aa r X i v : . [ m a t h . K T ] J u l ABOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD
MIRCEA VOINEAGU
Abstract.
We prove that, over a perfect field, Bredon motivic cohomologycan be computed by Suslin-Friedlander complexes of equivariant equidimen-sional cycles. Partly based on this result we completely identify Bredon motiviccohomology of a quadratically closed field and of a euclidian field in weights 1and σ . We also prove that Bredon motivic cohomology of an arbitrary field inweight 0 with integer coefficients coincides (as abstract groups) with Bredoncohomology of a point. Contents
1. Introduction 12. Preliminaries 33. Complexes of equivariant equidimensional cycles 74. Bredon motivic cohomology of a field in weight 0 105. Bredon motivic cohomology of a field in weights 1 and σ , , σ Introduction
Motivated by Voevodsky’s definition of motivic cohomology, we introduced in [7](together with J.Heller and P.A. Ostvaer) an equivariant generalization of motiviccohomology called Bredon motivic cohomology for G − equivariant smooth schemesover a field k and for a finite group G . In a subsequent paper [8] (together withJ.Heller and P.A. Ostvaer), we extended this definition to an equivariant motiviccohomology bigraded by a pair of virtual C − representations and defined in the Z / A − homotopy category.We proved in [8] that for a smooth C − scheme X over the field of complexnumbers there is a natural cycle map into Bredon cohomology of X ( C ), which in acertain range of indexes becomes an isomorphism with finite coefficients. This resultallows us to obtain partial computations of Bredon motivic cohomology groups ofa complex variety in terms of Bredon cohomology of its complex points. However,apart from applications of this theorem, there is a lack of computations of Bredonmotivic cohomology groups.In this paper we give complete computations of Bredon motivic cohomologygroups with integer coefficients in weight 0 of an arbitrary field. We also computeBredon motivic cohomology of a field of characteristic zero with integer coefficientsof bi-index given by one-dimensional Z / zero in weights 1 and σ . The strategy to prove the results in weights 1 and σ (seesection 5) is to show that Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles (see section 3). Thistogether with a result of Nie [17] gives a computation of the Bredon cohomologyof a field of characteristic zero in bidegree ( a + 2 qσ, b + qσ ) (see Proposition 3.4).Using this computation and after a careful analysis of the maps that appear in thel.e.s. given by the cofiber sequence C → S → S σ we complete the computationof Bredon motivic cohomology with integer coefficients in weights 1 and σ of certainfields (see section 5 Theorems 5.7, 5.10, 5.8, 5.11).A consequence of this computation is that Bredon motivic cohomology with Z / σ coin-cides (as abstract groups) with Bredon cohomology of a point with Z / − coefficients(but they have different computations with integer coefficients). For this see Corol-lary 5.12 and Corollary 5.9.For example, in weight 1 with Z / − coefficients we have the following (see Corol-lary 5.9): Theorem 1.1.
Let k be a field of characteristic zero. If k is a quadratically closedfield or a euclidian field we have (1.2) H a + pσ, C ( k, Z /
2) = Z / ⊕ k ∗ /k ∗ ≤ − a ≤ p − Z / ⊕ k ∗ /k ∗ < a ≤ − p Z / a = 2 ≤ − p, ≤ − a = pk ∗ /k ∗ a = 1 < p, < a = − p + 10 otherwise Notice that in the case of a quadratically closed field (for example any alge-braically closed field such as C ) k ∗ /k ∗ = 0 and in the case of a euclidian field (forexample R ) k ∗ /k ∗ ≃ Z / . We know that H , ( k, Z ) = K M ( k ) = k ∗ ([9], [11]). As a generalization of thisresult we have the following result for Bredon motivic cohomology (Corollary 5.6): Theorem 1.3.
For an arbitrary field k of characteristic zero we have H σ,σC ( k, Z ) = Z / , H ,σC ( k, Z ) = k ∗ , H σ, C ( k, Z ) = Z / and H , C ( k, Z ) = k ∗ . In section 4 we analyze the cohomology groups of the shift complexes Z top ( nσ )( k )and Z top ( n )( k ) for an arbitrary field k . We obtain computations for weight 0 Bredonmotivic cohomology with integer coefficients of a field.As a consequence, we obtain that Bredon motivic cohomology with integer coef-ficients in weight 0 of an arbitrary field coincides (as abstract groups) with Bredoncohomology of a point with integer coefficients. In conclusion, with Z / Theorem 1.4.
For an arbitrary field k we have (1.5) H a + pσ, C ( k, Z /
2) = Z / ≤ − a ≤ p Z / < a ≤ − p otherwise In section 3 we show that Suslin-Friedlander complexes of C − equivariant equidi-mensional cycles compute Bredon motivic cohomology over a perfect field (see The-orem 3.1). In section 6 we discuss Borel motivic cohomology of a field in weights0,1 and σ . BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 3
Notation and Conventions.
We let G = Z / k via G = ⊔ Spec ( k ). Sometimes we write C for the same group. Welet GSch/k be the category of G − equivariant separated quasi-projective schemeswith G -equivariant morphisms of schemes and GSm/k be its subset of smoothG-equivariant quasi-projective schemes over k .We write A ( V ) = Spec ( Sym ( V ∨ )) for the affine k-scheme associated to a k-vector space V and P ( V ) = P roj ( A ( V )) for the projective scheme.We denote S x = { g ∈ G | gx = x } = G/Gx the set-theoretical stabilizer of a point x and Gx = G × S x { x } the orbit of a point x . In general for a Z ⊂ X we write GZ = ∪ g ∈ G gZ for the orbit of Z .For an abelian group G we write G := { g ∈ G | g = 0 } . We write k [ G ] = 1 ⊕ σ for the regular C − representation. We denote by σ the real C -representationgiven by R with the − S σ the sphere associated to the C − representation σ and in general by S V the topological sphere associated to the C -representation V .We use the following notation for indexes in RO ( G )-graded Bredon cohomology H p,qBr ( X, Z ) := H p + qσBr ( X, Z ) for any virtual C − representation V = p + qσ . Wewrite H n,q ( X ) for the motivic cohomology of the scheme X . Acknowledgements:
The author would like to thank Jeremiah Heller for manyuseful discussions. The author would also like to thank the University of Illinoisat Urbana-Champaign, where the final stages of this project were completed. Theauthor thanks the referee for many useful comments.2.
Preliminaries
Let V be a C -representation and T V := P ( V ⊕ / P ( V ) ≃ A ( V ) / A ( V ) \ { } be the motivic representation sphere with the isomorphism in the C − equivariant A − homotopy category ([1], [6], [8]). For n ≥ T nV = T V ∧ ... ∧ T V with n copies of T V in the right term. More generally we have T V ⊕ W = T V ∧ T W for any two C -representations V and W .We have by definition that A ( V ) := a eNis C ∗ A tr,C ( T V )[ − a − bσ ]is the Bredon motivic complex associated to the C − representation V = a + bσ .Here A tr,C ( X ) = Cor k ( − , X ) C ⊗ A and a eNis is a sheafification in equivariantNisnevich topology ([1], [7], [4]). Here Cor k ( X, Y ) C is the free abelian groupgenerated by elementary equivariant correspondences. An equivariant elementarycorrespondence from X to Y is a correspondence of the form ˜ Z = Z + g Z + g Z + ... + g n Z where Z is an elementary correspondence (finite and surjective over aconnected component of X ) and g i range over a set of coset representatives for Stab ( Z ) = { g ∈ G | gZ = Z } . MIRCEA VOINEAGU
The equivariant Nisnevich topology is the smallest Grothendieck topology con-taining the elementary Nisnevich covers { U → X, Y → X } associated to the equi-variant distinguished square in C Sch/kW / / (cid:15) (cid:15) Y p (cid:15) (cid:15) U (cid:31) (cid:127) i / / X. Here p : Y → X is an equivariant etale morphism and U ֒ → X is an equivariantopen embedding such that ( X \ U ) red ≃ ( Y \ W ) red . According to [7] and [4] anequivariant etale map p : Y → X is a C − equivariant Nisnevich cover if for any x ∈ X there is y ∈ Y such that p ( y ) = x , k ( x ) ≃ k ( y ) and the set-theoretical stabilizerscoincide S x ≃ S y . According to [7] the points in the equivariant Nisnevich topologyare Hensel semi-local affine C − schemes with a single closed orbit. Any semilocalHenselian affine C − scheme over k with a single orbit is equivariantly isomorphicto Spec ( O hA,Gx ), the Spec of the henselization of the semilocal ring O A,Gx where A is some affine C − scheme and x ∈ A .For a different generalization of Nisnevich topology in the equivariant settingcalled the fixed point Nisnevich topology see [5]. The fixed point Nisnevich topologyis a topology finer than the equivariant Nisnevich topology, but coarser than theequivariant etale topology used by Thomasson [10]. For example if X is a G − schemewith free action and X e is the same scheme with trivial action then X e × G → X is a cover in fixed point topology, but not in the equivariant Nisnevich topology [7].The shift in the definition of Bredon motivic complexes is given by the tensorproduct in D − ( C Cor k ) with the invertible complexes Z top ( a + bσ ) associated toany virtual C − representation V = a + bσ [7]. We have that Z top ( σ ) := Cone ( Z tr ( C ) C → Z tr ( k ) C ) ∈ D − ( C Cor k )is the complex associated to the sign sphere S σ . Here D − ( C Cor k ) is the derivedcategory of bounded above chain complexes of equivariant Nisnevich sheaves withequivariant transfers on C Sm/k and it has a tensor product induced by Z tr ( X ) C ⊗ tr Z tr ( Y ) C := Z tr ( X × Y ) C . For an abelian group A we have that { M A n = Z tr,C ( T nk [ C ] ) } n defines the Bredonmotivic cohomology spectrum M A in the stable C -equivariant motivic homotopycategory SH C [8]. The construction of SH C as a stabilization with respect to themotivic sphere T k [ C ] is recalled in the appendix of [8] and it initially appeared in[6] as a tool to study Hermitian K-theory of fields. We write [ − , − ] SH C for mapsin SH C .The equivariant A − homotopy was introduced initially by Voevodsky [1] in or-der to understand motivic Eilenberg-Mac Lane spaces. In the C -equivariant A -homotopy category (see [1], [6]) we have the following spheres: the usual sphere S with trivial action, the sign sphere S σ with the conjugation action, the Tatesphere S t = ( A \ { } ,
1) with the trivial action and the sign Tate sphere S σt =( A ( σ ) \ { } ,
1) with the action x → x − .Using the notation from [8] we define below the following motivic spheres bi-indexed by virtual C -representations: S a + pσ,b + qσ := S a − b ∧ S ( p − q ) σ ∧ S bt ∧ S qσt . BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 5
