Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic
aa r X i v : . [ m a t h . K T ] F e b FRAMED CORRESPONDENCES AND THE ZEROTH STABLE MOTIVICHOMOTOPY GROUP IN ODD CHARACTERISTIC
ANDREI DRUZHININ AND JONAS IRGENS KYLLING
Abstract.
We prove the finite descent for framed correspondences and Neshitov’s moving lemma overa perfect fields. This allows to extend the results of G. Garkusha and I. Panin on framed motives ofalgebraic varieties [11] to finite base fields, and extend the computation of the zeroth cohomology group H ( Z F (∆ • k , G ∧ nm )) = K MW n , n ≥
0, by A. Neshitov [16] to the case of a perfect field k of odd characteristic. Introduction
Framed correspondences and Morel’s theorem.
In the unpublished notes [20] V. Voevodsky intro-duced the theory of framed correspondences. This theory grew and blossomed into the the theory of framedmotives introduced and developed by G. Garkusha and I. Panin in [11], [12], [1], [13]. The theory of framedmotives gives an explicit fibrant resolution of motivic spectra of smooth algebraic varieties, and in particularof the motivic sphere spectrum. A consequence is the identification of the zeroth motivic homotopy groups π n,n ( S )(pt k ) over an infinite perfect base field k with the zeroth cohomology of the Suslin complex of thepresheaf of stable linear framed correspondences.(1.1) π − n, − n ( S )(pt k ) ≃ H ( Z F (∆ • k , G ∧ nm ))In [16] A. Neshitov computed the right hand side of (1.1) to be Milnor-Witt K -theory when the base fieldhas characteristic zero H ( Z F (∆ • k , G ∧ nm )) ≃ K MW n ( k ) , n ≥ . This recovers a remarkable theorem of F. Morel [18, Theorem 5.40] for fields of characteristic 0.Our work extends the results of [11] to finite fields, and extend Neshitov’s computation [16] to perfectfields k of odd characteristic. This recovers Morel’s theorem for perfect fields of odd characteristic.The assumptions on the base field in this paper are as follows: • In section 2 the base field can be arbitrary though the interesting case is only finite fields; • In section 4 the base field is assumed to be perfect; • In section 5 the base field is assumed to be perfect of odd characteristic. • In section 6 the base field is assumed to be perfect.1.2.
Additional ingredients and modifications.
The present text is written as a complement to theabove mentioned papers of G. Garkusha, I. Panin, and A. Ananyevskiy and A. Neshitov. We only give newproofs of the statements in [11] and [16] which require the assumptions on the base field being infinite or ofcharacteristic 0. Here is a list of the places that require the stronger restrictive assumptions with respect toour ones, and the list of modifications and additional arguments we use to improve the result.1. ( finite descent ) The reason of the assumption in [11] on the base field to be infinite are some geometricalconstructions of framed correspondences and homotopies used in [12], namely these constructions are neededfor the injectivity and excision isomorphism theorems for stable linear framed presheaves. This leads tothe restrictive assumption in the formations of strictly homotopy invariance theorem [12] and cancellationtheorem [1] for stable linear framed presheaves, and consequently in the main results of the theory. In Section2 we prove a variant of a descent for framed correspondences with respect to a set of coprime extensions.
Mathematics Subject Classification.
With the descent theorem we prove the properties required by [12], [1] and [11] for presheaves over finitefields.2. In Neshitov’s work the assumptions stronger then the perfect fields of characteristic different from twoare required because of the following.2.1. (
Steinberg relation ) Neshitov’s arguments provides the homomorphismΨ ∗ : K MW ∗ ( k ) → H ( Z F (∆ • k , G ∧∗ ))for any field k , char k = 2 ,
3. The assumption char k = 3 is because of the proof of the Steinberg relation in H ( Z F (∆ • k , G ∧ )) [16, Lemma 8.9] that uses a certain curve of degree 3 and traces with respect to extensionsof degree 3, which play an important part of the proof. It is the same curve as in the proof of the Steinbergrelation in motivic cohomology [21], but in the case of motivic cohomology this did not lead to the restrictiveassumption since derivative of the polynomial defining the curve has no effect for Cor -correspondences, andit has for framed ones.In the present text we replace this argument by the reference to the original geometrical proof of theSteinberg relation in π , by Po Hu and Igor Kritz [15] or the alternative one [19]. Let us refer also to [4],where the homomorphism Ψ ∗ is constructed for an arbitrary base scheme.2.2. ( moving lemmas ) The assumption on the base field to be of a characteristic zero and to be infiniteis needed in the moving lemmas [16, Lemma 4.11, Lemma 5.4], which are essential ingredients providing thesurjectivity Ψ ∗ : K MW ∗ ( k ) ։ H ( Z F (∆ • k , G ∧∗ )) . [16, Lemma 4.11] allows to move an element in Z F ∗ (pt k , Y ), with Y ⊂ A nk open, to a correspondence withthe (non reduced) support being sa set of points with the separable residue fields; moving lemma [16, Lemma5.4]. moves an element in Z F ∗ (pt k , pt k ) to a correspondence with the (non reduced) support a disjoint unionof rational points.[16, Lemma 4.11] is the main result of section 4 in [16]. The proof of lemma assumes that the base field isof characteristic zero because of the use of the generic smoothness theorem [14, III, Corollary 10.7] (in [16,Lemma 4.2, Lemma 4.6]). Also it is needed that the field is infinite because of the generic position lemmaabout hypersurfaces in projective space [16, Lemma 4.1].In the article we prove [16, Lemma 4.11] over a perfect fields of an arbitrary characteristic. The strategy ofthe proof is completely different to the original one. For a given framed correspondence the moving processconsists of two parts:Firstly, we move the framing functions to generic position, see Lemma 6.3, but do not modify the supportof the correspondence, nor the framing functions on the first order thickening of the support. Next, in Lemma6.4 we change the framing functions φ i from the n th to 1st function to obtain a so called ( i )- simple linearframed correspondence. Here ( i )- simpleness is a ”continuous version” of the notion of simpleness of framedcorrespondences, see Definition 6.1. In particular a (1)- simple framed correspondence is simple .[16, Lemma 5.4] requires the infiniteness of the base field, since the proof uses the existence of a separablemonic polynomial in one variable with rational roots of arbitrary degree. As shown in lemma 6.9 the originalproof of [16, Lemma 5.4] can easily be modified to cover the case of an arbitrary field. Alternatively, theassumption can be avoided by use of the finite descent theorem of Section 2.1.3. Characteristic two.
The main reason of the assumption the base field is of odd characteristic in thepresent proof is that the construction of the left inverse to the homomorphism Ψ ∗ uses the theory of Chow-Witt cohomology introduced by Barge-Morel [2] and developed by Fasel [9], [10]. Actually the main problemare pushforwards for the complexes C ( X, G n , L ) Z , see [10].If we replace the pushforwards of the complexes C ( X, G ∧ n , L ) Z by the pushforwards of the Rost-Schmidtcomplexes of [18, Chapter 5, Lemma 4.18] then the argument above would give a proof in arbitrary charac-teristic. However this would not be independent of Morel’s proof.With out using of mentioned pushforwards but with using of the traces for Milnor-Witt K-theory along finite separable field extensions the present proof would give a surjective homomorphismK MW n ( k ) ։ H ( Z F (∆ • k , G ∧ nm )).1.4. Other work on framed motives over finite fields.
Similar results on framed motives over finite fields(presented in Section 4) were simultaneously and independently obtained by other authors: In [8, AppendixB] similar results are obtained in terms of the conservativity property of the scalar extension functors. The
RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 3 basic construction on the level of correspondences is close to ours. Also, there is an alternative constructionof the descent map due to Alexey Tsybyshev based on the homomorphism GW ( k ) → Z F (pt k , pt k ).The results on framed motives over finite fields in this paper was submitted to the Arxiv e-Print archivein a preliminary form as the Appendix of [5]. This was to announce the results in the first part of this workwhile the final parts were still work in progress.1.5. Acknowledgment.
The authors are grateful to Ivan Panin for discussions about framed motives overfinite fields and to Alexander Neshitov for discussions about extending his work [16] to fields of positivecharacteristic. The first author thanks the university of Oslo for the hospitality, where the essential part ofthis work was done.1.6.
Notation and conventions.
Throughout the text we work with explicit framed correspondences con-sidered as classes in the category of linear framed correspondences Z F ∗ and framed correspondences of pairs.We refer the reader to [11, Definition 2.1, Definition 8.4], and [12, Definition 3.2, Definition 3.5] for thedefinitions of categories Z F ∗ , Z F ∗ , Z F pr ∗ , and Z F ∗ .We need to extent the categories to the essentially smooth schemes, i.e. schemes that are localisations ofsmooth schemes at a point. We use the following definition. Note that precisely in the text we never workwith the correspondences with the target being except the correspondences defined by regular morphisms ofessentially smooth schemes. Definition 1.1.
An essentially smooth scheme Y is a scheme such that there is a sequence of open embeddings Y i → Y i +1 , i ∈ Z , Y = lim ←− i Y i . Denote by EssSm k the category of essentially smooth schemes.For an essentially smooth scheme Y = lim −→ i Y i and a scheme X define Z F ∗ ( X, Y ) = lim −→ i Z F ∗ ( X, Y i ). Foressentially smooth schemes X = lim −→ i X j , Y = lim −→ i Y i define Z F ∗ ( X, Y ) = lim ←− j lim −→ i Z F ∗ ( X j , Y i ).Denote by Sm pairk the full subcategory in the category of arrows in Sm k spanned by the pairs ( X, U ) ofa smooth scheme X and open subscheme. An open pair of essentially smooth schemes ( Y, V ) is a morphismof essentially smooth schemes V → Y that is limit of a sequence open embeddings V i → Y i with respect toopen embeddings of pairs ( Y i , V i ) → ( Y i +1 , V i +1 ).Define Z F ∗ (( X, U ) , ( Y, V )) = lim ←− j lim −→ i Z F ∗ (( X j , U j ) , ( Y i , V i )) for pairs ( X, U ) = lim ←− j ( X j , U j ) ( Y, V ) =lim −→ i ( Y i , V i ), j, i ∈ Z . Example 1.2.
