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K-Theory 549: 1­50, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Oriented Cohomology Theories of Algebraic Varieties
I. PANIN (after I. PANIN and A. SMIRNOV)
St. Petersburg Branch, Laboratory of Algebra, V.A. Stecklov Institute of Maths (POMI), Russian Academy of Sciences, 191011 St. Petersburg, Russia. e-mail: Panin@pdmi.ras.ru (Received:) Abstract. This article contains proofs of the results announced in [21] in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples. Mathematics Subject Classifications: Key words:

Author's proof!

1. Introduction The concept of oriented cohomology theory is well known in topology [1, Part II, p. 37], [27, Chapter 1, 4.1.1]. An algebraic version of this concept was introduced in [21] and is considered here. So in this article we consider a field k and the category of pairs (X, U ) with a smooth variety X over k and its open subset U . By a cohomology theory we mean a contravariant functor A from this category to the category of Abelian groups endowed with a functor transformation : A(U ) A(X, U ) and satisfying the localization, Nisnevich excision and homotopy invariance properties (Definition 2.1). We consider three structures a ring cohomology theory A can be equipped with: an orientation on A, a Thom structure on A and a Chern structure on A. An orientation on A is a rule assigning to each variety X and to each vector bundle E/ X a two-sided A(X )-module isomorphism A(X ) A(E , E - X) satisfying certain natural properties (Definition 3.1) and called Thom isomorphisms. A Thom structure on A is a rule assigning to each smooth variety X and each line bundle L over X a class th(L) A(L , L - X) satisfying certain natural properties (Definition 3.3) and called the Thom class. A Chern structure on A is a rule assigning to each smooth variety X and each line bundle L over X a class c(L) A(X ) satisfying certain natural properties (Definition 3.2) and called the first Chern class (or some times called the Euler class [20]).

Pdf output

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It is proved in this paper that for a given A these structures are in natural bijections with each other. More precisely we construct the following diagram Orientations RR A on k5 RRR kkk RRR kkk RRR kk RRR kk kkk ) o Chern structures on A Thom structures on A


(1)

in which each arrow is a bijection and each round trip coincides with the identity (Theorems 3.5, 3.35 and 3.36). The constructions of these arrows are described briefly below in this section. One of the consequence of the theorem is this: the existence at least one of these structures on A implies the existence of an orientation on A; an orientation on A is never defined by the ring cohomology theory itself, even on usual singular cohomology there are plenty different orientations (see an example below Section 1). However in practice certain ring cohomology theories come equipped with either a specific Chern structure or with a specific Thom structure. Thus they are equipped with distinguished orientations. Say, usual singular cohomology with integral coefficients (on complex algebraic varieties) come equipped with the known Chern structure, algebraic K-theory is equipped with a Chern structure as well (L [1] - [L ]). The motivic cohomology H (-, Z()) is equipped with a Chern structure. The algebraic cobordism theory MGL, is equipped with a natural Thom structure. Thus these two theories are equipped with the corresponding orientations. The algebraic cobordism theory MGL, of Voevodsky [31] is one of the main motivating example for this article, but it is expected to be written in details later. An oriented ring cohomology theory is a cohomology theory equipped with an orientation in the sense above. As it was already mentioned to orient a ring theory A is the same as to fix a Thom structure on A or to fix a Chern structure on A. An orientation is usually denoted . The Thom structure corresponding to via is written often as L th (L). The Chern structure corresponding to via is written often as L c (L). An orientable ring cohomology theory is a ring cohomology theory which can be equipped with an orientation. An oriented cohomology theory is as well an oriented cohomology pretheory in the sense of [20] because the integration constructed in [21], [23] is perfect in the sense of [20]. An example of a non-standard Chern structure on the usual singular cohomology with rational coefficients is given by the assignment L 1 - exp(-c1 (L)). This Chern structure gives an integration on H (-, Q) which, via the Chern character, respects the Chern structure on the algebraic K-theory given by L [1] - [L ]. Following [19] and [24] each orientation on A gives rise to a commutative formal group law over the coefficient ring A(p t ) of the theory. This is a formal power series F A(p t )[[u1 ,u2 ]] in two variables such that for each smooth variety X and each pair of line bundles L1 ,L2 over X one has the relation