Notice that this notation is slightly different from the one used in [6] (see [8] for thetranslation between these two notations). This notion is suitable for comparison ofBredon motivic cohomology and Bredon cohomology as the complex realization of Re ( S a + pσ,b + qσ ) = S a + pσ , where the right side defines the usual topological sphereassociated to the C − representation V = a + pσ := R a ⊕ R − p .The definition of Bredon motivic cohomology in SH C is the following: Definition 2.1. (see [7]) The Bredon motivic cohomology of a motivic C -spectrum E with coefficients in an abelian group A is defined by e H a + pσ,b + qσ ( E, A ) := [
E, S a + pσ,b + qσ ∧ M A ] SH C . We call the virtual C -representation V = a + pσ the cohomology index and thevirtual C − representation W = b + qσ the weight index.According to [7] this definition for E = Σ ∞ T k [ G ] X + , where X is a smooth C − schemeand the motivic sphere T k [ G ] = T ∧ T σ = S ∧ S t ∧ S σ ∧ S σt ([6], [8]), coincideswith the definition of Bredon motivic cohomology given by the hypercohomology inequivariant Nisnevich topology (or equivariant Zariski topology [7]) of the Bredonmotivic complexes A ( V ) defined above.Based on an equivariant cancellation theorem proved in [7] we can conclude thatif V = b + qσ is a virtual representation and W = c + dσ is a representation suchthat V ⊕ W is an actual C − representation then for any C − scheme X we have H a + pσ,b + qσ ( X, Z ) = H aeNis ( X + ∧ T W , Z ( V ⊕ W )[2 c + (2 d + p ) σ ]) . For the relationship between the Bredon motivic cohomology and Edidin-Grahamequivariant higher Chow groups [3] (which are a generalization of Totaro’s Chowgroups of classifying spaces [12]) see [8].For a pointed motivic C − space χ we have that the map in the equivariant A -homotopy category C ∧ χ ≃ C ∧ χ e → χ e induces an isomorphism e H a + pσ,b + qσC ( C ∧ χ, Z ) = e H a + p,b + q ( χ e , Z ), with χ e beingthe motivic space with the forgotten action [8] and the right side is given by theusual motivic cohomology. If the C -scheme X has trivial action then according to[8] we have that H a,bC ( X, Z ) = H a,b ( X, Z ). Proposition 3.3 in section 3 gives a slightgeneralization of this last result. If X is a smooth quasi-projective C − scheme withfree C -action then H n,mC ( X, A ) = H n,m ( X/C , A ) [8].In the stable equivariant A − homotopy category there is the following motivicisotropy sequence E Z / + → S → e E Z / C − homotopy cofiber sequences A ( nσ ) \ { } + → S → S nσ,nσ = S nσt ∧ S nt . By definition E Z / colim n A ( nσ ) \{ } . The first map is induced by the projection A ( nσ ) \ { } → Spec ( k ). We have that e E Z / colim n S nσt ∧ S nt . The geometricclassifying space is defined as the quotient of E Z / C − action B Z / E Z / Z = colim n A ( nσ ) \ { } / Z / . In Proposition 2.9 and Theorem 5.4 in [8] we showed that Bredon motivic coho-mology of E Z / σ − , σ −
1) and that Bredon motiviccohomology of e E Z/ σ − ,
0) and ( σ − , σ − MIRCEA VOINEAGU
In the stable equivariant A − homotopy category we have the basic cofiber se-quence C ∧ M Z p −→ M Z i −→ S σ ∧ M Z which gives rise to the natural long exact sequence for any C − scheme X · · · → H a − p +1) σ,b + qσC ( X ) δ −→ H a + p,b + q M ( X ) p ∗ −→ H a + pσ,b + qσC ( X )(2.2) i ∗ −→ H a +( p +1) σ,b + qσC ( X ) δ −→ H a + p +1 ,b + q M ( X ) → · · · , (2.3)For a complex scheme X , the basic cofiber sequence of the classical equivarianthomotopy theory given by C → S → S σ induces a l.e.s. for the Bredon coho-mology of the complex points of X . The complex realization X → X ( C ) definedon C Sm/ C → C T op extends to a map SH C ( C ) → SH C between the stableequivariant A − homotopy category over C and the classical stable equivariant ho-motopy category. According to [8] the complex realization of the motivic spectrum M A gives the spectrum that defines Bredon cohomology. The realization functoralso induces a map between the two long exact sequences induced by the cofibersequence C → S → S σ in the motivic equivariant homotopy category and inthe classical equivariant homotopy category.The connecting map in the l.e.s. 2.2 is induced by the map S σ δ → C ∧ S σ = C ∧ S . Proposition 2.4.
The composition S σ δ → C ∧ S σ = C ∧ S p → S σ wherethe first map is the connecting map and the second map is induced by C p → S induces multiplication by 2 on Bredon motivic cohomology of a field i.e. H a +( p − σ,b + qσC ( k, Z ) p ∗ → H a + p − ,b + q ( k, Z ) δ ∗ → H a +( p − σ,b + qσC ( k, Z ) is multiplication by 2 for any field k .Proof. The map S σ δ → C ∧ S σ induces a stable map S → C which is Spanier-Whitehead dual to the stable map C p → S . (cid:3) Corollary 2.5.
The map on Bredon motivic cohomology induced by C → S hasa 2-torsion kernel.Proof. According to the above composition we have that if x ∈ Ker ( p ∗ ) then2 x = 0. (cid:3) We also recall that the topological Bredon cohomology groups of a point with Z / H a + pσBr ( pt, Z /
2) = Z / ≤ − a ≤ p Z / < a ≤ − p . With integer coefficients ([2]), we have that if p > p iseven then H a + pσBr ( pt, Z ) = Z a = − p Z / − p < a ≤ , a = even0 otherwise , BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 7 and if p > p is odd then H a + pσBr ( pt, Z ) = ( Z / − p < a ≤ , a = even0 otherwise , and if p < p is even then H a + pσBr ( pt, Z ) = Z / < a < − p, a = odd Z a = − p , and if p < p is odd then H a + pσBr ( pt, Z ) = ( Z / < a ≤ − p, a = odd0 otherwise . Complexes of equivariant equidimensional cycles
We can define a presheaf of C − equivariant equidimensional cycles on C Sm/k for any smooth C -scheme as in the non-equivariant case: z equi ( T, r ) C ( S ) = the free abelian group generated by elementaryequivariant r − equidimensional cycles over S in S × T .An elementary equivariant r − equidimensional cycle is of the form ˜ Z = Z + g Z + ... + g n Z , where Z is an irreducible closed subvariety of S × T which is dominant andequidimensional of relative dimension r over a connected component of S and g i area set of coset representatives of Stab ( Z ) = { g ∈ G | gZ = Z } . We conclude that for aprojective or proper smooth C − scheme T we have z equi ( T, C = Z tr ( T ) C . Thisis because z equi ( T, S ) is the free abelian group generated by closed irreduciblesubvarieties Z of S × T which are quasi-finite and dominant over a connected com-ponent of S and coincides for a proper or projective C − scheme T with Z tr ( T )( S ).We define Suslin-Friedlander equivariant motivic complexes for a C − representation V = a + bσ to be Z SF ( V ) = C ∗ z equi ( A ( V ) , C [ − a − bσ ] . This is a complex of sheaves in the equivariant Nisnevich topology on C Sm/k .The next theorem states that Suslin-Friedlander equivariant motivic complexescompute Bredon motivic cohomology.
Theorem 3.1.
Let k be a perfect field and V a C -representation. Then there isa quasi-isomorphism of complexes of sheaves in the equivariant Zariski topology on C Sm/k , Z ( V ) ≃ Z SF ( V ) . In particular H a + pσ,b + qσ ( X, Z ) = H aC Nis ( X, Z SF ( V )[ pσ ]) = H aC Zar ( X, Z SF ( V )[ pσ ]) for any C -representation V = b + qσ .Proof. Let V = b + qσ a C -representation. As in the nonequivariant case (Theorem16.8 [9]) we let F V ( U ) = the free abelian group generated by the equivariant 0-equidimensional (over U ) closedsubvarieties on U × A ( V ) that doesn’t touch U × . We have that F V ( U ) ⊂ z equi ( A ( V ) , C ( U ). MIRCEA VOINEAGU
We have a commutative diagram in the category of presheaves with equivarianttransfers:0 / / Z tr ( P ( V ⊕ \ { } ) C / Z tr ( P ( V )) C / / (cid:15) (cid:15) Z tr ( P ( V ⊕ C / Z tr ( P ( V )) C / / (cid:15) (cid:15) coker / / r (cid:15) (cid:15) / / F V / / z equi ( A ( V ) , C / / coker / / . The vertical maps are injective by construction. The middle map is injective be-cause we have an exact sequence0 → Z tr ( P ( V )) C → Z tr ( P ( V ⊕ C → z equi ( A ( V ) , C . We see that coker ( U ) is free abelian on equivariant correspondences Z ⊂ U × P ( V ⊕
1) which touch U × { } and coker ( U ) is free abelian on the equivariantclosed 0-equidimensional W ⊂ U × A ( V ) which touch U × { } . But the middle mapis injective on these generators so it implies that r is injective.We can prove that the map r is surjective on semilocal affine Hensel schemes witha single closed orbit. Let S be such a scheme and Z ⊂ S × A ( V ) a 0 − equidimensionalclosed equivariant over S . This means that Z is quasi-finite over S . This impliesthat Z/C → S/C is quasi-finite over the local affine Hensel scheme S/C . Thisimplies that the quotient Z/C = Z ′ ⊔ Z ′ is a disjoint union of a closed subscheme Z ′ which is finite and surjective over S/C and a closed subscheme Z ′ which doesn’tcontain any point over the closed point of the local Hensel scheme. Moreover Z ′ doesn’t intersect S/C × { } (see Lemma 16.11 [9]). Let p : Z → Z/C be thecanonical quotient map which is finite and surjective and Z = p − ( Z ′ ), Z = p − ( Z ′ ). This implies that Z = Z ⊔ Z . We also have that Z is a closed C -equivariant subscheme which doesn’t intersect S × { } and Z is a C − equivariantclosed subscheme which is finite and surjective over S .This implies that Z is an C − equivariant cycle in F V ( S ). It is also clear that Z comes from Z tr ( P ( V ⊕ C / Z tr ( P ( V )) C and Z = Z in coker . This implies r is surjective on semilocal affine Hensel schemes with a single closed orbit.We have that C ∗ coker ≃ C ∗ coker is a quasi-isomorphism of sheaves in equi-variant Zariski topology (see Proposition 3.2 below).We notice that P ( V ⊕ \ { } ֒ → P ( V ⊕
1) is C − equivariantly A − homotopyequivalent to the inclusion P ( V ) ֒ → P ( V ⊕
1) (see Page 5 [8]) so C ∗ ( Z tr ( P ( V ⊕ \ { } ) C / Z tr ( P ( V )) C )is a chain contractible complex of presheaves.To conclude the theorem, we notice that C ∗ F V is also a chain contractible com-plex of presheaves. Let H be the following map H X : F V ( X ) → F V ( X × A ) (withtrivial action on A ) given by the equivariant pull-back of cycles f : X × A ( V ) × A → X × A ( V ), f ( x, r, t ) = ( x, rt ) which is flat on X × A ( V ) \{ } . Then F V ( i ) ◦ H X = id and F V ( i ) ◦ H X = 0 which implies that C ∗ F V is a chain contractible complex. Here i is the inclusion i : X × A ( V ) → X × A ( V ) × A with i ( x, t ) = ( x, t,
1) andsimilarly i : X × A ( V ) → X × A ( V ) × A with i ( x, t ) = ( x, t, A - equivalence in order to prove thatcertain complexes of presheaves are chain contractible complexes.Using that C ∗ is an exact functor and using the five lemma we conclude that C ∗ Z tr ( P ( V ⊕ C / Z tr ( P ( V )) C ≃ C ∗ z equi ( A ( V ) , C . BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 9 (cid:3)
Proposition 3.2.
