A1-invariants in Galois cohomology and a claim of Morel
aa r X i v : . [ m a t h . K T ] S e p A -invariants in Galois cohomology and a claim of Morel Tom [email protected] 11, 2018
Abstract
We establish variants of the results in [1] for invariants taking values in a strictly homotopyinvariant sheaf. As an application, we prove the folklore result of Morel that π A ( B ´ et Fin bij ) + = GW . A -invariants of ´Etale Algebras This entire section is basically a minor variation of arguments from [1]. Related results were also obtainedby Morel (unpublished) and Hirsch [2, Theorem 2.3.12].Throughout, we fix a field k . Let Sm ( k ) denote the category of smooth k -schemes and P re ( Sm ( k ))the category of presheaves on Sm ( k ). As usual, if F ∈ P re ( Sm ( k )) and X is an essentially smooth k -scheme, then F ( X ) makes sense. Definition.
We call F ∈ P re ( Sm ( k )) homotopy invariant if F ( X ) = F ( X × A ) for all X ∈ Sm ( k ) . Definition.
We call F ∈ P re ( Sm ( k )) unramified if F ( X ` Y ) = F ( X ) × F ( Y ) , for all connected X ∈ Sm ( k ) the canonical map F ( X ) → F ( k ( X )) is injective, and moreover F ( X ) = \ x ∈ X (1) F ( X x ) (intersection in F ( k ( X )) ).Remark . Suppose that k is perfect and F ∈ P re ( Sm ( k )) is a strictly homotopy invariant sheaf ofabelian groups. Then F is unramified [3, Lemma 6.4.4].In this section we are interested in the presheaf Et n ∈ P re ( Sm ( k )) which assigns to X ∈ Sm ( k ) theset of isomorphism classes of ´etale X -schemes, everywhere of rank n . (This is neither homotopy invariantnor unramified, of course.) Definition.
A rank n versal ´etale scheme is an ´etale morphism X → Y , everywhere of rank n , suchthat if T → Spec ( l ) is any rank n ´etale algebra over a finitely generated field extension l/k , then T isobtained from X → Y by pullback along a morphism Spec ( l ) → Y . We will make good use of the following result.
Theorem 2 ([1], Proposition 24.6(2)) . There exists a smooth k -scheme X , an irreducible divisor ∆ ⊂ A n and a finite ´etale morphism p : X → A n \ ∆ of rank n , such that p is versal.Moreover, let η be the generic point of A n , L h ∆ the Henselization of A n in the generic point of ∆ , and η h the generic point of L h ∆ . The finite ´etale η h -scheme X η h splits as a disjoint union X η h = X ` X with X of rank two (unless n = 1 ).Proof. Let us review the construction of p . Let p : X = Spec ( k [ x , . . . , x n ][ t ] / ( t n + x t n − + · · · + x n )) → A n be the evident map. Let ∆ ⊂ A n be the branch locus of p ; the reference proves that this is anirreducible divisor. Since ´etale algebras over fields are simple, it is clear that X := X \ p − (∆) is versal.Let L ∧ ∆ be the completion of L h ∆ and write ˆ η for the generic point of L ∧ ∆ . The reference proves that X ˆ η = Y ` Y , with Y of rank two and Y unramified . What this means is that there exists a finite ´etalemorphism Y ′ → L ∧ ∆ with Y ∼ = Y ′ × L ∧ ∆ ˆ η [1, beginning of Section 11, Proposition 24.2(3) and Definition24.3]. Since L ∧ ∆ → L h ∆ is a morphism of henselian local rings inducing an isomorphism on residue fields,it follows that there exists a finite ´etale morphism Y ′′ → L h ∆ with Y ′ ∼ = Y ′′ × L h ∆ L ∧ ∆ [5, Tag 04GK].Lemma 3(2) below then furnishes us with a retraction ( Y ′′ ) η h → X η h → ( Y ′′ ) η h . Thus X η h splits as X ` ( Y ′′ ) η h , and X must have the same degree as Y , i.e. 2. This concludes the proof.1e used the following result, which is surely well-known. Lemma 3.