Any morphism of essentially smooth schemes in
EssSm k or a pair of essentially smoothschemes in the sense of above definition defines a morphism in the categories Z F ∗ and Z F pair ∗ . As notedabove this is only one example of correspondences with the target being an essentially smooth scheme we usein the text.In addition to this list we use the following. Definition 1.3.
Denote by Z F pair ∗ the factor-category of Z F pr ∗ obtained by annihilating of the ideal generatedby identity morphisms id ( X,X ) of pairs ( X, X ), X ∈ Sm k ( X ∈ EssSm k ).Denote by Z F pair ∗ the factor-category of Z F pair ∗ obtained by annihilating of the ideal generated by en-domorphisms [ i ◦ pr ] − [ id ( X,U ) × A ] of the objects ( X, U ) × A , X ∈ Sm k , U ⊂ X open, i : ( X, U ) → ( X, U ) × A is the zero section, pr : ( X, U ) × A → ( X, U ) is the projection. For a morphisms of pairs a ∈ Z F pair ∗ (( X, U ) , ( Y, V )) we denote the class of a in Z F pair ∗ by [ a ].For a pair of morphisms a, b ∈ Z F ∗ ( X, Y ) (or a, b ∈ Z F pair ∗ (( X, U ) , ( Y, V ))) we write a A ∼ b iff [ a ] = [ b ] in Z F ∗ ( X, Y ) (or Z F pair ∗ ( X, Y )).Finally, denote Z F ( − , Y ) = lim −→ i Z F n ( − , Y ), Z F ( − , Y ) = lim −→ i Z F n ( − , Y ), where the limits are withrespect to morphisms σ : Z F n ( − , Y ) → Z F n +1 ( − , Y ). Similarly Z F ( − , ( Y, V )) = lim −→ i Z F n ( − , ( Y, V )), Z F ( − , ( Y, V )) = lim −→ i Z F n ( − , ( Y, V )).
Remark . According to the above definition Z F pair ∗ precisely is the category with objects being pairs( X, U ), X ∈ Sm k , U ⊂ X open, and Z F pair ∗ (( X, U ) , ( Y, V )) is the homology group in the middle term of thecomplex Z F ∗ ( X, V ) → Z F ∗ ( X, Y ) ⊕ Z F ∗ ( U, V ) → Z F ∗ ( U, Y ) . ANDREI DRUZHININ AND JONAS IRGENS KYLLING
Remark . For two pairs (
X, U ), (
Y, V ) the group of morphisms Z F pair ∗ (( X, U ) , ( Y, V )) is equal toCoker( Z F pair ∗ (( X, U ) × A , ( Y, V )) i − i −−−→ Z F pair ∗ (( X, U ) , ( Y, V )) , where i , i : ( X, U ) → ( X, U ) × A are zero and unit sections.The category Z F pair ∗ is not equal to the category Z F pr ∗ in [12], but it is equal to the category Z F . Theonly difference is the notation for morphisms; so in [12] morphisms in Z F are denoted as [[ a ]], but morphismsin Z F pair ∗ we denote by [ a ]. Remark . Finally let us note that in the definitions of framed correspondences we usually mean im-plicitly the inverse image of the regular functions on A n × X to the neighbourhood V . So we mean( V , Z, v ∗ ( φ ) , . . . v ∗ ( φ n ) , g ) writing ( V , Z, φ , . . . φ n , g ) ∈ F r n ( X, Y ), where Z ⊂ A nX , v : V → A nX is an etaleneighbourhood of Z , φ i ∈ k [ A nX ], g : V → Y .2. Finite descent
In this section we prove that linear framed correspondences satisfy a descent property up to A -homotopywith respect to a set of a finite field extensions of co-prime degrees, which we call the finite descent. Themain results are corollaries 2.14 and 2.15. Definition 2.1.
Let
K/k , K = k ( α ) be a separable extension of finite fields. Denote by T K/k = (Spec K, V → A k , f, r : V →
Spec K ) ∈ F r (pt k , Spec K )the framed correspondence, where Spec K is considered as a closed subscheme of A k via the function α , f isthe monic irreducible polynomial of the extension K/k , Spec K = Z ( f ), V is an open subscheme in A K thatis complement V = A K − W to the closed subscheme W ⊂ A K , such that (Spec K ) = W ∐ ∆ K , where ∆ K is the graph of α : Spec K → A k , the ´etale morphism v : V → A k is given by the composition V ֒ → A K → A k and f is considered as a regular function on V under the inverse image along v , finally, r : V →
Spec K isgiven by the projection A K → Spec K . Definition 2.2.
Let Λ l ∈ F r (pt k , pt k ) be the framed correspondence defined by the function x l on A k (i.e.,(0 , A k = A k , x l , A k → Spec k )).Let Λ ′ l ∈ Z F (pt k , pt k ) be the framed correspondence given by hm if l = 2 m and hm + h i if l = 2 m + 1,where h = h i + h− i is the hyperbolic plane. Here h i ∈ F r (pt k , pt k ) is the framed correspondence definedby the function x on A (i.e., Λ ), and h− i ∈ F r (pt k , pt k ) is the framed correspondence defined by thefunction − x on A . Definition 2.3.
Let c = ( Z, V , φ, g ) ∈ F r n ( X, Y ) be an explicit framed correspondence such that
V ⊂ A nX is a Zariski neighbourhood of Z , and the regular functions φ i , i = 1 , . . . n on V are restrictions of globallydefined regular functions on A nX . Then to shorten the notations we often omit writing either the support Z or V , and write c = ( V , φ, g ) or c = ( Z, φ, g ).If moreover, V = A nX , we will write just c = ( φ, g ); or if Y = pt k then we omit the canonical map g andwrite c = ( V , φ ) or c = ( Z, φ ).We denote by h λ i ∈ F r (pt k , pt k ) the framed correspondence given by ( A k , λx ). Lemma 2.4.
The classes of Λ l and Λ ′ l in the group Z F (pt k , pt k ) coincide. For any positive integers l , l and any n , n such that l n − l n = 1 we have [Λ ′ l ◦ Λ ′ l − Λ ′ l ◦ Λ ′ n ] = [ h ( − l n i ] ∈ Z F (pt k , pt k ) .Proof. The first statement is [16, remark 7.8]. For the readers convenience we repeat the proof: Firstly wenote that in the group of linear framed correspondences Z F r (pt k , pt k ) we have( A k , x l (1 + x )) = ( A k − { } , x l (1 + x )) + ( A k − {− } , x l (1 + x )) ∼ Λ l + h ( − l i , where the first equality holds by the definition of the group of linear framed correspondences Z F (pt k , pt k ).Then by induction it follows that [Λ l ] = [ l − P i =0 h ( − i i ] = [Λ ′ l ] ∈ Z F r (pt k , pt k ).The second statement is then straightforward. (cid:3) Lemma 2.5.
For any separable finite extension
K/k we have pr K/k ◦ T K/k A ∼ Λ deg K/k , where pr K/k :Spec K → Spec k . RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 5 Proof.
The claimed equivalence is provided by the homotopy defined by the framed correspondence ( A k × A k , λf + (1 − λ ) x deg K/k , pr ) ∈ F r ( A k , pt k ), where f is the monic polynomial from the definition 2.1, λ denotes the second coordinate on A k × A k , which is the homotopy parameter, and pr : A k × A k → pt k is thecanonical projection. (cid:3) Definition 2.6.
We call by a small precategory
Γ = ( Ob Γ , M or Γ , U Γ , ◦ ) a set of vertices Ob Γ , a set of arrows M or Γ , a subset U Γ ∈ M or Γ × M or Γ , and a map − ◦ − : U → M or Γ such that f ◦ ( g ◦ h ) = ( f ◦ g ) ◦ h for any f, g, h ∈ M or Γ , ( f, g ) , ( g, h ) , ( f ◦ g, h ) , ( f, g ◦ h ) ∈ U . A finite precategory is a precategory Γ with finite sets Ob Γ and M or Γ .A functor of small precategories F : Γ → Γ is a pair of maps F Ob : Ob Γ → Ob Γ and F Mor : M or Γ → M or Γ . We call the functors F with F Ob and F Mor being injective as embedding of (small) precategories.The category of small precategories is a category with objects being small precategories and morphisms beingfunctors of small precategories.
Definition 2.7.
A Γ-diagram in the category F is a pairs of maps f Ob : Ob Γ → Ob F and f Mor : M or Γ → M or F such that for any arrows α , α , α ∈ M or Γ , α = α ◦ α , ( α , α ) ∈ U Γ , we have f Mor ( α ) ◦ f Mor ( α ) = f Mor ( α ). Definition 2.8.
Let Γ ′ → Γ be an embedding of a precategories (in sense of def. 2.6). We say that theembedding is good with respect to descent if and only if for each pair of morphisms γ , γ ∈ Γ such that thecomposite γ ◦ γ is defined, γ or γ is the image of a morphism in Γ ′ . Example 2.9.
1) Any small category Γ is a small precategory with U Γ = M or Γ . 2) Any graph Γ can beconsidered as a small precategory with U Γ = ∅ . 3) Main examples of the embeddings of small precategorieswe will work with are represented by the diagrams Ob f (cid:15) (cid:15) ✤✤✤ f " " ❊❊❊❊❊❊❊❊ Ob f / / Ob Ob g " " ❊❊❊❊❊❊ g / / g (cid:15) (cid:15) Ob g (cid:15) (cid:15) Ob g / / g ②②② < < Ob where the left diagram represents the embedding Γ → Γ with Ob Γ = Ob Γ = { V , V , V } , M or Γ = { f , f } , M or Γ = { f , f , f } , f ◦ f = f , and the embedding Γ → Γ with Ob Γ = Ob Γ = { V , V , V , V } , M or Γ = { g , g , g , g , g } , M or Γ = { g , g , g , g , g , g } , g ◦ g = g , g ◦ g = g , g ◦ g = g , g ◦ g = g . Definition 2.10.
Let Γ be a small precategory.