Let k be a perfect field. Let F be a presheaf with equivarianttransfers such that F eNis = 0 . Then C ∗ F eNis ≃ and C ∗ F eZar ≃ .Proof. We have that C ∗ F eNis ≃ H i ) eNis = 0 where H i = H i ( C ∗ F eNis ) = 0for any i. This is obviously true for i ≤
0, where for i = 0 we used the condition F eNis = 0. We prove by induction that ( H i ) eNis = 0 for any i . Suppose ( H j ) eNis =0 for any j < i . This means that the good truncation at level i , τ ( C ∗ F eNis ) is quasi-isomorphic to C ∗ F eNis . We also have that the presheaf H i is homotopy invariantand then ( H i ) eNis is homotopy invariant (see Theorem 1.2. [7]). This implies that Hom D − eNis ( k ) (( C ∗ F ) eNis , ( H i ) eNis [ i ]) = Hom D − eNis ( k ) ( F eNis , ( H i ) eNis [ i ]) = 0 , where D − eNis ( k ) is the derived category of bounded above complexes of equivariantNisnevich sheaves. This implies that the map in D − eNis ( k ), C ∗ F eNis ≃ τ C ∗ F eNis → ( H i ) eNis [ i ]which induces an isomorphism on the i th homology groups is zero. This impliesthat ( H i ) eNis = 0 and the induction is complete.We have that H i ( S ) = 0 for any semilocal Hensel affine with a single closed orbit.In particular H i ( ⊔ j Spec ( E j )) = 0 with a Z / E i finitely generated field extensions of k . For every semilocal affine with a singleclosed orbit S over k and S a Z / F ( S ) ֒ → F ( S ) for every homotopy invariant presheaf F with equivariant transfers(Theorem 7.13, [7]). Taking the intersection of all affine Z / − equivariant opensubsets of S we obtain a disjoint union of finitely generated field extensions of k .We conclude that H i ( S ) = 0 for every semi-local affine with a single closed orbit.Notice that any point on a quasi-projective Z / H i ) eZar ≃ C ∗ F ) eZar ≃ (cid:3) Part of the following proposition is in Proposition 3.15 [8].
Proposition 3.3.
Let X be a smooth scheme with trivial C -action and V = b + qσ a C -representation. Then φ : H a + pσ,b + qσ ( X, Z ) ≃ H a − bNis ( X, Z ( V )[ pσ ]) . Proof.
We also have the functor ( − ) e : ( C Sm/k ) → ( Sm/k ) that forgets the C -action and sends an equivariant Nisnevich cover into a Nisnevich cover of theschemes without action giving a morphism of sites ( − ) e : ( Sm/k ) Nis → ( C Sm/k ) C Nis .This induces a map on cohomology φ : H a + pσ,b + qσ ( X, Z ) → H a − bNis ( X, Z ( V )[ pσ ]).To construct the inverse of this map we look at the inclusion t : ( Sm/k ) → ( C Sm/k ) which gives a morphism of sites t : ( C Sm/k ) C Nis → ( Sm/k ) Nis with t ∗ an exact functor and t ∗ ( Z ( V )[ pσ ]) = Z ( V )[ pσ ] (see Proposition 3.15[8] and Lemma3.19[7]). From the Leray spectral sequence we have that ψ : H a − bNis ( X, Z ( V )[ pσ ]) ≃ H a + pσ,b + qσ ( X, Z ) = H a − beNis ( X, Z ( V )[ pσ ])is an isomorphism; this is the inverse of φ . (cid:3) According to Theorem 5.4 [17] we have that C ∗ z equi ( A ( nk [ C ]) , C = ⊕ n − j = n Z / j )[2 j ] ⊕ Z (2 n )[4 n ] is a decomposition in DM − ( k ), where k is a field of characteristic zero. Here k [ C ] = 1 ⊕ σ is the regular representation of C .We conclude that C ∗ z equi ( A ( nσ ) , C ( n )[2 n ] = ⊕ n − j = n Z / j )[2 j ] ⊕ Z (2 n )[4 n ] so C ∗ z equi ( A ( nσ ) , C = ⊕ n − j =0 Z / j )[2 j ] ⊕ Z ( n )[2 n ] . Proposition 3.4.
Let k be a field of characteristic zero and b, q ≥ . Then H a +2 qσ,b + qσC ( k, Z ) = ⊕ q − j =0 H a +2 j,j + b ( k, Z / ⊕ H a +2 q,b + q ( k, Z ) . Proof.
Using Theorem 3.1 we have that H a +2 qσ,b + qσC ( k, Z ) = H a − b ( k, C ∗ z equi ( A ( V ) , C ) == H a − b ( k, C ∗ z equi ( A ( qσ ) , C ( b )[2 b ]) = H a − b ( k, ⊕ q − j =0 Z / j + b )[2 j +2 b ] ⊕ Z ( q + b )[2 q +2 b ]) == ⊕ q − j =0 H a − b ( k, Z / j + b )[2 j + 2 b ]) ⊕ H a − b ( k, Z ( q + b )[2 q + 2 b ]) == ⊕ q − j =0 H a +2 j,j + b ( k, Z / ⊕ H a +2 q,b + q ( k, Z ) . We used that C ∗ z equi ( A ( b + qσ ) , C ≃ C ∗ z equi ( A ( qσ ) , C ⊗ tr C ∗ z equi ( A b ,
0) = C ∗ z equi ( A ( qσ ) , C ( b )[2 b ]for any b, q ≥ (cid:3) With Z / − coefficients (the same for Z /l ) we have (Theorem 5.7 [17]): C ∗ z ( A ( nσ )) Z / ⊗ Z / ⊕ n − j =0 ( Z / j )[2 j ] ⊕ Z / j )[2 j + 1]) ⊕ Z / n )[2 n ] . This implies the following proposition:
Proposition 3.5.
For any field k of characteristic zero and any b, q ≥ we have H a +2 qσ,b + qσ ( k, Z /
2) = ⊕ q − j =0 ( H a +2 j,j + b ( k, Z / ⊕ H a +2 j +1 ,j + b ( k, Z / ⊕⊕ H a +2 q,q + b ( k, Z / ≃ ⊕ qj =0 H a + jet ( k, µ ⊗ b + j ) . The statement is also valid with Z /l -coefficients.Proof. The first statement follows from V.Voevodsky [14] and the statement with Z /l coefficients follows from M.Rost and V.Voevodsky’s Bloch-Kato theorem [15],[16]. (cid:3) In the case of n = 1 we have C ∗ z equi ( A ( σ )) Z / ⊗ Z / Z / ⊕ Z / ⊕ Z / C ∗ z equi ( A ( σ )) Z / ⊗ Z / k ) = Z / ⊕ Z / ⊕ k ∗ /k ∗ [1] ⊕ Z / . We also havethat C ∗ z equi ( A ( σ )) Z / ≃ Z / ⊕ O ∗ [1]. These remarks are used in Proposition 5.2.Remark that Proposition 3.5 and Proposition 3.4 are valid for any smooth schemewith a trivial action.4. Bredon motivic cohomology of a field in weight 0
In this section we will compute Bredon motivic cohomology in weight less thanor equal to zero of an arbitrary field k .We have the following vanishing theorem for Bredon motivic cohomology groups: Proposition 4.1.
For any smooth C − variety X and any b + q < , b < we havethat H a + pσ,b + qσC ( X, Z ) = 0 . BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 11
Proof.
We use the motivic isotropy cofiber sequence E C → S → e E C and theperiodicity of Bredon motivic cohomology of e E C and E C to obtain H a,b ( e E C × X ) = 0 → H a + pσ,b + qσC ( X ) → H a +2 q +( p − q ) σ,b + qC ( E C × X ) = 0 . This implies the vanishing of the statement. The last equality follows from the factthat H a + pσ,b ( E C × X ) = 0 for any a, p and b <
0. This is the result of usinginductively the l.e.s. H a,b ( E C × X ) = 0 → H a,bC ( E C × X ) = 0 → H a + σ,bC ( E C × X ) → H a +1 ,b ( E C × X ) = 0and the l.e.s. H a − ,b ( E C × X ) = 0 → H a − σ,bC ( E C × X ) → H a,bC ( E C × X ) = 0 → H a,b ( E C × X ) = 0 . (cid:3) Proposition 4.2.
Suppose b + q < . Then H a + pσ,b + qσC ( k, A ) = H a +2 qσ,b + qσC ( k, A ) for any abelian group A . When A = Z / and a > b ≥ or a ≤ − we have that H a + pσ,b + qσC ( k, Z /
2) = H a +2 qσ,b + qσC ( k, Z /
2) = ( H ∗ et ( k, µ ∗ )[ τ, u, v ] / ( u = vτ )) ( a − ,b ) . Proof.
We restrict to the case q < , b ≥ q < b < C → S → S σ and b + q < H a + pσ,b + qσC ( k, A ) = H a +2 qσ,b + qσC ( k, A ) for any p because motivic cohomology is zero in negative weight.From the definition, it follows that H a +2 qσ,b + qσC ( k, A ) = e H aC ( T − qσ , A ( b )). Wehave a cofiber sequence A ( − qσ ) \ { } + → S → T − qσ which implies that H a,b ( k, A ) → H aC ( A ( − qσ ) \ { } , A ( b )) → e H a +1 C ( T − qσ , A ( b )) → H a +1 ,b ( k, A )with H aC ( A ( − qσ ) \ { } , A ( b )) = H a,b ( A ( − qσ ) \ { } /C , A ) because A ( − qσ ) \ { } has a free C -action.We know that H ∗ , ∗ ( A ( − qσ ) \ { } /C , Z /
2) is a H ∗ , ∗ ( k, Z /
2) = H ∗ et ( k, µ ∗ )[ τ ]-module generated by uv i and v i where u is a (1 ,
1) element and v is a (2 ,
1) elementand τ is a (0 ,
1) element (Theorem 6.10 [13]). We also have the relation u = vτ .We can write the equality as H a + pσ,b + qσC ( k, Z /
2) = H a +2 qσ,b + qσC ( k, Z /
2) = ( H ∗ et ( k, µ ∗ )[ τ, u, v ] / ( u = vτ )) ( a − ,b ) , for any b + q < a > b or a ≤ − (cid:3) Proposition 4.3.
We have e H a + pσ, C ( e E C , A ) = 0 for any a, p and any abeliangroup A .Proof. From the periodicities ( σ − , σ −
1) and ( σ − ,
0) of e E C (Proposition 2.9[8]) we have that e H a + pσ, C ( e E C ) = e H a, C ( e E C ). We have from the motivic isotropysequence that0 → e H a, C ( e E C ) → H a, C ( k ) ≃ H a, C ( E C ) → e H a +1 , C ( e E C ) → H a, C ( k ) = H a, ( k ) ≃ H a, C ( E C ) = H a, ( B C ) for any a (which alsoimplies that the left map of the sequence is injective). In the case of a = 0 bothsides of the isomorphism are zero. When a = 0 the isomorphism is given by the map H , ( k, Z ) = Z → H , ( B C , Z ) = Z that sends a generator to a generator.This implies that e H a, C ( e E C ) = 0 for any a . (cid:3) Proposition 4.4.
