aa r X i v : . [ m a t h . K T ] F e b ALGEBRAIC ATIYAH-SINGER INDEX THEOREM
NGUYEN LE DANG THI
Abstract.
The aim of this work is to give an algebraic weak version of the Atiyah-Singerindex theorem. We compute then a few small examples with the elliptic differential operatorof order ≤ coming from the Atiyah class in Ext O X ( O X , Ω X/k ) , where X −→ Spec( k ) is asmooth projective scheme over a perfect field k . We follow Grothendieck [EGA4, §16.8] to recall briefly the notion of differential operators.Let f : X −→ S be a morphism of schemes. Consider the Cartesian square X id id ( ( ∆ X/S $ $ X × S X q (cid:15) (cid:15) p / / X f (cid:15) (cid:15) X f / / S The diagonal ∆ X/S : X −→ X × S X is an immersion. One defines the n -th normal invariantof ∆ X/S as P nX/S = ∆ − X/S ( O X × S X ) /I n +1 f . It is clear that {P nX/S } n form a projective system. One defines P ∞ X/S = lim ←− n P nX/S . The first projection p : X × S X −→ X induces an O X -algebra structure on P nX/S and thesecond projection q : X × S X → X induces a morphism d nX/S : O X −→ P nX/S . If F ∈ O X − M od is an O X -module, one defines P nX/S ( F ) = P nX/S ⊗ O X F . Let F , G ∈ O X − M od be two O X -modules. Let D : F −→ G be a morphism of the underlyingabelian sheaves. D is called a differential operator of order ≤ n relative S , if there exists a Date : 06. 02. 2017.1991
Mathematics Subject Classification.
Key words and phrases. K -theory, motivic cohomology, differential operators. orphism of O X -modules u : P nX/S ( F ) → G , such that the following diagram commutes: F D / / d nX/S (cid:15) (cid:15) GP nX/S ( F ) u ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ The morphism D : F → G between the underlying abelian sheaves is not O X -linear, but itis f − O S -linear. Let us denote by Diff nX/S ( F , G ) the set of all differential operator of order ≤ n between F and G . One defines Diff
X/S ( F , G ) = [ n Diff nX/S ( F , G ) . It follows from the definition [EGA4, 16.8.3.1] that one has an isomorphism of abelian groups
Hom O X ( P nX/S ( F ) , G ) ∼ = −→ Diff nX/S ( F , G ) . If f : X → S is locally of finite type, then P nX/S is a quasi-coherent O X -module of finite type(cf. [EGA4, Prop. 16.3.9]). If f : X → S is locally of finite presentation, then P nX/S is a quasi-coherent O X -module of finite presentation (cf. [EGA4, Cor. 16.4.22]). Consequently, if F isa quasi-coherent O X -module of finite type resp. locally of finite presentation and f : X → S is locally of finite type resp. locally of finite presentation, then P nX/S ( F ) = P nX/S ⊗ O X F isalso a quasi-coherent O X -module of finite type resp. locally of finite presentation. If S islocally Noetherian and f : X → S is proper, then by [EGA3, Thm. 3.2.1] the higher directimage sheaf R q f ∗ ( F ) = a Zar ( U H qZar ( f − U, F )) is coherent for any F ∈
Coh ( X ) and ∀ q ≥ . Consequently, if X −→ Spec( A ) is properover a Noetherian ring A , then for all coherent sheaf F ∈
Coh ( X ) the Zariski cohomologygroups H qZar ( X, F ) are A -modules of finite type ∀ q ≥ . A differential operator D : F −→ G induces a homomorphism of abelian groups Γ ab ( X, F ) defn = Hom Ab ( Z , F ) −→ Hom Ab ( Z , G ) defn = Γ ab ( Z , G ) , where we write Ab for the category of abelian sheaves on X and Z is the terminal object in Ab . In general, we can not say anything about the kernel or cokernel of this homomorphism.However, D induces also a homomorphism of Γ mod ( X, O X ) -modules Γ mod ( X, P nX/S ⊗ O X F ) = Hom O X − Mod ( O X , P nX/S ⊗ O X F ) −→ Hom O X − Mod ( O X , G )= Γ mod ( X, G ) . This motivates us to give the following definition:
Definition 1. (1)
Let X −→ Spec( k ) be a k -scheme of finite type, where k is a field and D : F → G be a differential operator between two O X -modules of order ≤ n . D iscalled a Fredholm operator, if the k -linear morphism between k -vector spaces D ( X ) : Γ mod ( X, P nX/k ⊗ O X F ) −→ Γ mod ( X, G ) has finite-dimensional kernel and cokernel. Let X → Spec( A ) be a scheme of finite type over a Noetherian ring A and D : F → G be a differential operator between two O X -modules. D is called a Fredholm operator,if the kernel and cokernel of the A -module morphism D ( X ) : Γ mod ( X, P nX/A ⊗ O X F ) −→ Γ mod ( X, G ) are A -modules of finite type. For a proper A -scheme X , where A is a Noetherian ring, any differential operator D : F −→ G between coherent O X -modules is a Fredholm operator. Definition 2.
