A ) /S ] where T P ( A ) /S is the rela tive tange nt bundle. The class s d is additive, a nd the dimension of S is less tha n d , so we co nclude that s d ( P ( A )) = s d ( T P ( A ) /S ) . Now [ T P ( A ) /S ] = [ π ∗ ( A ) ⊗ O (1) P ( A ) /S ] − 1; applying additivity again, together with the deﬁnition of s d and the decomp osition of A and hence π ∗ ( A ) int o line bundles, we obtain s d ( P ( A )) = deg p − 1 X i =0 c 1 ( π ∗ J ⊗ i ⊗ O (1)) d . The pro jective bundle formula presents the Chow ring C H ∗ ( P ( A )) as: C H ∗ ( P ( A )) = C H ∗ ( S )[ y ] / ( p − 1 Y i =0 ( y − ix )) where x = − c 1 ( J ) ∈ C H 1 ( S ) and y = c 1 ( O (1)) ∈ C H 1 ( P ( A )) . Then s d ( P ( A )) is the degree of the following element o f the ring C H ∗ ( P ( A )): s ′ d ( P ( A )) = X p − 1 i =0 ( y − ix ) d = X p − 1 i =0 a i y i x d − i for some integer co eﬃcients a i . Since x ∈ C H 1 ( S ), w e hav e x r = 0 f or any r > dim( S ) = p n − p . It follows that s ′ d ( P ( A )) = a p − 1 y p − 1 x dim( S ) . By pa rt (6) of the Chain Lemma 0.1, the degr ee of x dim( S ) = ( − 1) dim( S ) c 1 ( J ) dim( S ) is prime to p . In addition, π ∗ ( y p − 1 ) = π ∗ ( c 1 ( O (1)) p − 1 ) = [ S ] ∈ C H 0 ( S ). By the pro jection formula s d ( P ( A )) = a p − 1 deg x dim( S ) . Th us to prov e the theorem, it suﬃces t o show that a p − 1 ≡ p (mo d p 2 ); this algebra ic ca lculation is achiev ed in Lemma 8.2 b e low.  lem:coeffi cient Lem ma 8.2 . In the r ing R = Z /p 2 [ x, y ] / ( Q p − 1 i =0 ( y − ix )) , t he c o eﬃcient of y p − 1 in u m = P p − 1 i =0 ( y − ix ) p m − 1 is px b , with b = p m − p . NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 23 Pr o of. Since u m is ho mogeneous of degree p m − 1 , it suﬃces to de ter mine the co eﬃcient of y p − 1 in u m in the ring R/ ( x − 1) = Z / p 2 [ y ] / ( Y p − 1 i =0 ( y − i )) ∼ = Y p − 1 i =0 Z /p 2 . If m = 1, then u 1 = P p − 1 i =0 ( y − i ) p − 1 is a p olynomial of degr ee p − 1 with leading term py p − 1 . Inductiv ely , we use the fact that for all a ∈ Z /p 2 , w e hav e a p 2 − p = ( 0 , if p | a 1 , else . Thu s for m ≥ 2, if we set k = ( p m − 1 − 1) / ( p − 1), then a p m − 1 = a ( p − 1)+ k ( p 2 − p ) = a p − 1 ∈ Z /p 2 , and therefor e u m = p − 1 X i =0 ( y − i ) p m − 1 = p − 1 X i =0 ( y − i ) p − 1 = u 1 holds in R / ( x − 1 ); the re s ult follows.  9. The N orm Principle sec:NP W e now turn to the Nor m Principle, which concerns the gro up A 0 ( X, K 1 ) asso - ciated to a v a r iety X . In the liter a ture, this gro up is also k nown as H − 1 , − 1 ( X ) and H d ( X, K d +1 ), where d = dim( X ). W e r ecall the deﬁnition from 0.2. def:A1 Deﬁnition 9. 1 . If X is a regular sc heme then A 0 ( X, K 1 ) is t he cokernel of the map ⊕ y K 2 ( k ( y )) ( ∂ xy ) − → ⊕ x k ( x ) × . In this expression, the ﬁrst sum is taken over all po ints y ∈ X of dimens ion 1, and the second sum is ov er all closed po int s x ∈ X . The map ∂ xy : K 2 ( k ( y )) → k ( x ) × is the tame symbol as so ciated to the discrete v aluation o n k ( y ) a sso ciated to x ; if x is not a sp ecialization of y then ∂ xy = 0. If x ∈ X is closed and α ∈ k ( x ) × we write [ x, α ] for the image of α in A 0 ( X, K 1 ). The group A 0 ( X, K 1 ) is cov ariant for proper morphisms X → Y , and clearly A 0 (Spec k , K 1 ) = k × for every ﬁeld k . Th us if X → Sp ec( k ) is prope r then there is a mo rphism N : A 0 ( X, K 1 ) → k × , whose restriction to the g roup of units of a closed p oint x is the norm map k ( x ) × → k × . That is, N [ x, α ] = N k ( x ) /k ( α ). Deﬁnition 9.2 . When X is smo oth and pr op er over k , we wr ite A 0 ( X, K 1 ) for the quotient o f A 0 ( X, K 1 ) by the re lation that [ x 1 , N x/x 1 ( α )] = [ x 2 , N x/x 2 ( α )] for every closed p oint x = ( x 1 , x 2 ) of X × k X and every α ∈ k ( x ) × . It is prov en in [10, 1 .5–1.7 ] that if X ha s a k - rational p oint then A 0 ( X, K 1 ) = k × ; if X ( k ) = ∅ , then b o th the kernel and c o kernel of N : A 0 ( X, K 1 ) → k × hav e exp onent n , wher e n is the g cd of the degr ees [ k ( x ) : k ] for closed x ∈ X . In addition, if x, x ′ are t wo p oints of X then for any ﬁeld map k ( x ′ ) → k ( x ) ov er k and any α ∈ k ( x ) × we have [ x, α ] = [ x ′ , N x/x ′ α ] in A 0 ( X, K 1 ). T o illustrate the adv an tage o f passing t o A 0 , co nsider a cy c lic ﬁeld extension E /k . Then A 0 (Spec E , K 1 ) = E × and by Hilb ert 90, there is a n exact sequence 0 → A 0 (Spec E , K 1 ) → k × → Br( K/ k ) → 0 . W e now supp ose that k is a p -sp ecial ﬁeld, so tha t the kernel and cokernel of N : A 0 ( X, K 1 ) → k × are p -gro ups , and tha t X is a Norm v ariety (a p -generic splitting v ariety of dimension p n − 1). The Norm Pr inciple is concer ned with reducing the 24 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) degrees o f the ﬁeld ex tens ions k ( x ) used to represent elemen ts o f A 0 ( X, K 1 ). F or this, the following de ﬁnitio n is useful. Deﬁnition 9.3 . Let e A 0 ( k ) deno te the subset o f elemen ts θ of A 0 ( X, K 1 ) r e pr esented by [ x, α ] wher e k ( x ) = k or [ k ( x ) : k ] = p . If E /k is a ﬁeld extension, e A 0 ( E ) denotes the corr esp onding subset of A 0 ( X E , K 1 ). tAsubgroup Lemma 9. 4. If k is p -sp e cial and X is a Norm variety, t hen e A 0 ( k ) is a sub gr oup of A 0 ( X, K 1 ) . Pr o of. By the Multiplication Pr inciple [10, 5.7 ], which depends up on the Cha in Lemma 0.1, we know that fo r each [ x, α ], [ x ′ , α ′ ] in e A 0 ( k ), there is a [ x ′′ , α ′′ ] ∈ e A 0 ( k ) so that [ x, α ]+[ x ′ , α ′ ] = [ x ′′ , α ′′ ] in A 0 ( X, K 1 ). Hence e A 0 ( k ) is c lo sed under addition. It is nonempty b ecause E = k [ p √ a 1 ] s plits the symbol and therefore X ( E ) 6 = ∅ . It is a subgroup b eca use [ x, α ] + [ x, α − 1 ] = [ x, 1] = 0.  lem:Nspeci al Lemma 9 . 5 ([10, 1.24]) . If k is p -sp e cial and X is a Norm variety, the r estriction of A 0 ( X, K 1 ) N − → k × to e A 0 ( k ) is an inje ction. Pr o of. Let [ x, α ] repre s ent θ ∈ e A 0 ( k ). If N ( θ ) = N k ( x ) /k ( α ) = 1 then α = σ ( β ) /β . for some β by Hilb ert’s Theo rem 90. But [ x, σ ( β )] = [ x, β ] in e A 0 ( k ); see [10, 1.5].  ex:trivial A0 Example 9.5.1 . If X has a k -p oint z , then the nor m map N of 0.2 is an iso morphism e A 0 ( k ) ∼ = A 0 ( X, K 1 ) ≃ − → k × , split b y α 7→ [ z , α ]. Indeed, for e very clo sed p oint x of X we have [ x, α ] = [ z , N k ( x ) /k α ] in A 0 ( X, K 1 ), by [10, 1 .5]. Our goal in the next section is to prove the following theor e m. Le t E / k b e a ﬁeld extension with [ E : k ] = p . Since k has p th r o ots o f unit y , we can write E = k ( ǫ ) with ǫ p ∈ k . Theorem5 Theorem 9 .6. S upp ose that k is p -sp e cial, { a } E 6 = 0 and that X is a Norm variety for { a } . F or [ z , α ] ∈ e A 0 ( E ) , ther e exist p oints x i ∈ X of de gr e e p over k , t i ∈ k and b i ∈ k ( x i ) such that N E ( z ) /E ( α ) = Q N E ( x i ) /E ( b i + t i ǫ ) . Theorem 9.6 is the key ingredient in the pro of of Theorem 9.7. Theorem4 Theorem 9 .7. If k is p -sp e cial and [ E : k ] = p then A 0 ( X E , K 1 ) N E /k − → A 0 ( X, K 1 ) sends e A 0 ( E ) to e A 0 ( k ) . Pr o of. If { a } E = 0 then the generic s plitting v ariety X has an E - po int x , and Theorem 9.7 is immediate