aa r X i v : . [ m a t h . K T ] F e b MOTIVES OF AZUMAYA ALGEBRAS
BRUNO KAHN AND MARC LEVINE
Abstract.
We study the slice filtration for the K -theory of a sheaf of Azu-maya algebras A , and for the motive of a Severi-Brauer variety, the latter inthe case of a central simple algebra of prime degree over a field. Using theBeilinson-Lichtenbaum conjecture, we apply our results to show the vanishingof SK ( A ) for a central simple algebra A of square-free index. Contents
Introduction 21. The motivic Postnikov tower in SH S ( k ) and DM eff ( k ) 51.1. Constructions in A stable homotopy theory 61.2. Postnikov towers for T -spectra and S -spectra 61.3. Postnikov towers for motives 71.4. Purity 101.5. The homotopy coniveau tower 101.6. The 0th slice 131.7. Connected spectra 141.8. Well-connected spectra 192. Birational motives and higher Chow groups 202.1. Birational motives 202.2. The Postnikov tower for birational motives 212.3. Cycles and slices 232.4. The sheaf Z K A and Z A K A : definition and first properties 283.2. The reduced norm map 293.3. The presheaf with transfers Z A K A for embedded schemes 313.6. The cycle complex 343.7. Elementary properties 353.8. Localization 363.9. Extended functoriality 373.10. Reduced norm 374. The spectral sequence 38 Date : December 31, 2007.2000
Mathematics Subject Classification.
Primary 14C25, 19E15; Secondary 19E08 14F42,55P42.
Key words and phrases.
Bloch-Lichtenbaum spectral sequence, algebraic cycles, Morel-Voevodsky stable homotopy category, slice filtration, Azumaya algebras, Severi-Brauer schemes. SK i ( A ) to ´etale cohomology 535. The motivic Postnikov tower for a Severi-Brauer variety 565.1. The motivic Postnikov tower for a smooth variety 565.2. The case of K -theory 605.3. The Chow sheaf 605.4. The slices of M ( X ) 626. Applications 646.1. A spectral sequence for motivic homology 656.2. Computing the boundary map 71Appendix A. Modules over Azumaya algebras 75Appendix B. Regularity 77References 79 Introduction
Voevodsky [40] has defined an analog of the classical Postnikov tower in thesetting of motivic stable homotopy theory by replacing the classical suspensionΣ := S ∧ − with t -suspension Σ t := P ∧ − ; we call this construction the motivicPostnikov tower . In this paper, we study the motivic Postnikov tower in the cat-egory of S -spectra, SH S ( k ), and its analog in the category of effective motives, DM eff ( k ). We concentrate on objects arising from a central simple algebra A overa field k . In the setting of S -spectra, we look at the presheaf of the K -theoryspectra K A : Y K A ( Y ) := K ( Y ; A ) , where K ( Y ; A ) is the K -theory spectrum of the category of O Y ⊗ k A -moduleswhich are locally free as O Y -modules. For motives, we study the motive M ( X ) ∈ DM eff ( k ), where X is the Severi-Brauer variety of A . In the case of the Severi-Brauer variety, we are limited to the case of A having prime degree.Of course, K A is a twisted form of the presheaf K of K -theory spectra Y K ( Y )and X is a twisted form of a projective space over k , so one would expect the layersin the respective Postnikov towers of K A and M ( X ) to be twisted forms of thelayers for K and M ( P n ). The second author has shown in [18] that the n th layerfor K is the Eilenberg-Maclane spectrum for the Tate motive Z ( n )[2 n ]; similarly,the direct sum decomposition M ( P N ) = ⊕ Nn =0 Z ( n )[2 n ]shows that n th layer for M ( P N ) is Z ( n )[2 n ] for 0 ≤ n ≤ N , and is 0 for n outsidethis range. The twisted version of Z ( n ) turns out to be Z A ( n ), where Z A ⊂ Z is OTIVES OF AZUMAYA ALGEBRAS 3 the subsheaf of the constant sheaf with transfers having value Z A ( Y ) ⊂ Z ( Y ) = Z equal to the image of the reduced norm mapNrd : K ( A ⊗ k k ( Y )) → K ( k ( Y )) = Z . Here Y is any smooth irreducible scheme over k . Letting s n and s motn denote the n layer of the motivic Postnikov tower in SH S ( k ) and DM eff ( k ), respectively, andletting EM : DM eff ( k ) → SH S ( k ) denote the Eilenberg-Maclane functor [27],our main results are Theorem 1.
Let A be a central simple algebra over a field k . Then s n ( K A ) = EM ( Z A ( n )[2 n ]) for all n ≥ . Theorem 2.
Let A be a central simple algebra over a field k of prime degree ℓ , X := SB( A ) the associated Severi-Brauer variety. Then s motn ( M ( X )) = Z A ⊗ n +1 ( n )[2 n ] for ≤ n ≤ ℓ − , otherwise. See theorems 4.5.5 and 5.4.2, respectively, in the body of the paper.Since s n K A and s motn M ( X ) are the layers in the respective motivic Postnikovtowers . . . → f n +1 K A → f n K A → . . . → f K A = K A f motℓ M ( X ) → f motℓ − M ( X ) → . . . → f mot M ( X ) = M ( X )our computation of the layers gives us some handle on the spectral sequences E p,q := π − p − q ( s − q K A ( Y )) = ⇒ π − p − q K A ( Y )and E p,q := H p + q ( Y, s mot − q M ( X )( n )) = ⇒ H p + q ( Y, M ( X )( n ))arising from the towers. In fact, we use a version of the first sequence to helpcompute the layers of M ( X ). Putting in our computation of the layers into the K A -spectral sequence gives us the spectral sequence E p,q := H p − q ( Y, Z A ( − q )) = ⇒ K − p − q ( Y ; A )generalizing the Bloch-Lichtenbaum/Friedlander-Suslin spectral sequence from mo-tivic cohomology to K -theory [6, 10]. In particular, taking Y = Spec k , we get K ( A ) = H ( k, Z A (1))and for A of square-free index K ( A ) = H ( k, Z A (2)) . See theorem 4.7.1 and theorem 4.8.2 .To go further, we must use the Beilinson-Lichtenbaum conjecture. Recall that itis equivalent to the Milnor-Bloch-Kato conjecture relating Milnor’s K -theory withGalois cohomology [36], [12]. It seems to be now a theorem (see [44]), thanks towork of Rost and Voevodsky; accepted proofs are certainly that of Merkurjev andSuslin in the special case of weight 2 [25] and that of Voevodsky at the prime 2 (in allweights) [41]. Since this seems important to some people, we shall specify in whatweights we need the Beilinson-Lichtenbaum (or Milnor-Bloch-Kato) conjecture forour statements. BRUNO KAHN AND MARC LEVINE
We thus use our knowledge of the layers of M ( X ), together with the Beilinson-Lichtenbaum conjecture, to deduce a result comparing H p ( k, Z A ( q )) and H p ( k, Z ( q ))via the reduced norm map Nrd : H p ( k, Z A ( q )) → H p ( k, Z ( q )) , this just being the map induced by the inclusion Z A ⊂ Z . By identifying Nrd withthe change of topologies map from the Nisnevich to the ´etale topology (using thefact that Z A ( n ) ´et = Z ( n ) ´et ), a duality argument leads to Corollary 1.
Let A be a central simple algebra of square-free index e over k . Let n ≥ and assume the Beilinson-Lichtenbaum conjecture in weights ≤ n + 1 at allprimes dividing the index of A . Then Nrd : H p ( k, Z A ( n )) → H p ( k, Z ( n )) is an isomorphism for p < n , and we have an exact sequence → H n ( k, Z A ( n )) Nrd −−→ H n ( k, Z ( n )) ≃ K Mn ( k ) ∪ [ A ] −−−→ H n +2´et ( k, Z /e ( n + 1)) → H n +2´et ( k ( X ) , Z /e ( n + 1)) . Here [ A ] ∈ H ( k, Z (1)) = H ( k, G m ) is the class of A in the Brauer group of k ,and the map ∪ [ A ] is shorthand for the composition H n ( k, Z ( n )) ∼ → H n ´et ( k, Z ( n )) ∪ [ A ] −−−→ H n +3´et ( k, Z ( n + 1))(note that this cup-product map lands into e H n +3´et ( k, Z ( n + 1)) ≃ H n +2´et ( k, Z /e ( n +1)), the latter isomorphism being a consequence of the Beilinson-Lichtenbaum con-jecture in weight n + 1.)See theorem 6.2.2 in the body of the paper for this result.Combining this result with our identification above of K ( A ) and K ( A ) as“twisted Milnor K -theory” of k , we have (see theorem 6.2.2) Corollary 2.
Let A be a central simple algebra over k of square-free index. Thenthe reduced norm maps on K ( A ) , K ( A ) and K ( A )Nrd : K n ( A ) → K n ( k ); n = 0 , , are injective; in fact, we have an exact sequence → K n ( A ) Nrd −−→ K n ( k ) = H n ( k, Z ( n )) ∪ [ A ] −−−→ H n +2´et ( k, Z /e ( n + 1)) → H n +2´et ( k ( X ) , Z /e ( n + 1)) for n = 0 , , . (For n = 2 we need the Beilinson-Lichtenbaum conjecture in weight .) The injectivity of Nrd on K ( A ) is Wang’s theorem [43], and it was proved for K ( A ) and A a quaternion algebra by Rost [31] and Merkurjev [24]. They used itas a step towards the proof of the Milnor conjecture in degree 3; conversely, theMilnor conjecture in degree 3 was used in [15, proof of Th. 9.3] to give a simpleproof of the injectivity in this case. This proof was one of the starting points of thepresent paper.For n = 0, the exact sequence reduces to Amitsur’s theorem that ker( Br ( k ) → Br ( k ( X )) is generated by the class of A [1]. For n = 1, the exactness at K ( k ) isdue to Merkurjev-Suslin [25, Th. 12.2] and the exactness at H ( k, Z /e (2)) could OTIVES OF AZUMAYA ALGEBRAS 5 be extracted from Suslin [35]. For n = 1 and A a quaternion algebra, the exactnessat H ( k, Z /
2) is due to Arason [2, Satz 5.4]. For n = 2 and a quaternion algebra,it is due to Merkurjev [23, Prop. 3.15].The injectivity for K ( A ) with A of arbitrary square-free index has also beenannounced recently by A. Merkurjev (joint with A. Suslin); their method also relieson the Beilinson-Lichtenbaum conjecture, using it to give a computation of themotivic cohomology of the “ ˇCech co-simplicial scheme” ˇC( X ).We begin our paper in section 1 with a quick review of the motivic Postnikovtower in SH S ( k ) and DM eff ( k ), recalling the basic constructions as well as someof the results from [18] that we will need. In section 2 we recall some of the firstauthor’s theory of birational motives as well as pointing out the role these motivesplay as the Tate twists of slices of an arbitrary t -spectrum. We proceed in section 3to define and study the special case of the birational motive Z A arising from acentral simple algebra A over k ; we actually work in the more general setting of asheaf of Azumaya algebras on a scheme. In section 4 we prove our first main result:we compute the slices of the “homotopy coniveau tower” for the G -theory spectrum G ( X ; A ), where A is a sheaf of Azumaya algebras on a scheme X . This result relieson some regularity properties of the functors K p ( − , A ) which rely on results dueto Vorst and generalized by van der Kallen; we collect and prove what we need inthis direction in the appendix B. We also recall some basic results on Azumayaalgebras in the appendix A. Specializing to the case in which X is smooth over afield k and A is the pull-back to X of a central simple algebra A over k , the resultsof [18] translate our computation of the slices of the homotopy coniveau tower togive theorem 1.We also give in Subsection 4.9 a construction of homomorphisms from SK and SK of a central simple algebra A to quotients of ´etale cohomology groups of k , inthe spirit of an idea of Suslin [34, 33], albeit with a very different technique (for SK we need the Beilinson-Lichtenbaum conjecture in weight 3).We turn to our study of the motive of a Severi-Brauer variety in section 5, prov-ing theorem 2 there. We conclude in section 6 with a discussion of the reducednorm map and the proofs of corollaries 1 and 2. Acknowledgements.
The first author would like to thank Philippe Gille forhelpful exchanges about Azumaya algebras and Nicolas Perrin for an enlighteningdiscussion about the Riemann-Roch theorem. We also thank Wilberd van derKallen for helpful comments. This work was begun when the second author wasvisiting the Institute of Mathematics of Jussieu on a “Poste rouge CNRS” in 2000,for which visit the second author expresses his heartfelt gratitude. In addition,the second author thanks the NSF for support via grants DMS-9876729, DMS-0140445 and DMS-0457195, as well the Humboldt Foundation for support throughthe Wolfgang Paul Prize and a Senior Research Fellowship.1.
The motivic Postnikov tower in SH S ( k ) and DM eff ( k )In this section, we assume that k is a perfect field. We review Voevodsky’sconstruction of the motivic Postnikov tower in SH ( k ) and SH S ( k ), as well as theanalog of the tower in DM eff ( k ). We also give the description of these towers interms of the homotopy coniveau tower, following [18], and recall the main results jointly with R. Sujatha BRUNO KAHN AND MARC LEVINE of [18] on well-connected theories and on various generalizations of Bloch’s higherChow groups to higher Chow groups with coefficients.1.1.
Constructions in A stable homotopy theory. We start with the unsta-ble A homotopy category over k , H ( k ), which is the homotopy category of thecategory P Sh ( k ) of pointed presheaves of simplicial sets on Sm /k , with respect tothe Nisnevich- and A -local model structure (see Østvær-R¨ondigs [27]). We let T denote the pointed presheaf ( P , ∞ ), and Σ T the operation of smash product with T . SH S ( k ) is the homotopy category of the model category Spt S ( k ) of presheavesof spectra on Sm /k , where the model structure is given as in [27]. SH ( k ) is thehomotopy category of the category Spt T ( k ) of T -spectra, where we will take thisto mean the category of T -spectra in Spt S ( k ), that is, objects are sequences E := ( E , E , . . . )with the E n ∈ Spt S ( k ), together with bonding maps ǫ n : Σ T E n → E n +1 We give
Spt T ( k ) the model structure of spectrum objects as defined by Hovey [13].Σ T (on both Spt S ( k ) and Spt T ( k )) has as a right adjoint the T -loops functorΩ T , which passes to the homotopy categories by applying Ω T to fibrant models.Concretely, for E ∈ Spt S ( k ) we haveΩ T E ( X ) := fib( E ( X × P ) → E ( X × ∞ )) . Note that Σ T is an equivalence on SH ( k ), with inverse Ω T , but this is not the caseon SH S ( k ).1.2. Postnikov towers for T -spectra and S -spectra. Voevodsky [40] has de-fined a canonical tower on the motivic stable homotopy category SH ( k ), which wecall the motivic Postnikov tower . This is defined as follows: Let SH eff ( k ) ⊂ SH ( k )be the localizing subcategory generated by the T -suspension spectra of smooth k -schemes, Σ ∞ T X + , X ∈ Sm /k . Acting by the equivalence Σ T of SH ( k ) gives us thetower of localizing subcategories · · · ⊂ Σ n +1 T SH eff ( k ) ⊂ Σ nT SH eff ( k ) ⊂ · · · ⊂ SH ( k )for n ∈ Z . Voevodsky notes that the inclusion i n : Σ nT SH eff ( k ) → SH ( k ) admitsa right adjoint r n : SH ( k ) → Σ nT SH eff ( k ); let f n := i n ◦ r n with counit f n → id.Thus, for each E ∈ SH ( k ), there is a canonical tower in SH ( k )(1.2.1) . . . → f n +1 E → f n E → . . . → E which we call the motivic Postnikov tower . The cofiber s n E of f n +1 E → f n E is the n th slice of E .Voevodsky also defined the motivic Postnikov tower on the category of S -spectra, using the T -suspension operator as above to define the localizing sub-categories Σ nT SH S ( k ), n ≥ · · · ⊂ Σ n +1 T SH S ( k ) ⊂ Σ nT SH S ( k ) ⊂ · · · ⊂ SH S ( k ) , with right adjoint r n : SH S ( k ) → Σ nT SH S ( k ) to the inclusion. This gives the T -Postnikov tower for S -spectra(1.2.2) . . . → f n +1 E → f n E → . . . → f E = E. OTIVES OF AZUMAYA ALGEBRAS 7
One difference from the tower for T -spectra is that the S tower terminates at f E = E , whereas the tower for T spectra is in general infinite in both directions.We write f n/n + r E for the cofiber of f n + r E → f n E ; for r = 1, we use the notation s n := f n/n +1 to denote the n th slice in the Postnikov tower. We will mainly beusing the S tower, so the overlap in notations with the tower in SH ( k ) should notcause any confusion.We have the pair of adjoint functors SH S ( k ) Σ ∞ T / / SH ( k ) Ω ∞ T o o respecting the two Postnikov towers. In addition, we haveΣ T ◦ f n = f n +1 ◦ Σ T Ω T ◦ f n +1 = f n ◦ Ω T for T -spectra (and all n ) and(1.2.3) Ω T ◦ f n +1 = f n ◦ Ω T for S -spectra (for n ≥ T spectra follow from the fact thatΣ T is an equivalence, while the result for S -spectra is more difficult, and is provedin [18, theorem 7.4.2]. Remark . The identity f n ◦ Ω T E = Ω T ◦ f n +1 E for an S spectrum E givesby adjointness a map (in SH S ( k )) ϕ E,n : Σ T f n Ω T E → f n +1 E which is not in general an isomorphism. If however E = Ω ∞ T E for a T -spectrum E ∈ SH ( k ), then the isomorphismΣ T f n Ω T E ∼ = f n +1 Σ T Ω T E ∼ = f n +1 E passes to E , showing that ϕ E,n is an isomorphism for all n .1.3. Postnikov towers for motives.
There is an analogous picture for motives;we will describe the situation analogous to the S -spectra. The correspondingcategory of motives is the enlargement DM eff ( k ) of DM eff − ( k ). We recall brieflythe construction.The starting point is the category SmCor ( k ), with objects the smooth quasi-projective k -schemes Sm /k , and morphisms given by the finite correspondences Cor k ( X, Y ), this latter being the group of cycles on X × k Y generated by theintegral closed subschemes W ⊂ X × k Y such that W → X is finite and surjectiveover some component of X . Composition is by the usual formula for compositionof correspondences: W ′ ◦ W := p XZ ∗ ( p ∗ XY ( W ) · p ∗ Y Z ( W ′ )) . Sending f : X → Y to the graph Γ f ⊂ X × k Y defines a functor m : Sm /k → SmCor ( k ).Next, one has the category P ST ( k ) of presheaves with transfer , this being simplythe category of presheaves of abelian groups on SmCor ( k ). Restriction to Sm /k gives the functor m ∗ : P ST ( k ) → P Sh ( Sm /k ); BRUNO KAHN AND MARC LEVINE we let Sh tr Nis ( k ) ⊂ P ST ( k ) be the full subcategory of P such that m ∗ ( P ) is aNisnevich sheaf on Sm /k . Such a P is a Nisnevich sheaf with transfers on Sm /k .The inclusion Sh tr Nis ( k ) → P ST ( k ) has as left adjoint the sheafification functor . P ST ( k ) is an abelian category with kernel and cokernel defined pointwise; as usual, Sh tr Nis ( k ) is an abelian category with kernel the presheaf kernel and cokernel thesheafification of the presheaf cokernel.Recall that the category DM eff − ( k ) is the full subcategory of the bounded abovederived category D − ( Sh tr Nis (( k )) with objects the complexes C ∗ for which the hy-percohomology presheaves X H n Nis ( X, C ∗ )are A homotopy invariant for all n , i.e., the pull-back map H n Nis ( X, C ∗ ) → H n Nis ( X × A , C ∗ )is an isomorphism for all n and for all X in Sm /k . Definition 1.3.1. DM eff ( k ) is the full subcategory of D ( Sh tr Nis ( k )) consistingof complexes C such that the hypercohomology presheaves H n Nis ( − , C ∗ ) are A homotopy invariant for all n .We note that there is a model structure on C ( Sh tr Nis ( k )) for which DM eff ( k ) isequivalent to the homotopy category of C ( Sh tr Nis ( k )) (see [8, 27]); we will often usethis result to lift constructions in DM eff ( k ) by taking fibrant models. Also, wehave the localization functor RC Sus ∗ : D ( Sh tr Nis ( k )) → DM eff ( k )extending Voevodsky’s localization functor RC Sus ∗ : D − ( Sh tr Nis ( k )) → DM eff − ( k )The equivalence of the subcategory of homotopy invariant objects in D ( Sh tr Nis ( k ))with the localization of D − ( Sh tr Nis ( k )) is proved in [8, § L ( X ) for the presheaf with transfers represented by X ∈ Sm /k ; thisis in fact a Nisnevich sheaf with transfers. We have the Tate object Z (1) definedas the image in DM eff ( k ) of the complex L ( P ) → L ( k )with L ( P ) in degree 2. Remark . The inclusion D − ( Sh tr Nis ( k )) → D ( Sh tr Nis ( k )) is a full embedding, soVoevodsky’s embedding theorem [11, V, theorem 3.2.6], that the Suslin complexfunctor RC Sus ∗ ◦ L : DM effgm ( k ) → DM eff − ( k )is a full embedding, yields the full embedding RC Sus ∗ ◦ L : DM effgm ( k ) → DM eff ( k )For X ∈ Sm /k we write M ( X ) for the image of L ( X ) in DM eff ( k ).The operation of the functor ⊗ Z (1)[2] on DM eff ( k ) gives the tower of localizingsubcategories (for n ≥ · · · ⊂ DM eff ( k )( n + 1) ⊂ DM eff ( k )( n ) ⊂ · · · ⊂ DM eff ( k ) OTIVES OF AZUMAYA ALGEBRAS 9 where DM eff ( k )( n ) is the localizing subcategory of DM eff ( k ) generated by objects M ( X )( n )[2 n ], X ∈ Sm /k . Just as for SH S ( k ), we have the right adjoint r motn : DM eff ( k ) → DM eff ( k )( n ) to the inclusion i motn . Thus, for E in DM eff ( k ), wethe motivic Postnikov tower in DM eff ( k )(1.3.1) . . . → f motn +1 E → f motn E → . . . → f mot E = E with f motn := i motn ◦ r motn .One has the pair of adjoint functors SH S ( k ) Mot / / DM eff ( k ) EM o o where EM is the Eilenberg-Maclane functor, associating to a presheaf of abeliangroups the corresponding presheaf of Eilenberg-Maclane spectra, and M ot asso-ciates to a presheaf of spectra E := ( E , E , . . . ), first of all, the presheaf of singularchain complexes Sing E := lim −→ n Sing E n [ n ] , where the maps Sing E n [ n ] → Sing E n +1 [ n + 1] are induced by the bonding mapsfor E and the natural map ΣSing E n → SingΣ E n . One then takes the associated freely generated complexes of presheaves with trans-fer, i.e.
M ot := Z tr ◦ Sing . These functors respect the two towers of subcategories, and hence commute withthe two truncation functors f n and f motn . It follows from work of Østvær-R¨ondigs[27] that the Eilenberg-Maclane functor EM is faithful and conservative.In particular, the identity (1.2.3) implies the identity(1.3.2) Ω T ◦ f motn +1 = f motn ◦ Ω T for the motivic truncation functors.Let E be in SH S ( k ), Y ∈ Sm /k and W ⊂ Y a closed subset. We let E W ( Y )denote the homotopy fiber of ˜ E ( Y ) → ˜ E ( Y \ W )where ˜ E is a fibrant model of E in Spt S ( k ). We make a similar definition for F ∈ C ( P ST ( k )). If E is homotopy invariant and satisfies Nisnevich excision,then the map of the homotopy fiber of E ( Y ) → E ( Y \ W ) to E W ( Y ) is a weakequivalence; we will sometimes use this latter spectrum for E W ( Y ) without explicitmention.Similarly, we lift the functors s n , f n to operations on Spt S ( k ) by taking thefibrant model of the corresponding object in SH S ( k ); we make a similar lifting to C ( Sh tr Nis ( k )) for the functors f motn , s motn . Purity.
Let i : W → Y be a closed immersion in Sm /k such that the normalbundle ν := N W/Y admits a trivialization ϕ : O qW → ν . This gives us the Morel-Voevodsky purity isomorphism [26, Theorem 2.23] in SH (1.4.1) θ ϕ,E : E W ( Y ) → (Ω qT E )( W )and the isomorphism on homotopy groups(1.4.2) θ ϕ,n,E : π n ( E W ( Y )) → π n ((Ω qT E )( W )) . In general, the θ ϕ,n,E depend on the choice of ϕ .For later use, we record the following result: Lemma 1.4.1.
Let W ⊂ Y be a closed subset, Y ∈ Sm /k , such that codim Y W ≥ q for some integer q ≥ .1. For E ∈ SH S ( k ) , the canonical map f q E → E induces a weak equivalence ( f q E ) W ( Y ) → E W ( Y )
2. For
F ∈ DM eff ( k ) , the canonical map f motq F → F induces a weak equivalence ( f motq F ) W ( Y ) → F W ( Y ) Proof.
We prove (1), the proof of (2) is the parallel. Note that π n ( E W ( Y )) ∼ = Hom SH S ( k ) (Σ ∞ ( Y /Y \ W ) , Σ − n E ) . and similarly for π n (( f q E ) W ( Y )).Suppose at first that W is smooth and has trivial normal bundle in Y , ν ∼ = O pW , p ≥ q . Then Σ ∞ ( Y /Y \ W ) ∼ = Σ pT (Σ ∞ W + )hence Σ ∞ ( Y /Y \ W ) is in Σ qT SH S ( k ). In general, since k is perfect, W admits afiltration by closed subsets ∅ = W − ⊂ W ⊂ . . . ⊂ W N = W such that W n \ W n − is smooth and has trivial normal bundle in Y \ W n +1 . Wehave the homotopy cofiber sequence( Y \ W n +1 ) / ( Y \ W n ) → Y / ( Y \ W n ) → Y / ( Y \ W n +1 )so by induction, Σ ∞ ( Y /Y \ W ) is in Σ qT SH S ( k ). Thus, the universal property of f n E → E impliesHom SH S ( k ) (Σ ∞ ( Y /Y \ W ) , Σ − n f q E ) → Hom SH S ( k ) (Σ ∞ ( Y /Y \ W ) , Σ − n E )is an isomorphism, as desired. (cid:3) The homotopy coniveau tower.Definition 1.5.1.
1. For X ∈ Sm /k , and q, n ≥ S ( q ) X ( n ) := { W ⊂ X × ∆ n | W is closedand codim X × F W ∩ X × F ≥ q for all faces F ⊂ ∆ n } OTIVES OF AZUMAYA ALGEBRAS 11
Set X ( q ) ( n ) := { w ∈ X × ∆ n | w is the generic point ofsome irreducible W ∈ S ( q ) X ( n ) }
2. For E ∈ Spt S ( k ), X ∈ Sm /k and integer q ≥
0, define f q ( X, n ; E ) = lim −→ W ∈S ( q ) X ( n ) E W ( X × ∆ n )3. For E ∈ Spt S ( k ), X ∈ Sm /k and integer q ≥
0, define s q ( X, n ; E ) = lim −→ W ∈S ( q ) X ( n ) W ′ ∈S ( q +1) X ( n ) E W \ W ′ ( X × ∆ n \ W ′ )For fixed q , n
7→ S ( q ) X ( n ) forms a simplicial set, and n f q ( X, n ; E ), n s q ( X, n ; E ) form simplicial spectra. We let f q ( X, − ; E ) and s q ( X, − ; E ) denote therespective total spectra.For F ∈ C ( P ST ( k )), we make the analogous definition yielding the simplicialcomplexes n f qmot ( X, n ; F ) and n s qmot ( X, n ; F ); we let f qmot ( X, ∗ ; F ) and s qmot ( X, ∗ ; F ) be the associated total complexes. Proposition 1.5.2 ([18, theorem 7.1.1]) . Take X ∈ Sm /k and q ≥ an integer.Let E ∈ Spt S ( k ) be homotopy invariant and satisfy Nisnevich excision. Thenthere are natural isomorphisms in SH α X,q ; E : f q ( X, − ; E ) ∼ −→ f q ( E )( X ) β X,q ; E : s q ( X, − ; E ) ∼ −→ s q ( E )( X ) Corollary 1.5.3.