A weak Γ -diagram in the category Z F ∗ ( k ) is a map γ : Γ → Z F ∗ ( k ) such that for any morphisms α ◦ α = α ∈ M or Γ ( a, c ), we have [ γ ( α ) ◦ γ ( α )] = [ γ ( α )] ∈ Z F ( γ ( a ) , γ ( c )). A weak homotopy Γ -diagram in Z F ∗ ( k ) is a weak Γ-diagram in Z F ∗ ( k ). Precisely it is a map γ : Γ → Z F ∗ ( k )such that for any morphisms α ◦ α = α ∈ M or Γ ( a, c ), we have [ γ ( α ) ◦ γ ( α )] = [ γ ( α )] ∈ Z F ( γ ( a ) , γ ( c )) = Z F ( γ ( a ) , γ ( c )) / ∼ A .Weak (homotopy) Γ-diagrams in the category of pairs Z F pair ∗ ( k ) are defined similarly. Definition 2.11.
Let j : Γ ′ → Γ be an embedding of precategories (in sense of def. 2.6) that is good withrespect to descent (in sense of def. 2.8), and let γ ′ be a Γ ′ -diagram in the category Z F ∗ (or the category ofpairs Z F pair ∗ ).We say that we have the weak lifting property in the category Z F ∗ with respect to j and γ ′ iff there is aweak Γ-diagram in Z F ∗ ( K ) (def. 2.10) that restriction on Γ ′ is γ ′ .Similarly we define weak lifting properties in Z F ∗ , and weak lifting properties in Z F pair ∗ , and Z F pair ∗ withrespect to a Γ ′ -diagram in the category of pairs. Lemma 2.12.
1) For any small (finite) precategory Γ (in sense of def. 2.6) there is a small (finite) pre-category Γ s such that the category of week Γ -diagrams in the category Z F ∗ (or Z F pair ∗ ) is equivalent to thecategory Γ s -diagrams in the category Z F ∗ ( k ) (or Z F pair ∗ ).2) For any small (finite) precategory Γ (in sense of def. 2.6) there is a small (finite) precategory Γ s such that the category of week Γ -diagrams in the category Z F pair ∗ is equivalent to a full subcategory in thecategory of weak Γ s -diagrams in the category Z F ∗ ( k ) , defined by the condition that some arrows in the (week) Γ s -diagram are open embeddings of schemes. ANDREI DRUZHININ AND JONAS IRGENS KYLLING
Proof.
We note the following: (1) An equality [Φ ] = [Φ ] ∈ Z F ∗ ( X , X ) is represented by a morphism Z F ∗ ( X × A , X ). (2) Any morphism in the category of pairs ( X, U ) → ( Y, V ) in the category Z F ∗ ( k ) isrepresented by a commutative square in the category of correspondences(2.1) X ❅❅❅❅ / / YU O O / / V O O An equality in the category of pairs is equivalent to the existence of the diagonal in the square above.The lemma is proven in two steps: 1) Firstly, using observation (1) we can replace all equalities in thediagram Z F pair ∗ ( k ) of the form γ ( α ) ◦ γ ( α ) = γ ( α ◦ α ) by equalities in Z F pair ∗ ( k ). Each source vertex( X, U ) of the arrow α is replaced by the triple ( X, U ) × → ( X, U ) × A ← ( X, U ) ×
1. We denote theresulting diagram by γ ′ .2) Next we replace each vertex ( X, U ) of γ ′ by the pair of vertices X ← U , we replace each arrow by asquare of the form (2.1), and add a diagonal arrow to the square for each relation of the form γ ′ ( α ) ◦ γ ′ ( α ) = γ ′ ( α ◦ α ). (cid:3) Lemma 2.13.
Let Γ ′ → Γ be an embedding of precategories (in sense of def. 2.6) that is good with respectto descent (def. 2.8), and let γ ′ be a Γ ′ -diagram in the category Z F ∗ (or the category of pairs Z F pair ∗ ).Let K , K be two finite field extensions of a finite field k , such that deg K /k and deg K /k are relativelyprime, i.e., (deg K /k, deg K /k ) = 1 . Suppose that there exist lift of the weak Γ ′ -diagram γ ′ (def. 2.10) toa weak Γ -diagram in Z F ∗ ( K ) (or Z F pair ∗ ( K ) ) for K = K , K .Then there exists a lift of γ ′ to a weak Γ -diagram in Z F ∗ ( k ) (or Z F pair ∗ ( k ) ) (see def. 2.10).Proof. The question on the diagrams in the categories Z F ∗ , the case of Z F pair ∗ follows form the case of Z F ∗ ,the case of Z F pair ∗ by the first point of lemma 2.12. Now consider the case of Z F ∗ , and the case of Z F pair ∗ issimilar.Let l = deg K , l = deg K . Since ( l , l ) = 1 there are integers n , n such that l n − l n = 1. Upto remuneration of K and K we can assume that n , n >
0. Let S = Spec K ∐ Spec K , with inclusions j i : Spec K i ֒ → S, i = 1 ,
2. Define a framed correspondence(2.2) L = h ( − l n +1 i ◦ ( j ◦ T K /k ◦ Λ n − j ◦ T K /k ◦ Λ n ) ∈ Z F (pt k , S ) , and let pr : S → Spec k be the projection. Then Lemma 2.4 and Lemma 2.5 imply that(2.3) [ pr ◦ L ] = [ h ( − l n +1 i ◦ pr ◦ ( j ◦ T K /k ◦ Λ n − j ◦ T K /k ◦ Λ n )] lm . h ( − l n +1 i ◦ (Λ l ◦ Λ n − Λ l ◦ Λ n )] lm .
4= [ σ ] ∈ Z F (pt k , pt k ) . Consider the base change γ ′ S : Z F ∗ ( S ) of the Γ ′ -diagram Γ ′ , where Z F ∗ ( S ) denotes the category of framedcorrespondences over S . By assumption there is a lift of γ ′ S to a Γ-diagram γ S : Γ → Z F ∗ ( S ). Denote by L S the base change of L .To define the required lift γ for any morphism α ∈ Γ that is not a morphism in Γ ′ we put(2.4) γ ( α ) = (cid:0) ( id Y ⊠ pr ) ◦ γ S ( α ) ◦ ( X × L ) (cid:1) · h ( − l n i , where X and Y are varieties (or pairs) that are the source and the target of γ S ( α ), X × L S is the basechange of L S with respect to X → pt k , and · is the external product of correspondences (if α ∈ Γ ′ we put γ ( α ) = γ ′ ( α )). Denote L X = L ⊠ id X , for X ∈ Sm k .Now let α, β ∈ Γ be a pair of morphisms such that the target of α is equal to the source of β . Let X and Y be images of the source and target of α in Z F ∗ ( K ), and let Y and Z be images of the sources and targetof β in Z F ∗ ( K ). Since the embedding Γ ′ → Γ is good with respect to descent, for any such a pair either α ∈ Γ ′ , or β ∈ Γ ′ . RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 7 Suppose β ∈ Γ ′ , then[ γ ( β ) ◦ γ ( α )] = [ γ ′ ( β ) ◦ γ ( α )] = [ γ ′ ( β ) ◦ pr Y ◦ γ S ( α ) ◦ L X ] =[ pr Z ◦ γ ′ S ( β ) ◦ γ S ( α ) ◦ L X ] = [ pr Z ◦ γ S ( β ) ◦ γ S ( α ) ◦ L X ] =[ pr Z ◦ γ S ( β ◦ α ) ◦ L X ] = [ γ ( β ◦ α )]since we have the diagram S × X γ S ( α ) / / S × Y γ S ( β ) γ ′ S ( β ) / / pr Y (cid:15) (cid:15) S × Z pr Z (cid:15) (cid:15) X γ ( α ) / / L X O O Y L Y O O γ ′ ( β ) γ ( β ) / / Z. Now suppose α ∈ Γ ′ , then[ γ ( β ) ◦ γ ( α )] = [ γ ( β ) ◦ γ ′ ( α )] = [ pr Z ◦ γ S ( β ) ◦ L Y ◦ γ ′ ( α )] =[ pr Z ◦ γ S ( β ) ◦ γ ′ S ( α ) ◦ L X ] = [ pr Z ◦ γ S ( β ) ◦ γ ′ S ( α ) ◦ L X ] = [ pr Z ◦ γ S ( β ) ◦ γ S ( α ) ◦ L X ] =[ pr Z ◦ γ S ( β ◦ α ) ◦ L X ] = [ pr Z ◦ γ S ( β ◦ α ) ◦ L X ] = [ γ ( β ◦ α )]since we have the diagram S × X γ S ( α ) γ ′ S ( α ) / / S × Y γ S ( β ) / / pr Y (cid:15) (cid:15) S × Z pr Z (cid:15) (cid:15) X γ ( α ) γ ′ ( α ) / / L X O O Y L Y O O γ ( β ) / / Z. (cid:3) Corollary 2.14.
Let Γ ′ → Γ be an embedding of precategories (in sense of def. 2.6) that is good with respectto descent (def. 2.8), and let γ ′ be a Γ ′ -diagram in the category Z F ∗ ( k ) (or the category of pairs Z F pair ∗ ( k ) ).Suppose that there is an integer N such that for all field extensions K/k of degree deg
K/k ≥ N there isa lift of γ ′ to a weak Γ -diagram (def. 2.10) in the category Z F ∗ ( K ) (or Z F pair ∗ ( K ) ).Then there is such a lift of γ ′ over k .Proof. Consider any two separable extensions K /k , K /k such that deg K , deg K > N , (deg K , deg K ) =1. Then by assumption the required lift of the diagram γ ′ over K and K , so the claim follows from lemma2.13. (cid:3) Corollary 2.15 (Tsybyshev, Panin) . Let Γ ′ → Γ be an embedding of finite precategories (in sense of def.2.6) that is good with respect to descent (def. 2.8), and let γ ′ be a Γ ′ -diagram in the category Z F ∗ ( k ) (or thecategory of pairs Z F pair ∗ ( k ) ).Suppose that for all infinite field extensions K/k there is a lift of γ ′ to a weak Γ -diagram (def. 2.10) inthe category Z F ∗ ( K ) (or Z F pair ∗ ( K ) ).Then there is such a lift of γ ′ over k .In other words, if the weak lifting property (in sense of def. 2.11) with respect to the diagram γ ′ and theembedding Γ ′ → Γ holds in the category Z F ∗ ( K ) (or Z F pair ∗ ( K ) ) for any infinite field extension K/k then itholds in Z F ∗ ( k ) (or Z F pair ∗ ( k ) ).Proof. The claim follows from lemma 2.13 if we consider the towers of extensions of degrees p and q for twodifferent prime numbers, prime to the characteristic.In detail, let K ′ = lim −→ l k ( ξ /p l ) and K ′ = lim −→ l k ( ξ /q l ) be the infinite extensions of k for a pair of differentprime numbers p, q , ( p, char k ) = 1 , ( q, char k ) = 1. Then by assumption there is a lift of the Γ ′ -diagram γ ′ to a weak Γ-diagram. Since the precategory Γ is finite it follows that there is a lift of γ ′ to a weak Γ-diagramover a finite field extensions K = k ( ξ /p l ) and K = k ( ξ /q l ). Now the claim follows from lemma 2.13. (cid:3) ANDREI DRUZHININ AND JONAS IRGENS KYLLING
Lemma 2.16.