from Example 9.5.1. Indee d, in this ca s e X E has an E - po int x ′ ov er x , every element of e A 0 ( E ) ∼ = E × has the form [ x ′ , α ], and N E /k [ x ′ , α ] = [ x, α ]. Hence we may ass ume that { a } E 6 = 0. This has the adv a nt age that E ( x i ) = E ⊗ k k ( x i ) is a ﬁeld for every x i ∈ X . Cho ose θ = [ z , α ] ∈ e A 0 ( E ) a nd let x i ∈ X , t i and b i be the data g iven b y Theorem 9.6. Each x i lifts to an E ( x i )-po int x i ⊗ E o f X E so we may c onsider the element θ ′ = θ − X [ x i ⊗ E , b i + t i ǫ ] ∈ A 0 ( X E , K 1 ) . By 9.4 ov er E , θ ′ belo ngs to the subg roup e A 0 ( E ). By Theor em 9.6, its norm is N ( θ ′ ) = N E ( z ) /E ( α ) / Y N E ( x i ) /E ( b i + t i ǫ ) = 1 . NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 25 By Lemma 9.5, θ ′ = 0. Hence N E /k ( θ ) = P  x i , N E ( x i ) /k ( x i ) ( b i + t i ǫ )  in A 0 ( X, K 1 ). Since e A 0 ( k ) is a group b y 9.4, this is an element of e A 0 ( k ).  Theorem2 Corollary 9.8 (Theo rem 0.7(3)) . If k is p -s p e cial then e A 0 ( k ) = A 0 ( X, K 1 ) , and N : A 0 ( X, K 1 ) → k × is an inje ction. Pr o of. W e may supp ose that X ( k ) = ∅ . F or every closed z ∈ X there is an int ermediate subﬁeld E with [ k ( z ) : E ] = p a nd a k ( z )-p o int z ′ in X E ov er z . Since [ z ′ , α ] ∈ e A 0 ( E ), Theor em 9 .7 implies that [ z , α ] = N [ z ′ , α ] is in e A 0 ( k ). This prov es the ﬁrst assertion. The second follows from this and Lemma 9.5.  The Norm Pr inciple of the Introduction follows from Theo rem 9 .7. Pro of of the Norm Prin ci pl e (Theorem 0.3). W e consider a gene r ator [ z , α ] of A 0 ( X, K 1 ). Since [ k ( z ) : k ] = p ν for ν > 0, there is a subﬁeld E of k ( z ) with [ k ( z ) : E ] = p , and z lifts to a k ( z )-p oint z ′ of X E . By constr uction, [ z ′ , α ] ∈ e A 0 ( E ) and A 0 ( X E , K 1 ) → A 0 ( X, K 1 ) sends [ z ′ , α ] to [ z , α ]. By Theo rem 9 .7, [ z , α ] is in e A 0 ( k ), i.e. , is represented by an element [ x, α ] with [ k ( x ) : k ] = p .  10. Expressing No rms Recall that E = k ( ǫ ) is a ﬁxed Kummer extension o f a p -sp ecial ﬁeld k , and X is a Norm v ariety ov er k for the symbol { a } . The purp ose o f this section is to prove Theorem 9.6, that if an elemen t w ∈ E is a norm for a Kummer p o int of X E then w is a pro duct of norms of the form sp eciﬁed in Theorem 9.6. Recall from 7.2 that Q ⊆ P ( A ⊕ O ) × A 1 k is the v ariety of all points ([ β , t ] , w ) such that N ( β ) = t p w , and let q : Q → A 1 k be the pro jection. E xtending the base ﬁeld to E and applying the W eil restr iction functor, we obtain a morphism Rq = Res E /k ( q E ) : RQ = Res E /k ( Q E ) → A E . Moreov er, c ho ose once and for all a resolution of singularities ˜ Q → Q , which is a n isomorphism where t 6 = 0. This is p ossible since Q is s mo oth where t 6 = 0, see 7.2. rem:norms R emark 1 0.1 . Since k is p -sp ecial, s o is E . As stated in Lemma 9 .5, the norm map e A 0 ( E ) → E × is injective; w e iden tify e A 0 ( E ) with its image. Thus [ z , α ] ∈ e A 0 ( E ) is identiﬁed with N E ( z ) /E ( α ) ∈ E × . By [10, Theo rem 5.5], ther e is a p o int s ∈ S such that E ( z ) = A s ⊗ E ; Under the cor resp ondence E ( z ) ∼ = A ( A ) s ( E ), we identif y α with a p oint o f A ( A )( E ), lying ov er s ∈ S . Then N E ( z ) /E ( α ) = R q ([ α, 1] , N ( α )). In other words, e A 0 ( E ) ⊆ E × is equal to q ( Q ( E )) − { 0 } . T o prove Theorem 9.6 it therefor e suﬃces to show that ¯ Y w ( k ) is non-empty when w = R q ([ β , 1] , w ). T o do this, we will pro duce a cor resp ondence Z → ¯ Y × A E R Q that is dominant and o f degree prime to p ov er R Q . W e construct the cor resp ondence Z using the Multiplication Principle of [1 0, 5.7] in the following form. lem:multpr inc Lemma 10 . 