Let X be a proper k -scheme and D : F −→ G be a differential operatorbetween coherent O X -modules. We define the index of D to be Ind ( D ) defn = dim k ( Ker ( D ( X )) − dim k ( Coker ( D ( X )) . Let X be any scheme and QsCoh ( X ) be the category of quasi-coherent O X -modules.Following Grothendieck [EGA2, §1.7] we define a vector bundle associated to a quasi-coherentsheaf E ∈
QsCoh ( X ) to be the X -scheme V ( E ) defn = Spec( Sym O X ( E )) . The structure morphism π : V ( E ) → X is an affine morphism. For any morphism f : Y → X ,we have Hom
Sch/X ( Y, V ( E )) ∼ = −→ Hom O X − Alg ( Sym O X ( E ) , f ∗ O Y ) ∼ = −→ Hom O X − Mod ( E , f ∗ O Y ) . We call s : X → V ( E ) the -section, if it corresponds to the -homomorphism E → O X .Now we assume E is an O X -module of finite type. Then π : V ( E ) → X is a morphism offinite type [EGA2, Prop. 1.7.11 (ii)]. So if X is a Noetherian scheme, then V ( E ) is alsoa Noetherian scheme. Consider the locally small abelian category Coh ( V ( E )) of coherentsheaves on V ( E ) . We let K ( V ( E )) defn = K ( Coh ( V ( E ))) to be the Grothendieck group of V ( E ) . By [EGA2, Prop. 1.7.15] the morphism s : X −→ V ( E ) is a closed immersion. Let j : V ( E ) − s ( X ) −→ V ( E ) be the complement openimmersion. By [SGA6, §IX, Prop. 1.1] one has a localization exact sequence(1) K ( X ) s ∗ −→ K ( V ( E )) j ∗ −→ K ( V ( E ) − s ( X )) −→ . If f : X → S is the structure morphism of X over a base scheme S , then any differentialoperator D ∈ Diff nX/S ( F , G ) of order ≤ n between O X -modules defines a two terms complexof the underlying abelian sheaves −→ F D −→ G −→ . Or equivalently, a two terms complex of O X -modules −→ P nX/S ⊗ O X F −→ G −→ . We obtain then a two terms complex of O V ( E ) -modules −→ π ∗ ( P nX/S ⊗ O X F ) −→ π ∗ G −→ . emark that if S is Noetherian and f : X −→ S is of finite type, then X is also Noetherian, so π ∗ ( F ) , π ∗ ( P nX/S ) and π ∗ ( G ) are coherent O V ( E ) -modules, if F and G are coherent. Moreover,by [EGA4, Prop. 16.4.5] there is a canonical isomorphism π ∗ ( P nX/S ) ∼ = −→ P n V ( E ) /X . Definition 3.
Let f : X → S be a morphism of finite type, where S is a Noetherian scheme.Let E ∈ O X − M od be an O X -module of finite type and π : V ( E ) → X be the associated vectorbundle. Let D ∈ Diff nX/S ( F , G ) be a differential operator between coherent O X -modules oforder ≤ n . Then D is called an elliptic operator with respect to E , if there is an isomorphism σ ( D ) : P n V ( E ) /X ⊗ O V ( E ) π ∗ F | V ( E ) − s ( X ) ∼ = −→ π ∗ G | V ( E ) − s ( X ) , The isomorphism σ ( D ) is called the symbol of D . By the localization exact sequence 1, an elliptic operator D of order ≤ n with respect toan O X -module of finite type E defines a class [ P n V ( E ) /X ] · [ π ∗ F ] − [ π ∗ G ] ∈ K ( V ( E )) , whichbecomes after restricting to the complement of the -section. If f : X → S a morphismlocally of finite type, then the sheaf of relative differentials Ω X/S = I f /I f is an O X -moduleof finite type. Definition 4.