Let B ⊂ A nk be a closed subscheme in affine n -space over some field k . Then there is N ∈ Z such that for all field extensions K/k of degree deg
K/k > N there is a K -rational point p ∈ A nK − B K , where B K = B × Spec K .Proof. Let f ∈ k [ t , . . . , t n ] be a function, f (cid:12)(cid:12) B = 0, f = 0. Assume f | A n − K × P = 0 for any P ∈ A k ( K ). Thisis impossible if f is nonzero and deg K/k ≫
0, since the number of rational roots of a nonzero one-variablepolynomial is not greater than its degree. (cid:3) Injectivity and excision theorems for framed presheaves over a field.
In the section we apply the result of the previous section to extend the injectivity and excision theoremsfor homotopy invariant linear stable framed presheaves [, ] to the finite base field case.
Theorem 3.1 (injectivity on local schemes) . For a field k let X ∈ Sm k , x ∈ X be a point, U = Spec( O X,x ) , i : D → X be a proper closed subset. Then there exists an integer N and a morphism r ∈ Z F N ( U, X − D ) such that [ r ] ◦ [ j ] = [ can ] ◦ [ σ NU ] in Z F N ( U, X ) with j : X − D → X the open inclusion and can : U → X thecanonical morphism. Theorem 3.2 (injectivity on affine line) . For a field k let U ⊂ A k be an open subset and let i : V → U be anon-empty open subset. Then there is a morphism r ∈ Z F ( U, V ) such that [ i ] ◦ [ r ] = [ σ U ] ∈ Z F ( U, U ) . Theorem 3.3 (Zariski excision on affine line) . For a field k let U ⊂ A k be an embedding. Let i : V → U be an open inclusion with V non-empty. Let S ⊂ V be a closed subset. Then there are morphisms r ∈ Z F (( U, U − S ) , ( V, V − S )) and l ∈ Z F (( U, U − S ) , ( V, V − S )) such that [ i ] ◦ [ r ]] = [ σ U ] and [ i ] ◦ [ l ] = [ σ V ] in Z F pair (( U, U − S ) , ( U, U − S )) and Z F pair (( V, V − S ) , ( V, V − S )) respectively. Theorem 3.4 (´etale excision) . Let S ⊂ X and S ′ ⊂ X ′ be closed subsets. Let V ′ / / (cid:15) (cid:15) X ′ Π (cid:15) (cid:15) V / / X be an elementary distinguished square with X and X ′ affine k -smooth. Let S = X − V and S ′ = X ′ − V ′ beclosed subschemes equipped with reduced structures. Let x ∈ S and x ′ ∈ S ′ be two points such that Π( x ′ ) = x .Let U = Spec( O X,x ) and U ′ = Spec( O X ′ ,x ′ ) . Let π : U ′ → U be the morphism induced by Π .Under the notation above there is an integer N and a morphism r ∈ Z F N (( U, U − S ) , ( X ′ , X ′ − S ′ )) suchthat [Π] ◦ [ r ] = [ can ] ◦ [ σ NU ] in Z F pairN (( U, U − S ) , ( X, X − S )) , where can : U → X is the canonical morphism.There are an integer N and a morphism l ∈ Z F N (( U, U − S ) , ( X ′ , X ′ − S ′ )) such that [ l ] ◦ [ π ] = [ can ′ ] ◦ [ σ N ′ U ] in Z F pairN (( U ′ , U ′ − S ′ ) , ( X ′ , X ′ − S ′ )) with can ′ : U ′ → X ′ the canonical morphism. Lemma 3.5.
Any theorem of the list 3.1-3.4 states a weak lifting property (in sense of def .2.11) in thecategories Z F ∗ (or Z F pair ∗ ) with respect to some diagram γ ′ : Γ ′ → Z F ∗ (or γ ′ : Γ ′ → Z F pair ∗ ) and anembedding Γ ′ → Γ good with respect to a descent (see def 2.8) for a finite precategories Γ and Γ ′ (in sense ofdef. 2.6). RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 9 Proof.
The list of diagrams follows. Th. 3.1 Th. 3.2
X X − D o o U [ σ N ] O O ; ; ✇✇✇✇✇ U V o o U [ σ ] O O ? ? ⑦⑦⑦⑦ Th. 3.3(
U, U − S ) ( V, V − S ) o o ( V, V − S )( U, U − S ) [ σ ] O O [ r ] ♣♣♣♣♣♣ ( U, U − S ) [ l ] ♣♣♣♣♣♣ ( V, V − S ) [ σ ] O O o o Th. 3.4(
X, X − S ) ( X ′ , X ′ − S ′ ) [Π] o o ( X ′ , X ′ − S ′ )( U, U − S ) [ can ◦ σ N ] O O [ r ] ♥♥♥♥♥♥ ( U, U − S ) [ l ] ♥♥♥♥♥♥ ( U ′ , U ′ − S ′ ) [ can ′ ◦ σ N ] O O [Π] o o Since there is no one pair of composible dashed arrows in the following diagrams all of them are good withrespect to a descent in sense of 2.8. (cid:3)
Proof of theorems 3.1-3.4.
The case of infinite base field for th. 3.1-3.3 is given by [12, th. 2.11, th 2.9, th.2.10]. The th. 3.4 for the case of infinite field case is given by [12, th. 2.13-2.14], and [7, th 3.8] (for thecharacteristic two case for the second claim)Now by lemma 3.5 and corollary 2.15 it follows that if for a field k the theorems 3.1-3.4 hold over anyinfinite field extension of k then theorems 3.1-3.4 hold over k .In detail, by lemma 3.5 any theorem of the list 3.1-3.4 states some weak lifting properties in the categories Z F ∗ (or Z F pair ∗ ). By the results of [12] the properties holds over any infinite field. Then by corollary 2.15any such a lifting property that holds over any infinite field holds over any finite field as well. So the claimfollows. (cid:3) Corollary 3.6 (see theorem 2.15 in [12] for the infinite field case) . Let F be a homotopy invariant stablelinear framed presheaf. Then the following properties holds:1) Under the assumptions of theorem 3.1 the homomorphisms η ∗ F ( U ) → F ( k ( X )) and ( η h ) ∗ F ( U hx ) →F (Spec k ( U hx ))) are injective.2) Under the assumptions of theorem 3.2 the restriction homomorphism F ( U ) → F ( V ) is injective.3) Under the assumptions of theorem 3.3 the homomorphism i ∗ : F ( U − S ) / F ( U ) → F ( V − S ) / F ( V ) isan isomorphism;4) Under the notation of theorem 3.4 the homomorphism pi ∗ : F ( U − S x ) / F ( U ) → F ( U ′ − S ′ x ′ ) / F ( U ′ ) isan isomorphism, where S x = Spec O S,x , S ′ x ′ = Spec O S ′ ,x ′ .Proof. The claim follows form theorems 3.1-3.4 by the same argument as in [12, theorem 2.15]. (cid:3)
Theorem 3.7.