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c (L1 L2 ) = F (c (L1 ), c (L2 )) in A(X ). The formal group law plays a key rule in constructing push-forwards on A [21, 23] and is described in Section 3.9. In the topological setting there is the universal oriented theory. It is the complex cobordism equipped with a distinguished orientation [16, 25]. The corresponding formal group law is the universal one [24]. Other examples of oriented theories are the singular cohomology, the complex K-theory, the Brown­Peterson theory, Morava K-theories, elliptic cohomology. Stable cohomotopy is a typical example of an unorientable theory. We now sketch the structure of the text. In Section 1.1 certain general notation are introduced. In Section 2 the notion of cohomology theory introduced in [21] is recalled and general properties of a cohomology theory are proved. The deformation to the normal cone construction is recalled in Section 2.2.7 as well. An analog of the purity theorem from [18] is proved (Theorem 2.2). The canonical isomorphism A(X, X - Z) A(N , N - Z) from Theorem 2.2 one should consider = as a replacement of the excision isomorphism for a tubular neighborhood (well known in the topology). Although Theorem 2.2 is not used in the present article, it is very useful for the construction of an integration on a given oriented cohomology theory. In the end of the section we recall the notion of a ring cohomology theory [21]. In Section 3 we construct the triangle of correspondences , , mentioned above (see Theorems 3.5, 3.35 and 3.36). Proofs of these three theorems use Theorem 3.9, the Splitting principle (Lemma 3.24) and higher Chern classes Theorem (3.27). In the very end of the section it is shown how an orientation on the theory A gives rise to the formal group law F over the coefficient ring of A. Since the text is rather long it is reasonable to sketch here our constructions of the assignments , and . Suppose we are given with a Thom structure L/ X th(L) A(L , L - X) on A. For the zero section z: X L of a line bundle L over X set c(L) = zA (th(L)) A(X ) (the pull-back of the element th(L) under the inclusion (X, ) (L , L - X)). The assignment L c(L) is the Chern structure on A corresponding via to the Thom structure (see Theorem 3.5). Suppose we are given with an orientation . For a line bundle L over a smooth X consider the image th(L) A(L , L - X) of the unit 1 A(X ) under the Thom isomorphism A(X ) A(L , L - X) determined by the orientation . The assignment L th(L) A(L , L - X) is the Thom structure corresponding via to the orientation (see Theorem 3.36). Suppose we are given with a Chern structure L c(L) on A. In this case the Projective bundle theorem holds (see Theorem 3.9) and there is a Chern class theory E cn (E ) with values in A. To produce an orientation on A we associate to each vector bundle E/ X its Thom class th(E ) A(E , E - X). Firstly for a rank n vector bundle E we consider the vector bundle F = 1 E , the projective bundle p : P(F ) X , the tautological line bundle OF (1) on it and set ¯ (E ) := cn (OF (1) p (E )) A(P(F )). th

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It turns out that the element ¯ (E ) belongs to the subgroup A(P(F ), P(F ) - P(1)) th of the group A(P(F )). The class th(E ) is defined as the image of the element ¯ (E ) th under the excision isomorphism identifying A(P(F ), P(F ) - P(1)) with A(E , E - X). The required orientation on A is given by the assignment which associate to a vector bundle p : E X the map ( th(E )) p A : A(X ) A(E , E - X). Details are given in the proof of Theorem 3.35. There is another class thnaive (E ) A(E , E - X) which is quite often used in literature. It is constructed as the image under the mentioned excision isomorphism of the class
n

¯ naive (E ) = th
k =0

(-1)

n-k

cn-k (E ) k A(P(F )),

where = c(OF (-1)) is the first Chern class of the tautological line bundle on P(F ). The assignment E thnaive (E ) gives an orientation too. However in this case the assignment L c (L) = zA (thnaive (L)) A(X ) is a Chern structure on A which is in general different of the one we began with. If the Chern structure L c(L) satisfies the additivity property, that is c(L1 L2 ) = c(L1 ) + c(L2 ), then thnaive (E ) = th(E ) and c(L) = c (L). To simplify technicalities a reader may assume through the text that · all cohomology theories in the sense of Definition 2.1 take values in the category of Z/2-graded Abelian groups and grade-preserving homomorphisms, the boundary operator is either grade-preserving or of the degree +1 and moreover, · all ring cohomology theories in the sense of Definition 2.13 are Z/2-gradedcommutative ring theories, i.e. for any a Ap (P ) and b Aq (Q) one has the relation a â b = (-1)pq b â a in Ap+q (P â Q), · all Thom isomorphisms in the sense of Definition 3.1 are grade-preserving and all Thom and Chern classes are of even degree, · `a universally central elements' is just `an even degree element'. Following these simplifications the reader should replace everywhere through the text the concept of `universally central elements' (see Definition 2.15) by the concept of `even degree elements'. For instance the reader should replace the ring Auc (X ) of all universally central elements by the ring Aev (X ) of all even degree elements.

1.1. T E R MI NOL OGY AND NOTAT I ON Let k be a field. The term `variety' is used in this text to mean a reduced quasiprojective scheme over k . If X is a variety and U X is a Zariski open then Z := X - U is considered as a closed subscheme with a unique structure of

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the reduced scheme, so Z is considered as a closed subvariety of X . We fix the following notation: · Ab ­ the · Sm ­ the SmOp ­ isms are category of Abelian groups; category of smooth varieties; the category of pairs (X, U ) with smooth X and open U in X . Morphmorphisms of pairs.