Let X → S be a morphism of finite type over a Noetherian scheme S . Adifferential operator D ∈ Diff
X/S ( F , G ) between coherent O X -modules is called elliptic, if itis elliptic with respect to Ω X/S . Let X be a Noetherian scheme. Denote by V ect ( X ) the locally small exact category oflocally free sheaves of finite rank on X . We let K ( X ) defn = K ( V ect ( X )) . For an arbitrary morphism of Noetherian schemes f : Y → X one has a functorial pullbackhomomorphism f ∗ : K ( X ) −→ K ( Y ) , [ E ] [ f ∗ E ] . For a proper morphism f : Y −→ X between Noetherian schemes there is a pushforwardhomomorphism f ∗ : K ( Y ) −→ K ( X ) , F 7→ X i ( − i [ R i f ∗ F ] , which is well-defined as an exact sequence of coherent O Y -modules −→ F ′ −→ F −→ F ′′ −→ gives rise to an exact sequence of coherent O X -modules · · · −→ R i f ∗ ( F ′ ) −→ R i f ∗ ( F ) −→ R i f ∗ ( F ′′ ) −→ R i +1 f ∗ ( F ′ ) −→ · · · The functoriality of the pushforward follows easily from the Leray-Grothendieck spectralsequence E p,q = R p g ∗ ( R q f ∗ F ) ⇒ R p + q ( g ◦ f ) ∗ F . If X is a regular Noetherian scheme, then the canonical homomorphism induced by theobvious exact embedding of categories K ( X ) δ −→ K ( X ) s an isomorphism, since every coherent sheaf on a regular scheme has a finite locally freeresolution.Let now S be an arbitrary scheme. We denote by SH ( S ) the stable motivic homotopycategory together with the formalism of six functors ( f ∗ , f ∗ , f ! , f ! , ∧ , Hom) as in [Ay08],[CD12] and [Hoy14, Appendix C]. Let E be a motivic ring spectrum parameterized byschemes. We will assume E is stable by base change, i.e. for any morphism of schemes f : X −→ Y there is an isomorphism of spectra in SH ( X ) f ∗ E Y ∼ = −→ E X . Let S be a scheme and E S ∈ SH ( S ) be a motivic ring spectrum. The unit ϕ S : S → E S gives rise to a class [ ϕ S ] ∈ e E , ( P S ) ∼ = E , ( P S ) . We have a tower of S -schemes given by the obvious embeddings P S −→ P S −→ · · · −→ P nS −→ · · · Let P ∞ S = colim n P nS , which is an object in the pointed unstable motivic homotopy categoryof Morel-Voevodsky Ho A , • ( S ) [MV99]. Let i : P S −→ P ∞ S be the obvious map in Ho A , • ( S ) ,which gives rise to a map Σ ∞ ( i ) : Σ ∞ ( P S ) + −→ Σ ∞ ( P ∞ S ) + . E S is called oriented, if there isa class e E , ( P ∞ S ) ∋ c S : Σ ∞ ( P ∞ S ) + ) −→ E S ∧ S , , such that i ∗ ( c S ) = [ ϕ S ] . We say E is oriented, if E S is oriented for any scheme S and for anymorphism f : T → S one has f ∗ ( c S ) = c T . If S is regular, by [MV99, §4, 1.15, 3.7] there isa canonical isomorphism B G m ∼ = P ∞ S in Ho A , • ( S ) , which gives to a canonical isomorphism P ic ( S ) ∼ = −→ [ S + , B G m ] ∼ = −→ [ S + , P ∞ S ] . For an oriented motivic ring spectrum E , one can define the first Chern class as c : P ic ( S ) ∼ = −→ [ S + , P ∞ S ] Σ ∞ −→ Hom SH ( S ) (Σ ∞ S + , Σ ∞ P ∞ S ) ( c S ) ∗ −→ Hom SH ( S ) (Σ ∞ S + , E S ∧ S , ) = −→ E , ( S ) . Assume S is a regular scheme. Let X −→ S be a smooth S -scheme. For a vector bundle V ( E ) associated to a locally free O X -module E of finite rank r , there is an isomorphism (cf.