Take X ∈ Sm /k and q ≥ an integer. Let F ∈ C ( P ST ( k )) behomotopy invariant and satisfy Nisnevich excision. Then there are natural isomor-phisms in D ( Ab ) α X,q ; F : f qmot ( X, ∗ ; F ) ∼ −→ f motq ( F )( X ) β X,q ; F : s qmot ( X, ∗ ; F ) ∼ −→ s motq ( F )( X )Indeed, the corollary follows directly from proposition 1.5.2 by using the Eilen-berg-Maclane functor. Remarks .
1. The isomorphisms in proposition 1.5.2(1) are natural with re-spect to flat morphisms X → X ′ in Sm /k and with respect to maps E → E ′ in Spt S ( k ), for E , E ′ which are homotopy invariant and satisfy Nisnevich excision.2. As each W ′ ∈ S ( q +1) X ( n ) is in S ( q ) X ( n ), we have the natural maps f q +1 ( X, n ; E ) → f q ( X, n ; E ), compatible with the simplicial structure. Similarly, we have the natu-ral restriction maps f q ( X, n ; E ) → s q ( X, n ; E ). Since E satisfies Nisnevich excision,the sequence f q +1 ( X, n ; E ) → f q ( X, n ; E ) → s q ( X, n ; E )is a homotopy cofiber sequence, giving the homotopy cofiber sequence f q +1 ( X, − ; E ) → f q ( X, − ; E ) → s q ( X, − ; E ) on the total spectra. In addition, the diagram f q +1 ( X, − ; E ) / / α X,q +1; E (cid:15) (cid:15) f q ( X, − ; E ) / / α X,q ; E (cid:15) (cid:15) s q ( X, − ; E ) β X,q ; E (cid:15) (cid:15) f q +1 ( E )( X ) / / f q ( E )( X ) / / s q ( E )( X )commutes.4. The analogous statements hold for F in C ( P ST ( k )) as in corollary 1.5.3.For E ∈ SH S ( k ), we have the diagram E τ q ←− f q E π q −→ s q E Lemma 1.5.5.
Take E ∈ SH S ( k ) , X ∈ Sm /k and integers q, n ≥ . For all p ≥ q the map τ q : f q E → E induces weak equivalences f p ( X, n ; f q E ) τ q −→ f p ( X, n ; E ) s p ( X, n ; f q E ) τ q −→ s p ( X, n ; E ) Proof.
That τ q : f p ( X, n ; f q E ) → f p ( X, n ; E ) is a weak equivalence follows fromlemma 1.4.1. We have the map of distinguished triangles f p +1 ( X, n ; f q E ) / / τ q (cid:15) (cid:15) f p ( X, n ; f q E ) / / τ q (cid:15) (cid:15) s p ( X, n ; f q E ) τ q (cid:15) (cid:15) f p +1 ( X, n ; E ) / / f p ( X, n ; E ) / / s p ( X, n ; E )hence τ q : s p ( X, n ; f q E ) → s p ( X, n ; E ) is also a weak equivalence. (cid:3) Proposition 1.5.6.
Take E ∈ SH S ( k ) , X ∈ Sm /k and integer q ≥ .1. For all p ≥ q , the map τ q : f q E → E induces weak equivalences f p ( X, − ; f q E ) τ q −→ f p ( X, − ; E ) s p ( X, − ; f q E ) τ q −→ s p ( X, − ; E )
2. The map π q : f q → s q induces a weak equivalence s q ( X, − ; f q E ) π q −→ s q ( X, − ; s q E ) Proof. (1) follows from lemma 1.5.5. For (2), we have the commutative diagram in SH s q ( X, − ; f q E ) β X,q ; fqE (cid:15) (cid:15) π q / / s q ( X, − ; s q E ) β X,q ; sqE (cid:15) (cid:15) s q ( f q E )( X ) s q ( π q ) / / s q ( s q E )( X )with vertical arrows isomorphisms. The bottom horizontal diagram extends to thedistinguished triangle s q ( f q +1 E ) → s q ( f q E ) s q ( π q ) −−−−→ s q ( s q E ) → s q ( f q +1 E )[1] OTIVES OF AZUMAYA ALGEBRAS 13 and we have the defining distinguished triangle for s q : f q +1 ( f q +1 E ) → f q ( f q +1 E ) → s q ( f q +1 E ) → f q +1 ( f q +1 E )[1]Since f q +1 E is in Σ q +1 T SH S ( k ) ⊂ Σ qT SH S ( k ), the canonical maps f q +1 ( f q +1 E ) → f q +1 E, f q ( f q +1 E ) → f q +1 E are isomorphisms, hence s q ( f q +1 E ) ∼ = 0 and s q ( π q ) is an isomorphism. (cid:3) Remark . Making the evident changes, the analogs of lemma 1.5.5 and propo-sition 1.5.6 hold for
F ∈ DM eff ( k ).1.6. The 0th slice.
Let F be a presheaf of spectra on Sm /k which is A -homotopyinvariant and satisfies Nisnevich excision. Then F is pointwise weakly equivalent toits fibrant model. In addition, these properties pass to H om ( X, F ) for X ∈ Sm /k .Furthermore, ( s F )( Y ) can be described using the cosimplicial scheme of semi-local ℓ -simplices ˆ∆ ℓ (denoted ∆ ℓ in [18]). In fact, for Y ∈ Sm /k , let O ( ℓ ) k ( Y ) ,v bethe semi-local ring of the set v of vertices of ∆ ℓk ( Y ) and setˆ∆ ℓk ( Y ) := Spec O ( ℓ ) k ( Y ) ,v . Clearly ℓ ˆ∆ ℓk ( Y ) forms a cosimplicial subscheme of ∆ ∗ k ( Y ) . It follows from proposi-tion 1.5.2 below that ( s F )( Y ) weakly equivalent to total spectrum of the simplicialspectrum ℓ F ( ˆ∆ ℓk ( Y ) ) , which we denote by F ( ˆ∆ ∗ k ( Y ) ).We have an analogous description of s mot F ( Y ) for F ∈ C ( P ST ( k )) which is A -homotopy invariant and satisfies Nisnevich excision. Using the Eilenberg-Maclanefunctor, it follows from the case of S spectra that s mot F ( Y ) is represented by thetotal complex associated to the simplicial object of C ( Ab ) ℓ
7→ F ( ˆ∆ ℓk ( Y ) ) , which we denote by F ( ˆ∆ ∗ k ( Y ) ). Remark . The 0th slice computes the q th slice with supports in codimension ≥ q , as follows. Let W ⊂ Y be a closed subset, Y ∈ Sm /k . We have shown in [18](1) Let W ⊂ W be an open subset of W such that W \ W has codimension > q on Y , let F := W \ W . Then the restriction s q ( E ) W ( Y ) → s q ( E ) W ( Y \ F )is a weak equivalence. This follows directly from lemma 1.4.1.(2) Suppose that W is smooth with trivial normal bundle ν in Y and thatcodim Y W = q . A choice of trivialization ψ : ν → O qW together with thepurity isomorphism (1.4.1) gives an isomorphism s q ( E ) W ( Y ) ∼ = Ω qT ( s q ( E ))( W )in SH . Combining with the de-looping isomorphism (1.2.3) gives us theisomorphism ˆ θ E,W,Y,q : s q ( E ) W ( Y ) → s (Ω qT E )( W )It is shown in [18, corollary 4.2.4] (essentially a consequence of proposi-tion 1.5.2) that ˆ θ is in fact independent of the choice of trivialization ψ . Let Y ( q ) W be the set of generic points of W of codimension exactly q on Y . Combining(1) and (2) we have, for each W ⊂ Y of codimension ≥ q , a natural isomorphism ρ E,W,Y,q : s q ( E ) W ( Y ) → ∐ w ∈ Y ( q ) W s (Ω qT E )( k ( w )) . Connected spectra.
We continue to assume the field k is perfect. Definition 1.7.1.
Call E ∈ SH S ( k ) connected if for each X ∈ Sm /k , the spec-trum ˜ E ( X ) is -1 connected, where ˜ E ∈ Spt S ( k ) is a fibrant model for E . Lemma 1.7.2.
Let E ∈ SH S ( k ) be connected. Then1. For each q ≥ , Ω qT E is connected.2. For X ∈ Sm /k and W ⊂ X a closed subset, the spectrum with supports E W ( X ) is -1 connected.3. Let j : U → X be an open immersion in Sm /k , W ⊂ X a closed subset .Then j ∗ : π ( E W ( X )) → π ( E W ∩ U ( U )) is surjective.Proof. For (1) it suffices to prove the case q = 1. Take X ∈ Sm /k . Since ∞ ֒ → P is split by P → Spec k , (Ω T E )( X ) is a retract of E ( X × P ). Since E ( X × P ) is-1 connected by assumption, it follows that (Ω T E )( X ) is also -1 connected, henceΩ T E is connected.For (2), suppose first that i : W → X is a closed immersion in Sm /k and thatthe normal bundle ν of W in X admits a trivialization, ν ∼ = O qW . We have theMorel-Voevodsky purity isomorphism (1.4.1) E W ( X ) ∼ = (Ω qT E )( W ) . By (1) (Ω qT E )( W ) is -1 connected, verifying (2) in this case.In general, we proceed by descending induction on codim X W , starting withthe trivial case codim X W = dim k X + 1, i.e. W = ∅ In general, suppose thatcodim X W ≥ q for some integer q ≤ dim k X . Then there is a closed subset W ′ ⊂ W with codim X W ′ > q such that W \ W ′ is smooth and has trivial normal bundle in X \ W ′ . We have the homotopy fiber sequence E W ′ ( X ) → E W ( X ) → E W \ W ′ ( X \ W ′ )thus the induction hypothesis, and the -1 connectedness of E W \ W ′ ( X \ W ′ ) impliesthat E W ( X ) is -1 connected.(3) follows from the homotopy fiber sequence E W \ U ( X ) → E W ( X ) → E W ∩ U ( U )and the -1 connectedness of E W \ U ( X ). (cid:3) Lemma 1.7.3.
Suppose E ∈ SH S ( k ) is connected. Then for X ∈ Sm /k andevery q, n ≥ , f q ( X, n ; E ) and s q ( X, n ; E ) are -1 connected.Proof. This follows from lemma 1.7.2(2), noting that f q ( X, n ; E ) and s q ( X, n ; E )are both colimits over spectra with supports E W ( X × ∆ n ), E W \ W ′ ( X × ∆ n \ W ′ ). (cid:3) OTIVES OF AZUMAYA ALGEBRAS 15
Proposition 1.7.4.
Suppose E ∈ SH S ( k ) is connected. Then for every q ≥ , f q E and s q E are connected.Proof. Take X ∈ Sm /k . We have isomorphism in SH : f q E ( X ) ∼ = f q ( X, − ; E ) , s q E ( X ) ∼ = s q ( X, − ; E )By lemma 1.7.3, the total spectra f q ( X, − ; E ) and s q ( X, − ; E ) are -1 connected,whence the result. (cid:3) Definition 1.7.5.
Fix an integer q ≥ W ⊂ Y be a closed subset with Y ∈ Sm /k and codim Y W ≥ q . For E ∈ SH S ( k ), define the comparison map ψ EW ( Y ) : π ( E W ( Y )) → π ( s q ( E ) W ( Y ))as the composition π ( E W ( Y )) ∼ ←− π (( f q E ) W ( Y )) → π ( s q ( E ) W ( Y ))noting that π (( f q E ) W ( Y )) → π ( E W ( Y )) is an isomorphism by lemma 1.4.1. Lemma 1.7.6.
Let w ∈ Y ( q ) be a codimension q point of Y ∈ Sm /k and let Y w := Spec O Y,w . Take E ∈ SH S ( k ) and suppose that E is connected. Then thecomparison map ψ Ew ( Y w ) : π ( E w ( Y w )) → π ( s q ( E ) w ( Y w )) is an isomorphism.Proof. Since ˆ∆ k ( Y ) = Spec k ( Y ), we have the natural map π ((Ω qT E )( k ( Y ))) → π ((Ω qT E )( ˆ∆ ∗ k ( Y ) ))which is an isomorphism. Indeed, by lemma 1.7.2(1), Ω qT E is connected for all q ≥
0. In particular, (Ω qT E )( ˆ∆ nk ( Y ) ) is -1 connected for all Y and all n ≥
0. Thuswe have the presentation of π ((Ω qT E )( ˆ∆ ∗ k ( Y ) )): π ((Ω qT E )( ˆ∆ k ( Y ) )) i ∗ − i ∗ −−−→ π ((Ω qT E )( k ( Y )) → π ((Ω qT E )( ˆ∆ ∗ k ( Y ) )) . By lemma 1.7.2(3) and a limit argument, the map π ((Ω qT E )(∆ k ( Y ) )) → π ((Ω qT E )( ˆ∆ k ( Y ) ))is surjective; since ∆ k ( Y ) = A k ( Y ) and Ω qT E is homotopy invariant, the map i ∗ − i ∗ is the zero map.Choose a trivialization of the normal bundle ν of w ∈ Y w , k ( w ) q ∼ = ν . This givesus the purity isomorphisms E w ( Y w ) ∼ = (Ω qT E )( w ), ( s q E ) w ( Y w ) ∼ = s (Ω qT E )( w ) ∼ =(Ω qT E )( ˆ∆ ∗ k ( w ) ), giving the commutative diagram π ( E w ( Y w )) ψ Ew ( Y w ) / / (cid:15) (cid:15) π ( s q ( E ) w ( Y w )) (cid:15) (cid:15) π (Ω qT E ( w )) / / π ((Ω qT E )( ˆ∆ ∗ k ( w ) ))with the two vertical arrows and the bottom horizontal arrow isomorphisms. Thus ψ Ew ( Y w ) is an isomorphism. (cid:3) Lemma 1.7.7.
Suppose E ∈ SH S ( k ) is connected. Fix an integer q ≥ andlet W ⊂ Y be a closed subset, with Y ∈ Sm /k and codim Y W ≥ q . Then thecomparison map ψ EW ( Y ) : π ( E W ( Y )) → π ( s q ( E ) W ( Y )) is surjective.Proof. Recall that Y ( q ) W denotes the set of generic points w of W with codim Y w = q .Let Y W := Spec O Y,Y ( q ) W . By remark 1.6.1, the restriction map s q ( E ) W ( Y ) → ∐ w ∈ Y ( q ) W s q ( E ) w ( Y W )is a weak equivalence. By lemma 1.7.6, ψ Ew ( Y W ) : π ( E w ( Y W )) → π ( s q ( E ) w ( Y W ))is an isomorphism for all w ∈ Y ( q ) W . Thus we have the commutative diagram π ( E W ( Y )) ψ EW ( Y ) / / (cid:15) (cid:15) π ( s q ( E ) W ( Y )) (cid:15) (cid:15) ⊕ w ∈ Y ( q ) W π ( E w ( Y W )) Σ w ψ Ew ( Y W ) / / ⊕ w ∈ Y ( q ) W π ( s q ( E ) w ( Y W )) . By remark 1.6.1, the right hand vertical arrow is an isomorphism; the bottomhorizontal arrow is an isomorphism by lemma 1.7.6. It follows from lemma 1.7.2(3)that the left hand vertical arrow is surjective, hence ψ EW ( Y ) is surjective as well. (cid:3) Lemma 1.7.8.
Suppose that E ∈ Spt S ( k ) is connected. Take Y ∈ Sm /k , w ∈ Y ( q ) and let Y w := Spec O Y,w . Then the purity isomorphism θ ϕ, ,E : π ( E w ( Y w )) → π (Ω qT E ( w )) is independent of the choice of trivialization ϕ .Proof. We have the commutative diagram of isomorphisms π ( E w ( Y w )) ψ Ew ( Y w ) / / θ ϕ, E (cid:15) (cid:15) π ( s q ( E ) w ( Y w )) θ ϕ, ,sqE (cid:15) (cid:15) π (Ω qT E ( w )) / / π ((Ω qT E )( ˆ∆ ∗ k ( w ) ))By [18, corollary 4.2.4], θ ϕ, ,s q E is independent of the choice of ϕ , whence theresult. (cid:3) Take E ∈ SH S ( k ) connected. For each closed subset W ⊂ Y , Y ∈ Sm /k , wehave the canonical map ρ E,Y,W : E W ( Y ) → EM ( π ( E W ( Y ))) . Definition 1.7.9.
Let E ∈ SH S ( k ) be connected. Let Y be in Sm /k and let W ⊂ Y be a closed subset of codimension ≥ q . The cycle map cyc WE ( Y ) : E W ( Y ) → EM ( ⊕ w ∈ Y ( q ) W π ((Ω qT E )( w ))) OTIVES OF AZUMAYA ALGEBRAS 17 is the composition E W ( Y ) ρ E,Y,W −−−−−→ EM ( π ( E W ( Y ))) res −−→ EM ( ⊕ w ∈ Y ( q ) W π ( E w ( Y W ))) θ ϕ, E −−−→ EM ( ⊕ w ∈ Y ( q ) W π ((Ω qT E )( w ))) . We let π (cyc WE ( Y )) : π ( E W ( Y )) → ⊕ w ∈ Y ( q ) W π ((Ω qT E )( w ))be the the composition π ( E W ( Y )) res −−→ ⊕ w ∈ Y ( q ) W π ( E w ( Y W )) θ ϕ, ,E −−−−→ ⊕ w ∈ Y ( q ) W π ((Ω qT E )( w )) . In other words, π (cyc WE ( Y )) is the map on π induced by cyc WE ( Y ). Definition 1.7.10.
Let E ∈ SH S ( k ) be connected. For X ∈ Sm /k and integers q, n ≥ z q ( X, n ; E ) := ⊕ w ∈ X ( q ) ( n ) π ((Ω qT E )( w )) . Taking the limit of the maps cyc W \ W ′ E ( X × ∆ n \ W ′ ) for for E ∈ SH S ( k )connected, W ∈ S ( q ) X ( n ), W ′ ∈ S ( q +1) X ( n ) we have the maps of spectracyc E ( X, n ) : s q ( X, n ; E ) → EM ( z q ( X, n ; E ))and the maps of abelian groups π (cyc E ( X, n )) : π ( s q ( X, n ; E )) → z q ( X, n ; E ) Lemma 1.7.11.
Let E ∈ SH S ( k ) be connected and let X be in Sm /k . Then π (cyc s q E ( X, n )) : π ( s q ( X, n ; s q E )) → z q ( X, n ; s q E ) is an isomorphism.Proof. First note that, by proposition 1.7.4, s q E is connected, hence all terms inthe statement are defined. By remark 1.6.1, the restriction map π (( s q E ) W ( Y )) → ⊕ w ∈ Y ( q ) W π (( s q E ) w ( Y W ))is an isomorphism; since π (cyc s q E ( X, n )) is constructed by composing restrictionmaps with purity isomorphisms, this proves the result. (cid:3)
Lemma 1.7.12.
Let E ∈ SH S ( k ) be connected and let X be in Sm /k . There isa unique structure of a simplicial abelian group n z q ( X, n ; E ) such that the maps π (cyc E ( X, n )) define a map of simplicial abelian groups [ n π ( s q ( X, n ; E ))] π (cyc E ( X, − )) −−−−−−−−−→ [ n z q ( X, n ; E )] . Proof.
Since E is connected, the cycle maps π ( E W ( Y )) res −−→ ⊕ w ∈ Y ( q ) W π ( E w ( Y W )) ∼ = ⊕ w ∈ Y ( q ) W π ((Ω qT E ( w ))are surjective. Thus π (cyc E ( X, n )) is surjective, which proves the uniqueness.
For existence, the map π (cyc E ( X, n )) is natural with respect to E . In addition,by proposition 1.7.4, both f q E and s q E are connected; applying π (cyc ? ( X, n )) tothe diagram E ← f q E → s q E gives the commutative diagram π ( s q ( X, n ; E )) π (cyc E ( X,n )) (cid:15) (cid:15) π ( s q ( X, n ; f q E )) π (cyc fqE ( X,n )) (cid:15) (cid:15) / / o o π ( s q ( X, n ; s q E )) π (cyc sqE ( X,n )) (cid:15) (cid:15) z q ( X, n ; E ) z q ( X, n ; f q E ) o o / / z q ( X, n ; s q E )By lemma 1.5.5, the left hand map in the top row is an isomorphism. The maps inthe bottom rows are induced by maps π ((Ω qT E )( w )) ← π ((Ω qT f q E )( w )) → π ((Ω qT s q E )( w ))By (1.2.3), Ω qT f q E = f (Ω qT f q E ) = Ω qT E and similarly Ω qT s q E = s (Ω qT E ). Thusthe bottom row is a sum of isomorphisms π ((Ω qT E )( w )) → π ( s (Ω qT E )( w )) . Finally, the right hand vertical map is an isomorphism by lemma 1.7.11. As thetop row is the degree n part of a diagram of maps of simplicial abelian groups, theisomorphisms π ( s q ( X, n ; s q E )) → z q ( X, n ; s q E ) ← z q ( X, n ; E )induce the structure of a simplicial abelian group from [ n π ( s q ( X, n ; s q E ))]to [ n z q ( X, n ; E )], so that the maps π (cyc E ( X, n )) define a map of simplicialabelian groups. (cid:3)
We use the above results to give a generalization of the higher cycle complexesof Bloch:
Definition 1.7.13.
Let E ∈ SH S ( k ) be connected. For X ∈ Sm /k , and q, n ≥ z q ( X, ∗ ; E ) be the complex associated to the simplicial abelian group n z q ( X, n ; E ). Similarly, for F ∈ C ( P ST ( Sm /k )) which is homotopy invariantand satisfies Nisnevich excision, we set z q ( X, n ; F ) = ⊕ w ∈ X ( q ) ( n ) H ((Ω qT F )( w )) , giving the simplicial abelian group n z q ( X, n ; F ). We denote the associatedcomplex by z q ( X, ∗ ; F ).For integers q, n ≥
0, setCH q ( X, n ; E ) := H n ( z q ( X, ∗ ; E ))and CH q ( X, n ; F ) := H n ( z q ( X, ∗ ; F )) OTIVES OF AZUMAYA ALGEBRAS 19
Well-connected spectra.
Following [18] we have
Definition 1.8.1. E ∈ SH S ( k ) is well-connected if(1) E is connected.(2) For each Y ∈ Sm /k , and each q ≥
0, the total spectrum (Ω qT E )( ˆ∆ ∗ k ( Y ) ) has π n ((Ω qT E )( ˆ∆ ∗ k ( Y ) )) = 0for n = 0. Remark . Under the Eilenberg-Maclane map, the corresponding notion in DM eff ( k ) is: Let F ∈ C ( P ST ( k )) be A homotopy invariant and satisfy Nis-nevich excision. Call F well-connected if(1) F is connected(2) For each Y ∈ Sm /k , the total complex (Ω qT F )( ˆ∆ ∗ k ( Y ) ) satisfies H n ((Ω qT F )( ˆ∆ ∗ k ( Y ) ))) = 0for n = 0. Remark . We gave a slightly different definition of well-connectedness in [18,definition 6.1.1], replacing the connectedness condition (1) with: E W ( Y ) is -1 con-nected for all closed subsets W ⊂ Y , Y ∈ Sm /k . By lemma 1.7.2, this condition isequivalent with the connectedness of E .The main result on well-connected spectra is: Theorem 1.8.4.
1. Suppose E ∈ SH S ( k ) is well-connected. Then cyc E ( X ) : s q ( X, − ; E ) → EM ( z q ( X, − ; E )) is a weak equivalence. In particular, there is a natural isomorphism CH q ( X, n ; E ) ∼ = π n (( s q E )( X )) ∼ = Hom SH S ( k ) (Σ ∞ T X + , Σ − ns s q ( E )) .
2. Suppose
F ∈ C ( P ST ( k )) is well-connected. Then cyc E ( X ) : s q ( X, ∗ ; F ) → z q ( X, ∗ ; F ) . is a quasi-isomorphism. In particular, there is a natural isomorphism CH q ( X, n ; F ) ∼ = H − n Nis ( X, s motq F ) ∼ = Hom DM eff ( k ) ( M ( X ) , s motq ( F )[ − n ]) . Proof.
We prove (1), the proof of (2) is the same. We have commutative diagram(in SH ) s q ( X, − ; E ) cyc E ( X ) (cid:15) (cid:15) s q ( X, − ; f q E ) cyc fqE ( X ) (cid:15) (cid:15) τ q o o π q / / s q ( X, − ; s q E ) cyc sqE ( X ) (cid:15) (cid:15) EM ( z q ( X, − ; E )) EM ( z q ( X, − ; f q E )) τ q o o π q / / EM ( z q ( X, − ; s q E ))By proposition 1.5.6, the arrows in the top row are isomorphisms. As we have seenin the proof of lemma 1.7.12 the arrows in the bottom row are also isomorphisms.Thus, it suffices to prove the result with E replaced by s q E .The map cyc s q E ( X ) is just the map on total spectra induced by the map on n -simplices cyc s q E ( X, n ) : s q ( X, n ; s q E ) → EM ( z q ( X, n ; s q E )) By lemma 1.7.11, the map on π , π (cyc s q E ( X, n )) : π ( s q ( X, n ; s q E )) → z q ( X, n ; s q E ) , is an isomorphism. However, since E is well-connected, and since s q ( X, n ; s q E ) ∼ = ∐ w ∈ X ( q ) ( n ) s (Ω qT E )( k ( w )) , it follows that s q ( X, n ; s q E ) = EM ( π ( s q ( X, n ; s q E ))) , and cyc s q E ( X, n ) is the map induced by π (cyc s q E ( X, n )). Thus cyc s q E ( X, n ) isa weak equivalence for every n , hence cyc s q E ( X ) is an isomorphism in SH , asdesired. (cid:3) Birational motives and higher Chow groups
Birational motives have been introduced and studied by Kahn-Sujatha [16] andHuber-Kahn [14]. In this section we re-examine their theory, emphasizing the re-lation to the slices in the motivic Postnikov tower. We also extend Bloch’s con-struction of cycle complexes and higher Chow groups: Bloch’s construction may beconsidered as the case of the cycle complex with constant coefficients Z whereas ourgeneralization allows the coefficients to be in a birational motivic sheaf . Finally, weextend the identification of Bloch’s higher Chow groups with motivic cohomology[11, 39] to the setting of birational motivic sheaves.2.1. Birational motives.Definition 2.1.1.
A motive
F ∈ DM eff ( k ) is called birational if for every denseopen immersion j : U → X in Sm /k and every integer n , the map j ∗ : Hom DM eff ( k ) ( M ( X ) , F [ n ]) → Hom DM eff ( k ) ( M ( U ) , F [ n ])is an isomorphism. If F is a sheaf, i.e., F ∼ = H ( F ) in D ( Sh tr Nis ( k )), we call F a birational motivic sheaf . Remarks .