We identify the category Sm with a full subcategory of SmOp assigning to a variety X the pair (X, ): · pt = Spec(k ); For a smooth X and an effective divisor D X we write L(D ) for a line bundle over X whose sheaf of sections is the sheaf LX (D ) (see [9, Chapter II, Section 6, 6.13]). P(V ) = Proj(S (V )) ­ the space of lines in a finite-dimensional k-vector space V ; LV = OV (-1) ­ the tautological line bundle over P(V ); 1X ­ the trivial rank 1 bundle over X , often we will write 1 for 1X ; · P(E ) ­ the space of lines in a vector bundle E ; LE = OE (-1) ­ the tautological line bundle on P(E ); E 0 ­ the complement to the zero section of E ; E ­ the vector bundle dual to E ; z: X E ­ the zero section of a vector bundle E ; · For a contravariant functor A on Sm set (2) A(P ) = lim A(P(V )), - where the projective system is induced by all the finite-dimensional vector subspaces V k . Similarly set A(P â P ) = lim A(P(V ) â P(W )), - where the projective system is induced by all the finite-dimensional subspaces V, W k . 2. Cohomology Theories DEFINITION 2.1. A cohomology theory is a contravariant functor SmOp - Ab together with a functor morphism : A(U ) A(X, U ) satisfying the following properties 1. Localization: the sequence A(X ) - A(U ) - A(X, U ) - A(X ) - A(U ) is exact for each pair P = (X, U ) SmOp , where j : U X and i : (X, ) (X, U ) are the natural inclusions. 2. Excision: the operator A(X, U ) A(X ,U ) induced by a morphism e: (X ,U ) (X, U ) is an isomorphism, if the morphism e is etale and for Z = X - U , Z = X - U one has e-1 (Z ) = Z and e: Z Z is an isomorphism.
P

A

j

A



i

A

j

A

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3. Homotopy invariance: the operator A(X ) A(X â A1 ) induced by the projection X â A1 X is an isomorphism. The operator P is called the boundary operator and is written usually as . A morphism of cohomology theories : (A, A ) (B , B ) is a functor transformation : A B commuting with the boundary morphisms in the sense that B A for every pair P = (X, U ) SmOp one has P U = P P . We write also AZ (X ) for A(X, U ), where Z = X - U , and call the group AZ (X ) cohomology of X with the support on Z . The operator AZ (X ) - A(X ) is called the support extension operator for the pair (X, U ). We do not assume at all in this text that cohomology theories are graded and the boundary operator is of degree +1. We do not assume this in particular because it is never used below and it is even inconvenient to assume this for certain points. One could replace in this definition the category of Abelian group by any Abelian category or even by additive one which is equipped with Kernels and Cokernels for projectors. We left such a replacement to a reader to avoid technicalities as much as it is possible.
i
A

(3)

2.1. EX A M PLES Consider a number of examples. 2.1.1. Classical Singular Cohomology Let k = C and let A be an Abelian group. Let (X, U ) H p (X (C), U (C); A) - be the usual singular cohomology (with coefficients in A) of the pair of the complex point sets with respect to the complex topology. Take as a boundary the usual boundary map (see for instance [29]). 2.1.2. A Generalized Cohomology Theory p Let k = C and let (X, U ) p =- E (X (C), U (C)) be a generalized cohomology theory say represented by a spectrum E with the usual boundary map (see for instance [1] or [29, 8.33]). 2.1.3. Singular Cohomology of the Real Point Sets Let k = R and let A be an Abelian group (X, U ) H p ((X (R), U (R); A) 0 be the usual singular cohomology (with coefficients in A) of the pair of real points set considered with respect to the strong topology. Take as a boundary the usual boundary map for the pair (X (R), U (R)).

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2.1.4. Witt Theory of Balmer p Let (X, X - Z) p =- WZ (X ) be the Witt functor defined in [2]. Clearly it is a cohomology theory in the sense of Definition 2.1. 2.1.5. Bloch's Higher Chow Groups p Let (X, X - Z) p =- q =- CHZ (X, q) be the higher Chow groups defined in [3]. Clearly it is a cohomology theory in the sense of Definition 2.1. 2.1.6. Motivic Cohomology of M. Levine Clearly it is a cohomology theory in the sense of Definition 2.1 [15]. 2.1.7. Etale Cohomology Let F be a locally constant torsion sheaf on the etale k -situs and assume n char(k ) is prime to the torsion of F . In this example An (X, U ) = HZ (Xet [17, 3.1] and is defined in [17, 3.1.25]. The localization property for the (A, ) is proved in [17, 3.1.25], the excision property is proved in [17, 3.1.27] the homotopy invariance is proved in [17, 6.4.20].

that ,F ) pair and

2.1.8. K-theory Algebraic K-theory also can be fitted to Definition 2.1. To do this use, for instance, K-groups with support Kn (X onZ ) (n 0) of [30]. So set An (X, U ) = K-n (X onZ ), where Z = X - U . Further set A(X, U ) = 0 An (X, U ). The n= definition of and the exactness of the localization sequence are contained in [30, Theorem 5.1] (except the surjectivity of the restriction A0 (X ) A0 (U )). If X Q is quasi-projective then K(X onX ) coincides with the Quillen's K-groups Kn (X ) n by [30, 3.9, 3.10]. This proves in particular the homotopy invariance A (X ) for smooth X . The excision property for A follows from [30, 3.19]. It remains now to check the surjectivity of the restriction A0 (X ) A0 (U ). Clearly A0 (X ) = Q K0 (X ) coincides with the Grothendieck group of the vector bundles on X . Since X is smooth the desired surjectivity follows from [4, Section 8, Proposition 7]. Thus (A, ) satisfies Definition 2.1. 2.1.9. Motivic Cohomology C,p p Here AZ (X ) = HZ (X, C) := HomDM - (k) (MZ (X ), C[p ]) is the motivic cohomology with coefficients in a motivic complex C DM - (k ) [28], where the motive MZ (X ) with supports on Z is defined in [28, the text just below the proof of Theorem 4.8]. The motive MZ (X ) is identified with the complex C (Ztr (X )/ C,p Ztr (X - Z)) in the proof of Lemma 4.11 in [28]. Set AC(X ) = AZ (X ).The - Z C is defined in [28, ??]. boundary operator which we denote in this example The homotopy invariance property holds by [28, Proposition 4.2]. The excision property is proved in [28, the proof of Lemma 4.11]. The localization property follows from the exactness of the complex 0 Ztr (X - Z) Ztr (X ) Ztr (X )/Ztr(X - Z) 0

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8 because the functor C Theorem 1.12].