Let b + q = 0 , b ≤ . Then H a + pσ,b + qσC ( k, Z ) = H a +2 q +( p − q ) σ, C ( k, Z ) . Proof.
We use the motivic isotropy cofiber sequence E C → S → e E C . UsingProposition 4.3 we get for b ≤ e H a,bC ( e E C ) = 0 → H a + pσ,b + qσC ( k ) → H a + pσ,b + qσC ( E C ) → e H a +1 ,bC ( e E C ) = 0so we have H a + pσ,b + qσC ( k ) = H a + pσ,b + qσC ( E C ) = H a +2 q +( p − q ) σ, C ( E C ) from the(2 σ − , σ −
1) periodicity of E C and b + q = 0. This means that H a + pσ,b + qσC ( k ) = H a +2 q +( p − q ) σ, C ( E C ) = H a +2 q +( p − q ) σ, C ( k ) . (cid:3) We will compute below the abelian groups H a + pσ, C ( k, Z ). We have the followingl.e.s. → H a + p, ( k, Z ) → H a + pσ, C ( k, Z ) → H a +( p +1) σ, C ( k, Z ) → H a + p +1 , ( k, Z ) → . It implies that H a + pσ, C ( k, Z ) ≃ H a − ( a +1) σ, C ( k, Z ) for any p ≤ − a − H a + pσ, C ( k, Z ) ≃ H a − ( a − σ, C ( k, Z ) for any p ≥ − a + 1. It is enough to compute thegroups H a − ( a +1) σ, C ( k, Z ), H a − aσ, C ( k, Z ) and H a − ( a − σ, C ( k, Z ) for any a . We havea l.e.s. between these groups for any a → H a − ( a +1) σ, C ( k, Z ) → H a − aσ, C ( k, Z ) → H , C ( k, Z ) →→ H a +1 − ( a +1) σ, C ( k, Z ) → H a +1 − aσ, C ( k, Z ) → . Notice that by definition H a − aσ, C ( k, Z ) = H aC ( k, Z top ( − aσ )), H a +1 − aσ, C ( k, Z ) = H a +1 C ( k, Z top ( − aσ )) and H a − ( a +1) σ, C ( k, Z ) = H aC ( k, Z top (( − a − σ )). Proposition 4.5.
We have that H a − ( a − σ, C ( k, Z ) = 0 for any a ≥ or a = − and Z / if a = 0 . Also H a − ( a +1) σ, C ( k, Z ) = 0 for a ≤ .Proof. The complex Z top ( nσ ) = Z top ( σ ) ⊗ tr ... ⊗ tr Z top ( σ ) with n > − n, ..., − , Sm/k . Here Z top ( σ ) is the complex Z tr ( C ) C → Z tr ( k ) C on thepositions − n = 0 then Z top (0) = Z on the position zero. If n < Z top ( nσ ) = Z top ( − σ ) ⊗ tr ... ⊗ tr Z top ( − σ ) on the positions 0 , ... − n with Z top ( − σ )being the complex Z tr ( k ) C → Z tr ( C ) C on the positions 0 ,
1. This implies that H a − ( a − σ, C ( k ) = H − b +1 C ( k, Z top ( bσ )) = 0 if b ≤ b = − a + 1). It impliesthat H a − ( a − σ, C ( k ) = 0 for a ≥ a = 0 then H σ, C ( k, Z ) = H ( Z top ( σ )( k ))where Z top ( σ )( k ) is the complex Z → Z on the positions − ,
0. It implies that H σ, C ( k, Z ) = Z /
2. If a = − H , C ( k, Z ) → H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) → H , C ( k, Z ) = 0so we conclude that H − σ, C ( k, Z ) = 0. BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 13
We also have that H a − ( a +1) σ, C ( k ) = H − b − C ( k, Z top ( bσ )) = 0 if b ≥ b = − a − H a − ( a +1) σ, C ( k ) = 0 for a ≤ −
1. If a = 0 then H − σ, C ( k, Z ) = H ( Z top ( − σ )( k )) = 0because Z top ( − σ )( k ) is the complex Z id → Z on the positions 0 and 1. If a = 1 thenwe have a sequence0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = 0 → H , C ( k, Z ) = H , ( k, Z ) = Z so we conclude that H − σ, C ( k, Z ) = 0. (cid:3) In the next proposition we will study the cohomology of the complexes Z top ( ± σ )and Z top ( ± σ ). Proposition 4.6.
We have that H a +2 σ, C ( k, Z ) = 0 if a = − , and H σ, C ( k, Z ) = Z / , H − σ, C ( k, Z ) = Z . We also have H − σ, C ( k, Z ) = Z and H a − σ, C ( k, Z ) = 0 for any a = 2 .We have H σ, C ( k, Z ) = Z / , H a + σC ( k, Z ) = 0 for a = 0 and H a − σ, C ( k, Z ) = 0 forany a .Proof. We have that H a − aσ, C ( k ) = H a ( Z top ( − aσ )( k )). If a = − Z → Z on the positions − H σ, C ( k, Z ) = H ( Z top ( σ )( k )) = Z / H a + σ, C ( k, Z ) = 0 for a = 0. If a = 1 then the complex is Z id → Z on the positions0 , H a − σ, C ( k, Z ) = H a ( Z top ( − σ )( k )) = 0 for any a .Let’s compute the differentials of the complex Z top (2 σ ). We have the projections q i : C × C → C for i = 1 , p : C → pt . They induce the maps q i : Z tr ( C × C ) C → Z tr ( C ) C and q ti : Z tr ( C ) C → Z tr ( C × C ) C and p : Z tr ( C ) C → Z tr ( k ) C and p t : Z tr ( k ) C → Z tr ( C ) C . The action of C on C = { , } switchesthe copies of k i.e. 0 → → C × C i.e (0 , → (1 , , → (1 , , → (0 , , → (0 , p : Z tr ( C ) C ( k ) → Z tr ( k )( k ) is p ( a ) = 2 a because the map is induced by p : Z tr ( C )( k ) → Z tr ( k )( k ) which is p ( a, b ) = ( a, b )(1 , T = a + b .The dual p t : Z tr ( k )( k ) → Z tr ( C )( k ) is given by p ( a ) = ( a, a ) which is givenby the transpose matrix (1 , q t : Z tr ( C )( k ) → Z tr ( C × C )( k )we see that is given by q t ( a, b ) = ( a, b, a, b ) = ( a, b )( I , I ) which on fixed pointsgives q t : Z tr ( C )( k ) C → Z tr ( C × C )( k ) C with q t ( a ) = ( a, a ). Looking at thetranspose we obtain that q i : Z tr ( C × C ) → Z tr ( C ), q i ( a, b, c, d ) = ( a + c, b + d ) = ( a, b, c, d )( I , I ) T implying that on fixed points we get q i : Z tr ( C × C ) G → Z tr ( C ) G , q i ( a, b ) = a + b .In conclusion we have the following complex0 → Z tr ( C × C ) C → Z tr ( C ) C ⊕ Z tr ( C ) C → Z tr ( k ) → q , − q ) i.e. g ( a, b ) = ( a + b, − a − b ) andthe second map is f ( a, b ) = 2 a + 2 b = p + p . It implies that H − σ, ( k, Z ) = Z , H − σ, ( k, Z ) = 0 and H σ, ( k, Z ) = H σ, ( k, Z ) = Z / Z top ( − σ ) is the following complex0 → Z tr ( k ) → Z tr ( C ) C ⊕ Z tr ( C ) C → Z tr ( C × C ) G → g ( a ) = ( a, a ) and the second map f ( a, b ) = ( a − b, a − b ).It implies that H − σ, ( k, Z ) = Z and H n − σ, ( k, Z ) = 0 for n = 2. Notice that we can compute the cohomology of the complexes Z top ( ± σ ) directlyfrom the cohomology of the complexes Z top ( ± σ ) (without the need to compute thedifferentials in Z top ( ± σ )). We have the following l.e.s.0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = 0 → H , C ( k, Z ) →→ H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = 0 → . It implies that H − σ, C ( k, Z ) = 0 and H − σ, C ( k, Z ) = Z . In conclusion H a − σ, C ( k, Z ) =0 if a = 2. We have the following l.e.s.0 → H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) → H , C ( k, Z ) = Z →→ H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) → . This implies that H − σ, C ( k, Z ) = Z and H − σ, C ( k, Z ) = 0. We also have H σ, C ( k, Z ) = H σ, C ( k, Z ) = Z / H a +2 σ, C ( k, Z ) = 0 if a = − , (cid:3) We compute below the negative cone of weight 0 Bredon motivic cohomology.
Proposition 4.7. If n = even ≥ then H m − nσ, C ( k, Z ) = Z / ≤ m < n, m = odd Z m = n otherwise . If n = odd ≥ then H m − nσ, C ( k, Z ) = ( Z / ≤ m ≤ n, m = odd otherwise . Proof.