Let X ∈ Sm k , x ∈ X be a point, W = Spec( O X,x ) . Let i : V ⊂ A W be an open subset, W ⊂ V . Then there are morphisms r ∈ Z F pair (( A W , A W − × W ) , ( V, V − × W )) and l ∈ Z F pair (( A W , A W − × W ) , ( V, V − × W )) such that [ i ] ◦ [ r ] = [ σ A W ] and [ l ] ◦ [ i ] = [ σ V ] in Z F pair (( A W , A W − × W ) , ( A W , A W − × W )) and Z F pair (( V, V − × W ) , ( V, V − × W )) respectively.Proof. We follow the scheme of the arguments of [3] or [6] translated to the case of framed correspondences.To prove the first claim we need to construct r ∈ Z F pair (( A W , A W − W ) , ( V, V − W )) , h r ∈ Z F pair (( A W , A W − W ) × λ A , ( A W , A W − W )) such that h r ◦ i = i ◦ r, h r ◦ i = id ( A W , A W − W ) . Consider the following sections: s ∈ Γ( P W × x A , O ( n )) ˜ s ∈ Γ( P W × x A × λ A , O ( n )) s ′ ∈ Γ( P W × x A , O ( n − s (cid:12)(cid:12) P × W × A × = s ˜ s (cid:12)(cid:12) P × W × A × = ( t − xt ∞ ) s ′ s (cid:12)(cid:12) (( P × W ) \ V ) × A = t n ˜ s (cid:12)(cid:12) ∞× W × A × A = t n s ′ (cid:12)(cid:12) ∞× W × A = t n − s (cid:12)(cid:12) × W × A = t n − ∞ ( t − xt ∞ ) ˜ s (cid:12)(cid:12) × W × A × A = t n − ∞ ( t − xt ∞ ) s ′ (cid:12)(cid:12) × W × A = t n − ∞ s ′ (cid:12)(cid:12) Z ( t − xt ∞ ) × W = t n − ∞ where [ t : t ∞ ] denotes coordinates on P . Such sections s and s ′ exist for n large enough by the Serre theorem[14, Theorem 5.2] on sections of a powers of ample bundles, since U is affine or local, and consequently O (1)is ample on P × U × A and P × U × A × A . Having s and s ′ , we then put ˜ s = (1 − λ ) s + λ ( t − xt ∞ ) s ′ .Denote by t = t ∞ /t the coordinate on A = P − ∞ . Define the correspondence r as a correspondence ofpairs given by the pair of explicit framed correspondences( t V × x A , Z ( s ) , s/t n ∞ , pr t ) ∈ F r ( W × x A , t V ) , (( t V − W ) × ( x A − , Z ( s ) × x A ( x A − , s/t n ∞ , pr ′ t ) ∈ F r ( W × ( x A − , t V − W ) , where pr t : t V × x A → t V , pr ′ t : ( t V − × ( x A − → t V − W denote the projections. Let us write in short that r = [( t A × W × x A , Z ( s ) , s/t n ∞ , pr t )] ∈ Z F pair ( W × ( x A , x A − , ( t V , t V − W )) . In a similar way define the correspondence hh r, = [( t A × W × x A × λ A , Z ( e s ) , e s/t n ∞ , pr t )] ∈ Z F pair ( W × ( x A , x A − × λ A , W × ( t A , t A − , where pr t is the projection t A × W × x A × λ A → W × t A .Then the properties of s and ˜ s above implies that h r, ◦ i = i ◦ r ; h r, ◦ i =[( t A × W × x A , Z ( t − x ) , ( t − x ) g, pr t )] + [( t A × W × x A , Z ( g ) , ( t − x ) g, pr t )]where g = s ′ /t n − ∞ ∈ k [ A × A × U ]. By the definition of the correspondences of pairs we see that the secondsummand is trivial in Z F pair ( W × ( A , A − × λ A , W × ( A , A − Z F pair ( W × ( x A , x A − × λ A , W × ( t A , t A − h r, = [( t A × W × x A × λ A , Z ( t − x ) × λ A , ( t − x ) g (1 − λ ) + ( t − x ) , pr t )]using the fact that g (cid:12)(cid:12) Z ( t − x ) = 1. Then h r, ◦ i =[( t A × W × x A , Z ( t − x ) , ( t − x ) g, pr t )] ,h r, ◦ i =[( t A × W × x A , Z ( t − x ) , ( t − x ) , pr t )] = id ( A W , A W \ W ) Thus we put h = h r, + h r, − [( t A × W × x A , Z ( t − x ) , ( t − x ) g, pr t )], and the claim follows.2) To prove the second claim we need to construct l ∈ Z F pair (( A W , A W − W ) , ( V, V − W )) , h l ∈ Z F pair (( V, V − W ) × A , ( V, V − W ))such that h l ◦ i = l ◦ i and h l ◦ i = id ( V,V \ U ) . RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 11 For n large enough similarly as above using the Serre theorem [14, Theorem 5.2] we find the followingsections: s ∈ Γ( [ t : t ∞ ] P W × x A , O ( n )) ˜ s ∈ Γ( [ t : t ∞ ] P x V × λ A , O ( n )) s ′ ∈ Γ( [ t : t ∞ ] P x V , O ( n − s (cid:12)(cid:12) P × V × = f ˜ s (cid:12)(cid:12) P × V × = ( t − xt ∞ ) s ′ s (cid:12)(cid:12) (( P × W ) \ V ) × A = t n ˜ s (cid:12)(cid:12) ( A \ V ) × V × A = t n g (cid:12)(cid:12) ( A \ V ) × A = t n ( t − xt ∞ ) − s (cid:12)(cid:12) × W × A = t − xt ∞ ˜ s (cid:12)(cid:12) × V × A = t − xt ∞ s ′ (cid:12)(cid:12) × V = t n ∞ s ′ (cid:12)(cid:12) Z ( t − xt ∞ ) × W = t n − ∞ , where g = s ′ /t n − ∞ ∈ k [ A × V ].Next, under the same notation as above in the point (1) of the proof define l = ( t A × W × x A , Z ( s ) , s/t n ∞ , pr t ) ∈ Z F pair ( W × ( x A , x A − , ( t V , t V − W )) ,h l, = ( t V × W x V × λ A , Z (˜ s ) × x A V, ˜ s/t n ∞ , pr t ) ∈ Z F pair (( x V , x V − W ) × λ A , ( x V , x V − W )) , where t = t /t ∞ denotes the coordinate on A = P −
0, and pr t : t V × W x V × λ A → t V is the projection.Then the properties of s and s ′ above imply that h l ◦ i = i ◦ l ; h l ◦ i = [( t A × W × x A × λ A , Z ( t − x ) , ( t − x ) g, pr t )]+[( t A × W × x A × λ A , Z ( g ) , ( y − x ) g, pr t )] . The second summand is trivial by the definition of the group Z F ( W × ( A , A − × A , W × ( A , A − g (cid:12)(cid:12) Z ( t − x ) = 1 we can define h r, = [( t V × W x V × λ A , Z ( t − x ) × λ A , ( t − x ) g (1 − λ ) + ( t − x ) , pr t )] ∈ Z F pair (( x V , x V − W ) × λ A , ( t V , t V − . Finally we put h = h r, + h r, − [( t V × W x
V , Z ( t − x ) , ( t − x ) g, pr t )], and then h r ◦ i = l ; h r ◦ i = [( t V × W x
V , Z ( t − x ) , ( t − x ) g, pr t )]+[( t V × W x
V , Z ( g ) , ( t − x ) g, pr t )] − [( t V × W x
V , Z ( t − x ) , ( t − x ) g, pr t )]+[( t V × W x
V , Z ( t − x ) , ( t − x ) , pr t )] = id ( V,V − U ) So the claim is done. (cid:3)
Corollary 3.8 (see corollary 2.16 in [12] for the infinite base field case for the second claim) . Suppose that W ∈ Sm k is an affine scheme or a local scheme, and let V ⊆ V ⊆ A W be a pair of open subschemes suchthat W ∈ V . Let i : V ⊆ V denote the inclusion. Then, for any homotopy invariant stable linear framedpresheaf F , the restriction homomorphism i ∗ induces an isomorphism i ∗ : F ( V \ W ) / F ( V ) ∼ = −→ F ( V \ W ) / F ( V ) . And consequently F ( A W \ W ) / F ( A W ) ∼ = −→ F ( V \ W ) / F ( V ) . where V = ( A W ) W is the local scheme corresponding to the closed point of the subscheme W ⊂ A W .Proof. It follows form the theorem 3.7 that F ( A W − W ) / F ( A W ) ∼ = −→ F ( V ′ \ W ) / F ( V ′ ) , for any open V ′ ⊂ A W , 0 W ⊂ V . Now the first claim follows since we have F ( V \ W ) / F ( V ) ≃ F ( A W − W ) / F ( A W ) ∼ = −→ F ( V ′ \ W ) / F ( V ′ ) . The second claim follows, since F ( V \ W ) / F ( V ) = lim −→ V ′ ⊂ A W , W ⊂ V ′ F ( V ′ \ W ) / F ( V ′ ) . (cid:3) Framed motives over finite fields
In this section we apply the finite descent from the previous section to extend the theory of framed motives[11] to the finite base field case. The results of the theory of framed motives are based on sequence on theoremsabout framed correspondences (and corresponding S -spectra of representable sheaves), and on (pre)sheaveswith framed transfers including the strictly homotopy invariance theorem proven in [12], cancellation theorem[1], and the so called ’cone’ theorem [13]. The assumption on the base field to be infinite is needed in thestrictly homotopy invariance theorem and cancellation theorem. So to extend the results of the theory tothe case if finite fields it is enough to prove the mentioned theorems for the case such fields Let us recall thistheorems.4.1. In [11] the theory of framed motives is constructed over an infinite perfect field of characteristic differentform 2. In this section we explain how the result of the previous section extend the theory to the finite basefield case.The reason of the restrictive assumption on the base field in [11] are the strictly homotopy invariance [12]and cancellation theorems [1], and up to the references to these results the assumptions on the base fieldare not needed. Moreover the proof of the cancellation theorem [1] does not use the infiniteness assumptionexcept the references to [12].In its turn for an arbitrary perfect field the strictly homotopy invariance by the arguments of [12] followsform the injectivity and excision theorems [12, theorems 2.9, 2.10, 2.11, 2.13, 2.14, 2.15]. This means thatthe text of the proofs in [12] uses the infiniteness assumption nowhere except the references to the injectivityand excision theorems [12, theorems 2.9, 2.10, 2.11, 2.13, 2.14, 2.15],.So whenever we have proven the injectivity and excision properties over finite fields, see ths 3.1-3.4 andcorollary 3.6, we have got the strictly homotopy invariance and cancellation theorems consequently the resultson framed motives [11].In what follows we recall the list of steps in the mentioned above reductions and give precise references tothe arguments in the mentioned sources. All the proofs form [12] and [1] we refer to hold word by word inthe general prefect base field case.4.2. Strictly homotopy invariance.Theorem 4.1 (strictly homotopy invariance, see theorem 1.1 in [12] for infinite base fields) . Any homotopyinvariant linear framed σ -stable presheaf F over a perfect field k is strictly homotopy invariant and σ -stable.Proof. As shown in [12, section 16] the claim follows from corollary 3.6 and corollary 3.8. (cid:3)
Remark . Let us list the sequence of lemmas and propositions by which [, ] is proven: Namely, it isreduction is given by the sequences of [12, 16.9 16.8 16.7 16.6 16.4 16.3 16.2] and [12, 16.12 16.4 16.3].4.3.
Cancellation theorem.Theorem 4.3 (linear cancellation theorem, see theorem C in [1] for infinite base fields) . For a prefect basefield k the natural homomorphism of complexes of abelian groups (4.1) Z F ( X × ∆ • , Y ) → Z F ( X × ∆ • ∧ ( G m , , Y ∧ ( G m , is an quasi-isomorphism.Proof. The claim follows from theorems 4.1 and 3.1-3.4 by the arguments as in [1, theorem C]. (cid:3)
RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 13 Remarks on more precise and shorter arguments.
Remark . The infiniteness assumption on the base field in [12] actually is needed only in theorems 2.11,2.13, 2.14, and corollary 2.16. So we could not to reprove theorems 3.2, 3.3 that are [12, th 2.9, th 2.10].