1. For X ∈ Sm /k and F ∈ DM eff ( k ) ⊂ D ( Sh tr Nis ( k )), there is anatural isomorphismHom DM eff ( k ) ( M ( X ) , F [ n ]) ∼ = H n Nis ( X, F )Thus a motive F ∈ DM eff ( k ) is birational if an only if the hypercohomologypresheaf U H n Nis ( U, F )on X Zar is the constant presheaf on each connected component of X .2. Let F be a Nisnevich sheaf with transfers that is birational and homotopyinvariant. Then F is a birational motivic sheaf. Indeed, since F is birational, therestriction of F to X Zar is a locally constant sheaf. We haveHom D ( Sh tr Nis ( k )) ( L ( X ) , F [ n ]) = H n Nis ( X, F ) = H n Zar ( X, F );the Zariski cohomology H n Zar ( X, F ) is zero for n > F is strictly homotopy invariant and thus an object of DM eff − ( k ) ⊂ DM eff ( k ). FinallyHom DM eff ( k ) ( M ( X ) , F [ n ]) = Hom D ( Sh tr Nis ( k )) ( L ( X ) , F [ n ]) = ( F ( X ) for n = 00 for n = 0 OTIVES OF AZUMAYA ALGEBRAS 21 hence F is a birational motive.Of course, this result also follows from Voevodsky’s theorem [11] that a Nisnevichsheaf with transfers that is homotopy invariant is also strictly homotopy invariant,but the above argument avoids having to use this deep result.2.2. The Postnikov tower for birational motives.
In this section, we give atreatment of the slices of a birational motive. These results are obtained in [16];here we develop part of the theory of [16] in a slightly different and independentway.Let F be in DM eff ( k ). Since f mot F → F is an isomorphism, we have thecanonical map π : F → s mot F . The following result is taken from [16] in slightly modified form:
Theorem 2.2.1.
For F in DM eff ( k ) , π : F → s mot F is an isomorphism if andonly if F is a birational motive. In particular, since s mot F = s mot ( s mot F ) , s mot F is a birational motive.Proof. Since we have the distinguished triangle f mot F → F π −→ s mot F → f mot F [1] π is an isomorphism if and only if f mot F ∼ = 0.Suppose that π is an isomorphism. Let j : U → X be a dense open immersionin Sm /k and let W = X \ U . We show thatHom DM eff ( k ) ( M ( X ) , F [ n ]) j ∗ −→ Hom DM eff ( k ) ( M ( U ) , F [ n ])is an isomorphism by induction on codim X W , starting with codim X W = dim k X +1, i.e., W = ∅ . We may assume that X is irreducible.By induction we may assume that W is smooth of codimension d ≥
1, giving usthe Gysin distinguished triangle M ( U ) j −→ M ( X ) → M ( W )( d )[2 d ] → M ( U )[1] . But as d ≥
1, we have0 = Hom DM eff ( k ) ( M ( W )( d )[2 d ] , f mot F [ n ]) ∼ = Hom DM eff ( k ) ( M ( W )( d )[2 d ] , F [ n ])hence j ∗ is an isomorphism.Now suppose that F is birational. We may assume that F is fibrant as a complexof Nisnevich sheaves, so thatHom DM eff ( k ) ( M ( X ) , F [ n ]) = H n ( F ( X ))for all X ∈ Sm /k .Take an irreducible X ∈ Sm /k . By remark 1.6.1(2) (applied with Y = W = X ),we have a natural isomorphismHom DM eff ( k ) ( M ( X ) , s mot F [ n ]) ∼ = H n ( F ( ˆ∆ ∗ k ( X ) ))Also, as F is birational, the restriction to the generic point gives an isomorphismHom DM eff ( k ) ( M ( X ) , F [ n ]) ∼ = H n ( F ( k ( X ))) , and the mapHom DM eff ( k ) ( M ( X ) , F [ n ]) π −→ Hom DM eff ( k ) ( M ( X ) , s mot F [ n ]) is given by the map on H n induced by the canonical map F ( k ( X )) = F ( ˆ∆ k ( X ) ) → F ( ˆ∆ ∗ k ( X ) ) . On the other hand, since F is birational, the map F (∆ nk ( X ) ) → F ( ˆ∆ nk ( X ) )is a quasi-isomorphism for all n , and hence the map of total complexes F (∆ ∗ k ( X ) ) → F ( ˆ∆ ∗ k ( X ) )is a quasi-isomorphism. Since F is homotopy invariant, the map F ( k ( X )) = F (∆ k ( X ) ) → F (∆ ∗ k ( X ) )is a quasi-isomorphism; thus the composition F ( k ( X )) → F (∆ ∗ k ( X ) ) → F ( ˆ∆ ∗ k ( X ) )is a quasi-isomorphism as well. Taking H n , we see thatHom DM eff ( k ) ( M ( X ) , F [ n ]) π −→ Hom DM eff ( k ) ( M ( X ) , s mot F [ n ])is an isomorphism for all X ∈ Sm /k . Since the localizing subcategory of DM eff ( k )generated by the M ( X ) for X ∈ Sm /k is all of DM eff ( k ), it follows that π is anisomorphism. (cid:3) Corollary 2.2.2.
Let F be a birational motive. Then f motm ( F ( n )) = ( for m > n F ( n ) for m ≤ n. Proof.
Suppose n ≥ m ≥
0. As F ( n ) is in DM eff ( k )( m ) , we have f motm ( F ( n )) = F ( n ).Now take m > n . As a localizing subcategory of DM eff ( k ), DM eff ( k )( m ) isgenerated by objects M ( X )( m ), X ∈ Sm /k . Thus it suffices to show thatHom DM eff ( k ) ( M ( X )( m ) , F ( n )[ p ]) = 0for all X ∈ Sm /k and all p . By Voevodsky’s cancellation theorem [38], we haveHom DM eff ( k ) ( M ( X )( m ) , F ( n )[ p ]) = Hom DM eff ( k ) ( M ( X )( m − n ) , F [ p ])But since m − n ≥
1, we haveHom DM eff ( k ) ( M ( X )( m − n ) , F [ p ]) ∼ = Hom DM eff ( k ) ( M ( X )( m − n ) , f mot F [ p ])which is zero by theorem 2.2.1. (cid:3) Remark . Let F be a birational motive. Then F ( n ) = s motn ( F ( n )) for all n ≥
0. Indeed, f motn ( F ( n )) = F ( n ) and f motn +1 ( F ( n )) = 0. Remark . Let F be a birational motive. Then for all G in DM eff ( k ) and allintegers m > n ≥
0, we haveHom DM eff ( k ) ( G ( m ) , F ( n )) = 0Indeed, the universal property of f motm F → F gives the isomorphismHom DM eff ( k ) ( G ( m ) , f motm ( F ( n ))) ∼ = Hom DM eff ( k ) ( G ( m ) , F ( n ))but f motm ( F ( n )) = 0 by corollary 2.2.2. OTIVES OF AZUMAYA ALGEBRAS 23
Cycles and slices. If F/k is a finitely generated field extension, we definethe motive M ( F ) in DM eff ( k ) as the homotopy limit of the motives M ( Y ) as Y ∈ Sm /k runs over all smooth models of F . Since we will really only be usingthe functor Hom DM eff ( k ) ( M ( F ) , − ), the reader can, if she prefers, view this as anotational short-hand for the functor on DM eff ( k ) M lim −→ Yk ( Y )= F Hom DM eff ( k ) ( M ( Y ) , M )This limit is just lim ←− Yk ( Y )= F H ( Y, M )in other words, just the stalk of the 0th hypercohomology sheaf of M at the genericpoint of Y . Lemma 2.3.1.
Let F be a homotopy invariant Nisnevich sheaf with transfers.Then Hom DM eff ( k ) ( M ( k ( Y )) , F ( n )[2 n + r ])) = 0 for r > and for all Y ∈ Sm /k .Proof. Let F = k ( Y ). F ( n )[2 n ] is a summand of F ⊗ M ( P n ), so it suffices to showthat Hom DM eff ( k ) ( M ( F ) , F ⊗ M ( P n )[ r ])) = 0for r >
0. We can represent
F ⊗ M ( P n ) by C ∗ ( F ⊗ tr L ( P n )). We have the canonicalleft resolution L ( F ) → F of F (as a Nisnevich sheaf with transfers), where the terms in L ( F ) are direct sumsof representable sheaves, so we can replace C ∗ ( F ⊗ tr L ( P n )) with the total complexof . . . → C ∗ ( L ( F ) n ⊗ L ( P n )) → . . . → C ∗ ( L ( F ) ⊗ tr L ( P n ))This in turn is a complex supported in degrees ≤ L ( Y ), Y ∈ Sm /k . But for any X ∈ Sm /k , we haveHom DM eff ( k ) ( M ( X ) , M ( Y )[ r ]) ∼ = H r Zar ( X, C ∗ ( Y ))Thus Hom DM eff ( k ) ( M ( F ) , M ( Y )[ r ]) ∼ = H r ( C ∗ ( Y )( F ))which is zero for r >
0, and thusHom DM eff ( k ) ( M ( F ) , F ( n )[2 n + r ])) ⊂ H r ( C ∗ ( L ( F ) ⊗ tr L ( P n ))) = 0for r > (cid:3) Proposition 2.3.2.
Let F be a birational motivic sheaf. Then F ( n )[2 n ] is well-connected.Proof. We first show that F is connected, i.e., that H r Zar ( X, F ( n )[2 n ]) = Hom DM eff ( k ) ( M ( X ) , F ( n )[2 n + r ]) = 0for all r > X ∈ Sm /k . We have the Gersten-Quillen spectral sequence E p,q = ⊕ x ∈ X ( p ) Hom DM eff ( k ) ( M ( k ( x ))( p )[2 p ] , F ( n )[2 n + p + q ])= ⇒ Hom DM eff ( k ) ( M ( X ) , F ( n )[2 n + p + q ]) . For p > n , E p,q = 0 by remark 2.2.4. Using lemma 2.3.1 and Voevodsky’s can-cellation theorem [38], we see that E p,q = 0 for p + q > p ≤ n , whence theclaim.Next, note thatΩ mT ( F ( n )[2 n ]) = ( F ( n − m )[2 n − m ] for 0 ≤ m ≤ n m > n. Indeed, note that, for
G ∈ DM eff ( k ),Hom DM eff ( k ) ( G , Ω mT ( F ( n )[2 n ])) ∼ = Hom DM eff ( k ) ( G ( m )[2 m ] , F ( n )[2 n ]) . For m ≤ n , we have the canonical evaluation map ev : F ( n − m )[2 n − m ] → Ω mT ( F ( n )[2 n ]); the above identity says that ev induces the Tate twist mapHom DM eff ( k ) ( G , F ( n − m )[2 n − m ])) → Hom DM eff ( k ) ( G ( m )[2 m ] , F ( n )[2 n ]) . Voevodsky’s cancellation theorem [38] implies that the Tate twist map is an isomor-phism; as G was arbitrary, it follows that ev is an isomorphism. For the case m > n ,the right-hand side Hom DM eff ( k ) ( G ( m )[2 m ] , F ( n )[2 n ]) is zero by remark 2.2.4.Thus s mot (Ω mT ( F ( n )[2 n ])) = ( m ≥ , m = n F for m = n. In fact, we need only check for 0 ≤ m ≤ n . If 0 ≤ m < n , then Ω mT ( F ( n )[2 n ]) isin DM eff ( k )(1), hence the s mot (Ω mT ( F ( n )[2 n ])) = 0. Finally, Ω nT ( F ( n )[2 n ]) = F ,and thus s Ω nT ( F ( n )[2 n ]) = s mot ( F ) = F by remark 2.2.3.In particular, s mot (Ω mT ( F ( n )[2 n ])) is concentrated in cohomological degree 0 forall m , which shows that F ( n )[2 n ] is well-connected. (cid:3) Theorem 2.3.3.
Let F be a birational motivic sheaf. Then there is a naturalisomorphism H q − p ( X, F ( q )) := Hom DM eff ( k ) ( M ( X ) , F ( q )[2 q − p ]) ∼ = CH q ( X, p ; F ( q )[2 q ]) Proof.
Since F ( q )[2 q ] is well-connected, it follows from theorem 1.8.4 that the slices s motq ( F ( q )[2 q ]) are computed by the cycle complexes, i.e., there is a natural isomor-phism Hom DM eff ( k ) ( M ( X ) , s motq ( F ( q )[2 q ])[ − p ]) ∼ = CH q ( X, p ; F ( q )[2 q ]) . But s motq ( F ( q )[2 q ]) = F ( q )[2 q ] by remark 2.2.3. (cid:3) Remark . Let F be a birational sheaf. For Y ∈ Sm /k , we can define the groupof codimension q cycles on Y with values in F as z q ( Y ) F := ⊕ w ∈ Y ( q ) F ( k ( w )) , that is, an F -valued cycle on Y is a formal finite sum P i a i W i with each W i acodimension q integral closed subscheme of Y and a i ∈ F ( k ( W i )). The canonicalidentification F ( k ( w )) ∼ = H (( F ( q )[2 q ]) W ( Y ))for W ⊂ Y a codimension q integral closed subscheme gives the F -valued cyclegroups the usual properties of algebraic cycles, including proper pushforward, andpartially defined pull-back. In particular, for F = Z , we have the identification z q ( Y ) Z = z q ( Y ); OTIVES OF AZUMAYA ALGEBRAS 25 we will show in the next section that this identification is compatible with theoperations of proper pushforward, and pull-back (when defined).In addition, we have s mot (Ω qT ( F ( q )[2 q ])) ∼ = s mot ( F ) ∼ = F hence z q ( X, n ; F ( q )[2 q ]) = ⊕ w ∈ X ( q ) ( n ) F ( k ( w )) . Thus we can think of z q ( X, ∗ ; F ( q )[2 q ]) as the cycle complex of codimension q F -valued cycles in good position on X × ∆ ∗ .2.4. The sheaf Z . The most basic example of a birational motivic sheaf is theconstant sheaf Z . Here we show that the constructions of the previous section arecompatible with the classical operations on algebraic cycles.Let W ⊂ Y be a closed subset with Y ∈ Sm /k . We let z qW ( Y ) be the subgroupof z q ( Y ) consisting of cycles with support contained in W . Definition 2.4.1.
The category of closed immersions Imm k has objects ( Y, W )with Y ∈ Sm /k and W ⊂ Y a closed subset. A morphism f : ( Y, W ) → ( Y ′ , W ′ ) isa morphism f : Y → Y ′ in Sm /k such that f − ( W ′ ) red ⊂ W . Let Imm k ( q ) ⊂ Imm k be the full subcategory of closed subsets W ⊂ Y such that each component of W has codimension ≥ q .Note that for each morphism f : ( W ⊂ Y ) → ( W ′ ⊂ Y ′ ), the pull-back of cyclesgives a well-defined map f ∗ : z qW ′ ( Y ′ ) → z qW ( Y ). Definition 2.4.2.
Let f : Y ′ → Y be a morphism in Sch k , with Y and Y ′ equi-dimensional over k . We let z q ( Y, ∗ ) f ⊂ z q ( Y, ∗ ) be the subcomplex defined byletting z q ( Y, n ) f be the subgroup of z q ( Y, n ) generated by irreducible W ⊂ Y × ∆ n , W ∈ z q ( Y, n ), such that for each face F ⊂ ∆ n , each irreducible component of( f × id F ) − ( W ∩ X × F ) has codimension q on Y ′ × F .Assuming that f ( Y ′ ) is contained in the smooth locus of Y , the maps ( f × id ∆ n ) ∗ thus define the morphism of complexes f ∗ : z q ( Y, ∗ ) f → z q ( Y ′ , ∗ )We recall Chow’s moving lemma in the following form: Theorem 2.4.3 (Bloch [4]) . Suppose that Y is a quasi-projective k -scheme, andthat f : Y ′ → Y has image contained in the smooth locus of Y . Then the inclusion : z q ( Y, ∗ ) f → : z q ( Y, ∗ ) is a quasi-isomorphism. One proves this by first using “moving by translation” to prove the result for Y = P n , then using the method of the projecting cone to prove the result for Y projective, and finally using Bloch’s localization theorem, applied to a projectivecompletion Y → ¯ Y , to prove the general case. Lemma 2.4.4.
Take Y ∈ Sm /k , W ⊂ Y a closed subset. Suppose that each irre-ducible component of W ⊂ Y has codimension ≥ q . Then there is an isomorphism ρ Y,W,q : H qW ( Y, Z ( q )) → z qW ( Y ) such that the ρ Y,W,q define a natural isomorphism of functors from
Imm op k,q to Ab .In addition, the maps ρ Y,W,q are natural with respect to proper push-forward.
Proof.
By definition, Z (1)[2] is the reduced motive of P , Z (1)[2] = ˜ M ( P ) ∼ = cone( M ( k ) i ∞∗ −−→ M ( P )) , and Z ( q )[2 q ] is the q th tensor power of Z (1)[2]. Via the localization functor RC Sus ∗ : D − ( Sh tr Nis ( Sm /k )) → DM eff − ( k )and using [11, V, corollary 4.1.8], we have the isomorphism Z ( q )[2 q ] ∼ = C Sus ∗ ( z q . fin ( A q ))and the natural identification H q + p ( Y, Z ( q )) ∼ = H p Nis ( Y, C
Sus ∗ ( z q . fin ( A q ))) ∼ = H p ( C Sus ∗ ( z q . fin ( A q ))( Y )) . In particular, we have the natural identification of the motivic cohomology withsupports H qW ( Y, Z ( q )) ∼ = H (cone( C Sus ∗ ( z q . fin ( A q ))( Y ) → C Sus ∗ ( z q . fin ( A q ))( Y \ W ))[ − . Set C Sus ∗ ( z q . fin ( A q ))( Y ) W := cone( C Sus ∗ ( z q . fin ( A q ))( Y ) → C Sus ∗ ( z q . fin ( A q ))( Y \ W ))[ − . In addition, from the definition of the Suslin complex, we have the evident in-clusion of complexes C Sus ∗ ( z q . fin ( A q ))( Y ) ⊂ z q ( Y × A q , ∗ ) f × id ⊂ z q ( Y × A q , ∗ ) . It follows from [11, VI, theorem 3.2, V, theorem 4.2.2] that the inclusion C Sus ∗ ( z q . fin ( A q ))( Y ) ⊂ z q ( Y × A q , ∗ )is a quasi-isomorphism; by theorem 2.4.3, the inclusion C Sus ∗ ( z q . fin ( A q ))( Y ) ⊂ z q ( Y × A q , ∗ ) f × id is a quasi-isomorphism as well.Let U = Y \ W , U ′ := Y ′ \ W ′ and let f U : U ′ → U be the restriction of f .Setting z q ( Y, ∗ ) W,f = cone( z q ( Y, ∗ ) f → z q ( U, ∗ ) f U )[ − , we thus have the quasi-isomorphism C Sus ∗ ( z q . fin ( A q ))( Y ) W → z q ( Y × A q , ∗ ) W × A q ,f × id . We have the commutative diagram C Sus ∗ ( z q . fin ( A q ))( Y ) W / / ( f ∗ ,f ∗ U ) (cid:15) (cid:15) z q ( Y × A q , ∗ ) W × A q ,f × id( f × id ∗ ,f U × id ∗ ) (cid:15) (cid:15) C Sus ∗ ( z q . fin ( A q ))( Y ′ ) W ′ / / z q ( Y ′ × A q , ∗ ) W ′ × A q Since the horizontal maps are quasi-isomorphisms, we can use the right-hand sideto compute f ∗ : H qW ( Y, Z ( q )) → H qW ′ ( Y ′ , Z ( q )).By the homotopy property for the higher Chow groups, and using the movinglemma again, the pull-back maps p ∗ : z q ( Y, ∗ ) W,f → z q ( Y × A q , ∗ ) W × A q ,f × id p ∗ : z q ( Y ′ , ∗ ) W ′ → z q ( Y × A q , ∗ ) W ′ × A q OTIVES OF AZUMAYA ALGEBRAS 27 are quasi-isomorphisms. Thus we can use f ∗ : z q ( Y, ∗ ) W,f → z q ( Y ′ , ∗ ) W ′ to compute f ∗ : H qW ( Y, Z ( q )) → H qW ′ ( Y ′ , Z ( q )).Let d = dim k Y . Chow’s moving lemma together with the localization distin-guished triangle z d − q ( W, ∗ ) → z d − q ( Y, ∗ ) → z d − q ( U, ∗ )shows that the inclusion z d − q ( W, ∗ ) ⊂ z d − q ( Y, ∗ ) f induces a quasi-isomorphism z d − q ( W, ∗ ) → z q ( Y, ∗ ) W,f . Similarly, the inclusion z d ′ − q ( W ′ , ∗ ) ⊂ z d ′ − q ( Y ′ , ∗ ), d ′ := dim k Y ′ , induces a quasi-isomorphism z d ′ − q ( W ′ , ∗ ) → z q ( Y ′ , ∗ ) W ′ . Since each component of W has codimension ≥ q on Y , it follows that the inclusion z d − q ( W ) = z d − q ( W, → z d − q ( W, ∗ )is a quasi-isomorphism. As z d − q ( W ) = z qW ( Y ), we thus have the isomorphism ρ Y,W,q : z qW ( Y ) → H qW ( Y, Z ( q ))In addition, the diagram z d − q ( W ) z qW ( Y ) / / f ∗ (cid:15) (cid:15) z q ( Y, ∗ ) W,ff ∗ (cid:15) (cid:15) z d ′ − q ( W ′ ) z qW ′ ( Y ′ ) / / z q ( Y ′ , ∗ ) W ′ commutes. Combining this with our previous identification of H qW ( Y, Z ( q )) with H ( z q ( Y, ∗ ) W,f ) and H qW ′ ( Y ′ , Z ( q )) with H ( z q ( Y ′ , ∗ ) W ′ ) shows that the isomor-phisms ρ Y,W,q are natural with respect to pull-back.The compatibility of the ρ Y,W,q with proper push-forward is similar, but easier,as one does not need to introduce the complexes z q ( Y × A q , ∗ ) f × id , etc., or useChow’s moving lemma. We leave the details to the reader. (cid:3) Now take X ∈ Sm /k , W ∈ S ( q ) X ( n ). We thus have the isomorphism ρ X × ∆ n ,W,q : H qW ( X × ∆ n , Z ( q )) → z qW ( X × ∆ n )In addition, if W ′ ⊂ W is a closed subset of codimension > q on X × ∆ n , then therestriction map H qW ( X × ∆ n , Z ( q )) → H qW \ W ′ ( X × ∆ n \ W ′ , Z ( q ))is an isomorphism. Noting that H (( Z ( q )[2 q ]) W ( X × ∆ n )) = H qW ( X × ∆ n , Z ( q ))it follows from the definition of z q ( X, n ; Z ( q )[2 q ]) that we have z q ( X, n ; Z ( q )[2 q ]) = lim −→ W ⊂ X × ∆ n W ∈S ( q ) X ( n ) H qW ( X × ∆ n , Z ( q )) . Thus taking the limit of the isomorphisms ρ X × ∆ n ,W,q over W ∈ S ( q ) X ( n ) gives theisomorphism ρ X,n : z q ( X, n ; Z ( q )[2 q ]) → z q ( X, n ) . Proposition 2.4.5.
For X ∈ Sm /k , the maps ρ X,n define an isomorphism ofcomplexes z q ( X, ∗ ; Z ( q )[2 q ]) ρ X −−→ z q ( X, ∗ ) natural with respect to flat pull-back.Proof. It follows from lemma 2.4.4 that the isomorphisms ρ X,W,n are natural withrespect to the pull-back maps in Imm k ( q ); in particular, with respect to flat pull-back and with respect to the face maps X × ∆ n − → X × ∆ n . Passing to the limitover W ∈ S ( q ) X ( n ) proves the result. (cid:3) The sheaves K A and Z A We apply the results of the previous sections to the K -theory of Azumaya al-gebras. The basic construction will be valid for a sheaf of Azumaya algebras overa fairly general base-scheme X ; as the general theory we have already discussed isonly available for X = Spec k , we are forced to repeat some of the constructions inthe more general setting before reducing the proof of the main result to the case X = Spec k .3.1. K A : definition and first properties. Fix a sheaf of Azumaya algebras A on an R -scheme of finite type X . For p : Y → X ∈ Sch X , we have the sheaf p ∗ A of Azumaya algebras on Y . We may sheafify the K -groups of p ∗ A for the Zariskitopology on Y , giving us the Zariski sheaves K A n on Sch X . Lemma 3.1.1.
Suppose that X is regular. Then (1) K A is an A homotopy invariant presheaf on Sm /X . (2) K A is a birational presheaf on Sm /X , i.e., for Y ∈ Sm /X , j : U → Y adense open subscheme, the restriction map j ∗ : K A ( Y ) → K A ( U ) is an isomorphism. Equivalently, K A is locally constant for the Zariskitopology on Sm /X , hence is a sheaf for the Nisnevich topology on Sm /X .Proof. The homotopy invariance follows from the fact that Y K ( Y ; A ) is ho-motopy invariant, and that the restriction map K ( Y, A ) | toK ( U, A ) is surjectivefor each open immersion U → Y in Sm /X .For the birationality property, we may assume that Y is irreducible. By corol-lary A.4, any object in the category P X ; A is locally A -projective, hence it sufficesto show that for each y ∈ Y , the map K ( A ⊗ O B O Y,y ) → K ( A ⊗ O B k ( Y ))is an isomorphism.Since Y is regular, surjectivity follows easily from corollary A.5. On the otherhand, since O Y,y is local, the category of finitely generated projective
A ⊗ O B O Y,y modules has a unique indecomposable generator ([9], [17, III.5.2.2]) and similarly,the category of finitely generated projective
A ⊗ O B k ( Y ) modules has a uniquesimple generator. Thus the map is also injective, completing the proof that K A isbirational.To see that K A is a sheaf for the Nisnevich topology, it suffices to check thesheaf condition on elementary Nisnevich squares; this follows directly from thebirationality property. (cid:3) OTIVES OF AZUMAYA ALGEBRAS 29
The reduced norm map.
Let Spec F → X be a point. We define a mapNrd F : Z ≃ K ( A F ) → K ( F ) = Z by mapping the positive generator of K ( A F ) to e F [ F ], where e F is the index of A F .Recall that, by definition, e F = [ D : F ] where D is the unique division F -algebrasimilar to A F . Lemma 3.2.1.
The assignment F Nrd F defines a morphism of sheaves Nrd : K A → Z which realizes K A as a subsheaf of the constant sheaf Z on Y . This is the reducednorm map attached to A .Proof. In view of lemma 3.1.1, it suffices to check that if L is a separable extensionof F , the diagram K ( A L ) Nrd L −−−−→ K ( L ) x x K ( A F ) Nrd K −−−−→ K ( F )commutes. This is classical: by Morita invariance, we may replace A F by a similardivision algebra D . Choose a maximal commutative subfield E ⊂ D which isseparable over F . First assume that L = E : then D L is split and Nrd L is anisomorphism by Morita invariance; on the other hand, the generator [ D ] of K ( D )maps to e times the generator of K ( D L ), which proves the claim in this specialcase. The general case reduces to the special case by considering a commutativecube involving the extension LE . (cid:3) The presheaf with transfers Z A . For a scheme X we let M X denote thecategory of coherent sheaves (of O X modules) on X . Given a sheaf of Azumayaalgebras A on X , we let M X ( A ) denote the category of sheaves of A -modules F which are coherent as O X -modules, using the structure map O X → A to define the O X -module structure on F . We let G ( X ; A ) denote the K -theory spectrum of theabelian category M X ( A ). If f : Y → X is a morphism, we often write G ( Y ; A ) for G ( Y ; f ∗ A ).Suppose that X is regular. Let f : Z → Y be a finite morphism in Sch X with Y in Sm /X . Restriction of scalars defines a map of sheaves f ∗ : f ∗ K ( Z ; A ) → G ( Y ; A ) . Using corollary A.5, we see that the natural map K ( Y ; A ) → G ( Y ; A )is an isomorphism, giving us the pushforward map f ∗ : K A ( Z ) → K A ( Y )Now take Y, Y ′ ∈ Sm /X and let Z ⊂ Y × X Y ′ be an integral subscheme whichis finite over Y and surjective onto a component of Y ; let p : Z → Y , p ′ : Z → Y ′ be the maps induced by the projections. Define Z ∗ : K A ( Y ′ ) → K A ( Y )by Z ∗ := p ∗ ◦ p ′∗ . For X regular, this operation extends to Cor X ( Y, Y ′ ) by linearity. Lemma 3.3.1.