[NSO09, Thm. 2.11]) r − M i =0 E ∗− i, ∗− i ( X ) ∼ = −→ E ∗ , ∗ ( P ( E )) , ( x , · · · , x r − ) r − X i =0 p ∗ ( x i ) ∪ c i ( O P ( E ) ( − , where P ( E ) = Proj( Sym O X ( E )) p −→ X is the projective bundle associated to E [EGA2, §4].So E ∗ , ∗ ( P ( E )) is a free module over E ∗ , ∗ ( X ) with the basis { , c ( O P ( E ) ( − , · · · , c ( O P ( E ) ( − r − ) } . So for a vector bundle V ( E ) of rank r one can define the higher Chern classes E i,i ( X ) ∋ c i ( V ( E )) , r X i =0 p ∗ ( c i ( V ( E )) ∪ ( − c ( O P ( E ) ( − r − i = 0 , here one puts c ( V ( E )) = 1 and c i ( V ( E )) = 0 for i / ∈ [0 , r ] . Recall that one has a homotopycofiber sequence [MV99, §3] P ( E ) i −→ P ( E ⊕ O X ) q −→ T h X ( V ( E )) , where T h X ( V ( E )) denotes the Thom space of the vector bundle V ( E ) . The homotopy cofibersequence induces a long exact sequence · · · −→ E ∗ , ∗ ( T h X ( V ( E ))) q ∗ −→ E ∗ , ∗ ( P ( E ⊕ O X )) i ∗ −→ E ∗ , ∗ ( P ( E )) −→ · · · The projective bundle formula tells us that i ∗ is a split epimorphism, so E ∗ , ∗ ( T h X ( V ( E ))) isa free module of rank over E ∗ , ∗ ( X ) , which is just Ker ( i ∗ ) . The Thom class is the uniqueclass in E r,r ( T h X ( V ( E ))) t V ( E ) = ( q ∗ ) − ( r X i =0 p ∗ ( c i ( V ( E )) ∪ ( − c ( O P ( E⊕O X ) ( − r − i ) , where p : P ( E ⊕ O X ) → X is the structure morphism. Let s : X −→ P ( E ⊕ O X ) be itssection. The Thom isomorphism is given by th : E ∗− r, ∗− r ( X ) ∼ = −→ E ∗ , ∗ ( T h X ( V ( E ))) , x x ∪ t V ( E ) . Its inverse is the composition E ∗ , ∗ ( T h X ( V ( E )) / / q ∗ / / E ∗ , ∗ ( P ( E ⊕ O X )) s ∗ / / E ∗− r, ∗− r ( X ) L ri =0 E ∗− i, ∗− i ( X ) ∼ = O O which sends r X i =0 p ∗ ( x i ) ∪ ( − c ( O P ( E⊕O X ) ( − r − i ) x r . If ξ is the universal quotient bundle on P ( E ⊕ O X ) , i.e. there is an exact sequence −→ O P ( E⊕O X ) ( − −→ p ∗ ( E ⊕ O X ) −→ ξ −→ . Then by Whitney sum formula we have q ∗ ( t V ( E ) ) = c r ( ξ ) . Now let
KGL S ∈ SH ( S ) be the motivic ring spectrum representing the algebraic K -theoryconstructed in [Rio10], [CD12, §13]. If S is regular and X ∈ Sm/S is a smooth S -scheme,then one has a natural isomorphism(2) Hom SH ( S ) (Σ ∞ X + , KGL S ) ∼ = −→ K ( X ) .KGL is an oriented motivic ring spectrum. Indeed, by Bott periodicity one has an isomor-phism KGL , ( S ) ∼ = K ( S ) ∼ = ^ KGL , ( P S , ∞ ) . The Bott element β is the image of under this isomorphism and β ∈ KGL − , − ( S ) . Theorientation of KGL is given by c KGL defn = β − · (1 − [ O P ∞ S (1)]) ∈ KGL , ( P ∞ S ) . or any line bundle L ∈ P ic ( S ) , its first Chern class is c KGL ( L ) = β − (1 − L ∨ ) . For an arbitrary morphism f : T → S of regular schemes one has a commutative diagram[CD12, §13.1] Hom SH ( S ) ( S , KGL S ) / / ∼ = (cid:15) (cid:15) Hom SH ( T ) ( f ∗ S , f ∗ KGL S ) Hom SH ( T ) ( T , KGL T ) ∼ = (cid:15) (cid:15) K ( S ) f ∗ / / K ( T ) Now assume E is a locally free sheaf of finite rank on a smooth scheme X over a regular base S . We have a homotopy cofiber sequence (cf. [MV99, §3]) V ( E ) − s ( X ) −→ V ( E ) −→ T h X ( V ( E )) , where T h X ( V ( E )) denotes the Thom space of the vector bundle V ( E ) . This homotopy cofibersequence