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takes short exact sequences to exact triangles [28,

2.1.10. Semi-topological Complex and Real K-theories [6] If the ground field k is the field R of reals then the semi-topological K-theory of real algebraic varieties K Rsemi defined in [6] is a cohomology theory as it is proved in [6]. 2.1.11. Representable Theories Here Ap (X, U ) = q E p,q (X/ U ), where E is a T -spectrum [31]. Set A(X, U ) = Ap (X, U ). The boundary operator is described in [31] and is defined via the - triangulated structure on the stable homotopy category [31]. In particular, in the case E = MGL [31, Section 6.3] we obtain the algebraic cobordism theory.

2.2. GE NE RAL P ROP E RT I E S OF COHOMOL OGY T H E ORI E S We specify here certain properties of an arbitrary cohomology theory A which are useful below in the text. 2.2.1. The localization property implies that A (X ) = A(X, X ) = 0. Therefore A() = A () = 0. 2.2.2. If any two of morphisms (X, U ) (Y , V ), X Y , U V , defined by a morphism f : (X, U ) (Y , V ), induce isomorphisms on A-cohomology then the third of these morphisms induces an isomorphism on A-cohomology. 2.2.3. Localization Sequence for a Triple Let T Y X be closed subsets of a smooth variety X . Let : A(X - T) AT (X ) be the boundary map for the pair (X, X - T). Consider the support extension map eA : AY -T (X - T) A(X - T) and set Y,T = eA : AY -Z (X - Z) AT (X ). We claim that the sequence ··· AT (X ) - AY (X ) - AY with the call this Y = X, (X, X -
-T

(X - T) - AT (X ) - AY (X ) ··· -



Y,T



obvious mappings , and is a complex and moreover it is exact. We sequence the localization sequence for the triple (X, X - T, X - Y). If then this sequence coincides with the localization sequence for the pair T).

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If U X - T is an open containing Y - T then the pull-back AY -T (X - T) AY -T (U ) is an isomorphism by the excision property. So replacing AY -T (X - T) by AY -T (U ) we get an exact sequence ··· AT (X ) - AY (X ) - AY
-T

(U ) - AT (X ) - AY (X ) ···. -



Y,T



We call it the localization sequence for the triple (X,U ,X - Y). 2.2.4. Mayer­Vietoris Sequence If X = U1 U2 is a union of two open subsets U1 and U2 and if Y is a closed subset in X , then set Ti = Y - Ui , Yi = Y Ui = Y - Ti , U12 = U1 U2 and Y12 = U12 Y . Consider the morphism of the localization sequences for the triples (X, U1 ,X - Y) and (U2 ,U12 ,U2 - Y) induced by the inclusion of the triples (U2 ,U12 ,U2 - Y2 ) (X, U1 ,X - Y) AY (X ) -- - AY1 (U1 ) --- AT1 (X ) -- - AY (X )
2 1

1

e

A

- AY12 (U12 ) --- AT1 (U2 ) -- - AY2 (U2 ). AY2 (U2 ) --
2





The map is an isomorphism by the excision property. Set d = eA : AY12 (U12) AY (X ). We claim that the sequence -- ··· An (X ) - An1 (U1 ) An2 (U2 ) Y Y Y - - An12 (U12) - An+1 (X ) ··· -- Y Y
(1 ,-2 ) d 1 +2

-1



is exact and call this sequence the Mayer­Vietoris sequence of the open covering X = U1 U2 . The proof of the exactness is straightforward and we skip it. The Mayer­Vietoris sequence is natural in the following sense. If f : X X is a morphism and X = U1 U2 is a Zariski covering of X such that f(Ui ) Ui and if Y is a closed subset in X containing f -1 (Y ), then the pull-back mappings f A : AY (X ) AY (X ), f A : AYi (Ui ) AYi (Ui ), f A : AY12 (U12 ) AY12 (U12 ) form a morphism of the corresponding Mayer­Vietoris sequences. 2.2.5. Let ir : Xr X1 X2 be the natural inclusion (r = 1, 2). Let Yr Xi be a closed subset for (r = 1, 2). Then the induced map AX1 X2 (X1 X2 ) AY1 (X1 ) AY2 (X2 ) is an isomorphism. Proof. This follows from the Mayer­Vietoris property and the fact that A () = 0. 2.2.6. Strong Homotopy Invariance Let p : T X be an affine bundle (i.e., a torsor under a vector bundle). Let Z X be a closed subset and let S = p -1 (Z ). Then the pull-back map p A : AZ (X )