Suppose X regular. For Z ∈ Cor X ( Y, Y ′ ) , Z ∈ Cor X ( Y ′ , Y ′′ ) wehave ( Z ◦ Z ) ∗ = Z ∗ ◦ Z ∗ Proof.
We already have a canonical operation of Cor X ( − , − ) on the constant sheaf Z making Z a sheaf with transfers; one easily checks that this action agrees withthe action we have defined above for A = O X . It is similarly easy to check that, for Z integral and f : Z → Y finite and surjective with Y smooth, f ∗ commutes withNrd. Since Nrd is injective, this implies that K A is also a sheaf with transfers, asdesired. (cid:3) Definition 3.3.2.
Let X be a regular R -scheme of finite type, A a sheaf of Azumayaalgebras on X . We let Z A denote the Nisnevich sheaf with transfers on Sm /X defined by K A . Remark . The reduced norm map Nrd : K A → Z defines a map of Nisnevichsheaves with transfers Nrd : Z A → Z . Lemma 3.3.4.
The subsheaf with transfers ( Z A , Nrd) of the constant sheaf (withtransfers) Z only depends on the subgroup of Br ( X ) generated by A . In particular,it is Morita-invariant.Proof. Indeed, if B generates the same subgroup of Br ( X ) as A , there exist integers r, s such that A ⊗ X s is similar to B and B ⊗ X s is similar to A . This implies readilythat A and B have the same splitting fields (say, over a point Spec F of X ), hencehave the same index (say, over any extension of F ). (cid:3) Remark . The maps K ( F ) → K ( A F ) given by extension of scalars also definea morphism of sheaves Z → Z A . But this morphism is not Morita-invariant.In case X is the spectrum of a field, lemma 3.1.1 yields Proposition 3.3.6.
Take X = Spec k , k a field, and let A be a central simplealgebra over k . Then the sheaf with transfers Z A on Sm /k is a birational motivicsheaf. Severi-Brauer varieties.
Let p : SB ( A ) → X be the Severi-Brauer varietyassociated to A . Lemma 3.4.1.
Suppose X = Spec k , k a field. Then the subgroup Nrd( K ( A )) ⊂ K ( k ) = Z is the same as the image p ∗ (CH ( SB ( A ))) ⊂ CH ( B ) = Z . Moreover, p ∗ : CH ( SB ( A )) → Z is injective.Proof. This is a theorem of Panin [28]. We recall the proof of the first statement.Let x = Spec K be a closed point of SB ( A ). Then K is a finite extension of F which is a splitting field of A . It is classical that K is a maximal commutativesubfield of some algebra similar to A ; in particular, [ K : F ] is divisible by the indexof A . Conversely, replacing A by a similar division algebra D , for any maximalcommutative subfield L ⊂ D , [ L : F ] equals the index of A . (cid:3) Now let us come back to the case where X is regular. Let us denote by CH ( SB ( A ) /X ) the sheafification (for the Zariski topology) of the presheaf on Sm /X U CH dim k U ( SB ( A ) × X U ) . OTIVES OF AZUMAYA ALGEBRAS 31
The push-forward p U ∗ : CH dim k U ( SB ( A ) × X U ) → CH dim k U ( U ) = Z defines the map deg : CH ( SB ( A ) /X ) → Z where Z is viewed as a constant sheaf on ( Sm /X ) Zar . Lemma 3.4.2.
The map deg identifies CH ( SB ( A ) /X )) with the locally constantsubsheaf Nrd( Z A ) ⊂ Z . In other words, there is a canonical isomorphism of sub-sheaves of Z ( Z A , Nrd) ≃ ( CH ( SB ( A ) /X )) , deg) . Proof.
As we have already remarked, the result is true at Spec F , F a field. For Y local, the restriction map j ∗ : CH dim X ( SB ( A ) × X Y ) → CH ( SB ( A ⊗ O B k ( Y ))(dim X := the Krull dimension) is surjective, from which the result easily follows. (cid:3) Remark . It is evident that the transfer structure of lemma 3.3.1 on Z A coin-cides with the natural transfer structure on CH ( SB ( A ) /X )) .3.5. K A for embedded schemes. Let k be a field. We fix a sheaf of Azumayaalgebras A on some finite type k -scheme X ; we do not assume that X is regular.As a technical tool, we extend the definition of the category Imm k (defini-tion 2.4.1) as follows: Definition 3.5.1.
The category of closed immersions Imm
X,k has objects (
Y, W )with Y ∈ Sm /k and W ⊂ X × k Y a closed subset. A morphism f : ( Y, W ) → ( Y ′ , W ′ ) is a morphism f : Y → Y ′ in Sm /k such that (id × f ) − ( W ′ ) red ⊂ W .Let Y be a smooth k -scheme, let i : W → X × k Y be a reduced closed subschemeof pure codimension. Letting W reg ⊂ W be the regular locus, we have the (constant)Zariski sheaf K A defined on W reg . We describe how to extend K A to W ⊂ X × k Y so that ( Y, W )
7→ K A ( W ⊂ X × k Y )defines a presheaf K A on Imm X,k .For this, we define K A on i : W → X × k Y to be K A ( W reg ), where j : W reg → W is the regular locus of W . The trick is to define the pull-back maps.We let G W ( X × k Y ; A ) denote the homotopy fiber of the restriction map G ( X × k Y ; A ) → G ( X × k Y \ W ; A ) Lemma 3.5.2.
Suppose that X is local, with closed point x . Let i : Y ′ → Y be a closed embedding in Sm ess /k , with Y local having closed point y . Let W ⊂ X × Y be a closed subset such that X × Y ′ ∩ W = ( x, y ) (as a closed subset). If codim X × Y W > codim X × Y ′ ( x, y ) , then the restriction map i ∗ : G W ( X × Y ; A ) → G ( x,y )0 ( X × Y ′ ; A ) is the zero map. Proof.
The proof is a modification of Quillen’s proof of Gersten’s conjecture. Mak-ing a base-change to k ( x, y ), and noting that G ( x,y )0 ( X × Y ; A )) = G (( x, y ); A ), wemay assume that k ( y ) = k ( x ) = k . Since K -theory commutes with direct limits (ofrings) we may replace Y and Y ′ with finite type, smooth affine k -schemes, and weare free to shrink to a smaller neighborhood of y in Y as needed.Let ¯ W ⊂ Y be the closure of p ( W ). Note that the condition codim X × Y W > codim X × Y ′ ( x, y ) implies that dim k W < dim k Y , hence ¯ W is a proper closed subsetof Y . Take a divisor D ⊂ Y containing ¯ W . Then there is a morphism π : Y → A n ,n = dim k Y −
1, such that π is smooth in a neighborhood of y and π : D → A n isfinite. Let W ′ := π − ( π ( W )) . Choosing π general enough, and noting thatcodim X × Y W ′ = codim X × Y W − ≥ codim X × Y ′ ( x, y ) = dim k X × Y ′ , we may assume that W ′ ∩ X × Y ′ is a finite set of closed points, say T . Let S ⊂ D be the finite set of closed points π − ( π ( y )) ∩ D .The inclusion D → Y induces a section s : D → Y × A n D to p : Y × A n D → D ;since π is smooth at y ′ , s ( D ) is contained in the regular locus of Y × A n D andis hence a Cartier divisor on Y × A n D . Noting that p : Y × A n D → Y is finite,there is an open neighborhood U of S in Y such that s ( D ) ∩ Y × A n U is a principaldivisor; let t be a defining equation. Let D U := D ∩ U .This gives us the commutative diagram Y × A n U q / / p (cid:15) (cid:15) UD Us O O i ; ; vvvvvvvvv with q finite. Thus we have, for M ∈ M D U ; A , the exact sequence0 → q ∗ ( p ∗ M ) q ∗ ( × t ) −−−−→ q ∗ ( p ∗ M )) → i ∗ M → M .Note that, if M is supported in W , then q ∗ ( p ∗ M ) is supported in W ′ . Letting i ′ : W → W ′ be the inclusion, our exact sequence gives us the identity[ i ′∗ M ] = 0 in G W ′ ( Y ; A ) , hence i ∗ ([ i ′∗ M ]) = 0 in G W ′ ∩ Y ′ ( Y ′ ; A ) . Let ¯ i : ( x, y ) → T be the inclusion. We have the commutative diagram G W ( X × Y ; A ) i ′∗ / / i ∗ (cid:15) (cid:15) G W ′ ( X × Y ′ ; A ) i ∗ (cid:15) (cid:15) G ( x,y )0 ( X × Y ′ ; A ) , ¯ i ∗ / / G T ( X × Y ′ ; A ) . Since T is a finite set of points containing ( x, y ), G T ( X × Y ′ ; A ) = G ( x,y )0 ( X × Y ′ ; A ) ⊕ G T \{ ( x,y ) } ( X × Y ′ ; A ) , OTIVES OF AZUMAYA ALGEBRAS 33 with ¯ i ∗ the inclusion of the summand G ( x,y )0 ( X × Y ′ ; A ), from which the resultfollows directly. (cid:3) For a closed immersion i : W → X × Y , restricting to the generic points of W and using the canonical weak equivalence G ( W ; A ) → G W ( X × Y ; A )gives the map ϕ W : G W ( X × Y ; A ) → K A ( W ) . Each map of pairs f : ( i ′ : W ′ → X × Y ′ ) → ( i : W → X × Y ) induces acommutative diagram of inclusions X × Y ′ \ W ′ / / (cid:15) (cid:15) X × Y ′ (cid:15) (cid:15) X × Y \ W / / X × Y ;Noting that id × f : X × Y ′ → X × Y is an lci morphism, we may apply G ( − ) tothis diagram, giving us the induced map on the homotopy fibers f ∗ : G W ( X × Y ; A ) → G W ′ ( X ′ ; A ) . Thus, we have the diagram G W ( X × Y ; A ) f ∗ / / ϕ W (cid:15) (cid:15) G W ′ ( X × Y ′ ; A ) ϕ W ′ (cid:15) (cid:15) K A ( W ) K A ( W ′ )In order that f ∗ descend to a map f ∗ : K A ( W ) → K A ( W ′ ) , it therefore suffices to prove: Lemma 3.5.3. (1) For each i : W → X × Y , the map ϕ W is surjective.(2) ϕ W ′ ( f ∗ (ker ϕ W )) = 0 .Proof. The surjectivity of ϕ W follows from Quillen’s localization theorem, whichfirst of all identifies K W ( X × Y ; A ) with G ( W ; A ) and secondly implies that therestriction map j ∗ : G ( W ; A ) → G ( k ( W ); A ) = K ( k ( W ); A )is surjective.For (2), we can factor f as a composition of a closed immersion followed bya smooth morphism. In the second case, f − ( W \ Spec k ( W )) is a proper closedsubset of W ′ , hence classes supported in W \ Spec k ( W ) die when pulled back by f and restricted to k ( W ′ ). Thus we may assume f is a closed immersion.Fix a generic point w ′ = ( x, y ) of W ′ . We may replace X with Spec O X,x andreplace Y with Spec O Y,y . Making a base-change, we may assume that k ( x, y ) isfinite over k . Since X × k Y is smooth, it follows thatcodim X × Y W ≥ codim X × Y ′ ( x, y ) . Let W ′′ ⊂ W is a closed subset of W containing no generic point of W . Thencodim X × Y W ′′ > codim X × Y ′ ( x, y ) , hence by lemma 3.5.2 the restriction map G W ′′ ( X × Y ; A ) → G ( x, y )( X × Y ′ ; A )is the zero map. By Quillen’s localization theorem we haveker ϕ W = lim −→ G W ′′ ( X × Y ; A )over such W ′′ , which proves the lemma. (cid:3) The cycle complex.
Let T be a finite type k -scheme. We let dim k T denotethe Krull dimension of T ; we sometimes write d T for dim k T .We fix as above a finite type k -scheme X and a sheaf of Azumaya algebras A on X . We have the cosimplicial scheme ∆ ∗ with∆ n := Spec k [ t , . . . , t n ] / X i t i − face F of ∆ n is a closedsubscheme of the form t i = . . . , = t i s = 0. We let S Xr ( n ) be the set of closedsubsets W ⊂ X × ∆ n withdim k W ∩ X × F ≤ r + dim k F for all faces F ⊂ ∆ n . We order S Xr ( n ) by inclusion. If g : ∆ m → ∆ n is the mapcorresponding to a map g : [ m ] → [ n ] in Ord , and W is in S Xr ( n ), then g − ( W ) isin S Xr ( m ), so n
7→ S Xr ( n ) defines a simplicial set. We let X r ( n ) ⊂ S Xr ( n ) denotethe set of irreducible W ∈ S Xr ( n ) with dim k W = r + n . Definition 3.6.1. z r ( X, n ; A ) := ⊕ W ∈ X r ( n ) K ( k ( W ); A ) . Remark . Let W ⊂ X × ∆ n be a closed subset. Then restriction to the genericpoints of W gives the isomorphism K A ( W ⊂ X × ∆ n ) ∼ = ⊕ w ∈ W (0) K ( k ( w ); A ) . Thus, we can identify z r ( X, n ; A ) with the quotient: z r ( X, n ; A ) ∼ = lim −→ W ∈S Xr ( n ) K A ( W ⊂ X × ∆ n )lim −→ W ′ ∈S Xr − ( n ) K A ( W ′ ⊂ X × ∆ n )Suppose each irreducible W ′ ∈ S Xr − ( n ) is contained in some irreducible W ∈S Xr ( n ) with dim k W = r + n ; as the map K A ( W ′ ) → K A ( W )is in this case the zero-map, it follows that z r ( X, n ; A ) ∼ = lim −→ W ∈S Xr ( n ) K A ( W ⊂ X × ∆ n )if this condition is satisfied, e.g., for X quasi-projective over k . OTIVES OF AZUMAYA ALGEBRAS 35
Let g : ∆ m → ∆ n be a map corresponding to g : [ m ] → [ n ] in Ord . Bylemma 3.5.3 and the above remark, we have a well-defined pullback mapid × g ∗ : z r ( X, n ; A ) → z r ( X, m ; A ) , giving us the simplicial abelian group n z r ( X, n ; A ). We let ( z r ( X, ∗ ; A ) , d )denote the associated complex, i.e., d n := n X i =0 ( − i (id × δ n − i ) ∗ : z r ( X, n ; A ) → z r ( X, n − A ) . Definition 3.6.3.
We define the higher Chow groups of dimension r with coeffi-cients in A as CH r ( X, n ; A ) := H n ( z r ( X, ∗ ; A )) . Elementary properties.
The standard elementary properties of the cyclecomplexes are also valid with coefficients in A , if properly interpreted. Projective pushforward . Let f : X ′ → X be a proper morphism. For Y ∈ Sm /k and W ⊂ X ′ × Y , we have the pushfoward map f × id ∗ : G W ( X ′ × Y, f ∗ A ) → G f × id( W )0 ( X × Y ; A )Also, if g : Y ′ → Y is a morphism in Sm /k , then the diagram X ′ × Y ′ f × id / / id × g (cid:15) (cid:15) X × Y ′ id × g (cid:15) (cid:15) X ′ × Y f × id / / X × Y is cartesian and Tor-independent, and the vertical maps are lci morphisms, fromwhich it follows that the diagram (with W ′ = (id × g ) − ( W )) G W ′ ( X ′ × Y ′ ) f × id / / G f × id( W ′ )0 ( X × Y ′ ) id × g (cid:15) (cid:15) G W ( X ′ × Y ) id × g ∗ O O f × id / / G f × id( W )0 ( X × Y ) id × g ∗ O O is commutative.Thus, then maps ( f × id ∆ n ) ∗ induce a map of complexes f ∗ : z r ( X ′ , ∗ ; f ∗ A ) → z r ( X, ∗ ; A )with the evident functoriality. Flat pullback . Let f : X ′ → X be a flat morphism. For Y ∈ Sm /k and W ⊂ X × k Y ,we have the pull-back map f × id ∗ : G W ( X × Y, A ) → G ( f × id) − ( W )0 ( X ′ × Y, f ∗ A )commuting with the pull-back maps id × g ∗ for g : Y ′ → Y a map in Sm /k . Since f is flat, the codimension of W is preserved, hence the pullback maps f × id ∗ ∆ n induce a map of complexes f ∗ : z r ( X, ∗ ; A ) → z r ( X ′ , ∗ ; f ∗ A ) functorially in f . Elementary moving lemmas and homotopy property . Definition 3.7.1.
Fix a Y ∈ Sm /k and let C be a finite set of locally closed subsetsof Y . Let X × Y C r ( n ) be the set of irreducible dimension r + n closed subsets W of X × Y × ∆ n such that W is in X × Y r ( n ) and for each C ∈ C W ∩ X × C × ∆ n is in S X × Cr ( n ) . We have the subcomplex z r ( X × Y, ∗ ; F ) C of z r ( X × Y, ∗ ; F ), with z r ( X × Y, n ; F ) C = ⊕ W ∈ X × Y C r ( n ) K A ( W ) . Exactly the same proof as for [5, lemma 2.2], using translation by GL n , givesthe following: Lemma 3.7.2.
Let C be a finite set of locally closed subsets of Y , with Y = A n or Y = P n − . Then the inclusion z r ( X × Y, ∗ ; A ) C → z r ( X × Y, ∗ ; A ) is a quasi-isomorphism. Similarly, we have
Lemma 3.7.3.
The pull-back map z r ( X, ∗ ; A ) → z r +1 ( X × A ; A ) is a quasi-isomorphism. Localization.
Let j : U → X be an open immersion with closed complement i : Z → X . Let Y be in Sm /k . If W ⊂ X × Y is an irreducible closed subsetsupported in Z × Y , then i × id induces an isomorphism i × id ∗ : G W ( X × Y, i ∗ A ) → G W ( X × Y ; A ) , which in turn induces the isomorphism i ∗ : K i ∗ A ( W ) → K A ( W )Similarly, if the generic point of W lives over U × Y , then we have the surjection j × id ∗ : G W ( X × Y ; A ) → G W ∩ U × Y ( U × Y, j ∗ A )inducing an isomorphism j ∗ : K A ( W ) → K j ∗ A ( W ∩ U × Y )This yields the termwise exact sequence of complexes(3.8.1) 0 → z r ( Z, ∗ , i ∗ A ) i ∗ −→ z r ( X, ∗ ; A ) j ∗ −→ z r ( U, ∗ , j ∗ A )It follows from the main result of [21] that Lemma 3.8.1.
The inclusion j ∗ ( z r ( X, ∗A )) ⊂ z r ( U, ∗ , j ∗ A ) is a quasi-isomorphism hence OTIVES OF AZUMAYA ALGEBRAS 37
Corollary 3.8.2.
The sequence (3.8.1) thus determines a canonical distinguishedtriangle in D − ( Ab ) , and we have the long exact localization sequence . . . → CH r ( Z, n, i ∗ A ) i ∗ −→ CH r ( X, n ; A ) j ∗ −→ CH r ( U, n, j ∗ A ) → CH r ( Z, n − , i ∗ A ) → . . . This in turn yields the
Mayer-Vietoris distinguished triangle for X = U ∪ V , U, V ⊂ X Zariski open subschemes(3.8.2) z r ( X, ∗ ; A ) → z r ( U, ∗ ; A U ) ⊕ z r ( V, ∗ ; A V ) → z r ( U ∩ V, ∗ ; A U ∩ V ) → z r ( X, ∗ − A )3.9. Extended functoriality.
Assume now that X is equi-dimensional over k (butnot necessarily smooth). A modification of the method of [19], derived from Chow’smoving lemma, yields a functorial model for the assignment Y z r ( X × Y, ∗ ; A );as we will not need the functoriality in this paper, we omit a further discussion ofthis topic.3.10. Reduced norm.
For X ∈ Sch k , A = k , the complex z r ( X, ∗ ; k ) is justBloch’s cycle complex z r ( X, ∗ ). Indeed, for a field F , we have the canonical iden-tification of K ( F ) with Z by the dimension function, giving the isomorphism z r ( X, n ; k ) = ⊕ w ∈ X ( r ) ( n ) K ( k ( w )) ∼ = ⊕ w ∈ X ( r ) ( n ) Z = z r ( X, n ) . In addition, if W ⊂ X × ∆ n is an integral closed subscheme of dimension d , i :∆ n − → ∆ n is a codimension one face and if W is not contained in X × i (∆ n − ),then it follows directly from Serre’s intersection multiplicity formula that the imageof (id × i ) ∗ ([ O W ]) in ⊕ w ∈ ( X × ∆ n − ) ( d − K ( k ( w )) goes to the pull-back cycle (id × i ) ∗ ([ W ]) under the isomorphism ⊕ w ∈ ( X × ∆ n − ) ( d − K ( k ( w )) ∼ = z d − ( X × ∆ n − ) . Now take A to be a sheaf of Azumaya algebras on X . The collection of reducednorm maps Nrd A k ( w ) : K ( k ( w ); A ) → K ( k ( w ))thus defines the homomorphismNrd X,n ; A : z r ( X, n ; A ) → z r ( X, n ) . Lemma 3.10.1.
The maps
Nrd
X,n ; A define a map of simplicial abelian groups n [Nrd X,n ; A : z r ( X, n ; A ) → z r ( X, n )] . Proof.
We note that the maps Nrd X ′ ,n ; A for X ′ → X ´etale define a map ofpresheaves on X ´et . Both z r ( X, n ; A ) and z r ( X, n ) are sheaves for the Zariskitopology on X and Nrd X,n ; A defines a map of sheaves, so we may assume that X is local. If X ′ → X is an ´etale cover, then z r ( X, n ; A ) → z r ( X ′ , n ; A ) and z r ( X, n ) → z r ( X ′ , n ) are injective, so we may replace X with any ´etale cover. Since A is locally a sheaf of matrix algebras on X ´et , we may assume that A = M n ( O X ).In this case, Nrd X,n ; A is just the Morita isomorphism; we thus may extend Nrd X,n ; A to the Morita isomorphismNrd W : G W ( X × ∆ n ; A ) → G W ( X × ∆ n ) for every W ∈ S X ( r ) ( n ). But the pull-back maps g ∗ : z r ( X, n ; A ) → z r ( X, m ; A ) and g ∗ : z r ( X, n ) → z r ( X, m ) for g : [ m ] → [ n ] in Ord are defined by lifting elementsin z r ( X, n ; A ) (resp. z r ( X, n )) to G W ( X × ∆ n ; A ) (resp. G W ( X × ∆ n )) for some W , applying (id × g ) ∗ and mapping to z r ( X, m ; A ) (resp. z r ( X, m )). Thus themaps Nrd
X,n ; A define an isomorphism of simplicial abelian groups, completing theproof. (cid:3) Thus we have maps Nrd X, A : z r ( X, ∗ ; A ) → z r ( X, ∗ )Nrd X ; A : CH r ( X, n ; A ) → CH r ( X, n )The naturality properties of Nrd show that the maps Nrd X, A are natural withrespect to flat pull-back and proper push forward (on the level of complexes).4. The spectral sequence
We are now ready for the first of our main constructions and results. We beginby discussing the homotopy coniveau tower associated to the G -theory of sheafof Azumaya algebras A on a scheme X . Our main result (theorem 4.1.3) is theidentification of the layers in the homotopy coniveau tower with the Eilenberg-Maclane spectra associated to the twist cycle complex z p ( X, ∗ ; A ). The proof isexactly the same as for standard K -theory K ( X ) (see [20, 18]), except that at onepoint we need to use an extension of some regularity results from K ( − ) to K ( − ; A );this extension is given in Appendix B.We then turn to the case X = Spec k , where we have the motivic Postnikovtower for the presheaf K A . We show how our computation of the layers in thehomotopy coniveau tower for K A ( X ) = K ( X ; A ⊗ k O X ), for each X ∈ Sm /k , leadto a computation of of the layers in the motivic Postnikov tower for K A . Thiscompletes the proof of our first main theorem 1 (see theorem 4.5.5). We concludethis section with a comparison of the reduced norm maps in motivic cohomologyand K -theory, and some computations of the Atiyah-Hirzebruch spectral sequencein low degrees.4.1. The homotopy coniveau filtration.
Following [18] we define G ( p ) ( X, n ; A ) := lim −→ W ∈S Xp ( n ) G W ( X × ∆ n ; A )giving the simplicial spectrum n G ( p ) ( X, n ; A ), denoted G ( p ) ( X, − ; A ). Notethat, for all p ≥ d X , the evident map G ( p ) ( X, − ; A ) → G ( X × ∆ ∗ ; A )is an isomorphism. Remark . In order that n G ( p ) ( X, n ; A ) form a simplicial spectrum, oneneeds to make the G -theory with support strictly functorial. This is done by firstreplacing the categories M X × ∆ n ( A ) with the full subcategory M X × ∆ n ( A ) ′ of A modules which are coherent sheaves on X × ∆ n and are flat with respect to allinclusions X × F → X × ∆ n , F ⊂ ∆ n a face. Quillen’s resolution theorem showsthat K ( M X × ∆ n ( A ) ′ ) → K ( M X × ∆ n ( A )) OTIVES OF AZUMAYA ALGEBRAS 39 is a weak equivalence. One then uses the usual trick of replacing M X × ∆ n ( A ) ′ with sequences of objects together with isomorphisms (indexed by the morphismsin Ord ) to make the pull-backs strictly functorial.A similar construction makes Y G ( X × k Y, A ) strictly functorial on Sm /k ;we will use this modification from now on without further mention.Since G ( X × − ; A ) is homotopy invariant, the canonical map G ( X ; A ) → G ( d X ) ( X, − ; A )is a weak equivalence (on the total spectrum). This gives us the homotopy coniveautower (4.1.1) . . . → G ( p − ( X, − ; A ) → G ( p ) ( X, − ; A ) → . . . → G ( d X ) ( X, − ; A ) ∼ G ( X ; A ) . Setting G ( p/p − r ) ( X, − ; A ) equal to the homotopy cofiber of G ( p − r ) ( X, − ; A ) → G ( p ) ( X, − ; A ), the tower (4.1.1) yields the spectral sequence(4.1.2) E p,q = π − p − q ( G ( q/q − ( X, − ; A )) = ⇒ G − p − q ( X ; A ) Remarks .
1. Let T be a finite type k -scheme, W ⊂ T a closed subschemewith open complement j : U → T and A a sheaf of Azumaya algebras on T . Wehave the homotopy fiber sequence G W ( T ; A ) → G ( T ; A ) → G ( U ; j ∗ A )In addition, the spectra G ( T ; A ) and G ( U ; j ∗ A ) are -1 connected, and the restrictionmap j ∗ : G ( T ; A ) → G ( U ; j ∗ A )is surjective. Thus G W ( T ; A ) is -1 connected, hence the spectra G ( p ) ( X, n ; A ) are-1 connected for all n and p .2. Noting that S Xp ( n ) = ∅ for p + n <
0, the -1 connectedness of G ( p ) ( X, n ; A )implies that π N ( G ( p ) ( X, − ; A )) = 0for N < − p , i.e., that G ( p ) ( X, − ; A ) is − p − G ( p/p − r ) ( X, − ; A ) is − p − r ≥
0, that the natural map G ( X ; A ) → holim n G ( d X / − n ) ( X ; A )is a weak equivalence and that the spectral sequence (4.1.2) is strongly convergent.Our main result in this section is Theorem 4.1.3.
There is a natural isomorphism π n ( G ( p/p − ( X, − ; A )) ∼ = CH p ( X, n ; A ) . Corollary 4.1.4.
There is a strongly convergent spectral sequence E p,q = CH q ( X, − p − q ; A )) = ⇒ G − p − q ( X ; A )The proof is in three steps: we first define a natural “cycle map”cyc : π n ( G ( p/p +1) ( X, − ; A )) → CH p ( X, n ; A ) . which will define the isomorphism. We then use the localization properties of G ( p/p +1) ( X, − ; A ) and CH p ( X, ∗ ; A ) to reduce to the case X = Spec F , F a field,and finally we apply theorem 1.8.4 to complete the proof. The cycle map.