Remark . In the above argument we have extended [12, th 2.9, th. 2.10, th 2.11] to the finite field caseapplying corollary 2.15, and we have gave the independent argument for [12, corollary 5] for the case of suchfields. In the same time let us note that if we reformulate [12, corollary 5] in therms of the category ofcorrespondences of pairs, in a similar form to the theorem 3.7 or [12, theorem 2.12], then [12, corollary 5]could deduced in the case of finite fields form the infinite field case applying corollary 2.15 as well.
Remark . Using corollary 2.14 instead of corollary 2.15 theorems 3.1 and 3.4 could be deduced from theinner results of [12] . Namely we mean some inner statements in the proof of theorems 3.1 and 3.4 in [12].The corollary 2.15 and a short proof above was explained to the authors by I. Panin and A. Tsybyshev.Let us explain this argument. The assumption in the proofs is needed to satisfy some conditions of genericposition. Namely,(1) constructing a relative curve with a “good” compactification in both theorems it is needed to choosesome projection in affine space such that the restriction to some smooth subscheme of codimensionone is ´etale;(2) constructing of the morphism from the ´etale neighbourhood V of the support of the framed corre-spondence to the target Y of the framed correspondence, it is assumed to choose a generic projectionagain;(3) in the injective ´etale excision, when choosing a section of a line bundle on a projective curve thatdoes not satisfy some closed property.All these constructions require to find a k -rational point in a non-empty open subscheme U ⊂ P Nk for some N . By lemma 2.16 such point exists for all enough big field extensions K/k , and so theorem 3.4 holds forall fields K , k ⊂ K , such that such that deg K/k > L for some L ≫
0. Now the case of a finite base field k follows by lemma 2.14. Remark . The finite descent argument allow us to deduce theorem 4.3 over finite base fields from theinfinite base field case directly without references to the inner scheme of the proof in [1]. We show thesurjectivity of the homomorphism 4.1, the injectivity is similar and simpler.Let c ∈ Z F ( X × ∆ i ∧ ( G m , , Y ∧ ( G m , Z F ( X × ∆ • ∧ ( G m , , Y ∧ ( G m , c ′ ∈ Z F ( X × ∆ i , Y ) of the complex Z F ( X × ∆ • , Y ), and an element b ∈ Z F ( X × ∆ i − ∧ ( G m , , Y ∧ ( G m , c = θ ( c ′ ) + d i ( b ). By assumption such elements c ′ K and b K exist over an infinite field extension K/k . As in Corollary 2.15 this implies that there exist c ′ K , b K , c ′ K and b K for a pair of co-prime finite field extensions K /k , and K /k . Now we see that elements c ′ = pr ◦ ( c K ∐ c K ) ◦ ( L ⊠ id X × ∆ i ) and b = pr ′ ◦ ( b K ∐ b K ) ◦ ( L ⊠ id X × ∆ i − ) satisfy the requiredconditions, where pr : Y × S → Y , pr ′ : Y ∧ ( G m , × S → Y ∧ ( G m , S = Spec K ∐ Spec K , and L ∈ Z F (Spec k, Spec K ∐ Spec K ) is defined as in (2.2).4.5. Corollary on zeroth motivic homotopy groups.
By the above all results of [11] hold over anarbitrary perfect field. A particular consequence of [11, theorem 11.7] is the following result on the zerothmotivic homotopy groups
Theorem 4.8.
Let k be a perfect field, then [ pt + , G ∧ nm ] SH • ( k ) ≃ H ( Z F (∆ • k , G ∧ nm )) . Proof of the isomorphism K MW n ≃ H ( Z F (∆ • , G ∧ nm )) . In this section we recall Neshitov’s proof of the isomorphism (5.1) in the case characteristic zero [16], andextend the arguments to obtain a proof for perfect fields of characteristic different from two. Actually, herewe repeat the same arguments as in [16] replacing some references to the results that require alternativeproof. Namely the reference to the proof of Steinberg relation given in [16, subsection 8.3] is replaced bythe reference to [15], [19] or [4]; and the reference to the moving lemma [16, lemma 4.10] is replaced by thelemma 6.4 proven in the next section of the present article.
It is written at the beginning of [16] that throughout the text the base field is of characteristic 0. In thesame time the assumption is used only in few places and many of the arguments works in any characteristicor in any odd characteristic. Let us cite some of original lemmas from [16] indicating what are essentialassumptions on the base field in the proofs for each statement.
Theorem 5.1.
For a perfect field k of characteristic different form there is a graded ring isomorphism (5.1) K MW ≥ ( k ) ≃ H ( Z F (∆ • k , G ∧∗ m )) that takes the symbol [ a , . . . a n ] ∈ K MW n ( k ) , a i ∈ k × , to the class of the correspondences defined by the regularmap ( a , . . . a n ) : pt k → G ∧ nm .Proof. The lemmas 5.2, 5.3 give us the injective ring homomorphism K MW n ( k ) → H ( Z F (∆ • k , G ∧ nm )). Thehomomorphism is surjective by lemma 5.4. (cid:3) Lemma 5.2 (section 6.2, lemma 9.1, section 7 in [16]) . For a perfect field k , char k = 2 , there are homomor-phisms Φ n,k : H ( Z F (∆ • , G ∧ nm )) → K MW n ( k ) , n ≥ , that takes the class of the correspondences defined by the map ( a , . . . a n ) : pt k → G ∧ nm to the symbol [ a , . . . a n ] . Lemma 5.3.
For a field k , char k = 2 , there is a graded ring homomorphism Ψ ∗ ,k : K MW ≥ ( k ) → H ( Z F (∆ • k , G ∧∗ m )) that takes the symbol [ a , . . . a n ] ∈ K MW n ( k ) , a i ∈ k × , to the class of the correspondences defined by the regularmap ( a , . . . a n ) : pt k → G ∧ nm . The arguments of [16, section 7, section 8] proves the claim for a field k , char k = 2, char k = 3. Theassumption char k = 2 is needed for the proof of the relations in GW ( k ) [16, section 7], and the assumptionchar k = 3 for the Steinberg relation [16, subsection 8.2]. Nevertheless theorem 4.8 provides that the originalwork [15], where the Steinberg relation is proven in π , ( S ) over an arbitrary base scheme S , implies therelation in H ( Z F (∆ • k , G ∧ m )) for a prefect k . Thus repeating the arguments replacing the proof of Steinbergrelation by the reference to [15] we get the construction of Ψ ∗ over a perfect fields of odd characteristic. Letus note that as shown in the preprint [4] the required homomorphism of rings Ψ ∗ actually exists over anarbitrary base scheme. Proof of lemma 5.3.
To construct the homomorphism it is needed to prove the relations of the Milnor-WittK-theory ring K MW ∗ ( k ) in H ( Z F (∆ • , G ∧∗ m )). Due to theorem 4.8 this is equivalent to prove the relations inthe ring π ∗ , ∗ s (pt k ).Firstly, the arguments of [16, lemma 7.6] provides the homomorphism Ψ : GW ( k ) → H ( Z F (∆ • , pt k ).The moving lemma 6.9 proven in the next section yields immediate that Ψ is surjective. Then by lemma5.2 it follows that Ψ is an isomorphism.As shown in [16, subsection 8.3] the isomorphism Ψ and the Steinberg relation implies the rest relationsof K MW . In detail, [16, lemmas 8.5, corollary 8.14, lemma 8.15] provides the homomorphism Ψ : K MW1 ( k ) → H ( Z F (∆ • , G ∧ m )). Hence since by [18, remark 3.2] the Steinberg relation defines a factor algebra of thetensor algebra T GW ( k ) (K MW1 ( k )) we get the homomorphism Ψ ∗ . (cid:3) Lemma 5.4 (proposition 9.6 [16] for char k = 0) . The homomorphism Ψ ∗ ,k is surjective for any perfect field k , char k = 2 .Proof. As noted above the case of Ψ follows from the moving lemma proven in the next section 6.9. Thegeneral case follows form lemmas 5.5 and lemma 5.8 in a similar way to [16, proposition 9.6]. (cid:3) Lemma 5.5 (see lemma 9.5 and subsection 3.1 in [16]) . Let k be a field, char k = 2 . Let L/k be a finitefield extension of a field k , char k = 2 . There are a transfer map T r Lk : K MW n ( L ) → K MW n ( k ) given by [18,definition 4.26] and [18, theorem 4.27] and a transfer map tr Lk : H ( Z F (∆ • L , G ∧ nm )) → H ( Z F (∆ • k , G ∧ nm )) such that tr Lk ◦ Ψ n,L = Ψ n,k ◦ T r Lk for all integer n ≥ .Proof. The claim follows similar as in [16, lemma 9.5] using lemmas 5.6 and 5.7 and the Steinberg relation. (cid:3)
RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 15 Lemma 5.6 (Lemma 7.9 [16].) . Let k be a field, char k = 2 . Suppose L/k is separable field extension, [ L : k ] = l is a prime number, and k has no prime-to- l extensions. Then the transfer diagram is commutative K MW0 ( L ) T r Lk (cid:15) (cid:15) Ψ / / H ( Z F (∆ • L , pt L ) tr L/k (cid:15) (cid:15) K MW0 ( k ) Ψ / / H ( Z F (∆ • k , pt k ) where T r Lk is the transfer map homomorphism fro Milnor-Witt K-theory defined in [18, Definition 4.28] .Proof. We repeat the proof from [16].Take L = k ( α ). Since l is prime, we may assume that L = k ( α ). Then tr L/k (Ψ( h α i )) = tr L/k ( h α i ) =( A L , α ( x − α ) , pr k ). Let p ( x ) be the minimal monic polynomial of α . Then ( A L , α ( x − α ) , pr k ) ∼ ( U ′ , xp ′ ( x ) p ( x ) , pr k ), where U ′ ⊂ A L is an open subset does not contain any root of xp ′ ( x ) p ( x ) except α .Take U ⊂ A k to be the image of U ′ . Then tr L/k (Ψ( h α i )) = ( U ′ , xp ′ ( x ) p ( x ) , pr k ) = ( U, xp ′ ( x ) p ( x ) , pr k ) = h xp ′ ( x ) p ( x ) i− ( A k − Z ( p ) , xp ′ ( x ) p ( x ) , pr k ) = h c d x l + d +1 i− ( A k − Z ( p ) , xp ′ ( x ) p ( x ) , pr k ), where d = deg p ′ ( x ) < l ,and c d is the leading coefficient of p ′ ( x ). Since k has no prime-to- l extensions, all roots of p ′ ( x ) are rational,so xp ′ ( x ) = c d ( x − λ ) r ( x − λ g ) r g . Then by [16, Lemma 7.9]Φ ( tr L/k Ψ ( h α i )) = h c d i s ε − (cid:0) g X i =1 ( r i ) ǫ h c d p ( λ i ) λ i Y j = i ( λ i − λ j ) r j i (cid:1) , s = 1 + d + l. Let τ Lk be the geometrical transfer map defined in [18, 4.2]. Then T r Lk ( h α i ) = τ Lk ( α )( h p ′ ( α ) α i ) = − ∂ − /x ∞ ([ xp ′ ( x ) p ( x )] − P gi =1 [( x − λ i ) r i c d p ( λ i ) λ i Q j = i ( λ i − λ j ) r j ]) = h c d i s ǫ − (cid:0) P gi =1 ( r i ) ǫ h c d p ( λ i ) λ i Q j = i ( λ i − λ j ) r j i (cid:1) by [16, Lemma 7.9]. Thus Φ ( tr L/k Ψ ( h α i )) = T r Lk ( α ). Hence the diagram commutes since Ψ ◦ Φ is identity. (cid:3) Lemma 5.7 (lemma 9.3 [16]) . Let k be a field, char k = 2 . Suppose L/k is separable field extension, [ L : k ] = l is a prime number, and k has no prime-to- l extensions. Then is generated as abelian group by GW ( L )K MW1 ( k ) + S , where S = {h± ω ( a ) i [ a ] | a ∈ L × } , where for any a ∈ L × ω ( a ) = p ′ ( a ) and p ′ is aderivative of the minimal polynomial of a .Proof. The proof is the same as for [16, lemma 9.3] just without the equality d + m = l − (cid:3) Lemma 5.8.