2 (Multiplication Principle) . L et k b e a p -sp e cial ﬁeld . Then the set of values of t he map N : A ( A )( k ) → k is a multiplic ative subset of k × . Pr o of. Given Remar k 10.1, this is a consequence of Lemma 9.4.  lem:Zexist s Lem ma 10. 3. L et F = k ( ¯ Y ) b e the function ﬁeld. Then t her e exists a ﬁnite ex - tension L/F , of de gr e e prime to p , and a p oint ξ ∈ RQ ( L ) lying over t he generic p oint of A E . 26 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) Pr o of. Let F ′ be the ma x imal prime-to- p extension of F ; then the ﬁeld E F ′ = E ⊗ k F ′ is p -sp ecia l. W e may reg ard the generic p oint of ¯ Y as an element in ¯ Y ( F ). Applying the inclusion F ⊂ F ′ to this element, follow ed by the pro jection ¯ Y → A E , we obtain an element ω of A E ( F ′ ) = E F ′ . By 7 .8, ω is a pro duct of norms fro m A ( A )( E F ′ ). By the Multiplication Pr inciple 1 0.2, ther e exists β ∈ A ( A )( E F ′ ) such that N ( β ) = ω . Now let ξ be the point ([ β , 1] , ω ) ∈ R Q ( F ′ ). Then Rq ( ξ ) = ω and ξ is deﬁned over some ﬁnite intermediate extension F ⊆ L ⊆ F ′ , with [ L : F ] prime to p .  W rite η L for the p oint o f ¯ Y ( L ) deﬁned b y the inclusion F ⊆ L . W e can now deﬁne ¯ Y f ← Z g → RQ to be a (smoo th, pro jective) mo del of ( η L , ξ ) ∈ ( ¯ Y × A E RQ )( L ) . thm:degree Theorem 10 . 4. The morphism g : Z → R Q is pr op er and dominant (henc e onto) and of de gr e e prime to p . Pr o of. Let ω ∈ A E be the gener ic p o int, k ( ω ) the function ﬁeld a nd E ( ω ) = E ⊗ k ( ω ). As deg ree is a generic notio n and inv ar iant under extension o f the base ﬁeld, we may r e place ¯ Y ← Z → RQ b y its basechange alo ng the morphism Spec ( E ( ω )) → Sp ec( k ( ω )) ω − → A E , to obtain morphisms f : Z E ( ω ) → ¯ Y E ( ω ) and g : Z E ( ω ) → RQ E ( ω ) . Using the normal basis theor em, we can wr ite E ( ω ) = E ( ω 1 , . . . , ω p ) for tra nscendentals ω i that are p ermuted under the action of the cyclic group Ga l( E /k ). W e will a pply the DN Theo rem A.1 with bas e ﬁeld k ′ = E ( ω ). In the notatio n of Theorem A.1, we let r = p ; we write Y for some desingula rization of ¯ Y E ( ω ) ; we let X be R ˜ Q E ( ω ) , a nd we let W b e a model for Z E ( ω ) mapping to Y a nd X . Finally , we let u i = { a 1 , . . . , a n , ω i } ∈ K M n +1 ( k ′ ) /p . Observe that o ur base ﬁeld con tains E , so R ˜ Q E ( ω ) = Res E /k ( ˜ Q E ) × A E E ( ω ) splits as a pro duct R ˜ Q E ( ω ) = Q p i =1 ˜ Q ω i , where Q ω i is the ﬁb er of Q → A 1 ov er the po int ω i ∈ A 1 ( E ( ω )) = E ( ω ) . Therefor e w e hav e X = Q p i =1 X i where X i is ˜ Q ω i , the r e solution of singularities of Q ω i . By Remark 7.4, X i is a smo oth, geo metrically irreducible s plitting v ariety for the symbo l u i of dimensio n p n − 1 . Th us, hypo thes is (1) of the DN Theo rem A.1 is sa tisﬁed. By Theor em A.10, t d, 1 ( X i ) = t d, 1 ( P ( A )); by Lemma A.6, we conclude tha t s d ( X i ) ≡ v s d ( P ( A )) (mo d p 2 ) for s ome unit v ∈ Z /p . Since s d ( P ( A )) 6≡ 0 by Theorem 8.1, we conclude that hypothesis (3) of the DN Theorem A.1 is satisﬁed. F urthermor e, K = k ′ ( X 1 × · · · × X i − 1 ) is contained in a rational function ﬁeld ov er E ; in fact, the ﬁeld E ( ω j )( Q ω j ) b ecomes a r ational function ﬁeld once we adjoin p √ γ . Since E doe s not split { a } , K do es not split { a } either. It follows that K do es not split u i = { a } ∪ { ω i } , verifying hypo thesis (2) of Theorem A.1. W e