Let T be a finite type k -scheme, W ⊂ T a closed subschemewith open complement j : U → T and A a sheaf of Azumaya algebras on T . Wehave the homotopy fiber sequence G W ( T ; A ) → G ( T ; A ) → G ( U ; j ∗ A )In addition, the spectra G ( T ; A ) and G ( U ; j ∗ A ) are -1 connected, and the restrictionmap j ∗ : G ( T ; A ) → G ( U ; j ∗ A )is surjective. Thus G W ( T ; A ) is -1 connected. In particular, this implies that thespectra G ( p ) ( X, n ; A ) are all -1 connected. A similar argument shows that thespectra G ( p/p − r ) ( X, n ; A ) are all -1 connected.As we have seen in remark 4.1.2(1), the the spectra G ( p/p − ( X, n ; A ) are all-1 connected. Let EM ( π G ( p/p − ( X, n A )) denote the Eilenberg-Maclane spec-trum with π = π G ( p/p − ( X, n A ) and all other homotopy groups 0. Since G ( p/p − ( X, n ; A ) is -1 connected, we have the map of spectra ϕ n : G ( p/p − ( X, n ; A ) → EM ( π G ( p/p − ( X, n ; A ))natural in n . Letting EM ( π G ( p/p − ( X, − ; A )) denote the simplicial spectrum n EM ( π G ( p/p − ( X, n ; A )), this gives us the natural map of simplicial spectra ϕ : G ( p/p − ( X, − ; A ) → EM ( π G ( p/p − ( X, − ; A )) . Lemma 4.2.1.
There is a natural map ψ n : π ( G ( p/p − ( X, n ; A )) → z p ( X, n ; A ) , which is an isomorphism if X = Spec F , F a field.Proof. Let W ⊂ X × ∆ n be a closed subset with generic points w , . . . , w r . Wehave the evident restriction map G W ( X × ∆ n ; A ) = G ( W ; A ) → ⊕ i G ( k ( w i ); A ) . Since Z A ( W ) = ⊕ i G ( k ( w i ); A ), we may define ψ n : π ( G ( p/p − ( X, n ; A )) → z p ( X, n ; A )by projecting ⊕ i G ( k ( w i ); A ) on the factors coming from the generic points of W ∈ S Xp ( n ) having dimension n + r over k . By lemma 3.5.3, ψ n is natural in n .Suppose now that X = Spec F , F a field; making a base-change and replacing p with p − dim k X , we may assume that F = k (note that in this case we may assume p ≤ X × ∆ n ∼ = A nk . It is easy to see that, for each W ∈ S Xp ( n ),the intersection of − p hypersurfaces of sufficiently high degree, containing W , is in S Xp ( n ) and has pure dimension p + n . Thus the closed subsets W ∈ S Xp ( n ) of puredimension p + n are cofinal in S Xp ( n ).Identify z p ( X, n ; A ) with the direction sum ⊕ w G ( k ( w ); A ) as w runs over thegeneric points of dimension r + n W ∈ S Xp ( n ). From the localization sequence, wesee that the map lim −→ W ∈S Xp ( n ) G ( W ; A ) → ⊕ w G ( k ( w ); A )is surjective, with kernel the subgroup generated by the image of groups G ( W ′ ; A )with dim W ′ < p + n and W ′ ⊂ W for some W ∈ S Xp ( n ). The result thus followsfrom lemma 4.2.2 below. (cid:3) OTIVES OF AZUMAYA ALGEBRAS 41
Lemma 4.2.2.
Suppose that X = Spec k . Let W ′ ⊂ ∆ nk be a closed subset with W ′ ∈ S qX ( n ) and codim ∆ n W ′ > q . Then the natural map G ( W ′ ; A ) → lim −→ W ∈S qX ( n ) G ( W ; A ) is the zero-map.Proof. This is a modification of the proof of Sherman [32] that the Gersten complexfor A n is exact. We may assume that k is infinite. Take a general linear linearprojection π : ∆ nk = A nk → A n − k and let W = π − ( π ( W ′ )). Then π : W ′ → A n − k is finite and W is in S qX ( n ). In addition, π makes A n into a trivial A -bundle over A n − . Thus the canonical section s : W ′ → W ′ × A n − A n makes W ′ × A n − A n → W ′ into a trivial line bundle over W , hence s ( W ′ ) ⊂ W ′ × A n − A n is a principal Cartierdivisor. Letting t be a defining equation, we have the functorial exact sequence0 → p ∗ p ∗ ( M ) × t −−→ p ∗ p ∗ ( M ) → i ∗ ( M ) → p : W ′ × A n − A n → W ′ , p : W ′ × A n − A n → W ⊂ A n are the projectionsand i : W ′ → W is the inclusion. Thus i ∗ : G ( W ′ ; A ) → G ( W ; A )is the zero-map, completing the proof. (cid:3) We denote the composition EM ( ψ n ) ◦ ϕ n bycyc n : G ( p ) ( X, n ; A ) → EM ( z p ( X, n ; A ))and the map on the associated simplicial objects bycyc : G ( p ) ( X, − ; A ) → EM ( z p ( X, − ; A ))4.3. Localization.
Consider an open subscheme j : U → X with closed com-plement i : Z → X . We let S U X r ( n ) ⊂ S Ur ( n ) denote the set of closed subsets W ⊂ U × ∆ n such that(1) W is in S Ur ( n )(2) The closure ¯ W of W in X × ∆ n is in S Xr ( n ).Define the spectrum G ( r ) ( U X , n ; A ) by G ( r ) ( U X , n ; j ∗ A ) := lim −→ W ∈S UXr ( n ) G W ( U × ∆ n ; j ∗ A )giving us the simplicial spectrum G ( r ) ( U X , − ; A ). The restriction map j ∗ : G ( r ) ( U X , n ; A ) → G ( r ) ( U X , n ; A )factors through G ( r ) ( U X , n ; A ), giving us the commutative diagram G ( r ) ( X, − ; A ) ˆ j ∗ / / j ∗ ( ( QQQQQQQQQQQQ G ( r ) ( U X , − ; A ) ϕ (cid:15) (cid:15) G ( r ) ( U, − ; A ) Lemma 4.3.1.
The sequence G ( r ) ( Z, − ; i ∗ A ) i ∗ −→ G ( r ) ( X, − ; A ) ˆ j ∗ −→ G ( r ) ( U X , − ; A ) is a homotopy fiber sequence.Proof. In fact, it follows from the localization theorem for G ( − ; A ) that, for each n , the sequence G ( r ) ( Z, n ; i ∗ A ) i ∗ −→ G ( r ) ( X, n ; A ) ˆ j ∗ −→ G ( r ) ( U X , n ; A )whence the result. (cid:3) The localization results of [21] yield the following result:
Theorem 4.3.2.
The map ϕ : G ( r ) ( U X , − ; A ) → G ( r ) ( U, − ; A ) is a weak equivalence. Thus, we have
Corollary 4.3.3.
The sequences G ( r ) ( Z, − ; i ∗ A ) i ∗ −→ G ( r ) ( X, − ; A ) j ∗ −→ G ( r ) ( U, − ; A ) and G ( r/r − s ) ( Z, − ; i ∗ A ) i ∗ −→ G ( r/r − s ) ( X, − ; A ) j ∗ −→ G ( r/r − s ) ( U, − ; A ) are homotopy fiber sequences. In addition, we have
Lemma 4.3.4.
The diagram G ( r/r − ( Z, − ; i ∗ A ) i ∗ / / cyc (cid:15) (cid:15) G ( r/r − ( X, − ; A ) j ∗ / / cyc (cid:15) (cid:15) G ( r/r − ( U, − ; A ) cyc (cid:15) (cid:15) EM ( z r ( Z, − ; i ∗ A )) i ∗ / / EM ( z r ( X, − ; A )) j ∗ / / EM ( z r ( U, − ; j ∗ A )) defines a map of distinguished triangles in SH .Proof. It is clear the maps cyc n are functorial with respect to the closed immersion i and the open immersion j , hence the diagram π G ( r/r − ( Z, n ; i ∗ A ) i ∗ / / cyc n (cid:15) (cid:15) π G ( r/r − ( X, n ; A ) ˆ j ∗ / / cyc n (cid:15) (cid:15) π G ( r/r − ( U X , n ; A ) cyc n (cid:15) (cid:15) z r ( Z, n ; i ∗ A ) i ∗ / / z r ( X, n ; A ) j ∗ / / z r ( U X , n ; j ∗ A )commutes for each n . Similarly, the diagram π G ( r/r − ( U X , n ; A ) ϕ / / cyc n (cid:15) (cid:15) π G ( r/r − ( U, n ; A ) cyc n (cid:15) (cid:15) z r ( U X , n ; A ) ϕ / / z r ( U, n ; j ∗ A ) OTIVES OF AZUMAYA ALGEBRAS 43 commutes for each n . The result follows directly from this. (cid:3) Proposition 4.3.5.
Suppose that the map cyc( X ) : G ( r/r − ( X, − ; A ) → EM ( z r ( X, − ; A )) is a weak equivalence for X = Spec F , F a finitely generated field extension of k .Then cyc( X ) is a weak equivalence for all X essentially of finite type over k .Proof. This follows from corollary 3.8.2, corollary 4.3.3, lemma 4.3.4 and noetherianinduction. (cid:3)
The case of a field.
We have reduced to the case X = Spec k , where we mayapply the method of [18, § K A ∈ Spt S ( k ) be the presheaf of spectra X K ( X ; A ). We note that Lemma 4.4.1. (1) K A is homotopy invariant and satisfies Nisnevich excision. (2) K A is connected (3) K A ∼ = Ω T ( K A ) . We have already seen (1); (2) follows from the weak equivalence K ( − ; A ) → G ( − ; A ) on Sm /k and (3) follows from the projective bundle formula.In particular, for Y in Sm /k and integer q ≥
0, we have the simplicial abeliangroup z q ( Y, − ; K A ) and the cycle mapcyc K A : s q ( Y, − ; K A ) → EM ( z q ( Y, − ; K A )) . Lemma 4.4.2.
Let Y be in Sm /k , d = dim k Y . Fix an integer q ≥ and let r = d − q . There is a weak equivalence of simplicial spectra n ϕ n : s q ( Y, n ; K A ) → G ( r/r − ( Y, n ; A ) and an isomorphism of simplicial abelian groups n ψ n : z q ( Y, n ; K A ) → z r ( Y, n ; A ) such that the diagram s q ( Y, − ; K A ) ϕ / / cyc K A ( Y ) (cid:15) (cid:15) G ( r/r − ( Y, − ; A ) cyc( Y ) (cid:15) (cid:15) EM ( z q ( Y, n ; K A )) EM ( ψ ) / / EM ( z r ( Y, − ; A )) commutes in SH .Proof. We have the natural transformation of functors on Sm /kK ( − ; A ) → G ( − ; A )In particular, for T ∈ Sm /k and W ⊂ T a closed subset, we have the map ϕ T,W : K W ( T ; A ) → G W ( T ; A )defining a natural transformation of presheaves of spectra on Imm k . Applying ϕ − , − to the colimit of spectra with supports forming the definition of s q ( Y, n ; K A )and G ( r/r − ( Y, n ; A ) gives ϕ n . The map ψ n is defined similarly, using the maps π ( ϕ T,W ). The compatibility with the cycle maps follows directly from the defini-tions. (cid:3)
Thus, to prove that cyc( Y ) : G ( r/r − ( Y, − ; A ) → EM ( z r ( Y, − ; A )) is an isomor-phism in SH for all r and all Y ∈ Sm /k (in particular, for Y = Spec k ), it sufficesto show that cyc K A ( Y ) : s q ( Y, − ; K A ) → EM ( z q ( Y, n ; K A ))is an isomorphism in SH for all q and all Y . For this, it suffices by theorem 1.8.4to show that K A is well-connected.We have already seen that K A is connected (lemma 4.4.1(2)). By lemma 4.4.1(3)we need only show that π n ( K ( ˆ∆ ∗ k ( Y ) ; A )) = 0for n = 0.We have shown in [18, theorem 6.4.1] that the theory Y K ( Y ) is well-connected, in particular, that π n ( K ( ˆ∆ ∗ k ( Y ) ; A )) = 0 for n = 0 and for A = k .Using the results of appendix B, especially proposition B.5, the same argumentshows π n ( K ( ˆ∆ ∗ k ( Y ) ; A )) = 0 for n = 0 for arbitrary A . This completes the proof oftheorem 4.1.3. Remark . For later use, we record the fact we have just proved above, namely,that the presheaf Y K ( Y ; A ) is well-connected.4.5. The slice filtration for an Azumaya algebra.
By proposition 3.3.6, Z A isa birational motivic sheaf, hence the cycle complex z q ( X, ∗ ; Z A ( q )[2 q ]) is defined. Theorem 4.5.1.
Let A be a central simple algebra over a field k . For X ∈ Sm /k ,there is an isomorphism of complexes z q ( X, ∗ ; A ) ϕ X, A −−−→ z q ( X, ∗ ; Z A ( q )[2 q ]) , natural with respect to proper push-forward and flat pull-back.Proof. We first define for each n ≥ ϕ X, A ,n : z q ( X, n ; A ) ∼ = z q ( X, n ; Z A ( q )[2 q ])Indeed, by definition z q ( X, n ; A ) = ⊕ w ∈ X ( q ) ( n ) K A ( k ( w )) . By remark 2.3.4, we have z q ( X, n ; Z A ( q )[2 q ]) = ⊕ w ∈ X ( q ) ( n ) Z A ( k ( w )) . But Z A is just K A considered as a sheaf with transfers, giving us the desiredisomorphism.This isomorphism ϕ X, A ,n is clearly compatible with proper push-forward andflat pull-back. It thus suffices to show that the ϕ X, A ,n are compatible with the facemaps X × ∆ n − → X × ∆ n .Let k → k ′ be an extension of fields. z q ( X, n ; A ) → z q ( X k ′ , n ; A ( q )[2 q ]) z q ( X, n ; Z A ( q )[2 q ]) → z q ( X k ′ , n ; Z A ( q )[2 q ])are injective, it suffices to check in case A is a matrix algebra. By Morita equiva-lence, it suffices to check for A = k .Recall from proposition 2.4.5 the isomorphism of simplicial abelian groups n [ ρ X,n : z q ( X, n ; Z ( q )[2 q ]) → z q ( X, n ) OTIVES OF AZUMAYA ALGEBRAS 45 and from § n [Nrd X,n ; A : z q ( X, n ; A ) → z q ( X, n )] . In case A = k , the maps Nrd X,n ; A are isomorphisms. It is easy to check that (for A = k ) the diagram of isomorphisms z q ( X, n ; A ) ϕ X, A ,n / / Nrd
X,n ; A & & MMMMMMMMMM z q ( X, n ; Z ( q )[2 q ]) ρ X,n v v nnnnnnnnnnnn z q ( X, n )commutes. Since both the Nrd
X,n ; A and ρ X,n define maps of simplicial abeliangroups, it follows that the ϕ X, A ,n are define maps of of simplicial abelian groups aswell. (cid:3) Remark . We have the reduced norm map Nrd A : Z A → Z (as a map of Nis-nevich sheaves with transfers) inducing a reduced norm map Nrd A ( q ) : Z A ( q )[2 q ] → Z ( q )[2 q ] and thus a map of complexesNrd A ( q ) X : z q ( X, ∗ ; Z A ( q )[2 q ]) → z q ( X, ∗ ; Z ( q )[2 q ]) . We have as well the reduced norm map of § X ; A : z q ( X, ∗ ; A ) → z q ( X, ∗ ) . We claim that the diagram z q ( X, ∗ ; A ) ϕ X, A / / Nrd X ; A (cid:15) (cid:15) z q ( X, ∗ ) ϕ X,k (cid:15) (cid:15) z q ( X, ∗ ; Z A ( q )[2 q ]) Nrd A ( q ) X / / z q ( X, ∗ ; Z ( q )[2 q ])commutes. Indeed, on z q ( X, n ; A ) = ⊕ w K A ( k ( w )), both compositions are justsums of the reduced norm mapsNrd : K ( A k ( w ) ) → K ( k ( w )) = Z . Corollary 4.5.3.
Let A be a central simple algebra over a perfect field k , Y ∈ Sm /k . Then there is an isomorphism ψ p,q ; A : CH q ( Y, q − p ; A ) → H p ( Y, Z A ( q )) natural with respect to flat pull-back and proper push-forward, and compatible withthe respective reduced norm maps.Proof. This follows from theorem 2.3.3 and theorem 4.5.1. (cid:3)
Corollary 4.5.4.
Let A be a central simple algebra over a perfect field k , Y ∈ Sm /k . Then there is a strongly convergent spectral sequence E p,q = H p − q ( Y, Z A ( − q )) = ⇒ K − p − q ( Y ; A ) . Proof.
By corollary 4.1.4, we have the strongly convergent E spectral sequence E p,q = CH − q ( Y, − p − q ; A ) = ⇒ K − p − q ( Y ; A ) . By corollary 4.5.3 we have the isomorphismCH − q ( Y, − p − q ; A ) ∼ = H p − q ( Y, Z A ( − q ))yielding the result. (cid:3) In fact, we have
Theorem 4.5.5.
Let A be a central simple algebra over a perfect field k .Then thereis a natural isomorphism s n ( K A ) ∼ = EM ( Z A ( n )[2 n ]) Proof.
We first show
Claim.
The presheaf K A is the 0-space of a T -spectrum K ( A ) .Proof of claim. Take Y ∈ Sm /k . For a A ⊗ k O Y -module M , which is a locally freecoherent O Y module, we associate the complex of sheaves on Y × P p ∗ ( M ) ⊗ O ( − × X −−−→ p ∗ ( M ) . This gives a functor from the category P Y, A to the category of perfect complexesof A -modules on Y × P with support in Y × ∞ . Taking K -theory spectra, thisgives a natural map ǫ Y : K ( Y ; A ) → K Y ×∞ ( Y × P ; A ) = (Ω T K A )( Y )Using the maps ǫ Y , we thus build a T Ω spectrum K ( A ) := (( K A , K A , . . . ) , ǫ ) . Clearly K A = Ω ∞ T ( K ( A ))proving the claim. (cid:3) By the claim and remark 1.2.1, it suffices to prove the theorem for the case n = 0.We have the natural map K A → s ( K A )giving the map of Nisnevich sheaves ϕ : K A → π ( s ( K A )) . But we have the canonical isomorphism (see § s ( K A )( Y ) ∼ = K ( ˆ∆ ∗ k ( Y ) , A )Since K A is well-connected, we have π n ( K ( ˆ∆ ∗ k ( Y ) , A )) = ( n = 0 K ( k ( Y ); A ) for n = 0where the map K ( k ( Y ); A ) → π ( K ( ˆ∆ ∗ k ( Y ) , A )) is induced by the canonical map K ( k ( Y ); A ) → K ( ˆ∆ ∗ k ( Y ) , A ). In particular, ϕ is an isomorphism on function fields; OTIVES OF AZUMAYA ALGEBRAS 47 since both K A and π ( s ( K ( − , A ))) are birational sheaves, ϕ is an isomorphism.This gives us the isomorphism s ( K A ) ∼ = EM ( Z A ) , as desired. (cid:3) The reduced norm map.
Let A be a central simple algebra over k . We havealready mentioned the reduced norm mapNrd : K ( A ) → K ( k )in section 3.2; there are in fact reduced norm mapsNrd : K n ( A ) → K n ( k )for n = 0 , ,
2. For n = 0 ,
1, these may defined with the help of a splitting field L ⊃ k for A and Morita equivalence: Use the composition A ⊂ A ⊗ k L ∼ = M d ( L ) K n ( A ) → K n ( A L ) ∼ = K n ( M d ( L )) ∼ = K n ( L ) . For n = 0, the map K ( k ) → K ( L ) is an isomorphism; one checks that the resultingmap K ( A ) → K ( k ) is the reduced norm we have already defined. For n = 1, onecan take L to be Galois over k (with group say G ) and use that fact that there is a1-cocycle { ¯ g σ } ∈ Z ( G ; PGL d ( L )) such that A ⊂ M d ( L ) is the invariant subalgebraunder the G action ( σ, m ) ¯ g σσ m ¯ g − σ As det : K ( M d ( L )) → K ( L ) = L × the isomorphism given by Morita equivalence,one sees that the image of K ( A ) in L × lands in the G -invariants, i.e., in k × = K ( k ).For n = 2, the definition of the reduced norm map (due to Merkurjev-Suslin inthe square-free degree case [25, Th. 7.3] and to Suslin in general [35, Cor. 5.7]) ismore complicated; however, we do have the following result. Let Spl A be the setof field extensions L/k that split A . Proposition 4.6.1.
Let L ⊃ k be an extension field.1. For n = 0 , , , the diagram K n ( A L ) Nrd / / Nm AL/A (cid:15) (cid:15) K n ( L ) Nm L/k (cid:15) (cid:15) K n ( A ) Nrd / / K n ( k ) commutes. Here Nm A L /A : K n ( A L ) → K n ( A ) is the map on the K -groups inducedby the restriction of scalars functor, and similarly for Nm L/k .2. For n = 0 , , the map X Nm A L /A : ⊕ L ∈ Spl A K n ( A L ) → K n ( A ) is surjective. If A has square free index, P Nm A L /A is surjective for n = 2 as well. For a proof of the last statement, see [25, Th. 5.2].Let L ⊃ k be a field. Since CH m ( L, n ; A ) = 0 for m > n , due to dimensionalreasons, we have the edge homomorphism p n,L ; A : CH n ( L, n ; A ) → K n ( A L )coming from the spectral sequence of corollary 4.5.3.Let L/k be a finite field extension. We letNm
L/k : CH q ( Y L , p ; A ) → CH q ( Y, p ; A )denote the push-forward map for the finite morphism Y L → Y . Lemma 4.6.2.
Let
L/k be a finite field extension, f : Spec L → Spec k the corre-sponding morphism. Then the diagram CH n ( L, n ; A ) p n,L ; A / / Nm L/k (cid:15) (cid:15) K n ( A L ) Nm AL/A (cid:15) (cid:15) CH n ( k, n ; A ) p n,k ; A / / K n ( A ) commutes.Proof. Let w be a closed point of ∆ nL , not contained in any face. We have thecomposition K ( L ( w ); A ) ∼ = K w (∆ nL ; A ) ∼ = K w (∆ nL , ∂ ∆ nL ; A ) α −→ K (∆ nL , ∂ ∆ nL ; A ) ∼ = K n ( A L )The first isomorphism is via the localization sequence for K ( − ; A ). We have thecanonical map K w (∆ nL , ∂ ∆ nL ; A ) → K w (∆ nL ; A )which is a weak equivalence since w ∩ ∂ ∆ nL = ∅ , giving us the second isomorphism.The map α is “forget supports” and the last isomorphism follows from the homotopyproperty of K ( − ; A ). Denote this composition by β wn,L ; A : K ( k ( w ); A ) → K n ( A L ) . Since z n ( L, n ; A ) = ⊕ w K ( k ( w ); A ), where the sum is over all closed points w ∈ ∆ nL \ ∂ ∆ nL , the maps β wn,L ; A induce β n,L ; A : z n ( L, n ; A ) → K n ( A L );we have as well the canonical surjection γ n,L ; A : z n ( L, n ; A ) → CH n ( L, n ; A ) . It follows easily from the definitions that the diagram z n ( L, n ; A ) β n,L ; A ' ' OOOOOOOOOOOO γ n,L ; A / / CH n ( L, n ; A ) p n,L ; A (cid:15) (cid:15) K n ( A L )commutes. OTIVES OF AZUMAYA ALGEBRAS 49
On the other hand, it is also a direct consequence of the definitions that, for x ∈ ∆ nk the image of w under ∆ nL → ∆ nk , we haveNm L/k ◦ γ n,L ; A = γ n,k ; A ◦ Nm A L ( w ) /A k ( x ) Nm L/k ◦ β n,L ; A = β n,k ; A ◦ Nm A L ( w ) /A k ( x ) whence the result. (cid:3) Lemma 4.6.3.
For all n ≥ , the map X L Nm L/k : ⊕ L ∈ Spl A CH n ( L, n ; A ) → CH n ( k, n ; A ) is surjective.Proof. In fact, the map X L Nm L/k : ⊕ L ∈ Spl A z n ( L, n ; A ) → z n ( k, n ; A )is surjective. Indeed, let x be a closed point of ∆ nk \ ∂ ∆ nk . Then A k ( x ) = M n ( D )for some division algebra D over k ( x ). Letting L ⊂ D be a maximal subfield of D containing k ( x ), L splits D , hence L/k splits A . Since L ⊃ k ( x ), there is a closedpoint w ∈ ∆ nL \ ∂ ∆ nL lying over x with L ( w ) = w , i.e., w is an L -point.Since L is a maximal subfield of D , the degree of L over k ( x ) is exactly the indexof Nrd( K ( D )) ⊂ K ( k ( x )). Thus the norm mapNm L/k ( x ) : K ( A L ) → K ( A k ( x ) )is surjective, i.e. K ( A k ( x ) ) · x is contained in the image of Nm L/k ( z n ( L, n ; A )). As z n ( k, n ; A ) = ⊕ x K ( A k ( x ) )with the sum over all closed points x ∈ ∆ nk \ ∂ ∆ nk , this proves the lemma. (cid:3) Recall from § X ; A : z q ( Y, ∗ ; A ) → z q ( Y, ∗ ) . Lemma 4.6.4.
Let j : k ֒ → L be a finite extension field, Y ∈ Sm /k . Then thediagram z q ( Y L , − ; A ) Nrd
YL,A / / Nm L/k (cid:15) (cid:15) z q ( Y L , − ) Nm L/k (cid:15) (cid:15) z q ( Y, − ; A ) Nrd
Y,A / / z q ( Y, − ) commutes.Proof. Take w ∈ Y ( q ) L ( n ) and let x ∈ Y ( q ) ( n ) be the image of w under Y L × ∆ n → Y × ∆ n . It is easy to check that the diagram K ( A k ( w ) ) Nrd / / Nm Ak ( w ) /Ak ( x ) (cid:15) (cid:15) K ( k ( w )) Nm k ( w ) /k ( x ) (cid:15) (cid:15) K ( A k ( x ) ) Nrd / / K ( k ( x )) commutes, from which the lemma follows. (cid:3) Proposition 4.6.5.
For n = 0 , , the diagram CH n ( k, n ; A ) p n,k ; A / / Nrd (cid:15) (cid:15) K n ( A ) Nrd (cid:15) (cid:15) CH n ( k, n ) p n,k / / K n ( k ) commutes.Proof. Let j : k ֒ → L be a finite extension field of k . We have the diagramCH n ( L, n ; A L ) Nm L/k ( ( QQQQQQQQQQQQ p n,L ; A / / Nrd (cid:15) (cid:15) K n ( A L ) Nrd (cid:15) (cid:15) Nm L/k % % JJJJJJJJJ CH n ( k, n ; A ) p n,k ; A / / Nrd (cid:15) (cid:15) K n ( A ) Nrd (cid:15) (cid:15) CH n ( L, n ) p n,L / / Nm L/k ( ( QQQQQQQQQQQQ K n ( L ) Nm L/k % % JJJJJJJJJ CH n ( k, n ) p n,k / / K n ( k )The left hand square commutes by lemma 4.6.4, the right hand square commutesby proposition 4.6.1, the top and bottom squares commute by lemma 4.6.2.Now suppose that L splits A . Then, after using the Morita equivalence, the mapsNrd are identity maps, hence the back square commutes. Thus for b ∈ CH n ( L, n ; A ), a = Nm L/k ( b ) ∈ CH n ( k, n ; A ), we haveNrd( p n,k,A ( a )) = p n,k (Nrd( a )) . But by lemma 4.6.3, CH n ( k, n ; A ) is generated by elements a of this form, as L runs over all splitting fields of A , proving the result. (cid:3) Computations.Theorem 4.7.1 (see also theorem 4.8.2 ) . Let A be a central simple algebra over k .1. For n = 0 , , the edge homomorphism CH n ( k, n ; A ) p n,k ; A −−−−→ K n ( A ) is an isomorphism.2. The sequence → CH ( k, A ) d − , − −−−−→ CH ( k, A ) p ,k ; A −−−−→ K ( A ) → CH ( k, A ) → is exact. OTIVES OF AZUMAYA ALGEBRAS 51
Proof.