For any element in Z F (pt k , G ∧ nm ) the class [ c ] ∈ H ( Z (∆ • k , G ∧ nm )) is equivalent to the class ofsome ˜ c = ∈ Z F (pt k , G ∧ nm ) such that the support of c is smooth.Proof. Any framed correspondences is equivalent to a correspondences in Z F (pt k , G ∧ nm ) that support issmooth, by lemma 6.4 (with i = 0), which is proven in the next section. In other words this means thatthe support is a set of points (with the separable residue fields). Now since any simple correspondences in Z F (pt k , G ∧ nm ) is equivalent to a simple one in Z F (pt k , G ∧ nm ) by [16, Lemma 4.10] the claim follows. (cid:3) Moving lemma
In this section we prove Lemma 5.8. Throughout we assume that k is perfect. Definition 6.1.
Let c = ( Z ⊂ A n , v : V → A n , ϕ = ( ϕ i ) ∈ k [ V ] n , g : V → Y ) ∈ F r n (pt k , Y ) be a framedcorrespondence such that v is an open immersion, and Y ⊂ A e is an open subscheme. Then c is said to be an( i )- simple correspondence for i = 1 , . . . n iff there is a vector of sections ( s j ) j , s j ∈ Γ( P n , O ( d j )), such that1) v ∗ ( s j /t d j ∞ ) (cid:12)(cid:12) Z ( I ( Z ) ) = ϕ j (cid:12)(cid:12) Z ( I ( Z ) ) , s j (cid:12)(cid:12) P n − = t d j j , j = 1 , . . . , n ,2) Z ⊂ P n − B i , where B i = S ≤ j
Suppose k is perfect. Let s = ( s i ) i be a vector of sections s i ∈ Γ( P n , O ( d i )) such that s i (cid:12)(cid:12) P n − = t d i i , and denote Z = Z ( s ) . Then there is a vector of sections ◦ s = ( ◦ s i ) , ◦ s i ∈ Γ( P n , O ( d ′ i )) such that ◦ s i (cid:12)(cid:12) P n − = t d ′ i i , ◦ s (cid:12)(cid:12) Z ( I ( Z ( s )) ) = ( st d ′ i − d i ∞ ) (cid:12)(cid:12) Z ( I ( Z ( s )) ) , and such that Z red ( ◦ s , . . . ◦ s n − ) B n , where B n = [ ≤ i Let c = ( Z ⊂ A n , v : V → A n , φ, g : V → Y ) ∈ F r n (pt k , Y ) for some open Y ⊂ A e . Then forany i , ≤ i ≤ n , there exist c + , c − ∈ F r n (pt k , Y ) , such that c + and c − are ( i ) -simple correspondences, and [ c ] = [ c + ] − [ c − ] ∈ Z F r n (pt k , Y ) .Proof. The proof follows immediately from Lemma 6.5 and Lemma 6.6. (cid:3) Lemma 6.5. Let Y ⊂ A ek be an open subscheme, and c = ( Z ⊂ A n , v : V → A n , φ, g : V → Y ) ∈ F r n (pt k , Y ) .Then there exist c + , c − ∈ F r n (pt k , Y ) , such that c + and c − are ( n ) -simple correspondences, and [ c ] =[ c + ] − [ c − ] ∈ Z F r n (pt k , Y ) .Proof. By Serre’s theorem [14, Theorem 5.2] we can choose integers d i and sections s i ∈ Γ( P n , O ( d i )), i =1 . . . n , s i /t d i ∞ = φ i (cid:12)(cid:12) Z ( I ( Z ) ) , s i (cid:12)(cid:12) P n − = t d i ∞ , where P n − ⊂ P n is the subspace at infinity and t ∞ ∈ O (1), Z ( t ∞ ) = P n − . Similarly we can choose sections e i ∈ Γ( P n , O ( l i )), 1 ≤ i ≤ k , e i /t l i ∞ (cid:12)(cid:12) Z ( I ( Z ) ) = g i (cid:12)(cid:12) Z ( I ( Z ) ) , RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 17 where the g i ’s are the coordinates of the composition V g −→ Y ֒ → A e . The functions λv ∗ ( s i /t d i ∞ ) + (1 − λ )( ϕ i )and λv ∗ ( e i /t l i ∞ ) + (1 − λ ) g i gives a homotopy from c to the framed correspondence( A n − ( Z ( s , . . . s n ) − Z ) , Z, ( s i /t d i ∞ ) , ( e i /t l i ∞ )) . Then applying Lemma 6.3 we can change c in such way that for a new vector of sections ( s i ) we have(6.2) Z red ( s , . . . s n − ) [ ≤ j K > R ( d ), for some R ( d ) ∈ Z , we find a sections s − n ∈ Γ( C, L ( d ′ n )), s − n ∈ Γ( C, O ( d ′ n )) such that Z ( s + ) and Z ( s − ) are smooth, s + (cid:12)(cid:12) D K = ( s · s − ) (cid:12)(cid:12) D K , s − (cid:12)(cid:12) Z ( s ,...s n ) ∪ D = t d ′ n ∞ , and s + , s − are invertible on D K = D × Spec K .Define the correspondences c + K and c − K in F r n (pt K , Y × Spec K ) as c + = ( Z ( s , . . . s n − , s + ) , A n − Z ′ , ( s /t d ∞ , . . . s n − /t d n − ∞ , f + ) ,g ) ,c − = ( Z ( s , . . . s n − , s − ) , A n − Z ′ , ( s /t d ∞ , . . . s n − /t d n − ∞ , f − ) ,g ) . where f + ∈ O ( A n ) is a lift of a regular function s ′ · s + /t d ′ n + d n ∞ ∈ O ( C − ( C ∩ P n − )), and f − ∈ O ( A n ) is alift of s ′ · s · s − /t d ′ n + d n ∞ ∈ O ( C − ( C ∩ P n − )).Thus we see that c + K and c − K are ( n )-simple framed correspondences and [ c K ] = [ c + K ] − [ c − K ] ∈ Z F r n (pt K , Y × Spec K ), where c K is the image of c by base change from k to K/k . Now using the finite descent fromSection 2 we get that [ c ] = [ c + ] − [ c − ] ∈ Z F r n (pt k , Y ) for some ( n )-simple framed correspondences c + and c − . More precisely, we consider a pair of extensions such that (deg K /k, deg K /k ) = (deg K , char k ) =(deg K , char k ). Then in the notation of lemma 2.14 we can define c + = pr ◦ ( c + K ⊕ c + K ) ◦ L , and similarlyfor c − . (cid:3) Lemma 6.6. Let i = 1 . . . n − . Then for any ( i + 1) -simple framed correspondence c = ( Z ⊂ A n , v : V → A n , φ, g : V → Y ) ∈ F r n (pt k , Y ) for some open Y ⊂ A e , there exist c + , c − ∈ F r n (pt k , Y ) , such that c + and c − are ( i ) -simple correspondences, and [ c ] = [ c + ] − [ c − ] ∈ Z F r n (pt k , Y ) .Proof. Consider the closed subscheme of dimension one ˆ Z i = Z red ( s , . . . s i − ) ∩ Z ( s i +1 , . . . s n ) in P n . Wewill prove that ˆ Z i = Sing ˆ Z i .Since c is ( i )-simple Z ⊂ Z red ( s , . . . , s i ) − Sing ( Z red ( s , . . . , s i )), and so any closed point z ∈ Z is asmooth point of Z red ( s , . . . , s i ). Hence dim T = n − i , where T = T z ( Z red ( s , . . . s i )) is the tangent vectorspace. On the other hand, by assumption Z ′ is smooth, where Z ′ = Z red ( s , . . . s i ) ∩ Z ( s i +1 , . . . s n ). Hencethe gradients dφ j of the functions φ j = s j /t d j j , j > i , are linearly independent on T ; and consequently thegradients dφ j , j > i , are linearly independent on the tangent space T z ( Z red ( s , . . . s i − )) ⊃ T . Thus z is asmooth point on ˆ Z i , and hence there is a smooth Zariski neighbourhood of z in ˆ Z i .The rest of the proof is similar to the proof of Lemma 6.5. Namely, let C be the union of the irreduciblecomponents of ˆ Z i that intersect Z i . Then there are line bundles L and L ′ on C , such that L ⊗ L ′ = O ( d i ), and sections s ∈ Γ( C, L ), s ′ ∈ Γ( C, L ′ ), such that Z ( s ) = Z i , Z ( s ′ ) ⊂ Z ′ , where Z ∐ Z ′ = Z ( s . . . s n ). Denote D = C ∩ ( ˆ Z i − C ∪ [ j