hav e now check ed the hypotheses (1– 3 ) o f Theorem A.1. It remains to chec k that X and Y a re G -ﬁxed p oint equiv alent up to a prime-to- p factor . In fact, we prov ed in The o rem 7.13 that ¯ Y E ( ω ) is G -ﬁxed p oint equiv alent to ( p − 1 )! co pies of P ( A ) p , hence so is Y (since the ﬁxed points lie in the smo o th locus), and in Theorem 7.3 that X i is G -ﬁxed point eq uiv alent to P ( A ). That is, Y is G -ﬁxe d po int equiv alent to ( p − 1)! copies of X . Therefore the DN Theore m applies to show that g is dominant and of degree pr ime to p , as a sserted.  NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 27 Pr o of of The or em 9.6. W e have proved that there is a diagra m ¯ Y f ← Z g → RQ such that the degr e e of g is prime to p . By blowing up if necessary we may ass ume that g : Z → R Q factors thro ugh ˜ g : Z → R ˜ Q , with deg ( ˜ g ) prime to p . Let [ z , α ] ∈ e A 0 ( E ), and se t w = N E ( z ) /E ( α ). By Rema rk 10.1, ther e exists a p oint ([ β , 1] , w ) ∈ RQ ( k ). Lift this to a p oint in R ˜ Q ( k ) (recall that R ˜ Q → RQ is a n isomorphism wher e t 6 = 0 ). Since Z → R ˜ Q is a morphism of smooth pro jective v a r ieties o f degre e prime to p and k is p -sp ecial, w e can lift ([ β , 1] , w ) to a k -p oint of Z , and then a pply f : Z → ¯ Y to g et a k -p oint in ¯ Y w . By the deﬁnition o f ¯ Y and Corolla ry 7.1 0, this means that we ca n ﬁnd K ummer extensions k ( x i ) /k (cor r esp onding to p oints s i ∈ S , and deter mining points x i ∈ X b ecause X is a p -generic splitting v ariety), elements b i ∈ k ( x i ) and t i ∈ k suc h tha t w = Q i N E ( x i ) /E ( b i + t i ǫ ), as asser ted.  28 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) A. Appendix: The DN Theorem app:A In this app endix, we give a pro of of the following Degr ee theore m, whic h is used in the pro of of the Norm P rinciple. Throughout, k will be a ﬁxed ﬁeld of characteristic 0, p > 2 will be a prime, n ≥ 1 will be an in teger and we ﬁx d = p n − 1. Recall from D eﬁnition 7.1 that if X and Y are G -ﬁxed point equiv alent then dim( X ) = dim ( Y ), the ﬁxed p oints are 0 -dimensional a nd their tangent space representations are is o morphic (ov er ¯ k ). thm:DN Theorem A.1 (DN Theor em) . F or r ≥ 1 , let u 1 , ..., u r b e symb ols in K M n +1 ( k ) /p and let X = Q r 1 X i , wher e the X i ar e irr e ducible sm o oth pr oje ctive G -varieties o f dimension d = p n − 1 such that: (1) k ( X i ) splits u i ; (2) u i is non-zer o over k ( X 1 × · · · × X i − 1 ) ; and (3) p 2 ∤ s d ( X i ) L et Y b e a smo oth irr e ducible pr oje ctive G -variety which is G -ﬁx e d p oint e quivalent to the disjoint union of m c opies of X , w her e p ∤ m . L et F b e a ﬁnite extension of k ( Y ) of de gr e e prime to p , and Sp ec ( F ) → X a p oint, with mo del f : W → X . Then f is dominant and of de gr e e prime to p . Spec ( F ) / / ﬁnite   W g   f (domin ant)   @ @ @ @ @ @ @ @ Spec ( k ( Y ) / / Y X The pro of will use tw o ingredients: the degre e formulas A.2 and A.5 b elow, due to Levine and Morel; and a standar d lo calization result A.10 in (complex) cob o rdism theory . The former concern the a lgebraic cob or dism ring Ω ∗ ( k ), a nd the la tter concern the complex b or dism r ing M U ∗ . These are related via the Lazard r ing L ∗ ; combining Quille n’s theorem [1, I I.8] and the Morel-Levine theor em [2, 4.3.7], w e hav e graded ring iso mo rphisms: Ω ∗ ( k ) ∼ = L ∗ ∼ = M U 2 ∗ . Here is the Levine-More l g eneralized degr ee fo rmula for an ir reducible pro jective v ariety X , taken from [2, Theorem 4.4.15 ]. It concerns the ideal M ( X ) of