gives rise to an exact sequence · · · −→ Hom SH ( S ) (Σ ∞ T h X ( V ( E )) , KGL S ) −→ Hom SH ( S ) (Σ ∞ V ( E ) + , KGL S ) −→ Hom SH ( S ) (Σ ∞ ( V ( E ) − s ( X )) + , KGL S ) The natural isomorphism 2 gives rise to a commutative diagram with exact rows (cf. [CD12,§13.4, (K6a)]), where we abbreviate [ − , − ] for Hom SH ( S ) ( − , − ) : [ T h X ( V ( E )) , KGL S ] / / [ V ( E ) , KGL S ] ∼ = (cid:15) (cid:15) / / [ V ( E ) − s ( X ) , KGL S ] ∼ = (cid:15) (cid:15) K ( X ) / / K ( V ( E )) / / K ( V ( E ) − s ( X )) The bottom row is the exact sequence 1. For a Noetherian scheme X and an O X -module offinite type E the functor π ∗ : Coh ( V ( E )) −→ Coh ( X ) is exact, since π is an affine morphism, which implies that R i π ∗ F = 0 , ∀ i > , ∀F ∈ Coh ( V ( E )) . So we still can define the pushforward π ∗ : K ( V ( E )) −→ K ( X ) . Hence, one may try to define the topological index by applying the homomorphism K ( T h X ( V (Ω X/k ))) −→ K ( V (Ω X/k )) then composing with f ∗ π ∗ . However, π : V ( E ) → X is not necessary projective, so we mayrun into difficulties, when we apply later the Grothendieck-Riemann-Roch theorem. Now werestrict ourselves to the situation where f : X → Spec( k ) is a smooth proper scheme overa field k . Ω X/k is a locally free O X -module of finite rank. Let D : F → G be an ellipticoperator of order ≤ n . Let π : V (Ω X/k ) −→ X denotes the vector bundles associated to Ω X/k . The symbol σ ( D ) defines then an element [ P n V (Ω X/k ) /X ] · [ π ∗ F ] − [ π ∗ G ] ∈ K ( V (Ω X/k )) , hich lies in the image of the homomorphism K ( T h X ( V (Ω X/k ))) −→ K ( V (Ω X/k )) . Let us denote by [ σ ( D )] ∈ K ( T h X ( V (Ω X/k ))) a class, which is mapped to [ P n V (Ω X/k ) /X ] · [ π ∗ F ] − [ π ∗ G ] , such that its image in K ( V (Ω X/k ) − s ( X )) is trivial. We call this class [ σ ( D )] a symbolclass associated to the symbol σ ( D ) . Definition 5.
Let f : X −→ Spec( k ) be a smooth proper k -scheme. Let D : F −→ G be anelliptic operator, where F , G ∈
Coh ( X ) . The topological index of D is defined as ind top ( D ) = f ∗ ( th − ([ σ ( D )])) , where th − : K ( T h X ( V (Ω X/k ))) ∼ = −→ K ( X ) is the inverse Thom isomorphism of algebraic K -theory and f ∗ : K ( X ) → K (Spec( k )) = Z , E 7→ X i ( − i [ H iZar ( X, E )] = χ ( E ) If S is any scheme, let H Q denote the Beilinson rational motivic cohomology ring spectrumin SH ( S ) constructed in [Rio10], [CD12, Defn. 14.1.2]. Let ch t : KGL Q ∼ = −→ ∨ i ∈ Z H Q ∧ S i,i be the total Chern character, which is an isomorphism of rational spectra, if S = Spec( k ) ,where k is a perfect field [Rio10, Defn. 6.2.3.9, Rem. 6.2.3.10]. One has the following resultdue to Riou: Theorem 6. [Rio10, Thm. 6.3.1] (Grothendieck-Riemann-Roch) Let k be a perfect field. Let f : X → S be a smooth projective morphism of smooth k -schemes. There is a commutativediagram in SH ( S ) R f ⋆ KGL Q f ⋆ (cid:15) (cid:15) R f ⋆ ( ch · T d ( T f )) / / W i ∈ Z R f ⋆ H Q ∧ S i,if ⋆ (cid:15) (cid:15) KGL Q ch t / / W i ∈ Z H Q ∧ S i,i As explained in [CD12, §13.7] f ⋆ induces the usual pushforward f ∗ on K -theory. Now wecan compute Proposition 7.