Let R be a henselian DVR with completion ˆ R , fraction field K and completed fraction field ˆ K .1. Let A/K be an ´etale algebra with completion ˆ A/ ˆ K . If x ∈ ˆ A satisfies a separable polynomial withcoefficients in A , then x ∈ A .2. Let A, B be ´etale K -algebras. Then Hom K ( A, B ) = Hom ˆ K ( ˆ A, ˆ B ) .Proof. (1) We may assume that A = L is a field. The normalization R ′ of R in L is finite [5, Tag 032L],and hence R ′ , being a domain, is local henselian [5, Tag 04GH(1)]. We may thus replace R by R ′ andassume that A = K . Let π be a uniformizer of R ; then ˆ K = ˆ R [1 /π ]. It follows that π n x ∈ R for n sufficiently large and still satisfies a separable polynomial; hence we may assume that x ∈ R . Thisreduced statement is a well-known characterisation of henselian DVRs. (2) We may assume that A = K [ T ] /P , where P is a separable polynomial. Then Hom ˆ K ( ˆ A, ˆ B ) is theset of elements t ∈ ˆ B with P ( t ) = 0. By (1), such t lie in B . It follows that Hom K ( A, B ) → Hom ˆ K ( ˆ A, ˆ B )is surjective. Injectivity is clear since A → ˆ A etc. are all injective. Lemma 4.
Let X be the localisation of a smooth scheme in a point of codimension one. Write X h forthe Henselization, η ∈ X for the generic point and η h for the generic point of X h . If F ∈ P re ( Sm ( k )) is a Nisnevich sheaf, then the following diagram is cartesian: F ( X ) −−−−→ F ( η ) y y F ( X h ) −−−−→ F ( η h ) . Proof.
Let X ′ → X be an ´etale neighbourhood of the closed point, and η ′ the generic point of X ′ . Then η ′ −−−−→ X ′ y y η −−−−→ X is a distinguished Nisnevich square; hence applying F yields a cartesian square. Since X h is obtained asthe filtered inverse limit of the X ′ and filtered colimits (of sets) commute with finite limits, the resultfollows.Recall that an ´etale algebra A/k is called multiquadratic if it is a (finite) product of copies of k andquadratic separable extensions of k . Corollary 5.
Let F ∈ P re ( Sm ( k )) be a homotopy invariant, unramified Nisnevich sheaf of sets and a : Et n → F any morphism (of presheaves of sets). Assume there exists ∗ ∈ F ( ∗ ) such that for any field l/k and any multiquadratic ´etale algebra A/l we have a ( A ) = ∗| l . Then for any Y ∈ Sm ( k ) and any A ∈ Et n ( Y ) we have a ( A ) = ∗| l .Proof. This is essentially the same as the proof of [1, Theorem 24.4].Since F is unramified, it suffices to prove the claim when Y is the spectrum of a field. We proceedby induction on n . If n ∈ { , } there is nothing to do.Suppose now that l/k is a field, A/l ∈ Et n ( l ) and A ≈ A × A with A multiquadratic (but non-zero). Define an invariant a ′ of Et n − over l via a ′ ( B ) = a ( B × A ). By assumption a ′ ( B ) = ∗ if B ismultiquadratic, hence a ′ = ∗ by induction. We conclude that a ( A ) = ∗ .Now let X → A n \ ∆ be the versal morphism from Theorem 2. We consider a ( X η ) ∈ F ( K ), where K = k ( A n ). I claim that if x ∈ A n is a point of codimension one, then a ( X η ) is in the image of F ( A nx ) → F ( K ). If x ∆ this is clear, because then x ∈ A n \ ∆ and a ( X η ) = a ( X x ) | η . We thus need todeal with the case where x is the generic point of ∆. By Theorem 2, X η h splits off a quadratic factor.Thus a ( X η ) | η h = a ( X η h ) = ∗ , by the previous step. The claim now follows from Lemma 4.Since F is unramified, it follows that a ( X η ) ∈ F ( K ) lies in the image of F ( A n ) → F ( K ). But F ( A n ) = F ( ∗ ), and so there exists a ∈ F ( ∗ ) such that F ( X η ) = a | K . Since X η is versal this impliesthat a ( A/l ) = a | l for any l and any A . But a ( k/k ) = ∗ , so a = ∗ . This concludes the inductionstep. It is proved for example here: https://mathoverflow.net/q/105891 . Application: computing π A ( B ´ et Fin bij ) +Now let k be a perfect field of characteristic not two.We write Shv abNis, A ( k ) for the category of strictly homotopy invariant Nisnevich sheaves (of abeliangroups) on Sm ( k ), P re mon ( k ) for the category of presheaves of monoids, with morphisms the morphismsof monoids. We have obvious forgetful functors Shv abNis, A ( k ) → P re mon ( k ) → P re ( Sm ( k )) . We write U : Shv abNis, A ( k ) → P re ( k ), and U mon : Shv abNis, A ( k ) → P re mon ( k ). Then the functors U mon and U have (potentially partially defined) left adjoints denoted F mon and F . We make use of thefollowing result of Morel. Theorem 6 ([4], Theorem 3.46) . The morphism of presheaves of sets G m / → GW , [ a ]
7→ h a i exhibits GW as F ( G m / . Let Et ∗ ∈ P re mon ( k ) denote the presheaf of monoids X ` n ≥ Et n ( X ); the monoidal operation isgiven by disjoint union of ´etale schemes. For an ´etale algebra A/l , denote by tr ( A ) ∈ GW ( l ) the class ofits trace form. We shall now prove the result advertised in the heading, in the following guise. Proposition 7.