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AS (T ) is an isomorphism. If s : X T is a section then the induced operator s A : AS (T ) AZ (X ) is an isomorphism as well. Proof. First consider the case Z = X . Then S = T and we have to check that the pull-back map p A : A(X ) A(T ) is an isomorphism. Choose a finite Zariski open covering X = Ui such that Ti = p -1 (Ui ) is isomorphic to the trivial vector bundle over each Ui and then use the morphism of the Mayer­Vietoris sequences and the homotopy invariance property of the cohomology theory A. To prove the general case consider the localization sequences for the pairs (X, X - Z) and (T , T - S). The pull-back mappings form a morphism of these two long exact sequences. The 5-Lemma completes the proof. 2.2.7. Deformation to the Normal Cone The deformation to the normal cone is a well-known construction (for example, see [7]). Since the construction and its property (6) play an important role in what follows we give here some details. Let i : Y X be a closed imbedding of smooth varieties with the normal bundle N . There exists a smooth variety Xt together with a smooth morphism pt : Xt A1 and a closed imbedding it : Y â A1 Xt such that the map pt it coincides with the projection Y â A1 A1 and · the fiber of pt over 1 A1 is canonically isomorphic to X and the base change of it by means of the imbedding 1 A1 coincides with the imbedding i : Y X; · the fiber of pt over 0 A1 is canonically isomorphic to N and the base change of it by means of the imbedding 0 A1 coincides with the zero section Y N. Thus we have the diagram - (N , N - Y) - (Xt ,Xt - Y â A1 ) (X, X - Y).
i0 i1

(4)

Here and further we identify a variety with its image under the zero section of any vector bundle over this variety. Let us recall a construction of Xt , pt and it . For that take Xt to be the blow-up of ~ ~ X â A1 with the center Y â{0}.Set Xt = Xt - X,where X is the proper preimage 1 of X â{0} under the blow-up map. Let : Xt X â A be the restriction of the blow-up map : Xt X â A1 to Xt and set pt to be the composition of and the projection X â A1 A1 . The proper preimage of Y â A1 under the blow-up map is mapped isomorphically to Y â A1 under the blow-up map. Thus the inverse isomorphism gives the ~ desired imbedding it : Y â A1 Xt (observe that it (Y â A1 ) does not cross X ). It is not difficult to check that the imbedding it satisfies the mentioned two properties (the preimage of X â 0 under the map consists of two irreducible components: the proper preimage of X and the exceptional divisor P(N 1).Their

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intersection is P(N ) and it (Y â A1 ) crosses P(N 1) along P(1) = the zero section of the normal bundle N ). We claim that the diagram (4) consists of isomorphisms on the A-cohomology. THEOREM 2.2. The following diagram consists of isomorphisms AY (N ) AY -
A i0

âA

1

(Xt ) - AY (X ).

A i1

(5)

Moreover for each closed subset Z Y the following diagram consists of isomorphisms as well - AZ (N ) AZ
A i0

âA

1

(Xt ) - AZ (X ).

A i1

(6)

This theorem is analogous to the Homotopy Purity Theorem from [18, Theorem 3.2.3]. The proof is postponed until Section 2.3. Now we state and prove the following corollary. COROLLARY 2.3. Let j0 : P(1 N) Xt be the imbedding of the exceptional divisor into Xt and let j1 = et i1 : X Xt , where et : Xt Xt is the open inclusion. Then the diagram - AP(1)(P(1 N)) AY
A j0

âA

1

(Xt ) - AY (X )

A j1

(7)

consists of isomorphisms. Proof. Consider the commutative diagram -- AY âA1 (Xt ) AP(1)(P(1 N)) - A A
e et
A j0

AY (N )

- -- AY

A i0

âA

1

(Xt ),

where the vertical arrows are the obvious pull-backs. These vertical arrows are A isomorphisms by the excision property. The operator i0 is an isomorphism by the A first item of Theorem 2.2. Thus the operator j0 is an isomorphism. 2.2.8. Let X be a smooth variety and let L be a line bundle over X .Let E = 1 L and let ¯ iL : X = P(L) P(E ) be the closed imbedding induced by the direct summand L of E . Let AP(1)(P(E )) - A(P(E )) be the support extension operator and let ¯A iL : A(P(E )) A(P(L)) be the pull-back operator. We claim that the following sequence 0 AP(1)(P(E )) - A(P(E )) - A(P(L)) 0. is exact.
i
A

i

A

¯A iL

(8)

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To prove this consider U = P(E ) - P(1) with the open inclusion j : U P(E ) and observe that U becomes a line bundle over X by means of the linear projection q : U P(L) = X (the line bundle is isomorphic to L ) The obvious inclusion ¯ iL : P(L) U is just the zero section of this line bundle, iL = j iL and the A pull-back operator iL : A(U ) A(P(L)) is an isomorphism (the inverse to the one q A ). Now consider the pair (P(E ), U ). By the localization property (Definition 2.1) the following sequence ··· AP(1) (P(E )) - A(P(E )) - A(U ) ···
A is exact. If P(E ) - X is the natural projection then the operator p A iL : A(U ) A A A(P(E )) splits j . This implies the surjectivity of j and the injectivity of i A . To A proof that the sequence (8) is short exact it remains to recall that the operator iL is ¯ an isomorphism and iL = j iL . p i
A

j

A

2.2.9. We use here the notation from Section 2.2.7. Let et : Xt and let p : P(1 N) Y be the projection and let s : Y of the projection identifying Y with the subvariety P(1) commutative diagram will be repeatedly used below in - P(1 N) -- s Y
k0 j0

Xt be the open inclusion P(1 N) be the section in P(1 N). The following the text

Xt It

- -- X i
k1

j1

-- - Y â A1 - -- Y,

where It = et it and j0 is the inclusion of the exceptional divisor and j1 = et i1 and k0 , k1 are the closed imbedding given by y (y , 0) and y (y , 1) respectively. LEMMA 2.4. (Useful lemma). Under the notation from Section 2.2.7 let jt : Vt = Xt - Y â A1 Xt be the inclusion. If the support extension operator AP(1) (P(1 N)) A(P(1 N)) is injective then
A Ker(j0 ) Ker(jtA ) = (0),

in the other words the operator
A j0 jtA : A(Xt ) A(P(1 N)) A(Vt )

is monomorphism. In particular this holds if Y is a divisor on X .