Let f : X → Spec( k ) be a smooth projective scheme over a perfect field k . Let D : F → G be an elliptic operator between coherent O X -modules. Then one has aformula ind top ( D ) = Z X ch ( th − ([ σ ( D )])) ∪ T d ( T X/k ) , where R X means that we take the pushforward on motivic cohomology Z X : H ∗ , ∗M ( X, Q ) −→ H ∗− dim ( X ) , ∗− dim ( X ) M (Spec( k ) , Q ) = Q . roof. This is a trivial consequence of Thm. 6, where S = Spec( k ) . (cid:3) Now let D ∈ Diff nX/A ( F , G ) be an arbitrary differential operator of order ≤ n on a projectivescheme X → Spec( A ) over a Noetherian ring A and F , G ∈
Coh ( X ) . As D gives rise to an O X -module homomorphism P nX/A ⊗ O X F −→ G , we may take the twisting P nX/A ⊗ O X F ⊗ O X O X ( m ) −→ G ⊗ O X O X ( m ) , which in turn gives us a differential operator of order ≤ nD ( m ) : F ( m ) −→ G ( m ) . We call D ( m ) a twisting of D . Proposition 8.
Let X −→ Spec( k ) be a smooth projective scheme over a perfect field k and D : F −→ G be a differential operator of order ≤ n between coherent O X -modules. Thenthere exists a number N , such that for all N ≥ N one has Ind ( D ( N )) = Z X ch (([ P nX/k ] · [ F ] − [ G ])( N )) ∪ T d ( T X/k ) . Proof.
This is quite trivial. By the result of Serre (see e.g [EGA3, Thm. 2.2.1, Prop. 2.2.2]),there is a number n , such that for all q > and all a ≥ n H qZar ( X, P nX/k ⊗ O X F ( a )) = 0 , and a number m , such that for all q > and all b ≥ m H qZar ( X, G ( b )) = 0 . We take N = max ( n , m ) to be the maximal of n and m . We have trivially that for all N ≥ N Ind ( D ( N )) = dim k Γ mod ( X, P nX/k ⊗ O X F ( N )) − dim k Γ mod ( X, G ( N )) . The proposition follows now easily from the Hirzebruch-Riemann-Roch theorem dim k Γ mod ( X, E ( N )) = Z X ch ( E ( N )) ∪ T d ( T X/k ) . (cid:3) Now we are ready to state the following algebraic weak form of the Atiyah-Singer indextheorem
Theorem 9.
Let X −→ Spec( k ) be a smooth projective scheme over a perfect field k and D ∈ Diff nX/k ( F , G ) be an elliptic differential operator of order ≤ n , where F , G ∈
Coh ( X ) are coherent O X -modules. There exists a number N , such that for all N ≥ N there is anequality Ind ( D ( N )) = ind top ( D ( N )) . roof. It remains to prove that th − ([ σ ( D ( N ))]) = ([ P nX/k ] · [ F ] − [ G ])( N ) , which means that we have to show there is a commutative diagram K ( T h X ( V (Ω X/k ))) th − ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ / / K ( V (Ω X/k )) s ∗ (cid:15) (cid:15) K ( X ) By construction, we have a commutative diagram K ( T h X ( V (Ω X/k ))) th − ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ / / K ( P (Ω X/k ⊕ O X )) s ∗ (cid:15) (cid:15) K ( X ) So it remains to see that we have a commutative diagram K ( V (Ω X/k )) s ∗ (cid:15) (cid:15) / / K ( P (Ω X/k ⊕ O X )) s ∗ (cid:15) (cid:15) K ( X ) K ( X ) But this is quite obvious, since the closed immersion s : X −→ P (Ω X/k ⊕ O X ) is thecomposition of the -section s : X −→ V (Ω X/k ) and V (Ω X/k ) −→ P (Ω X/k ⊕ O X ) (cf.[EGA2, Prop. 8.3.2]). (cid:3) Finally, we will give in the last part of this paper a few small examples. For this purpose,we always restrict now ourselves to the case of a smooth projective k -scheme X −→ Spec( k ) of dimension dim( X ) = d , where k is a perfect field. Let us begin with the following lemma: Lemma 10. (de Jong and Starr [dJS06, Lem. 2.1] ) Let X ⊂ P n be a smooth completeintersection of type ( d , · · · , d c ) . Then ch ( T X ) = n − c + n − c X i =1 ( n + 1 − c X i =1 d ki ) c ( O X (1)) k k ! . Proof.