Let k be a perfect field of characteristic not two.The morphism of presheaves of monoids tr : Et ∗ → GW , A tr ( A ) exhibits GW as F mon ( Et ∗ ) .Proof. Let φ : Et ∗ → U mon F be any morphism, where F ∈ Shv abNis, A ( k ) is arbitrary. For a ∈ O ( X ) × let X a := X [ t ] / ( t − a ). Since we are in characteristic not two, X a → X is ´etale. Define t : G m / → U F by mapping a ∈ O ( X ) × to t ([ a ]) := φ ([ X a/ ]) − φ ([ X ]) + φ ([ X ]) ∈ F ( X ) . (The reason for this formula is that tr ([ X a ]) = h i + h a i , and hence tr ([ X a/ ]) − tr ([ X ])+ tr ([ X ]) = h a i .)By Theorem 6 this induces t ′ : GW → F . I claim that the following diagram commutes: Et ∗ φ −−−−→ F tr y (cid:13)(cid:13)(cid:13) GW t ′ −−−−→ F By Corollary 5 it suffices to show this for multi-quadratic algebras over fields l . But we are dealingwith morphisms of monoids into unramified sheaves, so it suffices to show this for l/l ∈ Et ( l ) and l a /l ∈ Et ( l ), where l/k is a finitely generated field extension. Now t ′ ( tr ([ l ])) = t ′ ( h i ) = t ([1]) = φ ([ l / ]) − φ ([ l ]) + φ ([ l ]) = φ ([ l ])since l and l / are isomorphic. Finally t ′ ( tr ([ l a ])) = t ′ ( h i + h a i ) = φ ([ l ]) − φ ([ l ])+ φ ([ l ])+ φ ([ l a ]) − φ ([ l ])+ φ ([ l ]) = φ ([ l a ])+ 2( φ ([ l ]) − φ ([ l ]))(note that [ l ] = 2[ l ]). Hence to prove the claim we need to show that 2( φ ([ l ]) − φ ([ l ])) = 0. Consider u : G m / → Et ∗ , a [ X a ]. Then, applying Theorem 6 again, we get a commutative diagram G m / u −−−−→ Et ∗ y φ y GW u ′ −−−−→ F. Since 2 h i = 2 ∈ GW ( k ) (indeed 2 x + 2 y = ( x + y ) + ( x − y ) ) and φ ([ l a ]) = φ ( u ( a )) = u ′ ( h a i ), thisproves the claim.Consequently we have proved that any morphism φ factors through tr : Et ∗ → GW . Since theimage of tr generates GW (as an unramified sheaf of abelian groups), this factorization is unique. Thisconcludes the proof. 3 eferences [1] Skip Garibaldi, Alexander Merkurjev, and Jean Pierre Serre. Cohomological invariants in Galoiscohomology . Number 28. American Mathematical Soc., 2003.[2] Christian Hirsch. Cohomological invariants of reflection groups. Master’s thesis, LMU Munich, 2009.[3] Fabien Morel. The stable A -connectivity theorems. K-theory , 35(1):1–68, 2005.[4] Fabien Morel. A -Algebraic Topology over a Field . Lecture Notes in Mathematics. Springer BerlinHeidelberg, 2012.[5] The Stacks Project Authors. Stacks Project . http://stacks.math.columbia.eduhttp://stacks.math.columbia.edu