We compute the negative cone of weight 0 Bredon motivic cohomology byinduction.If n = 3 we have a l.e.s0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) f → H , ( k, Z ) → H − σ, C ( k, Z ) → H − σ, ( k, Z ) = 0and because H − σ, C ( k, Z ) = 0 → H , ( k, Z ) = Z ≃ H − σ, C ( k, Z ) = Z → H − σ, C ( k, Z ) = 0we have, according to Proposition 2.4, that f is multiplication by 2. It implies H − σ, C ( k, Z ) = Z / H m − σ, C ( k, Z ) = 0 if m = 3. We also have the sequence0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = Z / → H , ( k, Z ) = Z →→ H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = 0which implies that H − σ, C ( k, Z ) = H − σ, C ( k, Z ) = H − nσ, C ( k, Z ) = Z / n ≥ H m − σ, C ( k, Z ) = H m − σ, C ( k, Z ) = 0 for m ≤ m ≥ H − σC ( k, Z ) = Z . In conclusion H n − σ, C ( k, Z ) = n ≤ , n ≥ Z / n = 3 Z n = 4 . We have 0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = Z f → H , ( k, Z ) = Z → BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 15 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = 0 → and that H , ( k, Z ) = Z ≃ H − σ, C ( k, Z ) = Z is an isomorphism from the abovel.e.s.. We conclude that f is multiplication by 2 (according to Proposition 2.4) so H − σ, C ( k, Z ) = 0 and H − σC ( k, Z ) = Z /
2. It also implies that H m − σ, C ( k, Z ) = 0for any m = 3 , H − σ, C ( k, Z ) = H − σ, C ( k, Z ) = Z / n = 6 and conclude by induction the general case.For n = 6 we have that0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = Z / → H , ( k, Z ) = Z →→ H − σC ( k, Z ) → H − σC ( k, Z ) = 0 . It implies that H − σ, ( k, Z ) ≃ H − σ, ( k, Z ) = Z / H , ( k, Z ) = Z ≃ H − σ ( k, Z ).We conclude that H − σ ( k, Z ) = Z , H − σ ( k, Z ) = Z / H − σ ( k, Z ) = 0, H − σ ( k, Z ) = Z / H m − σ ( k, Z ) = 0 for m ≤ m ≥ n = even ≥ H n − nσ ( k, Z ) = Z , H n − − nσ ( k, Z ) = Z / H n − − nσ ( k, Z ) = 0, H n − − nσ ( k, Z ) = Z / H − nσ ( k, Z ) =0 and H m − nσ ( k, Z ) = 0 for any m ≤ m ≥ n + 1.If n = odd ≥ H n − nσ ( k, Z ) = Z / H n − − nσ ( k, Z ) = 0, H n − − nσ ( k, Z ) = Z /
2, ... , H − nσ ( k, Z ) = Z / H m − nσ ( k, Z ) = 0 if m ≤ m ≥ n + 1. (cid:3) The proof of the above proposition computes completely the negative cone ofBredon motivic cohomology in weight 0 with integer coefficients.The positive cone of Bredon motivic cohomology in weight 0 with integer coef-ficients is completely computed in the next corollary:
Corollary 4.8. If n = even ≥ then H m + nσ, C ( k, Z ) = Z / − n < m ≤ , m = even Z m = − n otherwise . If n = odd ≥ then H m + nσ, C ( k, Z ) = ( Z / − n < m ≤ , m = even otherwise . Proof. If n = 3 we have0 → H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) →→ H , ( k, Z ) = Z g → H − σ, C ( k, Z ) = Z → H − σ, C ( k, Z ) → . We have that0 = H − σ ( k, Z ) → H − σ, C ( k, Z ) = Z ≃ H , ( k, Z ) = Z → H − σ ( k, Z ) = 0so we conclude that g is multiplication by 2. It implies that H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = 0. We also have H σ, C ( k, Z ) = H σ, C ( k, Z ) = Z / H m +3 σ, C ( k, Z ) = 0 if m = − ,
0. When n = 4 we have0 → H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) →→ H , ( k, Z ) = Z → H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) → , which implies that H − σ, C ( k, Z ) = Z , H − σ, C ( k, Z ) = 0, H − σ, C ( k, Z ) = H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = 0 and H σ, C ( k, Z ) = H σ, C ( k, Z ) = Z / H m +4 σ, C ( k, Z ) = Z m = − Z / − ≤ m ≤ , m = even otherwise . In the case of n = 5 we have the following0 → H − σ, C ( k, Z ) = 0 → H − σ, C ( k, Z ) →→ H , ( k, Z ) = Z g → H − σ, C ( k, Z ) = Z → H − σ, C ( k, Z ) → . The map g is multiplication by 2 because0 = H − σ, C ( k, Z ) → H − σ, C ( k, Z ) ≃ H , ( k, Z ) → H − σC ( k, Z ) = 0 . In conclusion H − σ, C ( k, Z ) = 0, H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = H − σ, C ( k, Z ) = 0, H − σ, C ( k, Z ) = H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = 0and H σ, C ( k, Z ) = Z / (cid:3) From the discussion above we also conclude:
Corollary 4.9.
We have H a − aσ, C ( k, Z ) = Z a = even a = odd , a ≤ Z / a = odd , a > . We also have H a − aσ, C ( k, Z /
2) = Z / for any a ≤ , a ≥ and zero for a = 1 . The above theorems conclude that Bredon motivic cohomology of a field with in-teger coefficients in weight 0 coincide (as abstract groups) with Bredon cohomologyof a point with integer coefficients. We also have the following theorem :
Theorem 4.10.
Let k be an arbitrary field. Then with Z / − coefficients we havethat H a + pσ, C ( k, Z /
2) = Z / < a ≤ − p Z / ≤ − a ≤ p otherwise . Bredon motivic cohomology of a field in weights 1 and σ In this section we assume that k is a field of characteristic zero.We have the following Bredon motivic cohomology of e E C in weight 1. Proposition 5.1.
We have e H a + pσ, C ( e E C , Z ) = 0 if a = 3 and e H pσ, C ( e E C , Z ) = Z / .Proof. We have that the motivic isotropy sequence gives a l.e.s0 → e H , C ( e E C ) → H , C ( k, Z ) = k ∗ α → H , C ( E C , Z ) = k ∗ → e H , C ( e E C , Z ) →→ H , C ( k, Z ) = 0 → H , C ( E C , Z ) → e H , C ( e E C , Z ) → H , C ( k, Z ) = 0 . BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 17
The map α : k ∗ = O ∗ ( k ) → k ∗ = O ∗ ( B C ) is the identity and H , C ( E C , Z ) = H , ( B C , Z ) = Z / e H m, C ( e E C , Z ) = 0 if m = 3 and e H , C ( e E C , Z ) = Z /
2. The general result follows from the periodicitiesof e E C . (cid:3) Proposition 5.2. a) H σ,σC ( k, Z ) = Z / , H σ − ,σC ( k, Z ) = k ∗ , H σ + n,σC ( k, Z ) = 0 for any n = 0 , − .b) H σ,σC ( k, Z /
2) = Z / , H σ − ,σC ( k, Z /
2) = Z / ⊕ k ∗ /k ∗ , H σ − ,σC ( k, Z /
2) = Z / and H σ + n,σC ( k, Z /
2) = 0 for any n = 0 , − , − .Proof. From Proposition 3.4 we have H σ,σC ( k, Z ) = H ( z equi ( A ( σ ) , C ( k )) = H ( Z / ⊕ k ∗ [1]) = Z / . With Z / H σ,σC ( k, Z /
2) = H ( z equi ( A ( σ ) , C ( k ) ⊗ Z /
2) = H (( Z / ⊕ k ∗ [1]) ⊗ Z /
2) == H ( Z / ⊕ Z / ⊕ k ∗ [1] ⊗ Z /
2) = H ( Z / ⊕ Z / ⊕ Z / ⊕ k ∗ /k ∗ [1]) = Z / . (cid:3) Proposition 5.3. a) We have H σ,σC ( k, Z ) = Z / and H σ + n,σC ( k, Z ) = 0 for any n = 0 . With Z / − coefficients we have H σ,σC ( k, Z /
2) = Z / and H σ − ,σC ( k, Z /
2) = Z / and H σ + n,σC ( k, Z /
2) = 0 for any n = 0 , − .b) H ,σC ( k, Z ) = k ∗ and H n,σC ( k, Z ) = 0 for any n = 1 ;With Z / − coefficients we have (5.4) H ,σC ( k, Z /
2) = (cid:26) Z / √− = ∅ otherwise and H ,σC ( k, Z /
2) = k ∗ /k ∗ .In the case √− = ∅ we have that k ∗ /k ∗ × ≃ k ∗ /k ∗ = H ,σC ( k, Z / . Also H n,σC ( k, Z /
2) = 0 for n = 0 , .Proof. a) We have the following l.e.s. H , ( k, Z ) = 0 → H σ − ,σC ( k, Z ) → H σ − ,σC ( k, Z ) = k ∗ → H , ( k, Z ) = k ∗ → H σ,σC ( k, Z ) → H σ,σC ( k, Z ) = Z / → H , ( k, Z ) = 0We have that id : H σ − ,σC ( k, Z ) = H C ( k, O ∗ ) = O ∗ ( k ) = k ∗ → H , ( k, Z ) = k ∗ . This is because the connecting map H σ − ,σC ( k, Z ) = H − C ( k, Z ( σ )[2 σ ]) = H − C ( k, O ∗ [1] ⊕ Z / H C ( k, O ∗ ) = k ∗ → H , ( k, Z ) = H σ − ,σC ( C , Z ) = H − C ( C , Z ( σ )[2 σ ]) = H C ( C , O ∗ C ) = k ∗ is the identity map being induced by the diagonal k ∗ → ( k ∗ × k ∗ ) C = k ∗ . It implies from the above l.e.s. that H σ − ,σC ( k, Z ) = 0 and H σ,σC ( k, Z ) ≃ H σ,σC ( k, Z ) = Z / . We also have that0 = H − , ( k ) → H σ − ,σC ( k ) → H σ − ,σC ( k, Z ) = 0which implies H σ − ,σC ( k, Z ) = 0. Similarly H σ + n,σC ( k, Z ) ≃ H σ + n,σC ( k, Z ) = 0 forany n ≥ n ≤ − From the split universal coefficients sequence we have that H σ,σC ( k, Z /
2) = H σ,σC ( k, Z ) ⊗ Z / Z / H σ − ,σC ( k, Z /
2) = H σ,σC ( k, Z ) = Z / . b) We have H , ( k, Z ) = 0 → H ,σC ( k, Z ) → H σ,σC ( k, Z ) = Z / α → H , ( k, Z ) = k ∗ δ → H ,σC ( k, Z ) → H σ +1 ,σC ( k, Z ) = 0 → H , ( k, Z ) = 0 → H ,σC ( k, Z ) → H σ,σC ( k, Z ) = 0 → . It implies that H n,σC ( k, Z ) = 0 for any n = 0 ,
1. The map α is either injective orzero. We prove that the map α is an injective map.We have H C ( S σ , O ∗ C ) = Z / H iC ( S σ , O ∗ C ) = 0 for any i = 1. This isbecause we have H n + σ,σC ( k, Z ) = H nC ( k, ( O ∗ C [1] ⊕ Z / − σ ]) == H nC ( S σ , O ∗ C [1] ⊕ Z /
2) = H n +1 C ( S σ , O ∗ C ) ⊕ H nC ( S σ , Z / . This is zero if n = 0 and Z / n = 0 from point a). We have that H nC ( S σ , Z /