Ω ∗ ( k ) generated by the classes [ Z ] of smo oth pro jective v a rieties Z suc h tha t there is a k -morphism Z → X , and dim( Z ) < dim( X ). thm:gendeg ree Theorem A. 2 (Genera lized Degree F o r mula) . L et f : Y → X b e a morphi sm of smo oth pr oje ctive k -varieties. If dim( X ) = dim( Y ) then [ Y ] − deg ( f )[ X ] ∈ M ( X ) . T rivia lly , if [ Z ] ∈ M ( X ) then M ( Z ) ⊆ M ( X ). W e a lso hav e: lem:Mideal Lemma A.3. Le t X b e a smo oth pr oje ctive k -variety. If Z and Z ′ ar e bir ational ly e quivalent, then [ Z ] ∈ M ( X ) holds if and only if [ Z ′ ] ∈ M ( X ) . Pr o of. By [2, 4 .4.17], the class of Z mo dulo M ( Z ) is a bira tional in v ariant. Thus [ Z ′ ] − [ Z ] ∈ M ( Z ). Because M ( Z ) ⊆ M ( X ), the result follows.  W e shall als o need the Levine- Morel “higher degree formula” A.5 , which is taken from [2, T he o rem 4.4.2 4], and concer ns the mo d p characteristic num ber s t d,r ( X ) of [2, 4.4], where p is prime, n ≥ 1 and d = p n − 1. NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 29 Cho ose a g raded ring homomorphism ψ : L ∗ → F p [ v n ] cor resp onding to some height n for mal gr o up law, w he r e v n has degree d ; many such group laws exist, and the class t d,r will dep end on this c hoice, but only up to a unit. def:todd Deﬁnition A. 4. F or r > 0, the homo morphism t d,r : Ω r d ( k ) ∼ = L r d → F p sends x to the c o eﬃcient of v r n in ψ ( x ). If X is a smo o th pro jective v ariety ov er k , of dimension rd , then X determines a class [ X ] in Ω r d ( k ), a nd t d,r ( X ) is t d,r ([ X ]). thm:higher degree Theorem A.5 (higher degr ee formula) . L et f : W → X b e a morphism of smo oth pr oje ctive varieties of dimension r d and supp ose that X admits a se quenc e of su r- je ctive morphisms X = X ( r ) → X ( r − 1) → · · · → X (0) = Sp ec( k ) such t hat (1) dim ( X ( i ) ) = i d. (2) If η is a zer o-cycle on X ( i ) × X ( i − 1) k ( X ( i − 1) ) , then p divides t he de gr e e of η . Then t d,r ( W ) = deg ( f ) t d,r ( X ) . Here are some pro p e r ties of this characteris tic num b er that we sha ll nee d. Rec all that if dim( X ) = d then p div ides s d ( X ), so that s d ( X ) /p is an integer. lem:todd Lemma A. 6. L et X/k b e a sm o oth pr oje ctive variety, and k ⊆ C and emb e dding. (1) F or r = 1 , ther e is a unit u ∈ F p such that t d, 1 ( X ) ≡ u s d ( X ) /p . (2) If X = Q r i =1 X i and dim( X i ) = d , then t d,r ( X ) = Q r i =1 t d, 1 ( X i ) . (3) t d,r ( X ) dep ends only on t he class of ( X × k C ) an in the c omplex c ob or dism ring. Pr o of. Part (1 ) is [2, Pr op osition 4.4.22.]. P art (2) is immediate fro m the deﬁnition of t d,r and the gra ded multiplicativ e structure on Ω ∗ ( k ). Finally , part (3 ) is a consequence of the fact that the na tural homo morphism Ω ∗ ( k ) → M U 2 ∗ is an isomorphism (since b oth rings are isomor phic to the Lazard ring).  R emark A.6.1 . The class called s d in this a rticle is the S d in [2 ]; the clas s called s d ( X ) in [2] is our class s d ( X ) /p . The next lemma is a v ariant of Theorem A.5. It uses the same h yp o theses. lem:M(X) Lemma A. 7. L et X b e as in The or em A .5. Then ψ ( M ( X )) = 0 . Pr o of. Consider Z with [ Z ] ∈ M ( X ). If d do es not divide dim( Z ), then ψ ([ Z ]) = 0 for degree rea s ons. If dim ( Z ) = 0 , then the image of Z is a closed p oint of X ; s inc e the degre e of such a close d po int is div is ible by p , we have ψ ([ Z ]) = 0. Hence w e may assume that dim ( Z ) = sd for s ome 0 < s < r . The cases r = 1 and s = 0 are immediate, so we pro ceed by induction o n r and s . Let f : Z → X b e a k -morphism