K > R for some integer R . The finitedescent of Section 2 finishes the proof as in the previous lemma. (cid:3) Lemma 6.7. Let C be a reduced projective curve over a perfect field k , and D ⊂ C a proper closed subschemesuch that C − D is smooth. Let L be an invertible sheaf on C , O (1) an ample invertible sheaf on C , and r and r ′ invertible sections of L and O on D . Then there exists an integer N such that for all d > N thereexists an integer R ( d ) such that for all field extensions K/k of degree deg k K > R ( d ) , there exists a section s ∈ Γ( C, L ( d )) such that Z ( s ) is smooth and s (cid:12)(cid:12) D = r · r ′ d .Proof. Consider the affine space Γ d ⊂ Γ( C, L ( d )), s ∈ Γ d iff s (cid:12)(cid:12) D = r · r ′ d . Consider the universal section˜ s ∈ Γ( C × Γ d , L ( d )), and its vanishing locus Z (˜ s ), which is a closed subscheme Z (˜ s ) ⊂ C × Γ d . The imageof Supp Ω Z (˜ s ) / Γ d ⊂ Z (˜ s ) under the projection to Γ d is the closed subscheme which parametrizes the set ofsections s such that Z ( s ) ⊂ C is non-reduced. Denote this image by B d and let U d ⊂ Γ d be the complementof B d .To find a section s satisfying the requirements of the lemma is equivalent to finding a rational point in U d for all d greater than some integer N . We want to prove that there exists R such that for all K of degreedeg K/k > R there exists s ∈ Γ d ( K ). Since U d is an open subscheme in an affine space over k , it suffices toshow that there exists some integer N such that for all d > N we have U d = ∅ . At the same time, to provethat U d = ∅ it suffices to prove this over an algebraic closure k/k , that is, ( U d ) k = ∅ .Thus we need to prove that U d = ∅ for all d greater than some N under the assumption that k isalgebraically closed. For an algebraically closed field k the property that Z ( s ) is non-reduced for some s ∈ Γ d means that there is some point p ∈ C , such that s (cid:12)(cid:12) Z ( I ( p ) ) = 0. Since the sections r and r ′ are invertible,we can assume in addition that p ∈ C ′ , where C ′ = C − D . Consider the closed subscheme E d ⊂ Γ d × C ′ , E = { ( s, p ) | s (cid:12)(cid:12) Z ( I ( p ) ) = 0 } . Then B d is the image of E d under the projection to Γ d .We claim that there exists some d ∈ Z such that for each point p ∈ C ′ we have(6.3) codim Γ d ( { s ∈ Γ d | s (cid:12)(cid:12) Z ( I ( p ) ) = 0 } ) = 2 . Then (6.3) implies the lemma. Indeed, if for all p ∈ C ′ , codim Γ d ( s ∈ Γ d | s (cid:12)(cid:12) Z ( I ( p ) ) = 0) = 2, then dim( E d ) ≤ dim( C ′ ) + dim(Γ d ) − 2, and hence codim Γ d ( B d ) = dim(Γ d ) − dim( B d ) ≥ dim Γ d − dim( E d ) ≥ d ∈ Z and for all p ∈ C ′ , the restriction homomorphism r dp : Γ( C, L ( d )) → Γ( Z ( I ( p ) ) ∐ D, L ( d )) is surjective. Consider the scheme C ′ × C as a relative curve over C ′ , and let ∆ ⊂ C ′ × C be the graph of the embedding C ′ ֒ → C . Then the set of points p ∈ C ′ such that r dp is not surjective is equal to W d = Supp pr ∗ (Coker( L ( d ) → j ∗ j ∗ L ( d ))) , where pr : C ′ × C → C ′ , i : C ′ × D → C ′ × C , j : Z → C ′ × C , Z = Z ( I (∆) ) ∪ C ′ × D , and j ∗ and j ∗ arethe direct and inverse images of coherent sheaves. Since O (1) is ample, it follows that for each p ∈ C ′ thereis N such that for all d > N the restriction homomorphism r dp is surjective. So (6.3) follows since C ′ is anoetherian scheme of finite Krull dimension. (cid:3) Lemma 6.8. Let C be a reduced projective curve over a perfect field k , D ∐ D = D ⊂ B ⊂ C be closedsubsets such that C − B is smooth and non-empty; let s ∈ Γ( C, L ) be a section in some invertible sheaf L on C such that s (cid:12)(cid:12) D is invertible; let O (1) be any ample bundle on C with a section t ∈ Γ( C, O (1)) such that Z red ( t ) ⊂ D .Then for all d > N , for some N ∈ Z , for all field extensions K/k, deg k K > R ( d ) , for some R ( d ) , thereexist s + ∈ Γ( C K , L ( d )) , s − ∈ Γ( C K , O ( d )) such that Z ( s + ) and Z ( s − ) are smooth, s + (cid:12)(cid:12) D K = ( s · s − ) (cid:12)(cid:12) D K , s − (cid:12)(cid:12) Z ( s ) ∪ D = t , and s + , s − are invertible on B K = B × Spec K (and consequently on D K = D × Spec K ). RAMED CORR. AND π n,ns ( k ) FOR PERFECT FIELDS OF ODD CHARACTERISTIC 19 Proof. Since D ∩ D = D ∩ Z ( s ) = D ∩ Z ( s ) = ∅ , and since B is a zero-dimensional scheme, it followsthat B splits into a disjoint union of B = B − ( D ∪ Z ( s )) , B = B − ( D ∪ Z ( s ) ∪ B ) ,B = B − ( D ∪ Z ( s ) ∪ B ) , B = B − ( D ∪ B ) . Let r be any invertible sections of O (1) on B , and let w denote any invertible section on L on B ∪ B .Applying Lemma 6.7 to the closed subset B , the line bundle O (1), and the invertible section r ⊕ t d (cid:12)(cid:12) B ∪ B ∪ B , we see that there exists N , such that for all d > N there exists R ( d ) such that for all K/k, deg K/k > R ( d ), there exists a section s − ∈ Γ( C K , O ( d )) such that s − (cid:12)(cid:12) Z ( s ) ∪ D = t d , s − (cid:12)(cid:12) D = r d , andsuch that Z ( s − ) is smooth.Applying Lemma 6.7 to the closed subset B , the line bundles L ( N ), O (1) and the invertible section( s · s − (cid:12)(cid:12) D ) ∐ ( w · t N ) ∈ Γ(( B ∪ B ) ∐ ( B ∪ B ) , L ( N )) ,r ∐ t ∈ Γ( B ∐ ( B ∪ B ∪ B ) , O ( d )) , we see that there exists N such that for all d > N there exists R ( d ) such that for all K, deg K/k > R ( d ),there exists a section s + ∈ Γ( C K , L ( d )) such that s + (cid:12)(cid:12) D = st d (cid:12)(cid:12) D , s + (cid:12)(cid:12) D = sr d (cid:12)(cid:12) D , s + (cid:12)(cid:12) B ∐ B = wt d , and Z ( s + ) is smooth.So we get the sections s + , s − with the required properties s + = ss − (cid:12)(cid:12) D and Z ( s ) ⊂ C − B , Z ( s ) issmooth. (cid:3) Lemma 6.9. For any framed correspondence c ∈ F r n (pt k , pt k ) there is a pair of standard framed correspon-dences c + , c − ∈ F r (pt k , pt k ) such that [ c ] = [ c + ] − [ c − ] ∈ Z F (pt k , pt k ) = H ( Z F (∆ • , pt k )) .Proof. Following the original strategy of the proof of [16, Lemma 5.4] we see that it suffices to consider thecase of(6.4) c = ( Z ( f ) , A k , f, pr ) ∈ F r (pt k , pt k ) , where f ∈ k [ A k ], pr : Z ( f ) → pt k is the projection. For completeness we recall the arguments from [16,Lemma 5.4].Firstly, by Lemma 6.4 and [16, Lemma 4.10] we can reduce to consider a simple correspondence c ∈ F r (pt k , pt k ). Now, let c = ( Z ( f ) , A k , f g, pr ) ∈ F r (pt k , pt k ), where f ∈ k [ A k ] = k [ t ] deg f = n , g ∈ k [ A k ] = k [ t ], Z ( g ) ∩ Z ( f ) = ∅ , pr : Z ( f ) → pt k is the projection. We prove that the class of c is a sum ofclasses of correspondences of the form (6.4). Actually, let g ′ ∈ k [ t ] be a polynomial of degree n − g (cid:12)(cid:12) Z ( f ) = g ′ (cid:12)(cid:12) Z ( f ) . Then because of the homotopy given by λg ′ + (1 − λ ) g we see that we can assume that g ′ = g . Then the class of the correspondence c ′ = ( Z ( g ) , A k , f g, pr ) ∈ F r (pt k , pt k ) satisfies the inductionassumption. Hence the claim follows, since [ c ] = [( Z ( f g ) , A k , f g, pr )] − [( Z ( g ) , A k , f g, pr )] ∈ Z F r (pt k , pt k ).Now all what is needed is to show that the class of (6.4) is standard. But this is true, since the homotopy λx n + (1 − λ ) f and Lemma 2.4 implies that [ c ] = mh , for n = 2 m , or [ c ] = mh + 1, for n = 2 m + 1. (cid:3) Remark . Alternatively we could say that the proof of the above lemma presented in [16, Lemma 5.4]holds for an arbitrary infinite field and the deduce the finite field case using the finite descent form the section2. References 1. A. Ananyevskiy, G. Garkusha, I. 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