We first note that CH m ( k, n ; A ) = 0 for m > n for dimensional reasons.In addition z ( k, − ; A ) is the constant simplicial abelian group n K ( A ), henceCH ( k, n ; A ) = 0 for n = 0. (1) follows thus from the spectral sequence of corol-lary 4.1.4.For (2), the same argument gives the exact sequence0 → CH ( k, A ) d − , − −−−−→ CH ( k, A ) p ,k ; A −−−−→ K ( A ) → CH ( k, A ) → . (cid:3) Codimension one.
We recall the computation of the codimension one higherChow groups due to Bloch:
Proposition 4.8.1 (Bloch [5, theorem 6.1]) . Let F be a field. Then CH ( F, n ) = ( F × for n = 10 for n = 1Note that CH ( F,
0) = 0 for dimensional reasons. To show that CH ( F, n ) = 0for n >
1, let D = P i n i D i be a divisor on ∆ nF , intersecting each face properly, i.e.,containing no vertex of ∆ nF in its support. Suppose that D represents an element[ D ] ∈ CH ( F, n ), that is, d n ( D ) = 0. Using the degeneracy maps to add “trivial”components, we may assume that D · ∆ n − j = 0 for all j , where ∆ n − j is the face t j = 0.As ∆ nF ∼ = A nF , the divisor D is the divisor of a rational function f on ∆ nF . Since D intersects each ∆ n − j properly, the restriction f j of f to ∆ n − j is a well-definedrational function on ∆ n − j ; as D · ∆ n − j = 0, Div( f j ) = 0, so f j is a unit on ∆ n − j ,that is, f j = a j for some a j ∈ k × . Since ∆ n − j ∩ ∆ n − l = ∅ for all j, l , all the a j are equal, thus f j = a ∈ k × for all j . Dividing f by a we may assume that f j ≡ j .Now let D be the divisor of g := tf − (1 − t ) on ∆ nF × A F , where A F := Spec F [ t ].As the restriction of g to ∆ n − j × A is 1, D defines an element [ D ] ∈ CH ( A F , n )with i ∗ ([ D ]) = [ D ], i ∗ ([ D ]) = 0. By the homotopy property, [ D ] = 0.We use essentially the same argument plus Wang’s theorem [43] to completetheorem 4.7.1 as follows: Theorem 4.8.2.
Let A be a central simple algebra over a field F . Suppose A hassquare-free index. Then CH ( F, n ; A ) = 0 for n = 1 , and the edge homomorphism CH ( k, A ) p ,k ; A −−−−→ K ( A ) is an isomorphism.Proof. We reduce as usual to the case where deg A = p is prime. As above, thecase n = 0 is trivially true. We mimic the proof for CH ( F, n ) in case n > A = M p ( k ), then CH ( F, n ; A ) = CH ( F, n ), so there is nothing to prove; wetherefore assume that A is a degree p division algebra over k . Then A admits asplitting field k ′ of degree p over k ; since CH ( F ⊗ k k ′ , n ; A ) = CH ( F ⊗ k k ′ , n ) = 0for n >
1, a norm argument shows that CH ( F, n ; A ) is p -torsion.We have seen in lemma 4.2.2 that the argument of Sherman [32, theorem 2.4]for the degeneration of the Quillen spectral sequence for K ( A nF ) goes through word This is where we use the hypothesis n > for word to give the degeneration of the Quillen spectral sequence for K ( A nF ; A ).We will use this fact throughout the remainder of the proof.Take a divisor D representing a class [ D ] ∈ CH ( F, n ; A ). Let M (1) ( X ; A )denote the category of A ⊗ k O X -modules which are coherent as O X -modules andare supported in codimension at least one on X . Since K (∆ nF ; A ) = K ( A ) by thehomotopy property, the localization sequence K ( A ⊗ F F (∆ n )) ∂ −→ K ( M (1) (∆ n ; A )) → K (∆ n ; A ) → K ( A ⊗ F F (∆ n ))for K (∆ nF ; A ) gives us an element f ∈ K ( A ⊗ F F (∆ n )) with ∂f = D. In fact, since D intersects each ∆ n − j properly, we may find such an f ∈ ( A ⊗O ∆ n ,∂ ∆ n ) × , where ∂ ∆ n := ∪ j ∆ n − j . We have the localization sequence0 → K (∆ n − j ; A ) → K ( k (∆ n − j ); A ) ∂ −→ K ( M (1) (∆ n − j ; A )) → By the degeneration of the Quillen spectral sequence on ∆ n − j , it follows that K ( M (1) (∆ n − j ; A )) = ⊕ w ∈ (∆ n − j ) (1) K ( A ⊗ k k ( w ))Thus, the fact that ∆ n − j · D = 0 implies that restriction of f to f j ∈ ( A ⊗ k (∆ n − j ) × lifts uniquely to K (∆ n − j ; A ) = K ( A ). Note that we have the surjection A × → K ( A ); similarly, we may lift f to an element ˜ f ∈ ( A ⊗ O ∆ n { v } ) × with restriction˜ f j to ∆ n − j .Let ∂ + ∆ n = ∪ nj =1 ∆ n − j . The degeneracy maps give compatible splittings to theinclusions ∆ n − j → ∆ n , so we can modify f so that ˜ f j = 1 ∈ ( A ⊗ k (∆ n − j ) × .Now let L := k (∆ n − ) and consider ˜ f ∈ ( A L ) × . Restricting to ∆ n − ∩ ∆ n − shows that f = 1 ∈ K ( A L ). Furthermore, the reduced norm mapNrd : K ( A L ) → K ( L ) = L × is injective [43], and finally, for a ∈ ( A L ) × , we haveNrd( a ) = ( a p for a ∈ L × Nm L ( a ) /L ( a ) for a ∈ A × L \ L × Now, L ( ˜ f ) is a subfield of A L of degree ≤ p over L . But since A is a divisionalgebra and L is a pure transcendental extension of k , A L is still a division algebra,and hence either L ( ˜ f ) = L or L ( ˜ f ) has degree exactly p over L . In the formercase, 1 = Nrd( ˜ f ) = f p , and since ˜ f j ≡
1, it follows that f ≡ L ( ˜ f ) has degree exactly p over L , then Nm L ( ˜ f ) /L ( ˜ f ) = 1. Let M ⊃ L ( ˜ f ) be the Galois closure of L ( ˜ f ) over L , let M ⊂ M be the unique sub-field of index p , and let σ ∈ Gal(
M/L ) be the generator for Gal(
M/M ). ThenNm M/M ( ˜ f ) = 1, so by Hilbert’s theorem 90, there is a ˜ g ∈ M × with˜ f = g σ g . Looking at the proof of Hilbert’s theorem 90, we see that we may take g in theintegral closure of O ∆ n − ,∂ + ∆ n − , with g ≡ ∂ + ∆ n − . OTIVES OF AZUMAYA ALGEBRAS 53
By the Skolem-Noether theorem, there is an element a g ∈ A × M with g σ = a − g ga g , i.e. ˜ f = a − g ga g g − . As above, we may take a g to be a unit in the integral closure of A ⊗ O ∆ n − ,∂ + ∆ n − .Let ˆ L := k (∆ n ), and let ˆ M ⊃ ˆ L ( ˜ f ) be the Galois closure of ˆ L ( ˜ f ) over ˆ L . Lift g to ˆ g ∈ ˆ M (or rather, in the integral closure ˆ R of O ∆ n , ∆ n − in ˆ M ), with ˆ g ≡ ∂ + ∆ n . Lift a g similarly to ˆ a g . Let d = [ ˆ M : ˆ L ]. We mayreplace ˜ f d with ˆ f := Nm A ˆ M /A ˆ L ( ˜ f ˆ a − g ˆ g ˆ a g ˆ g − )Then ˆ f restricts to 1 in A ⊗ k (∆ n − j ) for all j , giving a trivialization of d · [ D ] inCH ( F, n ). Since d is prime to p , it follows that [ D ] = 0 in CH ( F, n ; A ). (cid:3) Remark . We shall give in corollary 6.1.4 below a second proof of theorem 4.8.2, relying on the Merkurjev-Suslin theorem, by proving that H p ( k ; Z A (1)) = 0 for A of square-free index over a perfect field k and p = 1. Via the isomorphism ofcorollary 4.5.3 CH ( k, n ; A ) ∼ = H − n ( k, Z A (1))this shows a second time that CH ( k, n ; A ) = 0 for n = 1 in the square-free indexcase. We do not know if this holds for A of arbitrary index.4.9. A map from SK i ( A ) to ´etale cohomology. In this section, we use the ´etaleversion of the spectral sequence in the previous section to construct homomorphismsfrom SK i ( A ) to quotients of H i +2´et ( k, Q / Z ( i + 1)) for i = 1 , K A f n +1 K A → f n K A → . . . → f K A = K A induces by ´etale sheafification the ´etale version[ f n +1 K A ] ´et → [ f n K A ] ´et → . . . → [ f K A ] ´et = [ K A ] ´et with layers the ´etale sheafications s ´et n K A of the layers s n K A of the original tower.Since s n K A = EM ( Z A ( n )[2 n ]) (theorem 4.5.5), and Z A ( n ) ´et = Z ( n ) ´et , we have s ´et n K A = EM ( Z ( n ) ´et [2 n ]) . Evaluating at Spec k and taking the spectral sequence of this tower gives the ´etalemotivic Atiyah-Hirzebruch spectral sequence for A , with Bloch-Lichtenbaum num-bering: E p,q = H p − q ´et ( k, Z ( − q )) ⇒ K ´et − p − q ( A ) . Here is part of the corresponding E -plane: − − H ( k, Z ) 0 H ( k, Z )0 0 H ( k, Z (1)) 0 H ( k, Z (1)) H ( k, Z (2)) H ( k, Z (2)) H ( k, Z (2)) 0 H ( k, Z (2)) H ( k, Z (3)) H ( k, Z (3)) H ( k, Z (3)) 0 H ( k, Z (3)) H ( k, Z (4)) H ( k, Z (4)) H ( k, Z (4)) 0 H ( k, Z (4)) H ( k, Z (4)) H ( k, Z (4))For i = 1 ,
2, the composition K i ( A ) → K ´et i ( A ) ε → H i ´et ( k, Z ( i )) = K i ( k ) coincides with the reduced norm, where ε is the edge homomorphism of the spectralsequence and the isomorphism follows from the Beilinson-Lichtenbaum conjecturein weight i (that is, Kummer theory for i = 1 and the Merkurjev-Suslin theoremfor i = 2). Hence we get an induced map SK ( A ) → coker( K ( k ) ≃ H ( k, Z (2)) d → H ( k, Z (3))) . Note that the map H ( k, Q / Z (3)) → H ( k, Z (3))) is an isomorphism, indepen-dently of the Beilinson-Lichtenbaum conjecture.For SK ( A ), we get a priori a map to the quotient ofcoker( K M ( k ) ≃ H ( k, Z (3)) d → H ( k, Z (4)))by the image of a d differential starting from H ( k, Z (2)) ≃ K ( k ) ind . If k containsa separably closed field, this group is divisible, hence its image by the torsiondifferential d is 0. Note that we also have an isomorphism H ( k, Q / Z (4)) ∼ → H ( k, Z (4))) . Here, the isomorphism K M ( k ) ≃ H ( k, Z (3)) follows from the Beilinson-Lichtenbaumconjecture in weight 3; if one does not want to assume it, one gets a slightly moreobscure quotient.To compute d , we use the fact that this spectral sequence is a module on thecorresponding spectral sequence for K ´et F [29]. The latter is multiplicative [29] and d is obviously 0 on K ( F ) and K ( F ), hence on all K Mi ( F ). For the one above,we then have d ( x ) = x · d (1)where d (1) is the image of 1 ∈ K ( F ) in Br ( F ).When we pass to the function field K of the Severi-Brauer variety of A , A getssplit so d (1) K = 0. By Amitsur’s theorem, d (1) is a multiple δ [ A ] of [ A ].In fact, we have δ = 1. The computation is very similar to our compution of a re-lated boundary map for the motive of a Severi-Brauer variety (see proposition 6.2.1)so we will be a little sketchy in our discussion here. Proposition 4.9.1. d (1) = [ A ] .Proof. We begin by noting that by naturality, it suffices to restrict the presheaf Y K ( Y ; A ) to the small ´etale site over k . Fix a Galois splitting field L over k of A with group G . As the field extensions of L are cofinal in k ´et , it suffices toconsider the functor F K ( F ; A )on finite extensions F of k containing L ; denote this subcategory of k ´et by k ´et ( L ).For such an F , A F is isomorphic to a matrix algebra, say A F ∼ = M n ( F ), so byMorita equivalence, K ( F ; A ) is weakly equivalent to K ( F ). Similarly, Z A = Z on k ´et ( L ). Since H p ( F, Z ( n )) = 0for p < n , and since Z (1) ∼ = G m [ − s n K A ∼ = EM ( Z A ( n )[2 n ]) , that the cofiber f / K A of f K A → f K A is the same as the presheaf of cofibersof K A by its 1-connected cover τ ≤ K A := cofib[ τ ≥ K A → K A ] . OTIVES OF AZUMAYA ALGEBRAS 55
Thus, to compute d (1), we just need to apply the usual machinery of G -cohomologyto the fiber sequence Σ EM ( K A ) → τ ≤ K A → EM ( K A ) , similar to our computation in the proof of proposition 6.2.1.Let us choose a cocycle σ ¯ g σ ∈ PGL n ( L ) representing the class of A in H ( G, PGL n ( L )). Thus, if g σ ∈ GL n ( L ) is a lifting of ¯ g σ , we have the action of G on M n ( L ) ϕ σ ( m ) := g σ · σ m · g − σ where σ m is the usual action of G by conjugation of the matrix coefficients. A is isomorphic to the G -invariant k -subalgebra of M n ( L ). Also, the coboundaryin H ( k, G m ) of the class of A in H ( k, PGL n ) is represented by the 2-cocycle { c τ,σ } ∈ Z ( G, L × ) defined by c τ,σ · id L n = g τ τ g σ g − τσ . The ring homomorphism ϕ σ : M n ( L ) → M n ( L ) induces an exact functor ϕ σ ∗ : Mod M n ( L ) → Mod M n ( L ) sending projectives to projectives, hence a natural map ϕ σ ∗ : K ( L ; A ) → K ( L ; A )and thereby a map ϕ σ ∗ : τ ≤ K ( L ; A ) → τ ≤ K ( L ; A ). To compute d (1), we applythe following procedure: lift 1 ∈ K ( L ; A ) to a representing M n ( L )-module F . Foreach σ ∈ G , choose an isomorphism ψ σ : ϕ σ ∗ ( F ) → F , which gives us a path γ σ inthe 0-space of K ( L ; A ). The path γ ( τ, σ ) := γ τ · ϕ τ ∗ [ γ σ ] · γ − τσ is a loop in K ( L ; A ), giving an element c ′ σ,τ ∈ K ( L ; A ) = L × . This gives us acocycle { c ′ τ,σ } ∈ Z ( G ; L × ), which represents d (1) ∈ H ( k, Z (1)) = H ( k, G m ).To make the computation concrete, let F be a left M n ( L )-module. Then theisomorphism of abelian groups F → M n ( L ) ⊗ M n ( L ) F sending v to 1 ⊗ v identi-fies ϕ σ ∗ ( F ) with the M n ( L ) module with underlying abelian group F , and withmutliplication m · σ v := σ − [ g − σ mg σ ] · v. Under this identification, ϕ σ ∗ acts by the identity on morphisms.Take F = L n with the standard M n ( L )-module structure. One sees immediatelythat sending v to g σ · σ v gives an M n ( L )-module isomorphism ψ σ : ϕ σ ∗ ( F ) → F .The loop γ ( τ, σ ) is thus represented by the automorphism ψ τ ◦ ϕ τ ∗ ( ψ σ ) ◦ ψ − τσ : ψ τ ◦ ϕ τ ∗ ( ψ σ ) ◦ ψ − τσ ( v ) = ψ τ ◦ ϕ σ ∗ ( ψ σ )( ( τσ ) − [ g − τσ v ])= ψ τ ( g σ · τ − [ g − τσ · v ])= g τ τ g σ g − τσ Since the Morita equivalence Mod M n ( L ) → Mod L sends multiplication by c ∈ L on F to multiplication by c on L , we have the explicit representation of d (1) by thecocycle { c τ,σ } , completing the computation. (cid:3) The motivic Postnikov tower for a Severi-Brauer variety
Results of Huber-Kahn [14] give a computation of the sheaf H of the slices of M ( X ) for X any smooth projective variety and show that H n vanishes for n > X = SB( A ), we are able to show (in case A has prime degree ℓ over k ) that the negative cohomology vanishes as well. We dothis by comparing with the slices of the K -theory of X and using Adams operationsto split the appropriate spectral sequence, proving our second main result theorem 2(see theorem 5.4.2).5.1. The motivic Postnikov tower for a smooth variety.Lemma 5.1.1.
Let X be a smooth projective variety, M ( X ) ∈ DM eff ( k ) the mo-tive of X .1. f motn M ( X ) = 0 for n > dim k X .2. For dim k X ≤ n ≤ , Ω nT f motn X is represented by C Sus ( z equi ( X, n )) .Proof. (1) Since the objects M ( Z )[ p ] are dense in DM eff ( k ), it suffices to showthat Hom DM ( k ) ( M ( Z )( n )[ p ] , M ( X )) = 0for all p and all n > dim k . Since RC ∗ is full, it suffices to show the same vanishingfor the morphisms in DM effgm ( k ); since DM effgm ( k ) → DM gm ( k ) is faithful, it sufficesto show the vanishing for the morphisms in DM gm ( k ). X is smooth and projective, so we haveHom DM gm ( k ) ( M ( Z )( n )[ p ] , M ( X )) = Hom DM gm ( k ) ( M ( Z × X ) , Z ( d − n )[ − p ])where d = dim k X . ButHom DM gm ( k ) ( M ( Z × X ) , Z ( d − n )[ − p ]) = H − p ( Z × X, Z ( d − n ))which is zero for d − n < nT f motn M ( X ) = f mot Ω nT M ( X ) = Ω nT M ( X )By [11, V, theorem 4.2.2], we have the natural isomorphismHom DM eff − ( k ) ( M ( Y × P n ) , M ( X )[ m ]) ∼ = H m ( C Sus ∗ ( z equi ( X × P n , n ))( Y ))and one checks that the projectionHom DM eff − ( k ) ( M ( Y × P n ) , M ( X )[ m ]) → Hom DM eff − ( k ) ( M ( Y )( n )[2 n ] , M ( X )[ m ])corresponding to the summand M ( Y )( n )[2 n ] ⊂ M ( Y × P n ) corresponds to the mapinduced by the projection z equi ( X × P n , n )) → z equi ( X, n )This gives us the isomorphismΩ nT M ( X ) = R H om ( Z ( n )[2 n ] , M ( X )) ∼ = C Sus ∗ ( z equi ( X, n )) . (cid:3) OTIVES OF AZUMAYA ALGEBRAS 57
Remark . Suppose k admits resolution of singularities. Then for X ∈ Sm /k ,we have f motn M ( X ) = 0for n > dim k X . In fact, let U ⊂ Y be an open subscheme of some Y ∈ Sm /k suchthat the complement Z := Y \ U is smooth and of pure codimension n on Y . Wehave the Gysin distinguished triangle M ( U ) → M ( Y ) → M ( Z )( n )[2 n ] → M ( U )[1]Also, if Y is in addition projective, then so is Z ; since M ( Z )( n )[2 n ] is a summandof M ( Z × P n ), we see that M ( U ) is in the thick subcategory generated by smoothprojective varieties of dimension ≤ dim k Y . Using resolution of singularities, theexistence of a compactification of X with strict normal crossing divisor at infin-ity implies that every X ∈ Sm /k of dimension ≤ d is in the thick subcategoryof DM gm ( k ) generated by smooth projective varieties of dimension ≤ d . Thuslemma 5.1.1 shows that f motn M ( X ) = 0for n > dim k X .For later use, we make the following explicit computation: Lemma 5.1.3.
Let Y be in Sm /k . Let X be smooth, irreducible and projective ofdimension d over k . The canonical map f motd M ( X ) → f motd − M ( X ) induces the map(in D ( Ab ) ) [Ω d − T f motd M ( X )]( Y ) α −→ [Ω d − T f motd − M ( X )]( Y ) Then α is isomorphic to the map on Bloch’s cycle complexes p ∗ : z ( Y, ∗ ) → z ( X × Y, ∗ ) induced by the projection p : X × Y → Y .Proof. By (1.3.2), we haveΩ d − T f motd M ( X ) = f mot Ω d − T M ( X ) = f mot (Ω d − T f motd − M ( X ))By lemma 5.1.1(2), we haveΩ d − T f motd − M ( X ) = C Sus ∗ ( z equi ( X, d − d − T f motd M ( X ) ∼ = f mot C Sus ∗ ( z equi ( X, d − d − T f motd M ( X ) → Ω d − T f motd − M ( X ) is just the canonical map f mot C Sus ∗ ( z equi ( X, d − → C Sus ∗ ( z equi ( X, d − D ( Ab ) f mot C Sus ∗ ( z equi ( X, d − Y ) ∼ = f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − f mot C Sus ∗ ( z equi ( X, d − → C Sus ∗ ( z equi ( X, d − is isomorphic to f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − / / f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − C Sus ∗ ( z equi ( X, d − Y × ∆ ∗ ) . Now, for any T ∈ Sm /k , the inclusion C Sus ∗ ( z equi ( X, d − T ) ⊂ z ( T × X, ∗ )is a quasi-isomorphism. Thus, if W ⊂ T is a closed subset, we have the quasi-isomorphism C Sus ∗ ( z equi ( X, d − W ( T ) → cone( z ( T × X, ∗ ) → z ( T × X \ W × X, ∗ ))[ − W has pure codimension 1. By Bloch’s localization theorem, wehave the quasi-isomorphism z ( W, ∗ ) → cone( z ( T × X, ∗ ) → z ( T × X \ W × X, ∗ ))[ − z ( W ) = z ( W, → z ( W, ∗ ) is a quasi-isomorphism. If codim X W >
1, asimilar computation shows that C Sus ∗ ( z equi ( X, d − W ( T ) is acyclic. Applying thisto the computation of f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − D ( Ab ) ϕ : z ( Y, ∗ ) → f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − . Furthermore, the composition z ( Y, ∗ ) ϕ −→ f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − → f mot ( Y, ∗ ; z ( X × − , ∗ ))is the map W ⊂ Y × ∆ n X × W × ∆ ⊂ X × Y × ∆ n × ∆ It is then easy to see that the composition z ( Y, ∗ ) ϕ −→ f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − → f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − D ( Ab ) f mot ( Y, ∗ ; C Sus ∗ ( z equi ( X, d − ∼ = C Sus ∗ ( z equi ( X, d − Y ) ∼ = z ( X × Y, ∗ )is just the pull-back p ∗ : z ( Y, ∗ ) → z ( X × Y, ∗ ) . (cid:3) Let X be in Sm /k . For a presheaf of spectra E on Sm /k , we have the associatedpresheaf H om ( X, E ), defined by H om ( X, E )( Y ) := E ( X × Y ) . Applying H om ( X, − ) to a fibrant model defines the functor R H om ( X, − ) : SH S ( k ) → SH S ( k ) . We use the notation H om mot and R H om mot for the analogous operations on C ( P ST ( k )) and on DM eff ( k ).The operation R H om ( X, − ) does not in general commute with the truncationfunctors f n . However, we do have OTIVES OF AZUMAYA ALGEBRAS 59
Lemma 5.1.4.
Take m > dim k X . Then for all E ∈ SH S ( k ) , s R H om ( X, f m E ) ∼ = 0 . Proof.
Let F be a presheaf of spectra on Sm /k which is A -homotopy invariant andsatisfies Nisnevich excision. As we have seen in § SH ( s F )( X ) ∼ = F ( ˆ∆ ∗ k ( Y ) ) . Similarly, for E homotopy invariant and satisfying Nisnevich excision, the spec-trum H om ( X, f m E )( Y ) := f m E ( X × Y ) is weakly equivalent to the simplicialspectrum q f m E ( X × Y )( q ) with f m E ( X × Y )( q ) = lim −→ W ∈S ( m ) X × Y ( q ) E W ( X × Y × ∆ q )Using the moving lemma described in [19], we thus have the natural weak equiva-lence f m E ( X × ˆ∆ pk ( Y ) )( q ) ∼ = lim −→ W ∈S ( m ) X × ˆ∆ pk ( Y ) ( q ) C ( p ) E W ( X × Y × ∆ q )where C ( p ) is the set X × F , with F a face of ˆ∆ pk ( Y ) .Thus s H om ( X, f m E )( Y ) is weakly equivalent to the total spectral of the bi-simplicial spectra( p, q ) s H om ( X, f m E )( Y )( p, q ) = lim −→ W ∈S ( m ) X × ˆ∆ pk ( Y ) ( q ) C ( p ) E W ( X × ˆ∆ pk ( Y ) × ∆ q ) . We denote the total spectrum by s H om ( X, f m E )( Y )( − , − ).Let s H om ( X, f m E )( Y )( − , q ) be the total spectrum of the simplicial spectrum p s H om ( X, f m E )( Y )( p, q ) = lim −→ W ∈S ( m ) X × ˆ∆ pk ( Y ) ( q ) C ( p ) E W ( X × ˆ∆ pk ( Y ) × ∆ q ) . The face maps δ q ∗ i : s H om ( X, f m E )( Y )( − , q ) → s H om ( X, f m E )( Y )( − , q − i = 0 , . . . , q , q ≥ s H om ( X, f m E )( Y )( − , → s sHom ( X, f m E )( Y )( − , − )is a weak equivalence.Take W ∈ S ( m ) X × ˆ∆ pk ( Y ) (0) C ( p ) , so W is a closed subset of X × ˆ∆ pk ( Y ) of codimension ≥ m > dim k X , and W ∩ X × F has codimension ≥ m on X × F for all faces F ofˆ∆ pk ( Y ) . In particular, for each vertex v of ˆ∆ pk ( Y ) ,codim X × v W ∩ X × v > dim k X. Thus W ∩ X × v = ∅ . Since X is proper, the projection of W , p ( W ) ⊂ ˆ∆ pk ( Y ) , isa closed subset disjoint from all vertices v . Since ˆ∆ pk ( Y ) is semi-local with closedpoints the set of vertices, this implies that p ( W ) = ∅ , hence W = ∅ , i.e. S ( m ) X × ˆ∆ pk ( Y ) (0) C ( p ) = {∅} and thus s H om ( X, f m E )( Y )( − , ∼
0. Our description of s H om ( X, f m E )( Y )as a simplicial spectrum thus yields s H om ( X, f m E )( Y ) ∼ Y ∈ Sm /k , completing the proof. (cid:3) Thus, for X ∈ Sm /k , smooth and projective of dimension d over k , and for E ∈ SH S ( k ), we have the tower in SH S ( k )(5.1.1) 0 = s R H om ( X, f d +1 E ) → s R H om ( X, f d E ) → . . . → s R H om ( X, f E ) = s R H om ( X, E )gotten by applying s R H om ( X, − ) to the T -Postnikov tower of E . Since the func-tors s and R H om ( X, − ) are exact, the m th layer in the tower (5.1.1) is isomorphicto s R H om ( X, s m E ), m = 0 , . . . , dim k X . Evaluating at some Y ∈ Sm /k , we havethe strongly convergent spectral sequence(5.1.2) E a,b = π a + b ( s R H om ( X, s a E )( Y )) = ⇒ π a + b ( s R H om ( X, E )( Y )) . The case of K -theory. We take E = K , where K ( Y ) is the Quillen K -theory spectrum of the smooth k -scheme Y . By [18, theorem 6.4.2] we have thenatural isomorphism( s m K )( Y ) ∼ = EM ( z m ( Y, ∗ )) ∼ = EM ( Z ( m )[2 m ])( Y )In addition, we have natural Adams operations ψ k , k = 2 , , . . . acting on K andon the T -Postnikov tower of K , with ψ k acting on π ∗ ( s m K )( Y ) by multiplicationby k m for all Y ∈ Sm /k (see [20]).Thus we have Lemma 5.2.1.