with dim( Z ) = sd , and let f s : Z → X ( s ) be the obvious comp osition. As dim( Z ) = dim( X ( s ) ), the generalized deg ree fo r - m ula A.2 a pplies to show that [ Z ] − deg( f s )([ X ( s ) ]) ∈ M ( X ( s ) ). By induction on r , ψ ( M ( X ( s ) )) = 0, so ψ ([ Z ]) = deg ( f s ) ψ ([ X ( s ) ]). W e claim that deg( f s ) ≡ 0 (mo d p ), which yields ψ ([ Z ]) = 0, as desired. If f s is not dominant, then deg ( f s ) = 0 b y deﬁnition. O n the o ther hand, if f s is dominant, then the generic p oint of Z maps to a closed p oint η of X ( s +1) × X ( s ) k ( X ( s ) ). By condition (2) of Theorem A.5, p divides deg( η ) = deg ( f s ).  30 NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) W e will need to show that ψ ( M ( Y )) = 0 for the Y app ear ing in Theorem A.1. This is acco mplished in the next lemma. lem:M(Y) Lemma A. 8. Supp ose X , Y and W ar e smo oth pr oje ctive varieties of dimension rd over k , and f : W → X and g : W → Y ar e morphisms. Supp ose further that ψ ( M ( X )) = 0 and t hat p do es not divide deg ( g ) . Then ψ ( M ( Y )) = 0 . Pr o of. Supp ose [ Z ] ∈ M ( Y ). As g : W → Y is a prop er morphism of smo o th v arieties, of degree prime to p , we can lift the generic p oint Sp ec( k ( Z )) → Y to a po int q : Sp ec( F ) → W for some ﬁeld extension F / k ( Z ) o f degr ee e prime to p . L e t ˜ Z b e a smo oth pr o jective mo del o f F p osses s ing a morphism to Z and a mor phism to X extending the k -mo r phism f ◦ q : Sp ec( F ) → X . Hence [ ˜ Z ] ∈ M ( X ). B y the degree formula for the map ˜ Z → Z , e [ Z ] − [ ˜ Z ] ∈ M ( Z ). If dim( Z ) = 0, then M ( Z ) = (0). In gener a l, M ( Z ) is generated by the classes o f v arieties of dimension less than dim( Z ) that map to Z (hence a for tiori a lso map to Y ) ov er k . By induction on the dimension of Z , w e may assume that ψ ( M ( Z )) = 0. Moreo ver, ψ ([ ˜ Z ]) = 0 by ass umption; s ince p do es not divide e , w e c o nclude that ψ ([ Z ]) = 0 as asser ted.  Finally , w e will use the following standard b ordism lo ca lization result. Gbord Lemma A.9 . Supp ose that the ab elian p -gr oup G = µ n p acts without ﬁxe d p oints on an almost c omplex manifold M . Then ψ ([ M ]) = 0 in F p . Pr o of. By [1 1], [ M ] is in the ideal o f M U ∗ generated by { p, [ M 1 ] . . . , [ M n − 1 ] } , where dim C ( M i ) = p i − 1. Since p is the only generator of this idea l whose dimens ion is a mult iple of d = p n − 1, ψ is zero on every g enerator and hence on the ideal.  thm:bordlo c Theorem A.10. L et G b e µ n p and let X and Y b e c omp act c omplex G -manifolds which ar e G -ﬁxe d p oint e quivalent. Then ψ ([ X ]) = ψ ([ Y ]) . Pr o of. Remov e eq uiv ariantly iso morphic small balls ab out the ﬁxed points of X and Y , a nd let M = X ∪ − Y deno te the result o f joining the re s t of X and Y , with the oppo site o rientation on Y . Then M has a canonical almo st complex structure, G ac ts on M with no ﬁxed p oints, and [ X ] − [ Y ] = [ M ] in M U ∗ . By Lemma A.9, ψ ([ X ]) − ψ ([ Y ]) = ψ ([ M ]) = 0 .  W e can no w prove Theor em A.1. Note that the inclusion k ( Y ) ⊂ F induces a dominant ra tional map W → Y ; w e ma y replace W by a blowup to eliminate the po ints of indeterminacy and o btain a morphism g : W → Y , who se deg ree is prime to p , without a ﬀecting the statement of Theorem A.1. Pr o of of the DN The or em A.1. W e will a pply Theorem A.5 to X and the X ( t ) = Q t i =1 X i . W e m ust ﬁrst chec k that the h yp o theses are satisﬁed. The ﬁrst condition is ob vious. F or the second condition, it is conv e nie nt to ﬁx t a nd set F = k ( X 1 × · · · × X t − 1 ), X ′ = X ( t ) × X ( t − 1) F . By hypothese s (1–2 ) of Theor em A.1, the symbol u t is nonzero ov er F but splits over the gene r ic point of X ′ ; b y sp ecializa tion, it splits over all clos ed p oints. A transfer argument implies that the degr ee of a ny closed po int η of X ′ is divisible by p ; this is the second c o ndition. Hence Theorem A.5 applies and we hav e t d,r ( W ) = deg( f ) t d,r ( X ). By Lemmas A.8 and A.7, w e have that ψ ( M ( Y )) = 0; by the gener a lized de- gree formula A .2, we conclude that ψ ([ W ]) = deg( g ) ψ ([ Y ]), so that t d,r ( W ) = NORM V ARIETIES AND THE CHAIN LE MMA (AFTER MARK US ROST) 31 deg( g ) t d,r ( Y ) 6 = 0. Hence deg( f ) t d,r ( X ) = deg ( g ) t d,r ( Y ) . By Theorem A.10 and Lemma A.6(3), mt d,r ( X ) = t d,r ( Y ). Condition (3) of Theorem A.1 and Lemma A.6 imply that t d, 1 ( X i ) 6 = 0 for all i and hence that t d,r ( X ) 6 = 0. It fo llows that m deg ( g ) ≡ deg( f ) 6 = 0 mo dulo p , as required.  Ac knowledgemen ts . W e would like to state the obvious: we are deeply indebted to Markus Ro st for pr oving the results presented in this pap er, and for lecturing on them dur ing Spring 2000 and Spring 20 05 ter ms at the Institute fo r Adv anced Study . W e are also gra teful to the Ins titute fo r Adv anced Study for providing the conditions for these lectures . W e are also g r ateful to Marc Levine and Peter Landweber for their help with the cob or dism theory used in this pap er. References [1] J. F. Adams, stable homotopy and ge ner alize d homolo gy , Univ. C hi cago Pr ess, 1974. [2] M. Levine and F. Morel. Algebraic Cob ordism. Springe r Mono gr aphs in Mathematics , 2007. [3] A. M erkurjev and A. Suslin, K -cohomology of Sev er i-Brauer v arieties, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046 (Russian). English translation Math. USSR Izvest iya 21 (1983), 307–340. [4] J. Mi l nor and J. Stasheﬀ, Char acteristic Classes , Princeton Univ. Press, 1974. [5] M. Rost, Chow groups with co eﬃcient s, Do c. Math. J. D MV 1 (1996), 319–393. [6] M. Rost, Chain lemma for splitting ﬁelds of symbols, Pr eprint , 1998. Av ailable at h ttp://www. math. uni-bi el ef eld.de/ ∼ rost/c hain-l emma.ht ml [7] M. Rost, Construction of Spli tting V arieties, Pr eprint , 1998. Av ailable at http://www.math. uni-bielefeld.de/ ∼ rost/c hain-lemma.html [8] M. Rost, The chain lemma for Kummer elements of degree 3. C. R. Ac ad. Sci. Paris S´ e r. I Math. , 328(3):185–190, 1999. [Rost] M. Rost, Notes on Le ctur es g i ven at IAS , 1999-2000 and Spring T erm 2005. [9] R. Stong, Notes on Cob or dism The ory , Princeton Univ. Press, 1968. [10] A. Susli n and S. Joukho vitski, Norm V arieties, J. Pur e Appl. A lg. 206 (2006), 245–276. [11] T. tom Di ec k, A ctions of ﬁnite ab elian p -groups without stationary poi n ts, T op olo gy 9 (1970), 359-366. [12] V. V oevodsky , Reduced Po wer oper ations in Motivic Cohomology , Publ. IHES 98 (2003), 1–57. [13] V. V oevodsky , On Motivic Cohomology wi th Z /l co eﬃcien ts, Pr eprint , 2003. Av ailable at h ttp://www.math.uiuc. edu/K-theory/0639/. [14] C. W eib el, Axioms for the Norm Residue Isomorphism, Pr eprint , 2006. T o appear in Pr o- c e ed ings V al ladolid Confer e nc e , av ailable at htt p://www.math.uiuc. edu/K-theory/0809 /. [15] C. W eibel, Patc hing the Norm Residue Isomorphism Theorem, Pr eprint , 2007. Av ail able at h ttp://www.math.uiuc. edu/K-theory/0844/. [16] A. W eil, A deles and Algebr aic Gr oups , Birkh¨ auser, 1982. Based on lectures given 1959–1 960.

Norm Varieties and the Chain Lemma (after Markus Rost)

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