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Proof. Consider the commutative diagram A(P(1 N)) - --
A j0

A(Xt ) t
âA
1

AP(1)(P(1 N)) - -- AY

A j0

(Xt ),
A operators. The bottom operator j0 is an is injective by the very assumption (if Y A Since the composition j0 t coincides

where and t are the support extension isomorphism by Corollary 2.3. The map is a divisor in X then is injective by (8)). A with the one j0 it is injective as well. The localization sequence for the pair The lemma follows.

(Xt ,Vt ) shows that Ker(jtA ) = Im(t ).

2.2.10. Let i : P(V ) P(W ) and j : P(V ) P(W ) be two linear imbeddings (imbeddings induced by linear imbeddings V into W ). If the dimension of V is strictly less than the dimension of W , then i A = j A : A(P(W )) A(P(V )). In fact, in this case there exists a linear automorphism of W which has the determinant 1 and such that j = i . Since is a composite of elementary matrices and each elementary matrix induces the identity automorphism A(P(W )) (by the homotopy invariance of A) one gets the relation A = id. Therefore j A = iA A = iA. 2.3. P ROOF OF T H E ORE M 2. 2 Proof. Basically the proof mimics the arguments used for the proof of the Homotopy Purity Theorem [18, Theorem 3.2.3]. Since the proof of the second assertion will be left to the reader the proof of the first one will be given in details. We start with certain observations concerning elementary properties of the deformation to the normal cone construction. Namely, if U and V are Zariski open subsets of X , then the following holds (a) Ut Vt = (U V)t ; (b) Ut Vt = (U Vt ); ~~ ~ (c) if an etale morphism e: (X, X - Y) (X, X - Y) satisfies the hypotheses of ~~ the excision property (Definition 2.1), then the induced morphism et : (Xt , X - 1 1 ~ Y â A ) (Xt ,Xt - Y â A ) satisfies as well the hypotheses of the excision property. To prove the theorem, we will need the lemma and four claims below.
A LEMMA 2.5. If X = Y â An and Y = Y â{0} Y â An then i1,X : AY AY (X ) is an isomorphism. âA
1

(Xt )

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DEFINITION 2.6. An open subset U in X is called good if there exists a diagram (U , U - YU ) - -- (T , T - S) --- (YU â An ,YU â An - YU â{0}) with YU = Y U and with morphisms e and f satisfying the hypotheses of the excision property (Definition 2.1). NOTATION 2.7. We write below in this proof i1,U for the imbedding U Ut from the deformation to the normal cone construction for the pair (U , YU ). We will A A write below in the proof of theorem i1,U for the pull-back map i1,U : AYU âAn (Ut ) AYU (U ).
A CLAIM 2.8. If U X is good then the pull-back map i1,U : AYU AYU (U ) is an isomorphism. âA
n

e

f

(Ut )

CLAIM 2.9. If an open subset U in X is good then each open subset V in U is good as well. CLAIM 2.10. If an open subset U in X is good and if V X is an open subset A A such that the pull-back map i1,V is an isomorphism, then the pull-back map i1,U V is an isomorphism as well. CLAIM 2.11. For each point x X there exists a good Zariski open neighborhood U of the point x . Assuming for a moment Lemma 2.5 and these four claims one can complete the proof of theorem as follows. By the fourth claim there exists a finite Zariski open covering X = n=1 Ui with good open subsets Ui . Claim 2.8 states that the pulli A A back map i1,U1 is an isomorphism. Suppose for V = k=1 Ui the pull-back map i1,V i is an isomorphism. Since the open subset Uk+1 is good Claim 2.10 shows that the +1 A pull-back map i1,W is an isomorphism for W = k=1 Ui . The induction by k shows i A that the pull-back map i1 is an isomorphism. It remains to prove Lemma 2.5 and four claims. Proof of Lemma 2.5. Let F/ Y be a vector bundle and let F be the blow-up of F at the zero section. The variety F coincides with the total space of the line bundle OF (-1) over P(F ).Let qF : F P(F ) be projection of the line bundle to its base P(F ). If F = 1 E for a vector bundle E over Y then one has the following commutative diagram E E -- -
q