2) = 0for any n because the complexes Z top ( − σ ) have trivial cohomology (see proof ofProposition 4.5). Obviously H C ( S , O ∗ C ) = k ∗ and H nC ( S , O ∗ C ) = 0 for n = 1.The map α is the inclusion map being given by the forgetting action map Z / H C ( S σ , O ∗ C ) → e H C ( C ∧ S σ , O ∗ C ) = H C ( S , O ∗ C ) = k ∗ . The injectivity follows from the following diagram: H C ( k, O ∗ C ) / / r (cid:15) (cid:15) H C ( C , O ∗ C ) / / p (cid:15) (cid:15) H C ( S σ , O ∗ C ) / / α (cid:15) (cid:15) H C ( k, O ∗ C ) / / q (cid:15) (cid:15) H C ( C , O ∗ C ) (cid:15) (cid:15) H C ( C , O ∗ C ) / / e H C ( C ∧ C , O ∗ C ) / / e H C ( C ∧ S σ , O ∗ C ) / / H C ( C , O ∗ C ) / / e H C ( C ∧ C , O ∗ C ) . The map p is injective and the maps r and q are bijectives (Lemma 3.19, [7])which implies that the map α is injective.Because we have a short exact sequence of abelian groups0 → Z / → k ∗ × → k ∗ → H ,σC ( k, Z ) = k ∗ and H ,σC ( k, Z ) = 0.We have H ,σC ( k, Z /
2) = H ,σC ( k, Z ) = all elements x of order 2 in k ∗ .It implies that H ,σC ( k, Z /
2) = Z / − ∈ k ∗ ) or H ,σC ( k, Z /
2) = 0 (if − / ∈ k ∗ ). Also H − ,σC ( k, Z /
2) = H ,σC ( k, Z ) = 0 . With Z / − coefficients we have the l.e.s. → H − ,σC ( k, Z /
2) = 0 → H σ − ,σC ( k, Z /
2) = Z / ≃ H , ( k, Z /
2) = Z / → H ,σC ( k, Z / →→ H σ,σC ( k, Z /
2) = Z / α → H , ( k, Z /
2) = k ∗ /k ∗ δ → H ,σC ( k, Z / → . If − ∈ k ∗ then H ,σC ( k, Z /
2) = Z / α is zero which implies H ,σC ( k, Z /
2) = k ∗ /k ∗ . If − / ∈ k ∗ then H ,σC ( k, Z /
2) = 0 and the map α isinjective and from the short exact sequence0 → Z / → k ∗ /k ∗ → H ,σC ( k, Z / → , BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 19 we have that H ,σC ( k, Z /
2) = k ∗ /k ∗ with the connecting homomorphism δ beingmultiplication by 2. (cid:3) Proposition 5.5. a) H σ, C ( k, Z ) = Z / , H σ +1 , C ( k, Z ) = k ∗ /k ∗ , H σ + n, C ( k, Z ) = 0 for n = 0 , .b) H σ, C ( k, Z /
2) = Z / ⊕ k ∗ /k ∗ , H σ +1 , C ( k, Z /
2) = k ∗ /k ∗ , H σ − , C ( k, Z /
2) = Z / , H σ + n, C ( k, Z /
2) = 0 for n = 0 , − , .Proof. We have a l.e.s. H , ( k, Z ) = 0 → H , C ( k, Z ) = 0 → H σ, C ( k, Z ) → H , ( k, Z ) = k ∗ δ → δ → H , C ( k, Z ) = k ∗ → H σ, C ( k, Z ) →→ H , ( k, Z ) = 0 → H , C ( k, Z ) = 0 → H σ, C ( k, Z ) →→ H , ( k, Z ) = 0 → H , C ( k, Z ) = 0 → H σ, C ( k, Z ) → We obtain H σ + n, C ( k, Z ) = 0 for n = 1 , δ is the multiplication by 2. Accordingto Proposition 2.4 the composition H , C ( k, Z ) → H , ( k, Z ) δ ≃ H , C ( k, Z )is multiplication by 2. We have the following l.e.s.0 → H − σ, C ( k, Z ) → H , C ( k, Z ) → H , ( k, Z ) → H − σ, C ( k, Z ) → . Using the motivic isotropy sequence and the periodicity of Bredon motivic coho-mology of E Z / H − σ, C ( k, Z ) = H − σ, C ( E Z / , Z ) = H σ − ,σC ( E Z / , Z ) = H σ − ,σC ( k, Z ) = 0 . We also have that0 → H − σ, C ( k, Z ) → H − σ, C ( E Z / → Z / → H − σ, C ( k, Z ) = 0with H − σ, C ( E Z /
2) = H σ,σC ( E Z /
2) = H σ,σC ( k, Z ) = Z /
2. It implies that H − σ, C ( k, Z ) =0 and thus the map H , C ( k, Z ) → H , ( k, Z ) is an isomorphism. This implies thatthe connecting map δ is multiplication by 2.We obtain H σ +1 , C ( k, Z ) = k ∗ /k ∗ and H σ, C ( k, Z ) = Z / Z / − coefficients we have that H σ, C ( k, Z /
2) = H σ, C ( k, Z ) ⊗ Z / ⊕ H σ, C ( k, Z ) = Z / ⊕ k ∗ /k ∗ . We also have H σ, C ( k, Z /
2) = H σ, C ( k, Z ) ⊗ Z / ⊕ H σ, C ( k, Z ) = k ∗ /k ∗ and H − σ, C ( k, Z /
2) = H − σ, C ( k, Z ) ⊗ Z / ⊕ H σ, C ( k, Z ) = Z / . The rest of groupsare obviously zero. (cid:3)
In conclusion we have:
Corollary 5.6.
For any field k of char ( k ) = 0 we have H σ,σC ( k, Z ) = Z / , H ,σC ( k, Z ) = k ∗ , H σ, C ( k, Z ) = Z / , H , C ( k, Z ) = k ∗ .With Z / − coefficients we have H σ,σC ( k, Z /
2) = Z / , H ,σC ( k, Z /
2) = k ∗ /k ∗ , H σ, C ( k, Z /
2) = Z / ⊕ k ∗ /k ∗ , H , ( k, Z /
2) = k ∗ /k ∗ . In the following theorem we compute the positive cone of Bredon motivic coho-mology of a quadratically closed field or a euclidian field (of characteristic zero).
Theorem 5.7.
Let k be a quadratically closed field or a euclidian field. If n ≥ iseven, then H nσ − m, C ( k, Z ) = k ∗ m = n − k ∗ /k ∗ − ≤ m < n − , m = odd Z / ≤ m < n − , m = even otherwise . If n ≥ is odd, then H nσ − m, C ( k, Z ) = k ∗ /k ∗ − ≤ m < n − , m = odd Z / ≤ m ≤ n − , m = even otherwise . Proof.
We have the following l.e.s.0 → H σ − , C ( k, Z ) → H , ( k, Z ) = k ∗ α → H σ, C ( k, Z ) = Z / → H σ, C ( k, Z ) → H , ( k, Z ) = 0 . Here H σ, C ( k, Z ) = Z / H σ, C ( k, Z ) → H σ, C ( C , Z ) → H σ, C ( k, Z ) is multiplication by 2 (the vertical maps in the diagrambelow are given by forgetting the action): e H σ, C ( S σ ) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) e H σ, C ( C ∧ S ) α / / (cid:15) (cid:15) e H σ, C ( S σ ) (cid:15) (cid:15) (cid:127) _ (cid:15) (cid:15) e H , M ( S ) / / e H , M ( S ∨ S ) / / e H , M ( S ) . The first upper horizontal map is induced by the fold map C ∧ S σ → S σ and thesecond is induced by the pinch map S σ → C + ∧ S .It implies that the homomorphism α : k ∗ → Z / α (1) = α ( −
1) = 1. If k is quadratically closed (i.e. k = k ) then α is the zero map. If k is a euclidian field (i.e. formally real and k ∗ /k ∗ ≃ Z /
2) it implies that the map α is zero. Indeed suppose that α is surjective. Then H = Ker ( α ) is a subgroupof index 2 of k ∗ that contains k ∗ . It implies that H = k ∗ , but − ∈ H because α ( −
1) = 0. This is contradiction because k is formally real.Then we have H σ, C ( k, Z ) ≃ H σ, C ( k, Z ) = Z / H σ − , C ( k, Z ) = k ∗ . Byinduction we have H nσ, C ( k, Z ) ≃ H σ, C ( k, Z ) = Z / n ≥
1. We also have H nσ +1 , C ( k, Z ) = H σ +1 , C ( k, Z ) = H σ +1 , C ( k, Z ) = k ∗ /k ∗ .According to the sequence H n − s, C ( k, Z ) → H ( n − σ + s, C ( k, Z ) → H nσ + s, C ( k, Z ) → H n + s, ( k, Z )and by induction we obtain that H nσ + s, C ( k, Z ) = 0 if s ≥ n ≥
1. This followsfrom H nσ + s, C ( k, Z ) = 0 if n = 0 and s ≥
2. We also have0 → H σ − , C ( k, Z ) = 0 → H σ − , C ( k, Z ) → H , ( k, Z ) ψ → H σ − , C ( k, Z ) → H σ − , C ( k, Z ) → . BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 21
The map ψ is multiplication by 2 because we have from the above that H σ − , C ( k, Z ) = 0 → H σ − , C ( k, Z ) ≃ H , ( k, Z ) → , − motivic cohomology induced by S ∧ S → S ∧ S ∨ S ∧ S → S ∧ S is given by multiplication by 2. This is shown in the diagrambelow where the vertical maps are forgetting action maps and the lower horizontalcomposition is the multiplication by 2. e H σ, C ( S σ ∧ S ) ∼ = (cid:15) (cid:15) ∼ = / / e H σ, C ( S σ ∧ S ∧ C ) ψ / / (cid:15) (cid:15) e H σ, C ( S σ ∧ S ) ∼ = (cid:15) (cid:15) e H , M ( S ∧ S ) / / e H , M ( S ∧ S ∨ S ∧ S ) / / e H , M ( S ∧ S ) . In conclusion we have H σ − , C ( k, Z ) = Z / H σ − , C ( k, Z ) = k ∗ /k ∗ = H nσ − , C ( k, Z )for any n ≥
3. According to the above sequences we have H σ − s, C ( k, Z ) = H σ − s, C ( k, Z ) = 0for any s ≥
2. We have H σ − s, C ( k, Z ) = H σ − s, C ( k, Z ) = 0 for any s ≥
3. We have0 → H σ − , C ( k, Z ) = 0 → H σ − , C ( k, Z ) → H , C ( k, Z ) → H σ − , C ( k, Z ) → H σ − , C ( k, Z ) → H , C ( k, Z ) = k ∗ → H σ − , C ( k, Z ) = Z / H σ − , C ( k, Z ) = 0 → H σ − , C ( k, Z ) → H , ( k, Z )is injective and that we have the following commutative diagram with the lowerhorizontal composition being multiplication by 2. e H σ − , C ( S σ ) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) e H σ − , C ( C ∧ S ) / / (cid:15) (cid:15) e H σ − , C ( S σ ) (cid:15) (cid:15) (cid:127) _ (cid:15) (cid:15) e H , M ( S ) / / e H , M ( S ∨ S ) / / e H , M ( S ) . Then we have H σ − , C ( k, Z ) = k ∗ and H σ − , C ( k, Z ) = Z /
2. We have H σ − s, C ( k, Z ) = H σ − s, C ( k, Z ) = 0 for any s ≥ n be even. By induction we have H nσ − m, C ( k, Z ) = k ∗ /k ∗ if − ≤ m < n − m odd and H nσ − n +1 , C ( k, Z ) = k ∗ and H nσ − m, C ( k, Z ) = Z / ≤ m < n − m even.Let n be odd. Then H nσ − m, C ( k, Z ) = k ∗ /k ∗ if − ≤ m ≤ n − m odd and H nσ − m, C ( k, Z ) = Z / − ≤ m ≤ n , m even. (cid:3) The negative cone in weight 1 of Bredon motivic cohomology of a formally realfield or a quadratically closed field of characteristic zero is computed below.
Theorem 5.8.