The proof is so elementary, so we reproduce the proof. Let i : X −→ P n denote theclosed embedding. Consider the short exact sequence −→ T X −→ i ∗ T P n −→ c M i =1 O X ( d i ) −→ . Therefore, ch ( T X ) = ch ( i ∗ T P n ) − c − n − c X k =1 c X i =1 d ki c ( O X (1)) k k ! . Consider the Euler sequence −→ O P n −→ O P n (1) ⊕ ( n +1) −→ T P n −→ . ne has ch ( T P n ) = n + n X k =1 ( n + 1) c ( O P n (1)) k k ! . The lemma follows easily. (cid:3)
Consider the Atiyah class in
Ext O X ( O X , Ω X/k )0 −→ Ω X/k −→ P X/k u −→ O X −→ . The morphism u : P X/k −→ O X defines a differential operator of order ≤ D u ∈ Diff X/k ( O X , O X ) . Lemma 11. D u is an elliptic operator.Proof. Let π : V (Ω X/k ) −→ X be the vector bundle associated to Ω X/k . We pullback theAtiyah class via π to obtain an exact sequence −→ Ω V (Ω X/k ) /X −→ P V (Ω X/k ) /X −→ O V (Ω X/k ) −→ . For each point P ∈ V (Ω X/k ) we have by [EGA4, Cor. 16.4.12] P V (Ω X/k ) /X,P = O V (Ω X/k ) ,P / m P . The morphism π ∗ ( u ) : P V (Ω X/k ) /X −→ O V (Ω X/k ) becomes obviously an isomorphism at eachpoint P ∈ V (Ω X/k ) − s ( X ) . (cid:3) From now on we always consider the elliptic differential operator D u . We write H = c ( O X (1)) and for a curve X deg ( H ) = Z X c ( O X (1)) . Proposition 12.
Let C ⊂ P be a smooth curve of genus , which is a smooth completeintersection of two quadrics. For all integer number n ≥ one has ch (Ω C/k ( n )) ∪ T d ( T C/k ) ∈ H ∗ , ∗M ( C, Z ) . For
N >> one has Ind ( D u ( N )) = 4 N. Proof. C is smooth complete intersection of two quadrics. By the lemma 10 we have ch (Ω C/k ) = 1 . This implies ch (Ω C/k ( n )) = ch ( O C ( n )) = (1 + n · H ) , for any number n ≥ , where we write H = c ( O C (1)) for the hyperplane section. As g C = 1 ,the canonical divisor K C = 0 is trivial. So we have T d ( T C/k ) = 1 and hence ch (Ω C/k ( n )) ∪ T d ( T C/k ) = (1 + n · H ) ∈ H ∗ , ∗M ( C, Z ) . This implies easily that
Ind ( D u ( N )) = N · deg ( H ) = 4 N for N >> . (cid:3) roposition 13. Let X −→ P be a flat projective morphism over a perfect field K , suchthat the generic fiber X η is a smooth curve of genus one in P . Then there exists a number N >> , such that Ind ( D u ) ≡ . Proof.
We have for
N >> : Ind ( D u ( N )) = dim K Γ mod ( X, Ω X/K ( N )) . We take two embeddings i : P −→ P M and i : X −→ P M , such that they are compatiblewith f X f / / i & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ P i (cid:15) (cid:15) P M Let N >> be a big integer number. By [EGA3, Cor. 7.9.13, Seconde partie] f ∗ (Ω X/K ( N )) is a locally free O P -module of rank χ ( X η , Ω X η ,η ( N )) . By flat base change for Zariski coho-mology of coherent sheaves it is enough to compute χ ( X ¯ η , Ω X ¯ η ( N )) , where X ¯ η is the basechange of X η to an algebraic closure ¯ η . X ¯ η must be an elliptic curve. For an elliptic curve ( E, P ) , we know that Ω E ∼ = O E and L (3 P ) is very ample, i.e. L (3 P ) ∼ = O E (1) . The lastisomorphism means simply that we can embed | P | : E −→ P . By Riemann-Roch theorem for curves we also have dim H ( E, L ( nP )) = n . Now we canapply the theorem of Grothendieck for the decomposition of vector bundles on P and wehave f ∗ (Ω X/K ( N )) ∼ = N M i =1 O P ( a i ) , a i ∈ Z . Let N ′ >> be another big integer number. We have f ∗ (Ω X/K ( N ))( N ′ ) ∼ = N M i =1 O P ( a i + N ′ ) . So dim K f ∗ (Ω X/K ( N ))( N ′ ) = N X i =1 (1 + a i + N ′ ) = 3 N ( N ′ + 1) + N X i =0 a i . If this dimension is already odd, then there is nothing to prove. So we assume N ( N ′ + 1) + N X i =1 a i ≡ . Let N ′′ >> be another big integer number. After twisting by O ( N ′′ ) we have dim K f ∗ (Ω X/K ( N ))( N ′ + N ′′ ) = 3 N ( N ′ + N ′′ + 1) + N X i =0 a i . ow we take simply N ≡ , N ′ ≡ and N ′′ ≡ . We obtain dim K f ∗ (Ω X/K ( N ))( N ′ + N ′′ ) ≡ By projection formula we have for any number n > an isomorphism f ∗ (Ω X/K ) ⊗ O P O P ( n ) ∼ = −→ f ∗ (Ω X/K ) ⊗ O P i ∗ O P M ( n ) ∼ = −→ f ∗ (Ω X/K ⊗ O X f ∗ i ∗ O P M ( n )) ∼ = −→ f ∗ (Ω X/K ⊗ O X i ∗ O P M ( n )) ∼ = −→ f ∗ (Ω X/K ⊗ O X O X ( n )) . This implies dim K Γ mod ( X, Ω X/K ( N + N ′ + N ′′ )) = dim K Γ mod ( P , f ∗ (Ω X/K ( N + N ′ + N ′′ ))) ≡ , where the first equality follows easily from the adjunction f ∗ : O P − M od ⇆ O X − M od : f ∗ . Therefore, we can conclude that there exists a number N = N + N ′ + N ′′ >> such that Ind ( D u ( N )) ≡ , which finishes the proof. (cid:3) Proposition 14.
Let X ⊂ P be a smooth K -surface, which is a smooth complete inter-section, over a perfect field k . (1) If deg ( X ) = 4 , then for a big number N >> one has Ind ( D u ( N )) = 4 N − N + 1 . (2) If deg ( X ) = 6 , then for a big number N >> one has Ind ( D u ( N )) = 6 N − N − . Proof.
We write c ( O X (1)) = H and K for the canonical divisor. For an algebraic surfaceone has T d ( T X/k ) = 1 − K + 112 ( K + c ) . For a smooth projective K -surface one has K = 0 and K = 0 . Ω X/k is the dual bundle of T X/k . So we have by lemma 10 ch (Ω X/k ) = 2 − (5 + d + d ) · H + (5 − d − d ) · H . For any number n ≥ one has ch ( O X ( n )) = 1 + n · H + 12 n · H . If deg ( X ) = 4 then ( d , d ) = (2 , and if deg ( X ) = 6 then ( d , d ) = (2 , . The resultfollows now easily, since deg ( X ) = R X H and R X c = 24 . (cid:3) References [Ay08] J. Ayoub, Les six opération de Grothendieck et le formalisme des cycles évanescents dans lemonde motivique I and II. Astérisque , , (2008).[CD12] C. D. Cisinski, F. Déglise, Triangulated category of mixed motives, arXiv:0912.2110v3[math.AG], Preprint (2012).[dJS06] A. J. de Jong, J. Starr, Low degree complete intersections are rationally simply connected,Preprint, 74 pages.[EGA2] A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique. II. Étude globale élémentairede quelques classes de morphismes, Publ. Math. IHÉS (1961). EGA3] A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique. III. Étude cohomologiquedes faisceaux cohérents, Publ. Math. IHÉS (1961) and (1963).[EGA4] A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique. IV. Étude locale des schémaset des morphismes de schémas IV, Publ. Math. IHÉS , , , (1964-1967).[Hoy14] M. Hoyois, A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula, Algebr.Geom. Topol. 14, no. , (2014).[MV99] F. Morel, V. Voevodsky, A -homotopy theory of schemes, IHÉS Publ. Math., ( ) , 45-143,(1999).[NSO09] N. Naumann, M. Spitzweck, Paul Arne Østvær, Chern classes, K -theory and Landweber exact-ness over nonregular base schemes, in Motives and Algebraic Cycles: A Celebration in Honourof Spencer J. Bloch, Fields Inst. Comm., Vol. , (2009).[Rio10] J. Riou, Algebraic K -theory, A -homotopy and Riemann-Roch theorems, J. of Topology ,(2010), pp. 229-264.[SGA6] P. Berthelot, A. Grothendieck, L. Illusie, Théorie des intersections et théorème de Riemann-Roch, Lect. notes in Math. , Springer-Verlag, (1971). E-mail address : [email protected]@gmail.com