q

F F

- --

F -E q

- -- P(1) â A pr P(1),

1

P(E ) -- - P(F ) - -- P(F ) - P(E ) - --

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in which all the vertical arrows are the projections of the line bundles to their bases. Section 2.2.6 shows that the pull-back map q A : AP(1) (P(F ) - P(E )) AP(1)âA1 (F - E ) is an isomorphism. The projection q has two sections s0 and s1 . The section s0 is the zero section and the section s1 is given by x (x , 1). Since q s0 = id the pull-back map A s0 : AP(1)âA1 (F - E ) AP(1) (P(F ) - P(E )) is an isomorphism. Since q s1 = id A the pull-back map s1 : AP(1)âA1 (F - E ) AP(1) (P(F ) - P(E )) is an isomorphism as well. Now take X = E and Y = the zero section of E . Observe that the space Xt coincides with the variety F - E , the imbedding i1 : X Xt coincides with the section s1 : E F - E . The normal bundle N = NE/ Y to Y in E coincides with bundle E itself and the imbedding i0 : N Xt coincides with the section s0 : E Xt . Finally the variety Y âA1 coincides with P(1)âA1 and the imbedding Y â A1 Xt coincides with the imbedding P(1) â A1 F - E . Therefore both maps in the diagram - AY (E ) = AY (N ) AY
A i0

âA

1

(Xt ) - AY (E )

A i1

are isomorphisms. In particular these two maps are isomorphisms for the case of the trivial bundle E = An â Y . Thus we proved lemma. Proof of Claim 2.8. The claim follows immediately from lemma and the property (c) of the deformation to the normal cone construction. Proof of Claim 2.9. To prove this claim consider a diagram -- (T , T - S) --- (YU â An ,YU â An - YU â{0}) (U , U - YU ) - with morphisms e and f satisfying the hypotheses of the excision property. Let V U be an open subset. Set T = e-1 (V ) f -1 (YU â An ) and S = f -1 (YU ). Then S = f -1 (YU â{0}) and in the diagram -- (T , T - S) --- (YV â An ,YV â An - YV â{0}) (V , V - YV ) -
V

e

f

eV

f

the morphisms eV and fV satisfy the hypotheses of the excision property as well. Thus the open subset V is good. Proof of Claim 2.10. This claim follows immediately from the properties (a) and (b) and the first claim comparing the Mayer­Vietoris sequence for U V with the one for Ut Vt . Proof of Claim 2.11. This claim is proved in [32, Lemma ??]. Comment to the second assertion of theorem. Recall that a Nisnevich neighborhood of a closed subset Y in X is an etale morphism : X X such that for Y = -1 (Y ) the restriction map : Y Y

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is an isomorphism. Clearly if is a Nisnevich neighborhood of Y then for each closed subset Z in Y the map is a Nisnevich neighborhood of the subset Z as well. Recall as well that for any vector bundle p : E X and any its section s and any closed subset Z X the two pull-back maps p A : AZ (X ) Ap-1 (Z )(E ) and s A : Ap-1 (Z ) (E ) AZ (X ) are isomorphisms inverse of each other. These two observations shows that the proof of the first assertion of theorem works as well for the second assertion of theorem. 2.4. RI NG COHOMOL OGY T H E ORI E S DEFINITION 2.12. Let P = (X, U ), Q = (Y , V ) SmOp . Set P â Q = (X â Y, X â V U â Y) SmOp . This product is associative with the obvious associativity isomorphisms. The unit of this product is the variety pt . This product is commutative with the obvious isomorphisms P â Q Q â P . = DEFINITION 2.13. One says that a cohomology theory A is a ring cohomology theory if for every P, Q SmOp there is given a natural bilinear morphism called cross-product â: A(P ) â A(Q) A(P â Q) which is functorial in both variables and satisfies the following properties 1. associativity: (a â b) â c = a â (b â c) A(P â Q â R) for a A(P ), b A(Q), c A(R ); 2. there is given an element 1 A(p t ) such that for any pair P SmOp and any a A(P ) one has 1 â a = a = a â 1 A(P ); 3. partial Leibnitz rule: P âY (a â b) = P (a ) â b A(X â Y, U â Y) for a pair P = (X, U ) SmOp , smooth variety Y and elements a A(U ), b A(Y ). Given cross-products define cup-products : AZ (X ) â AZ (X ) AZ by ab =
A Z

(X ) (9)

(a â b),

where : (X, U V) (X â X, X â V U â X) is the diagonal. Clearly cup-products thus defined are bilinear and functorial in both variables. These cupproducts are associative as well: (a b) c = a (b c); the element p A (1) A(X ), (here p is the projection X pt ) is the unit for the cup-products : AZ (X ) â A(X ) AZ (X ) and : A(X ) â AZ (X ) AZ (X ); and a partial Leibnitz rule holds: (a b) = (a ) b for a A(U ), b A(X ). A A Given cup-products one can construct cross-products by a âb = pX (a ) pY (b) for a A(X, U ) and b A(Y , V ). Clearly these two constructions are inverse each to other. Thus having products of one kind we have products of the other kind and can use both products in the same time.