Suppose X has dimension p − over k for some prime p . Supposethat for Y ∈ Sm /k the differentials in the spectral sequence (5.1.2) , for E = K , areall zero after localising at p . Then the spectral sequence (5.1.2) degenerates at E .Proof. The Adams operations act on the spectral sequence and ψ k acts by multi-plication by k a on E ra,b . Thus the differential d r : E ra,b → E ra − r,q + r − is killed byinverting k a − k a − r . Since d ra,b = 0 if a + r ≥ p , we need only invert the numbers τ a,r := g.c.d. { k a (1 − k r ) | k = 2 , , . . . } for a = 0 , , , . . . , p − r = 1 , . . . , p − a −
1, which only involve primes q < p . (cid:3) The Chow sheaf.
For a smooth projective variety X , we have the Nisnevichsheaf with transfers CH n ( X ) on Sm /k , this being the sheaf associated to thepresheaf Y CH n ( X × Y ) . It is shown in [14, theorem 2.2] that CH n ( X ) is a birational motivic sheaf. We canalso label with the relative dimension, defining CH n ( X ) := CH dim k X − n ( X ) . For our next computation, we need:
OTIVES OF AZUMAYA ALGEBRAS 61
Lemma 5.3.1.
Take
F ∈ C ( Sh tr Nis ( k )) which is homotopy invariant and sat-isfies Nisnevich excision. Suppose in addition that F is connected. Then thesheaf H Nis0 ( s mot R H om ( X, s motn F )) is the Nisnevich sheaf associated to the presheaf H ( s mot R H om ( X, s motn F )) with value at Y ∈ Sm /k given by the exactness of lim −→ W ′ ∈S n +1 X × Y (1) W ∈S nX × Y (1) H ( F W \ W ′ ( X × Y × ∆ \ W ′ )) i ∗ − i ∗ −−−→ lim −→ W ′ ∈S n +1 X × Y (0) W ∈S nX × Y (0) H ( F W \ W ′ ( X × Y \ W ′ )) → H ( s mot R H om ( X, s motn F ))( Y ) → Proof.
From proposition 1.5.2 , R H om ( X, s motn F )( Y ) = ( s motn F )( X × Y ) is isomor-phic in D ( Ab ) to s nmot ( X × Y, − ; F ), the total complex of the simplicial complex m s nmot ( X × Y, m ; F ) := lim −→ W ∈S ( n ) X × Y ( m ) W ′ ∈S X × Y ( n +1)( n ) F W \ W ′ ( X × Y × ∆ m \ W ′ ) . By lemma 1.7.3, the spectra s nmot ( X × Y, m ; F ) are all -1 connected. Thus we havethe exact sequence H ( s n ( X × Y, F )) i ∗ − i ∗ −−−→ H ( s nmot ( X × Y, F )) → H ( s nmot ( X × Y, − ; F )) . In any case R H om ( X, s motn F ) is in DM eff ( k ), hence the homology presheaf Y H ( R H om ( X, s motn F )( Y )) = H ( s motn ( X × Y, − ; F ))is a homotopy invariant presheaf with transfers. Thus, by [11, III, corollary 4.18],if Y is local, the restriction map(5.3.1) H ( s motn ( X × Y, − ; F )) → H ( s motn ( X k ( Y ) , − ; F ))is injective. In addition, R H om ( X, s motn F ) is connected. Indeed, s motn F is con-nected by proposition 1.7.4, and this implies that R H om ( X, s motn F ) is connected.Thus the restriction map (5.3.1) is also surjective, hence an isomorphism.By theorem 2.2.1, s mot R H om ( X, s motn F ) is also birational, and is connected byproposition 1.7.4, hence the same argument shows that H ( s mot R H om ( X, s motn F )( Y )) → H ( s mot R H om ( X, s motn F )( k ( Y )))is an isomorphism.We now return to the situation Y ∈ Sm /k . As in the proof of lemma 5.1.4, s mot R H om ( X, s motn F )( Y ) is given by evaluating R H om ( X, s motn F ) on ˆ∆ ∗ k ( Y ) . Since R H om ( X, s motn F ) is connected by proposition 1.7.4, it follows that we have theexact sequence H ( R H om ( X, s motn F ))( ˆ∆ k ( Y ) ) i ∗ − i ∗ −−−−→ H ( R H om ( X, s motn F ))( ˆ∆ k ( Y ) ) → H ( s mot R H om ( X, s motn F )( Y )) → . But since R H om ( X, s motn F ) is connected, the restriction map H ( R H om ( X, s motn F )(∆ k ( Y ) )) → H ( R H om ( X, s motn F )( ˆ∆ k ( Y ) )) is surjective, which shows that H ( R H om ( X, s motn F )( k ( Y )) ∼ = H ( s mot R H om ( X, s motn F )( Y )) . Since the restriction map (5.3.1) is an isomorphism for Y local, it follows that thecanonical map H ( R H om ( X, s motn F )( Y )) → H ( s mot R H om ( X, s motn F )( Y ))is an isomorphism for Y local.Putting this together with our description above of H ( R H om ( X, s motn F )( Y ))proves the result. (cid:3) Lemma 5.3.2.
Let X be a smooth projective variety of dimension d . There is anatural isomorphism H Nis0 ( s mot R H om ( X, Z ( n )[2 n ])) ∼ = CH n ( X ) Proof.
Since Z is a birational motive, we have (remark 2.2.3) Z ( n )[2 n ] ∼ = s motn ( Z ( n )[2 n ]) . We can now use lemma 5.3.1 to compute H Nis0 ( s mot R H om ( X, s motn ( Z ( n )[2 n ]))).By lemma 2.4.4, for W ⊂ Y a closed subvariety of codimension n , Y ∈ Sm /k ,there is a natural isomorphism H (( Z ( n )[2 n ]) W ( T )) = H nW ( Y, Z ( n )) ρ Y,W,n −−−−→ z nW ( Y ) . From this, it follows from lemma 5.3.1 that H Nis0 ( s mot R H om ( X, s motn ( Z ( n )[2 n ])))is just the sheafification of Y CH n ( X × Y ) , i.e., H Nis0 ( s mot R H om ( X, s motn ( Z ( n )[2 n ]))) ∼ = CH n ( X ) . (cid:3) The slices of M ( X ) . Before proving our main theorem on the slices of themotive of a Severi-Brauer variety, we first use duality to shift the computation ofthe n th slice to a 0th slice of a related motive. 0th slices are easier to handle,because their cohomology sheaves are birational sheaves. Lemma 5.4.1.
Let X be smooth and projective of dimension d over k . Then for ≤ n ≤ d there is a natural isomorphism s motn M ( X ) ∼ = s mot (cid:0) R H om mot ( X, Z ( d − n )) (cid:1) ( n )[2 d ] Proof.
By [14] f motn M ( X ) = H om DM eff ( k ) ( Z ( n ) , M ( X ))( n )= H om DM eff ( k ) ( Z ( d )[2 d ] , M ( X )( d − n )[2 d ])( n )= H om DM eff ( k ) ( M ( X ) , Z ( d − n ))( n )[2 d ]In addition, using the isomorphism (1.2.3), we have f motn − ◦ H om DM eff ( Z (1) , − ) = H om DM eff ( Z (1) , − ) ◦ f motn ;this plus Voevodsky’s cancellation theorem [38] implies f motn ( F (1)) ∼ = f motn − ( F )(1) OTIVES OF AZUMAYA ALGEBRAS 63 and similarly for the slices s motn . Thus s motn M ( X ) = s motn ( f motn ( M ( X ))= s motn (cid:0) H om DM eff ( k ) ( M ( X ) , Z ( d − n ))( n )[2 d ] (cid:1) = s mot (cid:0) H om DM eff ( k ) ( M ( X ) , Z ( d − n )) (cid:1) ( n )[2 d ]= s mot (cid:0) R H om mot ( X, Z ( d − n )) (cid:1) ( n )[2 d ] (cid:3) Theorem 5.4.2.
Let X be a Severi-Brauer variety of dimension p − , p a prime,associated to a central simple algebra A of degree p over k . Then (1) s motn M ( X ) ∼ = CH n ( X )( n )[2 n ] for n = 0 , . . . , p − , s motn M ( X ) = 0 for n ≥ p . (2) There is a canonical isomorphism ⊕ p − n =0 CH n ( X ) ∼ = ⊕ p − n =0 Z A ⊗ n (3) For n = 0 , . . . , p − , we have CH n ( X ) ∼ = Z A ⊗ n ∼ = ( Z A for n = 1 , . . . , p − Z for n = 0 Proof.
We first note that X satisfies the conditions of lemma 5.2.1. If X = P p − ,then the projective bundle formula gives the weak equivalence R H om ( P p − , f m K ) ∼ = ⊕ p − i =0 f m − i K from which the degeneration of the spectral sequence at E for all Y ∈ Sm /k easily follows. In general, there is a splitting field L for A of degree p over k , so X L ∼ = P p − L , and thus the differentials are all killed by × p .We recall that s n K ∼ = EM ( Z ( n )[2 n ]) [18, theorem 6.4.2]. By Quillen’s compu-tation of the K -theory of Severi-Brauer varieties, R H om ( X, K ) ∼ = ⊕ p − n =0 K ( − ; A ⊗ n ) . Finally, the fact that K ( − ; A ⊗ n ) is well-connected (remark 4.4.3) implies s ( K ( − ; A ⊗ n )) = EM ( Z A ⊗ n ) . Since our spectral sequence degenerates at E , we therefore have the isomorphism ⊕ p − n =0 π Nis ∗ s ( R H om ( X, EM ( Z ( n )[2 n ])) ∼ = ⊕ p − n =0 π Nis ∗ EM ( Z A ⊗ n )Also, we have s ◦ EM = EM ◦ s mot , and R H om ( X, EM ( F )) = EM ( R H om mot ( X, F )) π Nis m ( EM ( F )) = H − m Nis ( F )for F ∈ DM eff ( k ). Thus we see that H m Nis ( s mot ( R H om mot ( X, Z ( n )[2 n ]))) = 0for m = 0 and(5.4.1) ⊕ p − n =0 H (cid:0) s mot ( R H om mot ( X, Z ( n )[2 n ]) (cid:1) ∼ = ⊕ p − n =0 Z A ⊗ n . In particular, s mot ( R H om mot ( X, Z ( n )[2 n ])) is concentrated in degree 0. Thus, itfollows from lemma 5.3.2 that s mot ( R H om mot ( X, Z ( n )[2 n ])) ∼ = CH n ( X )for n = 0 , . . . , p −
1, which together with (5.4.1) proves (2).Together with lemma 5.4.1, this gives s motn M ( X ) ∼ = s mot ( R H om mot ( X, Z ( p − − n ))( n )[2 p − ∼ = CH n ( X )( n )[2 n ]proving (1).For (3), take a finite Galois splitting field L/k for A with Galois group G . Wehave the natural map π ∗ : CH n ( X ) → CH n ( X L ) G ∼ = Z with kernel and cokernel killed by p . By (2), CH n ( X ) is torsion-free. Similarly, wehave the inclusion π ∗ : Z A ⊗ n → ( Z A ⊗ nL ) G ∼ = Z . We thus have compatible inclusions ⊕ p − n =0 CH n ( X ) ∼ = (cid:127) _ (cid:15) (cid:15) ⊕ p − n =0 Z A ⊗ n (cid:127) _ (cid:15) (cid:15) ⊕ p − n =0 CH n ( X L ) G ∼ = ⊕ p − n =0 ( Z A ⊗ nL ) G Clearly CH ( X ) ∼ = Z . For y ∈ Y ∈ Sm /k the quotient( ⊕ p − n =0 ( Z A ⊗ nL ) G ) y / ( ⊕ p − n =0 Z A ⊗ n ) y has order p p − if A y is not split, and 1 otherwise. Thus, for n = 1 , . . . , p − CH n ( X ) y ⊂ CH n ( X L ) Gy = Z has index p if A y is not split and index 1 if A y is split.Thus we can write CH n ( X ) ∼ = Z A ⊗ n for n = 0 , . . . , p −
1, completing the proof. (cid:3) Applications
In this section, we let X be the Severi-Brauer variety X := SB( A ) associated toa central simple algebra A of prime degree ℓ over k . We use our computations of thelayers for M ( X ), together with a duality argument and the Beilinson-Lichtenbaumconjecture, to study the reduced norm mapNrd : H p ( k, Z A ( q )) → H p ( k, Z ( q ))and prove the first of our main applications corollary 1 (see theorem 6.1.3). Com-bining these results with our identification of the low-degree K -theory of A with thetwisted Milnor K -theory of k gives us our main result on the vanishing of SK ( A )for A of square-free degree (corollary 2, see also theorem 6.2.2). OTIVES OF AZUMAYA ALGEBRAS 65
A spectral sequence for motivic homology.
We have the motivic Post-nikov tower for M ( X )(6.1.1) 0 = f motℓ M ( X ) → f motℓ − M ( X ) → . . . → f mot M ( X ) → f mot M ( X ) = M ( X )with slices s motb M ( X ) ∼ = Z A ⊗ b +1 ( b )[2 b ]; b = 0 , . . . , ℓ − . Let α ∗ : DM eff ( k ) → DM eff ( k ) ´et be the change of topologies functor, withright adjoint α ∗ : DM eff ( k ) ´et → DM eff ( k ). The functors α ∗ and α ∗ are exact,and applying α ∗ to Z A ( n ) → Z ( n ) gives an isomorphism α ∗ Z A ( n ) → α ∗ Z ( n )).Thus, we have the tower(6.1.2) 0 = α ∗ α ∗ f motℓ M ( X ) → α ∗ α ∗ f motp − M ( X ) → . . . → α ∗ α ∗ f mot M ( X ) → α ∗ α ∗ f mot M ( X ) = α ∗ α ∗ M ( X )with slices α ∗ α ∗ s motb M ( X ) ∼ = α ∗ α ∗ Z ( b )[2 b ]; b = 0 , . . . , ℓ − . Since α ∗ is right adjoint to α ∗ , the unit η of the adjunction gives the naturaltransformation of towers η : (6.1.1) → (6.1.2). Defining ¯ M ( X ), f motn ¯ M ( X ) and¯ Z A ⊗ b +1 ( a ) by the distinguished triangles M ( X ) → α ∗ α ∗ M ( X ) → ¯ M ( X ) → M ( X )[1] f motn M ( X ) → α ∗ α ∗ f motn M ( X ) → f motn ¯ M ( X ) → f motn M ( X )[1] Z A ⊗ b +1 ( a ) → α ∗ α ∗ Z ( a ) → ¯ Z A ⊗ b +1 ( a ) → ¯ Z A ⊗ b +1 ( a )[1]we have the tower(6.1.3) 0 = f motp ¯ M ( X ) → f motp − ¯ M ( X ) → . . . → f mot ¯ M ( X ) → f mot ¯ M ( X ) = ¯ M ( X )with slices s motb ¯ M ( X ) ∼ = ¯ Z A ⊗ b +1 ( b )[2 b ]; b = 0 , . . . , p − . This last tower thus gives rise to the strongly convergent spectral sequence(6.1.4) E p,q = ⇒ Hom DM eff ( k ) ( Z ( a )[ b ] , ¯ M ( X )( a ′ )[ p + q ])with E p,q = ( Hom DM eff ( k ) ( Z ( a )[ b ] , ¯ Z A ⊗− q +1 ( a ′ − q )[ p − q ]) for 0 ≤ − q ≤ ℓ −
10 else.
Lemma 6.1.1.
Hom DM eff − ( k ) ( Z ( r ′ ) , Z A ( r )[ q ]) = 0 for1. r = 0 , r ′ > and all q r = 0 = r ′ and q = 0 r ′ = 0 and ≤ r < q Proof. Z A is a homotopy invariant Nisnevich sheaf with transfers, soHom DM eff − ( k ) ( Z ( r ′ ) , Z A [ q − r ′ ]) = ker[ H q Zar ( P r ′ , Z A ) → H q Zar ( P r ′ − , Z A )]Since Z A is a constant sheaf in the Zariski topology H q Zar ( P r ′ , Z A ) = ( q = 0 Z A ( k ) for q = 0 , the last identity following from the homotopy invariance of Z A . This proves (1)and (2).For (3), we have seen thatHom DM eff − ( k ) ( M ( X ) , Z A ( r )[2 r + n ]) ∼ = CH r ( X ; A, n )for all n . Taking X = Spec k , (3) follows from the fact that z r (Spec k ; A, n ) = 0 for n < r for dimensional reasons. (cid:3)
Lemma 6.1.2.
The Beilinson-Lichtenbaum conjecture for weight n + 1 implies Hom DM eff ( k ) ( Z ( d )[2 d ] , ¯ M ( X )( n + 1)[ m ]) = 0 for m ≤ n + 2 and that the sequence → H n +3 ( X, Z ( n + 1)) → H n +3´et ( X, Z ´et ( n + 1)) → H n +3´et ( k ( X ) , Z ( n + 1)) is exact.Proof. The Beilinson-Lichtenbaum conjecture for weight n + 1 says that the coho-mology sheaves of ¯ Z ( n + 1) are 0 in degree ≤ n + 2, hence the natural map H m ( X, Z ( n + 1)) → H m ´et ( X, Z ´et ( n + 1))is an isomorphism for m ≤ n + 2 and there is an exact sequence0 → H n +3 ( X, Z ( n + 1)) → H n +3´et ( X, Z ´et ( n + 1)) → H ( X, H n +3´et ( Z ( n + 1)))since the cohomology sheaves of Z ( n + 1) vanish in degrees > n + 1. By the Gerstenconjecture for H n +3´et ( Z ( n + 1))), the map H n +3´et ( Z ( n + 1))) → H n +3´et ( k ( X ) , Z ( n + 1))is injective, which gives the exact sequence in the statement of the lemma.In terms of morphisms in DM eff ( k ) and DM eff ( k ) ´et , this says that the changeof topologies mapHom DM eff ( k ) ( M ( X ) , Z ( n + 1)[ m ]) → Hom DM eff ( k ) ´et ( α ∗ M ( X ) , α ∗ Z ( n + 1)[ m ])is an isomorphism for m ≤ n + 2 and an injection for m = n + 3. M ( X ) has dual M ( X ) ∨ = M ( X )( − d )[ − d ], hence the ´etale version α ∗ M ( X )has dual α ∗ M ( X )( − d )[ − d ]. In other words, we have natural isomorphismsHom DM eff ( k ) ( M ( X ) , Z ( n + 1)[ m ]) ∼ = Hom DM eff ( k ) ( Z ( d )[2 d ] , M ( X )( n + 1)[ m ])andHom DM eff ( k ) ´et ( α ∗ M ( X ) ,α ∗ Z ( n + 1)[ m ]) ∼ = Hom DM eff ( k ) ´et ( α ∗ Z ( d )[2 d ] , α ∗ M ( X )( n + 1)[ m ]) ∼ = Hom DM eff ( k ) ( Z ( d )[2 d ] , α ∗ α ∗ M ( X )( n + 1)[ m ]) . OTIVES OF AZUMAYA ALGEBRAS 67
Thus, the natural map M ( X ) → α ∗ α ∗ M ( X ) induces an isomorphismHom DM eff ( k ) ( Z ( d )[2 d ] , M ( X )( n + 1)[ m ]) → Hom DM eff ( k ) ( Z ( d )[2 d ] , α ∗ α ∗ M ( X )( n + 1)[ m ])for m ≤ n + 2 and an injection for m = n + 3, hence the lemma. (cid:3) Theorem 6.1.3.
Let A be a central simple algebra over k of prime degree ℓ . Let n ≥ , and assume that the Beilinson-Lichtenbaum conjecture holds in weights ≤ n + 1 and for the prime ℓ .1. For m < n , the reduced norm Nrd : H m ( k, Z A ( n )) → H m ( k, Z ( n )) is an isomorphism.2. There is an exact sequence → H n ( k, Z A ( n )) Nrd −−→ H n ( k, Z ( n )) ∂ n −→ H n +3´et ( k, Z ( n + 1)) → H n +3´et ( k ( X ) , Z ( n + 1)) where X is the Severi-Brauer variety of A .Proof. We proceed by induction on n . (1) in case n = 0 follows from lemma 6.1.1(2).For (2), the reduced norm identifies H ( k, Z A ) with ℓH ( k, Z ) in case [ A ] = 0, andis an isomorphism if [ A ] = 0; since ℓ [ A ] = 0, (2) follows.Now assume the result for all n ′ < n . By the Beilinson-Lichtenbaum con-jecture in weight n ′ , the map Z ( n ′ ) → α ∗ α ∗ Z ( n ′ ) induces an isomorphism onHom DM eff ( k ) ( Z , − [ m ]) for all m ≤ n ′ + 1 and an injection for m = n ′ + 2. ThusHom DM eff ( k ) ( Z , ¯ Z ( n ′ )[ m ]) = 0 for m ≤ n ′ + 1Similarly, applying (1) and (2) to the distinguished triangle defining ¯ Z A ( n ′ ), ourinduction assumption gives(6.1.5) Hom DM eff ( Z , ¯ Z A ( n ′ )[ m ]) = 0 for m < n ′ . Finally, by lemma 6.1.2, the Beilinson-Lichtenbaum conjecture for weight n + 1gives(6.1.6) Hom DM eff ( k ) ( Z ( d )[2 d ] , ¯ M ( X )( n + 1)[ m ]) = 0 for m ≤ n + 2 . Now consider our spectral sequence (6.1.4) with a = d , b = 2 d − n − a ′ = n +1,where d = dim k X = ℓ −
1. We haveHom( Z ( d )[2 d − n − , ¯ M ( X )( n + 1)[ p + q ])= Hom( Z ( d )[2 d ] , ¯ M ( X )( n + 1)[ n + 2 + p + q ])so by (6.1.6) the spectral sequence converges to 0 for p + q ≤ E p,q term is E p,q = Hom( Z , ¯ Z A ⊗− q +1 ( n + 1 − d − q )[ n + 2 − d + p − q ]) for 0 ≤ − q ≤ d and 0 otherwise. For 0 ≤ − q < d − p + q ≤
0, we have n ′ := n + 1 − d − q < nn + 2 − d + p − q < n ′ For − q = d , A ⊗− q +1 is a matrix algebra, hence ¯ Z A ⊗− q +1 ( N ) = ¯ Z ( N ). Thus E p, − d = Hom( Z , ¯ Z ( n + 1)[ n + 2 − d + p ])If p + q ≤
0, then p ≤ d , so n + 2 − d + p ≤ n + 2. Thus our induction hypothesisplus Hilbert’s theorem 90 in weight n + 1 yields E p,q = 0 for 0 ≤ − q ≤ d, − q = d − , p + q ≤ . hence there is exactly one E term that is possibly non-zero, namely E p, − d = Hom( Z , ¯ Z A ⊗ d ( n )[ n + 1 − d + p ])The d differential is E p, − d d −→ E p +2 , − d and since p + 2 − d ≤ E p +2 , − d = 0. Since E ∗ ,q = 0 for q < − d , there are no higherdifferentials coming out of E p, − d . Similarly, our induction hypothesis implies thatthere are no d r differentials going to E p, − dr . Thus E p, − d = E p, − d ∞ = 0.Now take p + q = 0. The abutment of the spectral sequence is still 0 and thereis still only one possibly non-zero E term, E d − , − d = Hom( Z , ¯ Z A ( n )[ n ]) . There is a single possibly non-trivial d differential, since E d +1 , − d = Hom( Z , ¯ Z ( n + 1)[ n + 3]) . As above, there are no other non-zero d r differentials, hence d d − , − d : E d − , − d → E d +1 , − d is an injection. Moreover, all d r differentials abutting to E d +1 , − dr have a sourceequal to 0, hence E d +1 , − d = E d +1 , − d ∞ .Let us collect the information obtained so far: • E p,q = 0 for p + q ≤
0, except possibly ( p, q ) = ( d − , − d ). • The differential d d − , − d induces an exact sequence(6.1.7) 0 → E d − , − d → E d +1 , − d → Hom( Z ( d )[2 d ] , ¯ M ( X )( n + 1)[ n + 3]) . Since E p, − d = 0, we get that the mapHom( Z , Z A ⊗ d ( n )[ n + 1 − d + p ]) → Hom( Z , α ∗ α ∗ Z A ⊗ d ( n )[ n + 1 − d + p ])is an isomorphism for p < d − p = d −
1. Since Z A ∼ = Z A ⊗ ℓ − ,we have Hom( Z , Z A ⊗ d ( n )[ n + 1 − d + p ]) ∼ = H n +1+ p − d ( k, Z A ( n ))Hom( Z , α ∗ α ∗ Z A ⊗ d ( n )[ n + 1 − d + p ]) ∼ = H n +1+ p − d ´et ( k, Z ( n ))hence the canonical map α A : H m ( k, Z A ( n )) → H m ´et ( k, Z ( n )) OTIVES OF AZUMAYA ALGEBRAS 69 is an isomorphism for m < n and an injection for m = n . Since α A factors as H m ( k, Z A ( n )) α A / / Nrd (cid:15) (cid:15) H m ´et ( k, Z ( n )) H m ( k, Z ( n )) α nnnnnnnnnnnn and α : H m ( k, Z ( n )) → H m ´et ( k, Z ( n ) ´et ) is an isomorphism for m ≤ n by theBeilinson-Lichtenbaum conjecture in weight n , it follows that Nrd is an isomor-phism for m < n and an injection for m = n , proving (1) and the injectivity of Nrdin (2).From the distinguished triangles defining ¯ Z A and ¯ Z , we have exact sequences → H n − ( k, Z A ( n )) → H n − ( k, Z ( n )) → E d − , − d → H n ( k, Z A ( n )) → H n ´et ( k, Z ( n )) → E d − , − d → H n +1 ( k, Z A ( n )) → and → H n +3 ( k, Z ( n + 1)) → H n +3´et ( k, Z ( n + 1)) → E d +1 , − d → H n +4 ( k, Z ( n + 1)) → But we have already shown E d − , − d = 0. Also, H n +1 ( k, Z A ( n )) = H n +3 ( k, Z ( n + 1)) = H n +4 ( k, Z ( n + 1)) = 0for dimensional reasons and H n ( k, Z ( n )) = H n ´et ( k, Z ( n )) by Bloch-Kato in weight n . Thus we get an exact sequence0 → H n ( k, Z A ( n )) Nrd → H n ( k, Z ( n )) → E d − , − d → H n +3´et ( k, Z ( n + 1)) ∼ → E d +1 , − d . Putting this together with (6.1.7), we get the exact sequence0 → H n ( k, Z A ( n )) Nrd −−→ H n ( k, Z ( n )) ∂ n −→ H n +3´et ( k, Z ( n + 1)) → Hom( Z ( d )[2 d ] , ¯ M ( X )( n + 1)[ n + 3]) , where ∂ n is the map induced by d d − , − d . By comparing the spectral sequence forHom( Z ( d )[2 d ] , M ( X )(( n + 1)[ ∗ ]) , Hom( Z ( d )[2 d ] , α ∗ α ∗ M ( X )(( n + 1)[ ∗ ])and Hom( Z ( d )[2 d ] , ¯ M ( X )(( n + 1)[ ∗ ]) , we see that H n +3´et ( k, Z ( n +1)) → Hom( Z ( d )[2 d ] , ¯ M ( X )( n +1)[ n +3]) factors throughthe image ofHom( Z ( d )[2 d ] , α ∗ α ∗ M ( X )( n + 1)[ n + 3]) → Hom( Z ( d )[2 d ] , ¯ M ( X )( n + 1)[ n + 3]) . By the exact sequence of lemma 6.1.2, we thus have the exact sequence0 → H n ( k, Z A ( n )) Nrd −−→ H n ( k, Z ( n )) ∂ n −→ H n +3´et ( k, Z ( n + 1)) → H n +3´et ( k ( X ) , Z ( n + 1)) . The resulting map H n +3´et ( k, Z ( n + 1)) → H n +3´et ( k ( X ) , Z ( n + 1)) is induced by an edge homomorphism of our spectral sequence, hence equals theextension of scalars map. This completes the proof. (cid:3) Corollary 6.1.4.