Let k be a formally real field or a quadratically closed field.If n ≥ is even then H m − nσ, C ( k, Z ) = k ∗ m = n + 1 k ∗ /k ∗ < m ≤ n, m = even Z / < m ≤ n, m = odd otherwise . If n ≥ is odd then H m − nσ, C ( k, Z ) = k ∗ /k ∗ < m ≤ n + 1 , m = even Z / < m ≤ n + 1 , m = odd otherwise . If n = 0 then H m, C ( k, Z ) = H m, ( k, Z ) = k ∗ if m = 1 and H m, C ( k, Z ) = 0 if m = 1 .Proof. We have that H m − nσ, C ( k, Z ) = H m, C ( k, Z ) = 0 for any m ≤ n ≥ → H − σ, C ( k, Z ) = 0 → H , C ( k, Z ) = k ∗ ≃ H , ( k, Z ) = k ∗ → H − σ, C ( k, Z ) = 0 → , from the proof of Proposition 5.5. It implies that H m − σ, C ( k, Z ) = 0 for any m . Wehave H − σ, C ( k, Z ) = 0 → H , ( k, Z ) → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = k ∗ and H m − σ, C ( k, Z ) = 0 for m = 3. It also impliesthat H − nσ, C ( k, Z ) = 0 for any n ≥ → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = k ∗ → H , ( k, Z ) = k ∗ → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = k ∗ → H , ( k, Z ) = k ∗ ≃ H − σ, C ( k, Z ) = k ∗ is multiplication by 2 from Proposition 2.4.In conclusion we obtain H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = k ∗ /k ∗ and H n − σ, C ( k, Z ) = 0 for n = 3 , n = 4 we have that H m − σ, C ( k, Z ) = H m − σ, ( k, Z ) for any m ≤ m ≥ → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = k ∗ /k ∗ α → H , ( k, Z ) = k ∗ → H − σ, C ( k, Z ) → . The composition S σ → C ∧ S → S σ gives multiplication by 2 (i.e. zero map)on the composition H − σ, C ( k, Z ) = k ∗ /k ∗ α → H , ( k, Z ) = k ∗ → H − σ, C ( k, Z ) = k ∗ /k ∗ from Theorem 2.4 and the second map is surjective because H , ( k, Z ) = k ∗ → H − σ, C ( k, Z ) = k ∗ /k ∗ → H − σC ( k, Z ) = 0. The map α is zero when k is aquadratically closed field. Suppose k is a formally real field. Then α ( x ) = 1 whichimplies Im ( α ) ⊂ Z /
2. Because Im ( α ) ⊂ k ∗ and k is a formally real field itimplies that α is the zero map.We conclude that H − σ, C ( k, Z ) = k ∗ /k ∗ , H − σ, C ( k, Z ) = H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = k ∗ and H n − σ, C ( k, Z ) = 0 for n ≤ n ≥ n = 5 we have the following sequence0 → H − σ, C ( k, Z ) → H − σ, C ( k, Z ) = k ∗ α → H , ( k, Z ) = k ∗ → H − σ, C ( k, Z ) → . From the above we have that0 → H , ( k, Z ) = k ∗ ≃ H − σ, C ( k, Z ) = k ∗ → H − σ, C ( k, Z ) = 0 BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 23 so by Theorem 2.4 we obtain that α is multiplication by 2. We conclude that H − σ, C ( k, Z ) = Z / H − σ, C ( k, Z ) = k ∗ /k ∗ . We also have H n − σ, C ( k, Z ) = H n − σ, C ( k, Z ) for n = 5 , n > H n +1 − nσ, C ( k, Z ) = k ∗ and H m − nσ, C ( k, Z ) = k ∗ /k ∗ for2 < m ≤ n even and H m − nσ, C ( k, Z ) = Z / < m < n , m odd.If n > H m − nσ, C ( k, Z ) = k ∗ /k ∗ for 2 < m ≤ n + 1, m even and H m − nσ, C ( k, Z ) = Z / < m ≤ n + 1, m odd.The case n = 0 follows from the computation of motivic cohomology of a fieldin weight 1. (cid:3) Corollary 5.9.
Let k be a field of characteristic zero. If k is a euclidian field or aquadratically closed field we have that the positive cone ( n ≥ ) is H nσ − m, C ( k, Z /
2) = k ∗ /k ∗ ⊕ Z / ≤ m ≤ n − Z / ≤ m = nk ∗ /k ∗ m = − < n otherwise . and the negative cone ( − n < ) H m − nσ, C ( k, Z /
2) = k ∗ /k ∗ ⊕ Z / ≤ m ≤ n Z / m = 2 ≤ nk ∗ /k ∗ m = n + 1 , n ≥ otherwise . We conclude that for a quadratically closed field the Bredon motivic cohomol-ogy of weight 1 with Z / − coefficients coincides as abstract groups with Bredoncohomology of a point. For a euclidian field like R we have that k ∗ /k ∗ = Z / σ of a formally realfield or a quadratically closed field is computed below. For a quadratically closedfield it coincides with the negative cone in weight 1 (see Theorem 5.8). It doesn’tcoincide for a formally real field. Theorem 5.10.
Suppose that k is a quadratically closed or a formally real field( char ( k ) = 0 ). Then for n ≥ we have:If n ≥ is even then H m − nσ,σC ( k, Z ) = k ∗ m = n + 1 k ∗ /k ∗ ≤ m ≤ n, m = even Z / < m ≤ n, m = odd otherwise . If n ≥ is odd then H m − nσ,σC ( k, Z ) = k ∗ /k ∗ ≤ m ≤ n + 1 , m = even Z / < m ≤ n, m = odd otherwise . Proof.
We have that H nσ + m,σC ( k, Z ) = H σ + m,σ ( k, Z ) = 0 from l.e.s. for any n ≥ m ≥
0. We also have H − m, C ( k, Z ) → H − mσ,σC ( k, Z ) → H ( − m +1) σ,σC ( k, Z ) → H − m +1 , C ( k, Z ) . We have H − mσ,σC ( k, Z ) = 0 = H ,σC ( k, Z ) = 0 for any m ≥
0. Also H n − mσ,σC ( k, Z ) =0 for any n ≤ m ≥
0. We have that H , C ( k, Z ) = 0 → H − σ,σC ( k, Z ) → H ,σC ( k, Z ) = k ∗ β →→ H , C ( k, Z ) = k ∗ → H − σ,σC ( k, Z ) → H ,σC ( k, Z ) = 0 . The map β is an injective map because H − σ,σC ( k, Z ) = 0. This follows from H − σ,σC ( k, Z ) = H − σ,σC ( E Z / , Z ) = H − σ, C ( E Z / , Z ) and the motivic isotropysequence gives H − σ, C ( E Z / , Z ) = H ,σC ( k, Z ) = 0 → Z / → H − σ, C ( k, Z ) = Z / → H − σ, C ( E Z / , Z ) → . It implies that H − σ, C ( E Z / , Z ) = 0.In conclusion β is injective and H − σ,σC ( k, Z ) = k ∗ /k ∗ . We conclude that H n − σ,σC ( k, Z ) = k ∗ /k ∗ for n = 2 and zero otherwise. We also notice that H − σ,σC ( k, Z ) ≃ H − nσ,σC ( k, Z ) = 0for any n ≥ → H − σ,σC ( k, Z ) → H − σ,σC ( k, Z ) = k ∗ /k ∗ τ → H , C ( k, Z ) = k ∗ → H − σ,σC ( k, Z ) → H − σ,σC ( k, Z ) = 0so H − σ,σC ( k, Z ) = k ∗ /k ∗ and H − σ,σC ( k, Z ) = k ∗ because the map τ is the zeromap because k is either a formally real field or a quadratically closed field.We also have H n − σ,σC ( k, Z ) = 0 for any n = 2 , → H − σ,σC ( k, Z ) → H − σ,σC ( k, Z ) = k ∗ τ → H , C ( k, Z ) = k ∗ → H − σ,σC ( k, Z ) → τ is the multiplication by 2 as above so H − σ,σC ( k, Z ) = Z / H − σ,σC ( k, Z ) = k ∗ /k ∗ . Also H − σ,σC ( k, Z ) = H − σ,σC ( k, Z ) = k ∗ /k ∗ and H n − σ,σC ( k, Z ) =0 if n = 2 , , n = 4 we have0 → H − σ,σC ( k, Z ) → H − σ,σC ( k, Z ) = k ∗ /k ∗ τ → H , C ( k, Z ) = k ∗ → H − σ,σC ( k, Z ) → H − σC ( k, Z ) = 0so H − σ,σC ( k, Z ) = k ∗ /k ∗ and H − σ,σC ( k, Z ) = k ∗ . This is because the map τ isthe zero map as above. Also H − σ,σC ( k, Z ) = H − σ,σC ( k, Z ) = Z / H − σ,σC ( k, Z ) = H − σ,σC ( k, Z ) = k ∗ /k ∗ and the rest are zero.By induction we have that if n is even then H − nσ,σC ( k, Z ) = k ∗ /k ∗ , H − nσ,σC ( k, Z ) = Z / H ( n − − nσ,σC ( k, Z ) = Z / H n − nσ,σC ( k, Z ) = k ∗ /k ∗ , H n +1 − nσ,σC ( k, Z ) = k ∗ .If n odd then H − nσ,σC ( k, Z ) = k ∗ /k ∗ , H − nσ,σC ( k, Z ) = Z / H n − − nσ,σC ( k, Z ) = k ∗ /k ∗ , H n − nσ,σC ( k, Z ) = Z / H n +1 − nσ,σC ( k, Z ) = k ∗ /k ∗ . (cid:3) In the following theorem we compute the positive cone of Bredon motivic coho-mology of weight σ of a quadratically closed field and a euclidian field. BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 25
Theorem 5.11.
Let k be a quadratically closed field or a euclidian field. Then if n ≥ is even H nσ − m,σC ( k, Z ) = k ∗ m = n − k ∗ /k ∗ ≤ m < n − , m = odd Z / ≤ m < n − , m = even otherwise. . If n ≥ is odd then H nσ − m,σC ( k, Z ) = k ∗ /k ∗ ≤ m < n − , m = odd Z / ≤ m ≤ n − , m = even otherwise . If n = 0 then H ,σ ( k, Z ) = k ∗ and H m,σ ( k, Z ) = 0 for any m = 1 .Proof. From Proposition 3.4 we have that H σ − ,σC ( k, Z ) = k ∗ , H σ,σC ( k, Z ) = Z / H σ + m,σC ( k, Z ) = 0 for m = − ,
0. We have0 → H σ − ,σC ( k, Z ) = 0 → H σ − ,σC ( k, Z ) →→ H , C ( k, Z ) = k ∗ γ → H σ − ,σC ( k, Z ) = k ∗ → H σ − ,σC ( k, Z ) → γ is given by multiplication by 2. This follows from the fact that H σ − ,σC ( k, Z ) → H , C ( k, Z ) is the identity map (Proposition 5.3) and from thefollowing diagram e H σ − ,σC ( S σ ) ∼ = / / ∼ = (cid:15) (cid:15) e H σ − ,σC ( C ∧ S ) γ / / (cid:15) (cid:15) e H σ − ,σC ( S σ ) (cid:15) (cid:15) ∼ = (cid:15) (cid:15) e H , M ( S ) / / e H , M ( S ∨ S ) / / e H , M ( S ) . Then H σ − ,σC ( k, Z ) = Z / H σ − ,σC ( k, Z ) = k ∗ /k ∗ . We also have H σ,σC ( k, Z ) = H σ,σC ( k, Z ) = Z /
2. For n = 4 we have0 → H σ − ,σC ( k, Z ) = 0 → H σ − ,σC ( k, Z ) →→ H , C ( k, Z ) = k ∗ α → H σ − ,σC ( k, Z ) = Z / → H σ − ,σC ( k, Z ) → . The map α is zero for a quadratically closed field or a euclidian field because H σ − ,σC ( k, Z ) = 0 → H σ − ,σC ( k, Z ) = Z / → H , C ( k, Z ) is injective and we havethe following diagram e H σ − , C ( S σ ) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) e H σ − , C ( C ∧ S ) α / / (cid:15) (cid:15) e H σ − , C ( S σ ) (cid:15) (cid:15) (cid:127) _ (cid:15) (cid:15) e H , M ( S ) / / e H , M ( S ∨ S ) / / e H , M ( S ) . This implies that H σ − ,σC ( k, Z ) = k ∗ and H σ − ,σC ( k, Z ) = Z /
2. Also H σ,σC ( k, Z ) = H σ,σC ( k, Z ) = Z / H σ − ,σC ( k, Z ) = H σ − ,σC ( k, Z ) = k ∗ /k ∗ . For n = 5 we have0 → H σ − ,σC ( k, Z ) = 0 → H σ − ,σC ( k, Z ) →→ H , C ( k, Z ) = k ∗ β → H σ − ,σC ( k, Z ) = k ∗ → H σ − ,σC ( k, Z ) → with the map β multiplication by 2 because H σ − ,σC ( k, Z ) = 0 → H σ − ,σC ( k, Z ) ≃ H , ( k, Z ) and we have the following diagram e H σ − ,σC ( S σ ) ∼ = / / ∼ = (cid:15) (cid:15) e H σ − ,σC ( C ∧ S ) β / / (cid:15) (cid:15) e H σ − ,σC ( S σ ) (cid:15) (cid:15) ∼ = (cid:15) (cid:15) e H , M ( S ) / / e H , M ( S ∨ S ) / / e H , M ( S ) . with the lower horizontal composition being multiplication by 2. It implies that H σ − ,σC ( k, Z ) = Z / H σ − ,σC ( k, Z ) = k ∗ /k ∗ . By previous computations weobtain H σ − ,σC ( k, Z ) = Z / H σ − ,σC ( k, Z ) = k ∗ /k ∗ , H σ,σC ( k, Z ) = Z / n is even then H nσ,σC ( k, Z ) = Z / H nσ − ,σC ( k, Z ) = k ∗ /k ∗ , ..., H nσ − n +2 ,σC ( k, Z ) = Z / H nσ − n +1 ,σC ( k, Z ) = k ∗ . If n is odd then H nσ,σC ( k, Z ) = Z / H nσ − ,σC ( k, Z ) = k ∗ /k ∗ , ..., H nσ − n +2 ,σC ( k, Z ) = k ∗ /k ∗ , H nσ − n +1 ,σC ( k, Z ) = Z / n = 0 was discussed in Proposition 5.3. (cid:3) Corollary 5.12.