Let A be a central simple algebra of square-free index over k .For n = 1 , H n ( k, Z A (1)) = 0 . Of course, we have already proved this by a direct argument (theorem 4.8.2 ).This second argument uses our main result on the reduced norm, theorem 6.1.3,which, in the weight one case, relies on the Merkurjev-Suslin theorem to proveBeilinson-Lichtenbaum in weight two (using in turn [36] or [12]).
Proof.
We first reduce to the case of A of prime degree ℓ . Writedeg( A ) = Y ℓ i = d. where the ℓ i are distinct primes. Write A = M n ( D ) for some division algebra D of degree d over k , and let F ⊂ D be a maximal subfield. Then F has degree d over k and splits D . Let ℓ = ℓ i for some i , let k ( ℓ ) ⊃ k be the maximal prime to ℓ extension of k and let F ( ℓ ) := F k ( ℓ ). Then clearly F ( ℓ ) has degree ℓ over k ( ℓ )and splits A k ( ℓ ) ; since k ( ℓ ) has no prime to ℓ extensions, F ( ℓ ) is Galois over k ( ℓ ).Passing from k to the Gal( k ( ℓ ) /k ) invariants alters only the prime to ℓ torsion.Thus we may replace k with k ( ℓ ) and assume that A is split by a degree ℓ Galoisextension of k . But then A is Morita equivalent to an algebra of degree ℓ , whichachieves the reduction.It follows from [5, theorem 6.1] that0 = CH ( k, − n ) ∼ = H n ( k, Z (1))for n = 1. By theorem 6.1.3(1), this implies that H n ( k, Z A (1)) = 0 for n < H n ( k, Z A (1)) ∼ = CH ( k, − n ; A )by corollary 4.5.3. Since CH ( k, m ; A ) = 0 for m < ( k, A ) = 0 fordimensional reasons, the proof is complete. (cid:3) Corollary 6.1.5.
Let A be a central simple algebra of square-free index over k .The the edge homomorphism p ,k ; A : CH ( k, A ) → K ( A ) is an isomorphismProof. From corollary 6.1.4, CH ( k, n ; A ) = 0 for n = 1. From theorem 4.7.1(2),we have the exact sequence0 → CH ( k, A ) d − , − −−−−→ CH ( k, A ) p ,k ; A −−−−→ K ( A ) → CH ( k, A ) → , hence the edge-homomorphism p ,k ; A : CH ( k, A ) → K ( A ) is an isomorphism. (cid:3) Finally, here is a global version of theorem 6.1.3:
Corollary 6.1.6.
Let ˜ Z A denote the cokernel of the reduced norm map Nrd : Z A → Z . Suppose that A has square-free index and assume the Beilinson-Lichtenbaumconjecture. Then, (1) For all n ≥ , the complex ˜ Z A ( n ) ∈ DM eff ( k ) is concentrated in degree n . OTIVES OF AZUMAYA ALGEBRAS 71 (2)
Let F n = H n (˜ Z A ( n )) . Then the stalk of F n at a function field K is iso-morphic to ker( H n +3´et ( K, Z ( n + 1) → H n +3´et ( K ( X ) , Z ( n + 1))) where X is the Severi-Brauer variety of A . (3) For any smooth scheme U we have a Gersten resolution → F n → M x ∈ U (0) ( i x ) ∗ ( F n ) → M x ∈ U (1) ( i x ) ∗ ( F n − ) → · · · → M x ∈ U ( p ) ( i x ) ∗ ( F n − p ) → . . . Proof.
As in the proof of corollary 6.1.4, it suffices to handle the case of A of primedegree over k .Clearly, Z A ( n ) has no cohomology in degrees > n ; by Voevodsky’s form of Ger-sten’s conjecture, the vanishing of H i (˜ Z A ( n )) for i < n reduces to theorem 6.1.3.The computation of the stalks of H n (˜ Z A ( n )) also follows from theorem 6.1.3.For (3), we first show (with the notation of [37, 3.1], that the Zariski sheaf asso-ciated to the presheaf ( F n ) − is F n − . This follows immediately from Voevodsky’scancellation theorem [38]: by definition( F n ) − ( U ) = coker (cid:0) F n ( U × A ) → F n ( U × ( A − { } ) (cid:1) = coker (cid:16) H n ( U × A , ˜ Z A ( n )) → H n ( U × ( A − { } ) , ˜ Z A ( n )) (cid:17) . By purity, the localization sequence for U × ( A − { } ) ⊂ U × A , and part (1) ofthe corollary, the latter cokernel is isomorphic toker( H n − ( U, ˜ Z A ( n − → H n +1 ( U, ˜ Z A ( n )) ≃ H ( U, F n )hence the Zariski sheaf associated to ( F n ) − is the sheaf associated to U H n − ( U, ˜ Z A ( n − ≃ F n − ( U ) . The statement on the Gersten complex follows from this and loc. cit. theorem4.37. (cid:3)
Computing the boundary map.
To finish our study of H n ( k, Z A ( n )), weneed to compute the boundary map ∂ n in theorem 6.1.3. As above, we fix a centralsimple algebra A over k of prime degree ℓ , let d = ℓ − X be the Severi-Brauer variety SB( A ). We let [ A ] ∈ H ( k, G m ) denote the class of A in the(cohomological) Brauer group of k .Concentrating on f motd − M ( X ) gives us the distinguished triangle s motd M ( X ) → f motd − M ( X ) → s motd − M ( X ) → s motd M ( X )[1]which by theorem 5.4.2 is Z ( d )[2 d ] → f motd − M ( X ) → Z A ( d − d − → Z ( d )[2 d + 1]Applying Ω d − T gives Z (1)[2] → Ω d − T f motd − M ( X ) → Z A → Z (1)[3]Applying the ´etale sheafification α ∗ and noting that Z ´et A ∼ = Z ´et gives the distin-guished triangle(6.2.1) Z (1) ´et [2] → α ∗ Ω d − T f motd − M ( X ) → Z ´et ∂ −→ Z (1) ´et [3] Thus ∂ : Z ´et → Z (1) ´et [3] gives us the element β A ∈ H ( k, Z (1) ´et ) = H ( k, G m ) . Proposition 6.2.1. β A = [ A ] .Proof. To calculate β A , it suffices to restrict (6.2.1) to the small ´etale site on k .By lemma 5.1.3, (6.2.1) on k ´et is isomorphic (in D ( Sh ´et ( k ))) to the sheafificationof the sequence of presheaves(6.2.2) L (cid:16) z ( L, ∗ ) p ∗ −→ z ( X L , ∗ ) → cone( p ∗ ) → z ( L, ∗ )[1] (cid:17) . Here, and in the remainder of this proof, we consider the cycle complexes as coho-mological complexes: z ( Y, ∗ ) n := z ( Y, − n ) . We recall that z ( X L , ∗ ) has non-zero cohomology only in degrees 0 and -1, andthat H − ( z ( X L , ∗ )) = Γ( X L , O × X L ) ,H ( z ( X L , ∗ )) = CH ( X L ) . Similarly, H − ( z ( L, ∗ )) = L × and all other cohomology of z ( L, ∗ ) vanishes. Since X is geometrically irreducible and projective, p ∗ : L × → Γ( X L , O × X L )is an isomorphism, and thus the cone of z ( L, ∗ ) p ∗ −→ z ( X L , ∗ ) has only cohomologyin degree 0, namely H (cone( p ∗ )) = CH ( X L ) . Thus the sequence (6.2.2) is naturally isomorphic (in D ( P Sh ´et ( k ))) to the canonicalsequence(6.2.3) L (cid:0) H − ( z ( X L , ∗ ))[1] → τ ≥− z ( X L , ∗ ) → H ( z ( X L , ∗ )) → H − ( z ( X L , ∗ ))[2] (cid:1) . We can explicitly calculate a co-cycle representing β A as follows: Take L/k tobe a Galois extension with group G such that A L is a matrix algebra over L . Then(6.2.3) gives a distinguished triangle in the derived category of G -modules, so wehave in particular the connecting homomorphism ∂ L : H ( G, H ( z ( X L , ∗ ))) → H ( G ; H − ( z ( X L , ∗ ))) = H ( G ; L × )Also X L ∼ = P dL . As H ( z ( X L , ∗ )) = CH ( X L ), H ( z ( X L , ∗ )) has a canonical G -invariant generator 1, namely the element corresponding to c ( O (1)). We canapply ∂ L to 1, giving the element ∂ L (1) ∈ H ( G ; L × ) which maps to β A under thecanonical map H ( G, L × ) → H ( k, G m ) . Since A L is a matrix algebra over L , A is given by a 1-cocycle { ¯ g σ | σ ∈ G } ∈ Z ( G, PGL ℓ ( L ))and X is the form of P d defined by { ¯ g σ } . This mean that there is an L isomorphism ψ : X L → P dL such that, for each σ ∈ G , we have¯ g σ := ψ ◦ σ ψ − , OTIVES OF AZUMAYA ALGEBRAS 73 under the usual identification Aut L ( P dL ) = PGL d +1 ( L ).Lifting ¯ g σ to g σ ∈ GL ℓ ( L ) and defining c τ,σ ∈ L × by c τ,σ id := g τ τ g σ g − τσ we have the co-cycle { c τ,σ } ∈ Z ( G, L × ) representing [ A ].For a G -module M , let ( C ∗ ( G ; M ) , ˆ d ) denote the standard co-chain complexcomputing H ∗ ( G ; M ), i.e., C n ( G ; M ) is a group of n co-chains of G with values in M . We will show that ∂ L (1) = { c τ,σ } in H ( G, L × ) by applying C ∗ ( G ; − ) to thesequence (6.2.3) and making an explicit computation of the boundary map.Fix a hyperplane H ⊂ P dk . Then D := ψ ∗ ( H L ) ∈ z ( X L , ∗ ) represents thepositive generator 1 ∈ CH ( X L ) ∼ = Z . As the class of D in CH ( X L ) is G -invariant,there is for each σ ∈ G a rational function f σ on X L such thatDiv( f σ ) = σ D − D. Given τ, σ ∈ G , we thus haveDiv( f τσ f − τσ f τ ) = τσ D − τ D − ( τσ D − D ) + τ D − D = 0so there is a c ′ τ,σ ∈ Γ( X L , O × X L ) = L × with c ′ τ,σ = f τσ f − τσ f τ . Using the fact that σ D = ψ ∗ (¯ g σ ( H L ))one can easily calculate that c ′ τ,σ = c τ,σ . Indeed, take a k -linear form L so that H is the hyperplane defined by L = 0. Let F σ := L ◦ g − σ L so Div( F σ ) = ¯ g σ ( H ) − H . Letting f σ := ψ ∗ F σ , we haveDiv( f σ ) = ψ ∗ (Div( F σ )) = ψ ∗ (¯ g σ ( H ) − H ) = σ D − D, and τ f σ = ψ ∗ ( L ◦ τ g − σ ◦ g − τ L ◦ g − τ ) . Thus c ′ τ,σ = ψ ∗ (cid:18) L ◦ τ g − σ ◦ g − τ L ◦ g − τ (cid:19) · ψ ∗ (cid:18) L ◦ g − τσ L (cid:19) − · ψ ∗ (cid:18) L ◦ g − τ L (cid:19) = ψ ∗ (cid:18) L ◦ τ g − σ g − τ L ◦ g − τσ (cid:19) = c τ,σ On the other hand, we can calculate the boundary ∂ L (1) by lifting the gen-erator 1 = [ D ] ∈ CH ( X L ) G to the element D ∈ z ( X L , ∗ ) and taking ˇCechco-boundaries. Explicitly, let Γ σ ⊂ X L × ∆ be the closure of graph of f σ , afteridentifying (∆ , ,
1) with ( P \{ } , , ∞ ). Define Γ c σ,τ ∈ z ( L, ∗ ) − similarly as thepoint of ∆ L corresponding to c τ,σ ∈ A ( k ) ⊂ P ( k ), and let δ denote the boundaryin the complex z ( X L , ∗ ). For σ ∈ G , we have δ − (Γ σ ) = σ D − D = ˆ d ( D ) σ . Since H − ( z ( X L , ∗ )) = Γ( X L , O × X L ) = L × , there is for each σ, τ ∈ G , an element B σ,τ ∈ z ( X L ,
2) with p ∗ Γ c σ,τ = τ Γ σ − Γ τσ + Γ τ + δ − ( B σ,τ )= ˆ d ( σ Γ σ ) τ,σ ∈ τ ≥− z ( X L , ∗ ) − . Thus ∂ L ([ D ]) = { c σ,τ } ∈ H ( G, H − ( z ( L, ∗ ))) = H ( G, L × ) . This completes the computation of ∂ L (1) and the proof of the proposition. (cid:3) Theorem 6.2.2.
Let A be a central simple algebra over k of square-free index e .Let n ≥ , and assume that the Beilinson-Lichtenbaum conjecture holds in weights ≤ n + 1 at all primes dividing e .1. For m < n , the reduced norm Nrd : H m ( k, Z A ( n )) → H m ( k, Z ( n )) is an isomorphism.2. We have an exact sequence → H n ( k, Z A ( n )) Nrd −−→ H n ( k, Z ( n )) ≃ K Mn ( k ) ∪ [ A ] −−−→ H n +2´et ( k, Z /e ( n + 1)) → H n +2´et ( k ( X ) , Z /e ( n + 1)) .
3. ( n = 1 ) SK ( A ) = 0 . More precisely, we have an exact sequence → K ( A ) Nrd −−→ K ( k ) ∪ [ A ] −−−→ H ( k, Z /e (2)) → H ( k ( X ) , Z /e (2)) .
4. ( n = 2 ) SK ( A ) = 0 . More precisely, we have an exact sequence → K ( A ) Nrd −−→ K ( k ) ∪ [ A ] −−−→ H ( k, Z /e (3)) → H ( k ( X ) , Z /e (3))To explain the map ∪ [ A ] in (2), (3) and (4): We have isomorphisms K ( k ) = k × ∼ = H ( k, Z (1)) K ( k ) ∼ = H ( k, Z (2)) H n ´et ( k, G m ) ∼ = H n +1´et ( k, Z (1) ´et ) . Thus we have [ A ] ∈ H ( k, Z (1) ´et ) and cup product maps H n ( k, Z ( n )) → H n ´et ( k, Z ( n ) ´et ) ∪ [ A ] −−−→ H n +3´et ( k, Z ( n + 1) ´et )which obviously land into e H n +3´et ( k, Z ( n +1) ´et . On the other hand, the exact triangle Z ( n + 1) ´et e → Z ( n + 1) ´et → Z /e ( n + 1) +1 → and the Beilinson-Lichtenbaum conjecture in weight n + 1 give an isomorphism H n +2´et ( k, Z /e ( n + 1)) ∼ → e H n +3´et ( k, Z ( n + 1) ´et ) . OTIVES OF AZUMAYA ALGEBRAS 75
Proof.
As in the proof of corollary 6.1.4, it suffices to handle the case of A of primedegree over k . Thus, (1) follows from theorem 6.1.3(1).For (2), applying α ∗ to the distinguished triangle Z (1)[2] → Ω d − T f motd − M ( X ) → Z A → Z (1)[2]we have Z (1) ´et [2] → α ∗ Ω d − T f motd − M ( X ) → Z ´et ∂ −→ Z (1) ´et [3]It follows from proposition 6.2.1 that the ∂ is given by cup product with [ A ] ∈ H ( k, Z (1) ´et ). Since the map ∂ n in theorem 6.1.3 is just the map induced by ∂ after tensoring with Z ( n ) ´et [ n ], (2) is proven in the form of an exact sequence0 → H n ( k, Z A ( n )) Nrd −−→ H n ( k, Z ( n )) ∪ [ A ] −−−→ H n +3´et ( k, Z ( n + 1) ´et ) → H n +3´et ( k ( X ) , Z ( n + 1) ´et ) . But the Beilinson-Lichtenbaum conjecture in weight n + 1, applied both to k and k ( X ), shows that in the commutative diagram H n +2´et ( k, Z /e ( n + 1)) −−−−→ H n +2´et ( k ( X ) , Z /e ( n + 1)) ∂ y ∂ y H n +3´et ( k, Z ( n + 1) ´et ) −−−−→ H n +3´et ( k ( X ) , Z ( n + 1) ´et )both horizontal maps have isomorphic kernels, hence the form of (2) appearing inTheorem 6.2.2.For (3) and (4), we have the isomorphism (corollary 4.5.3) ψ p,q ; A : H p ( k, Z A ( q )) → CH q ( k, q − p ; A )compatible with the respective reduced norm maps. From corollary 6.1.5, the edge-homomorphism p ,k ; A : CH ( k, A ) → K ( A ) is an isomorphism. It follows fromtheorem 4.7.1(1) that the edge homomorphism p ,k ; A : CH ( k, A ) → K ( A ) is anisomorphism as well. Together with proposition 4.6.5, this gives us the commutativediagram for n = 1 , H n ( k, Z A ( n )) ψ n,n ; A / / Nrd (cid:15) (cid:15) CH n ( k, n ; A ) p n,k ; A / / Nrd (cid:15) (cid:15) K n ( A ) Nrd (cid:15) (cid:15) H n ( k, Z ( n )) ψ n,n ; k / / CH n ( k, n ) p n,k ; k / / K n ( k )with all horizontal maps isomorphisms. Thus, in the sequence (1), we may replace H n ( k, Z A ( n )) with K n ( A ) and H n ( k, Z ( n )) with K n ( k ) for n = 1 ,
2, proving (3)and (4). (cid:3)
Appendix A. Modules over Azumaya algebras
We collect some basic results for use throughout the paper.Let R be a commutative ring and A an Azumaya R -algebra. Lemma A.1. If R is Noetherian, A is left and right Noetherian.Proof. Indeed, A is a Noetherian R -module, hence a Noetherian A -module (on theleft and on the right). (cid:3) Lemma A.2.
For an A − A -bimodule M , let M A = { m ∈ M | am = ma. } Then the functor M M A is exact and sends injective A − A -bimodules to injective R -modules.Proof. Let A e = A ⊗ R A op be the enveloping algebra of A . We may view M as aleft A e -module. A special A − A -bimodule is A itself, and we clearly have M A = Hom A e ( A, M ) . Since A is an Azumaya algebra, the map A e → End R ( A ) is an isomorphism of R -algebras; via this isomorphism, Hom A e ( A, M ) may be canonically identified with A ∗ ⊗ End R ( A ) M , with A ∗ = Hom R ( A, R ). Hence M A is the transform of M underthe Morita functor from End R ( A )-modules to R -modules; since this functor is anequivalence of categories, it is exact and preserves injectives. (cid:3) Proposition A.3.
For any two left A -modules M, N and any q ≥ , we have Ext qA ( M, N ) ≃ Ext qR ( M, N ) A . (Note that Ext qR ( M, N ) is naturally an A − A -bimodule, which gives a meaning tothe statement.)Proof. The bifunctor (
M, N ) Hom A ( M, N ) is clearly the composition of the twofunctors (
M, N ) Hom R ( M, N )(from left A -modules to A − A -bimodules) and Q Q A (from A − A -bimodules to R -modules). Note also that, if P is A -projective and I is A -injective, then Hom R ( P, I ) is an injective A − A -bimodule. The conclusiontherefore follows from lemma A.1. (cid:3) Corollary A.4.
Let M be a left A -module. Then M is A -projective if and only ifit is R -projective.Proof. If M is A -projective, it is R -projective since A is a projective R -module.The converse follows from proposition A.3. (cid:3) Corollary A.5.
Suppose R regular of dimension d . Then any finitely generatedleft A -module M has a left resolution of length ≤ d by finitely generated projective A -modules. In particular, A is regular.Proof. Since R is regular, it is Noetherian and so is A by lemma A.1. Proposi-tion A.3 also shows that Ext d +1 A ( M, N ) = 0 for any N . The conclusion is nowclassical [7, Ch. VI, Prop. 2.1 and Ch. V, Prop. 1.3]. (cid:3) OTIVES OF AZUMAYA ALGEBRAS 77
Appendix B. Regularity
We prove the main result on the regularity of the functor K ( − ; A ) that we needto compute the layers in the homotopy coniveau tower for G ( X ; A ) in section 4.Fix a noetherian commutative ring R . We let R -alg denote the category ofcommutative R -algebras which are localizations of finitely generated commutative R -algebras.Following Bass [3, Ch. XII, §
7, pp. 657–658], for an additive functor F : R -alg → Ab , we let N F : R -alg → Ab be the functor N F ( A ) := ker ( F ( A [ t ]) → F ( A [ t ] / ( t )))where A [ t ] is the polynomial algebra over A . We set N q F := N ( N q − F ); F iscalled regular if N q F = 0 for all q > N F = 0).For f ∈ A the morphism A [ X ] → A [ X ] , X f · X induces a group endo-morphism N F ( A ) → N F ( A ). So N F ( A ) becomes a Z [ T ] module. We denoteby N F ( A ) [ f ] the Z [ T, T − ] module Z [ T, T − ] ⊗ Z [ T ] N F ( A ). With these notationsVorst proves the following theorem in [42]. Theorem B.1.
Let A ∈ R -alg and let a , . . . , a n be elements of A which generatethe unit ideal. Suppose further that the map N F ( R [ T ] a i ,..., c a ij ,...,a ip ) [ a ij ] → N F ( A [ T ] a i ,...,a ip ) is an isomorphism, for each set of indexes ≤ i < · · · < i p ≤ n . Then thecanonical morphism ǫ : N F ( A ) → n M j =1 N F ( A a j ) is injective.Proof. Compare [42, Theorem 1.2] or [22, Lemma 1.1]. (cid:3)
This is extended by van der Kallen, in the case of the functor A K n ( A ), toprove an ´etale descent result, namely, Theorem B.2.
Let A be a noetherian commutative ring such that each zero-divisorof A is contained in a minimal prime ideal of A . Let A → B be an ´etale andfaithfully flat extension of A . Then the Amitsur complex → N q K n ( A ) → N q K n ( B ) → N q K n ( B ⊗ A B ) → . . . is exact for each q and n . In fact, one can abstract van der Kallen’s argument to give conditions on a func-tor F : R -alg → Ab as above so that the conclusion of theorem B.2 holds for theAmitsur complex for N F . For this, we recall the big Witt vectors W ( A ) of a com-mutative ring A , with the canonical surjection W ( A ) → A and the multiplicative Teichm¨uller lifting A → W ( A ) sending a ∈ A to [ a ] ∈ W ( A ). We have as well theWitt vectors of length n , with surjection W ( A ) → W n ( A ); we let F n W ( A ) ⊂ W ( A )be the kernel. If M is a W ( A )-module, we say M is a continuous W ( A ) module if M is a union of the submodules M n killed by F n W ( A ). Then one has Theorem B.3.
Let F : R -alg → Ab be a functor. Suppose that F satisfies: (1) Given a ∈ A ∈ R -alg , the natural map F ( A a ) → F ( A ) [ a ] is an isomor-phism. (2) Sending a ∈ A to the endomorphism [ a ] : N F ( A ) → N F ( A ) extends toa continuous W ( A ) -module structure on N F ( A ) , natural in A , with theTeichm¨uller lifting [ a ] ∈ W ( A ) acting by [ a ] : N F ( A ) → N F ( A ) . (3) F commutes with filtered direct limitsLet A ∈ R -alg be such that each zero-divisor of A is contained in a minimal primeideal of A . Let A → B be an ´etale and faithfully flat extension of A . Then theAmitsur complex → N F ( A ) → N F ( B ) → N F ( B ⊗ A B ) → N F ( B ⊗ A B ⊗ A B ) → . . . is exact. The main example of interest for us is the following: Let A be a noetheriancentral R -algebra, and let K n ( A ) be the n th K -group of the category of finitelygenerated projective (left) A -modules. Corollary B.4.
Let F : R -alg → Ab be the functor F ( A ) := N q K n ( A ⊗ R A ) . Then F satisfies the conditions of theorem B.3, hence (assuming A satisfies thehypothesis on zero-divisors) if A → B is an ´etale and faithfully flat extension of A ,then the Amitsur complex → N q K n ( A ⊗ R A ) → N q K n ( A ⊗ R B ) → N q K n ( A ⊗ R B ⊗ A B ) → . . . is exact.Proof. Weibel [46] has shown that N q K n ( A ) admits a W ( A )-module structure,satisfying the conditions (1) and (2) of theorem B.3. Since K -theory commuteswith filtered direct limits, this proves that the given F satisfies the conditions oftheorem B.3, whence the result. (cid:3) Now let X be an R -scheme and let A be a sheaf of Azumaya algebras over O X . We have the category P X ; A of left A -Modules E which are locally free as O X -Modules. We let K ( X ; A ) denote the K -theory spectrum of P X ; A . We extend K ( X ; A ) to a spectrum which is (possibly) not ( − KB ( X ; A ). For f : Y → X an X -scheme,we write K ( Y ; A ) for K ( Y ; f ∗ A ), and similarly for KB .The spectra KB ( X ; A ) have the following properties:(1) There is a canonical map K ( X ; A ) → KB ( X ; A ), identifying K ( X ; A ) withis the -1-connected cover of KB ( X ; A ).(2) There is the natural exact sequence0 → KB p ( X ; A ) → KB p ( X × A ; A ) ⊕ KB p ( X × A ; A ) → KB p ( X × G m ; A ) → KB p − ( X ; A ) → fundamental exact sequence .(3) If X is regular, then K ( X ; A ) → KB ( X ; A ) is a weak equivalence.From now on, we will drop the notation KB ( X ; A ) and write K ( X ; A ) for the(possibly) non-connected version. Proposition B.5.
Let X be a noetherian affine R -scheme such that O X has nonilpotent elements, and let p : Y → X be an ´etale cover. Let ˜ A be a sheaf ofAzumaya algebras over O X . For each point y ∈ Y , let Y y := Spec O Y,y and let
OTIVES OF AZUMAYA ALGEBRAS 79 p y : Y y → X be the map induced by p . Fix and integer q ≥ . Suppose there is an M such that, for each smooth affine k -scheme T , N q K n ( T × k Y y , ( p y ◦ p ) ∗ A ) = 0 for each y ∈ Y and each n ≤ M . Then N q K n ( T × k X ; A ) = 0 for each smoothaffine T and each n ≤ M .Proof. Write X = Spec A . Then Q p ∗ y : A → B := Q y O Y,y is faithfully flat and´etale. Since X is affine, ˜ A is the sheaf associated to a central A -algebra A and since˜ A is a sheaf of Azumaya algebras, each finitely generated projective left A module isfinitely generated and projective as an A -module. Thus N q K n ( X, ˜ A ) = N q K n ( A ).Similarly, N q K n ( Y y , p ∗ y ˜ A ) = N q K n ( p ∗ y A ). By corollary B.4, N q K n ( A ) = 0 for n ≥
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