Let k be a field of characteristic zero and n > . If k is a euclidianfield or a quadratically closed field then the negative cone is H m − nσ,σC ( k, Z /
2) = k ∗ /k ∗ × Z / ≤ m ≤ nk ∗ /k ∗ ≤ m = n + 1 , m = 10 otherwise . and the positive cone is H nσ − m,σC ( k, Z /
2) = k ∗ /k ∗ × Z / ≤ m ≤ n − Z / ≤ m = n, m = 00 otherwise . If n = 0 we have that H ,σC ( k, Z /
2) = ( Z / − ∈ k ∗ − / ∈ k ∗ . and H ,σ ( k, Z /
2) = k ∗ /k ∗ and H n,σ ( k, Z /
2) = 0 for n = 0 , .Proof. Universal coefficients sequence gives a split s.e.s0 → H nσ − m,σC ( k, Z ) ⊗ Z / → H nσ − m,σC ( k, Z / → H nσ − m +1 ,σC ( k, Z ) → . We have H nσ − m,σC ( k, Z / ≃ H nσ − m,σC ( k, Z ) ⊗ Z / ⊕ H nσ − m +1 ,σC ( k, Z ).Suppose n is even. If 1 ≤ m < n − m odd then 0 ≤ m − < n − m − H nσ − m,σC ( k, Z / ≃ k ∗ /k ∗ ⊕ Z / m = n − H nσ − m,σC ( k, Z / ≃ k ∗ ⊗ Z / ⊕ Z / ≃ k ∗ /k ∗ ⊕ Z / ≤ m < n − m even then 1 ≤ m − ≤ n − m − H nσ − m,σC ( k, Z / ≃ Z / ⊕ k ∗ /k ∗ . If 1 ≤ m = n then H nσ − m,σC ( k, Z / ≃ k ∗ ≃ Z / m = 0 then H nσ − m,σC ( k, Z / ≃ Z / n is odd. If 1 ≤ m < n − m odd then 0 ≤ m − < n − m − H nσ − m,σC ( k, Z / ≃ k ∗ /k ∗ ⊕ Z / ≤ m < n − m even then 1 ≤ m − ≤ n − m − BOUT BREDON MOTIVIC COHOMOLOGY OF A FIELD 27 decomposition is in this case H nσ − m,σC ( k, Z / ≃ Z / ⊕ k ∗ /k ∗ . If 1 ≤ m = n thenthe decomposition is H nσ − m,σC ( k, Z / ≃ Z / m = 0 the decomposition is H nσ − m,σC ( k, Z / ≃ Z / H m − nσ,σC ( k, Z / ≃ H m − nσ,σC ( k, Z ) ⊗ Z / ⊕ H m +1 − nσ,σC ( k, Z ) . The result follows from Theorem 5.10. If n ≥ ≤ m ≤ n − m even then 2 < m + 1 ≤ n , m + 1 odd and thedecomposition is H m − nσ,σC ( k, Z / ≃ k ∗ /k ∗ ⊕ Z /
2. If 2 < m < n , m odd then 2 2) = k ∗ /k ∗ ⊕ Z / 2. If 2 < m ≤ n , m odd then the decomposition is H m − nσ,σC ( k, Z / 2) = Z / ⊕ k ∗ /k ∗ . If m = 1 then H m − nσ,σC ( k, Z / 2) = k ∗ /k ∗ and if 1 ≤ m = n + 1 then H m − nσ,σC ( k, Z / 2) = k ∗ /k ∗ . (cid:3) Notice that for a quadratically closed field the Bredon motivic cohomology groups H a + pσ, C ( k, Z / 2) and H a + pσ,σC ( k, Z / 2) are computed as Bredon cohomology group H a,pBr ( pt, Z / σ of a quadratically closed field coincide (this is true also for the negative cones).6. Borel equivariant motivic cohomology in weights , , σ In [8] we extended Edidin-Graham equivariant higher Chow groups to an equi-variant theory represented in the stable equivariant A − homotopy category andbigraded by virtual C − representations. In Proposition 4.1 [8] we showed that CH bC ( X, b − a, A ) ≃ H a,bC ( X × E C , A )for any integers a, b and abelian group A .It implies that Borel equivariant motivic cohomology groups H a + pσ,b + qσC ( X × E C , A ) are an extension of Edidin-Graham equivariant higher Chow groups in-troduced in [3]. For weights 0 , , σ and an abelian group A we have the followingcomputation of the Borel equivariant motivic cohomology of a field: Proposition 6.1. a) H a + pσ, C ( E C , A ) = H a + pσ, C ( k, A ) .b) H a + pσ,σC ( E C , A ) = H a + pσ,σC ( k, A ) .c) H a + pσ, C ( E C , A ) = H a + pσ, C ( k, A ) if a = 2 , .d) H a + pσ, C ( E C , A ) = H a − p +2) σ,σC ( E C , A ) = H a − p +2) σ,σC ( k, A ) if a =2 , .Proof. For a) we have that the equivariant motivic isotropy sequence E C → S → e E C gives the following l.e.s. e H a + pσ, C ( e E C , A ) = 0 → H a + pσ, C ( k, A ) → H a + pσ, C ( E C ) → e H a +1+ pσ, C ( e E C ) = 0 . The same motivic isotropy sequence gives H pσ, C ( E C , A ) → e H pσ, C ( e E C , A ) → H pσ, C ( k, A ) → H pσ, C ( E C ) . with e H pσ, C ( e E C , A ) = Z / e H n + pσ, C ( e E C , A ) = 0 for n = 3 (Proposition5.1).We have from the same isotropy sequence that H a + pσ,σC ( E C , A ) = H a + pσ,σC ( k, A )because of the periodicities of Bredon motivic cohomology of e E C . This impliespart b).We also have a (2 σ − , σ − 1) periodicity for Bredon motivic cohomology groupsof E C . This implies part d). (cid:3) Edidin-Graham higher Chow groups of E C in weight 0 and 1 are CH C ( E C , , Z ) = CH C ( B C ) = Z / H σ,σC ( k, Z ) (from d) above), CH C ( E C , , Z ) = k ∗ = H , C ( k, Z ) (from c) above) and CH C ( E C , , Z ) = Z = H , C ( k, Z ), CH C ( E C , − a, Z ) =0 = H a, C ( k, Z ), for a = 0 (from a) above).Notice that the canonical map H a + pσ,b + qσ ( k, Z ) → H a + pσ,b + qσ ( EC , Z ) is notnecessarily an isomorphism (for example for a = 2, b = 1, q = 0, p = 0 the map is0 → Z / References [1] P. Deligne. Lectures on motivic cohomology. arXiv:0805.4436 , 2000.[2] Daniel Dugger. An Atiyah-Hirzebruch spectral sequence for KR -theory. K -Theory , 35(3-4):213–256 (2006), 2005.[3] D. Edidin and W. Graham. Equivariant intersection theory. Invent. Math. , 131(1):595–634,1998.[4] Ostvaer P.A Heller J., Krishna A. Motivic homotopy theory of group scheme actions. Journalof Topology , 8(4):1202–1236, 2015.[5] P. Herrmann. Equivariant motivic homotopy theory. Ph.D Thesis , 2013.[6] P. Hu, I. Kriz, and K. Ormsby. The homotopy limit problem for Hermitian K-theory, equi-variant motivic homotopy theory and motivic Real cobordism. Adv. Math. , 228(1):434–480,2011.[7] P.A. Ostvaer J. Heller and M. Voineagu. Equivariant cycles and cancellation for motiviccohomology. Doc. Math , 2015.[8] P.A. Ostvaer J. Heller and M. Voineagu. Topological comparisons for bredon motivic coho-mology. Trans AMS , 2019.[9] Carlo Mazza, Vladimir Voevodsky, and Charles Weibel. Lecture notes on motivic cohomology ,volume 2 of Clay Mathematics Monographs . American Mathematical Society, Providence, RI,2006.[10] Thomasson. Equivariant algebraic vs. topological k-homology atiyah-segal-style. Duke MathJ , 56(3):589–636, 1988.[11] B. Totaro. Milnor K-theory is the simplest part of algebraic K-theory. K-theory , 6, 1992.[12] B. Totaro. Chow ring of a classifying space. Proc. Symp. Pure Math. (K-Theory, 1997) , 1997.[13] V. Voevodsky. Reduced power operations in motivic cohomology. Publications Math´ematiquesde l’Institut des Hautes ´Etudes Scientifiques , 98(1):1–57, 2003.[14] Vladimir Voevodsky. Motivic cohomology with Z / Publ. Math. Inst. Hautes´Etudes Sci. , (98):59–104, 2003.[15] V.Voevodsky. On motivic cohomology with z/l coefficients. Annals of Maths , 174:401–438,2011.[16] C. Weibel. Patching the norm residue isomorphism. Journal of Topology , 2(2):346–372, 2009.[17] Z.Nie. Karoubi’s construction for motivic cohomology operations. Amer. J. Math. ,130(3):713–762, June 2008. E-mail address : [email protected]@unsw.edu.au