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DEFINITION 2.14. A ring morphism : (A, A , âA , 1A ) (B , B , âB , 1B ) of ring cohomology theories is a morphism : (A, A ) (B , B ) of the underlying cohomology theories which takes the unit 1A to the unit 1B and commutes with the cross-products: (1A ) = 1B B(p t ) and for every pairs P, Q SmOp and every elements a A(P ), b A(Q) one has P âQ (a â b) = (a ) â (b) B(P â Q). DEFINITION 2.15. Let A be a ring cohomology theory and let X be a smooth variety. An element a A(X ) is called universally central if for any smooth variety ~ ~ ~ X and any morphism f : X X the element f A (a ) is central in A(X). uc We will write below in the text A (X ) for the subring of A(X ) consisting of all ¯ universally central elements and we set Auc := Auc (p t ). Remark 2.16. Note, that if the theory A takes values in the category of Z/2graded Abelian groups and grade-preserving homomorphisms, and is moreover a Z/2-graded-commutative ring theory, i.e. for any a Ap (P ) and b Aq (Q) one has the relation a âb = (-1)pq b âa , then each even degree element is a universally central element. One should remark as well that in the graded commutative case the second partial Leibnitz rule holds (if we assume that for every pair (X, U ) the operator X,U is a graded operator of degree +1). Namely, if a Ap (U ) and b Aq (Y ) and U is open in a smooth X , then the relation Y âX,Y âU (bâa) = (-1)q bâX,U (a ) in A(Y â X, Y â U). If A is a ring cohomology theory, then for each pair (X, U ) SmOp the localization sequences from Section 2.2.3 are sequences of the A(X )-modules (partial Leibnitz rule). By the same reason for each open covering X = U1 U2 the Mayer­Vietoris sequence from Section 2.2.4 is a sequence of the A(X )-modules. Thus the following two propositions hold. PROPOSITION 2.17. Let f : (X, U ) (X ,U ) be a morphism of pairs, let A(X ) and let |U = |U A(U ). Denote the composition operator ( f A : A(X ) A(X )) (respectively (U f A : A(U ) A(U )) and ( f A : A(X ,U ) A(X, U ))) by (respectively U , and ). Then these operators form a morphism of the localization sequences for the pairs (X ,U ) and (X, U ), that is the diagram commutes A(X, U ) --- A(X ) --- A(U ) A(X ) --- A(U ) --- U U A(X ) --- A(U ) --- A(X ,U ) --- A(X --- A(U )). PROPOSITION 2.18. Let X = U1 U2 and let X = U1 U2 be open coverings. Let f : X X be a morphism such that f(Ui ) Ui for i = 1, 2. Let A(X ) be

X ,U



X,U

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an element, let i = |Ui A(Ui ) and let 12 = |U12 A(U12 ). Denote the composition operator ( f A : A(X ) A(X )) (respectively (i f A : A(Ui ) A(Ui )) and (12 f A : A(U12 ) A(U12 ))) by (respectively i and 12 ). Then these operators form a morphism of the Mayer­Vietoris sequences corresponding to the coverings X = U1 U2 and X = U1 U2 , that is the diagram commutes
A(U1 ) A(U2 ) - ---- A(U12 ) - ---- A(X) - ---- A(U1 ) A(U2 ) - ---- A(U1 2)
(1 ,2 )



12







(1 ,2 )



12



---- A(U12 ) - ---- A(X ) - ---- A(U1 ) A(U2 ) - ---- A(U12 )). A(U1 ) A(U2 ) -

The definition of a ring cohomology theory is equivalent to the following more technical but pretty useful one. DEFINITION 2.19. A ring cohomology theory is a weak morphism (A,µ,e): (SmOp, â,p t ) (Ab, , Z) of the monoidal categories together with a functor transformation such that the pair (A, ) is a cohomology theory (Definition 2.1) and the boundary operator satisfies the partial Leibnitz rule saying that P âY (µU,Y (a b)) = µP,Y (P (a ) b) A(X â Y, U â Y). (Under this variant of the notation the cross-product c â d A(P â Q) of elements c A(P ) and d A(Q) is the element µP,Q (a b) A(P â Q)). One could replace in this form of the definition the monoidal category (Ab, , Z) by any other Abelian monoidal category (C, C, 1C) reformulating the partial Leibnitz rule as follows: for every pair P SmOp and a smooth variety Y the relation holds
P âY

µU,Y (idU C idY ) = µP,Y (P C idA(Y ) ).

Once again we left such a replacement to a reader to avoid technicalities as much as it is possible. 2.5. EX A M PLES Consider following examples. 2.5.1. Etale Cohomology q Let A (X ) = + HZ (X, µm ) be the etale cohomology theory, where m is q =- Z an integer prime to char(k ). The cup-products are described in [17, Chapter V, Section 1, 1.17].

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2.5.2. K-theory Let A be the algebraic K-theory from Section 2.1.8. So A(X, U ) = 0 K-n n= (X onZ ), where Z = X - U . The idea of the definition of the products is given in [30]. 2.5.3. Motivic Cohomology Let AZ (X ) = 0 p =- H q=

p M,Z

(X, Ztr (q)) be the motivic cohomology [28]. with 0 AZtr (X ). Take q= text just below Lemma 3.3] Ztr (s ) Ztr (r + s). The unit 1 of this product is the
Z (q)

Under the notation of Section 2.1.9 AZ (X ) coincides = 0 Ztr (q) . The products are defined in [28, the q= and are induced by the canonical pairings Ztr (r ) tr products are associative and graded commutative, the 0 element 1 HM(p t , Ztr (0)) = Z [28].

2.5.4. Semi-topological Complex and Real K-theories [6] If the ground field k is the field R of reals then the semi-topological K-theory of real algebraic varieties K Rsemi defined in [6] is a ring cohomology theory as it is proved in [6]. 2.5.5. Algebraic Cobordism Theory To introduce a ring structure on the algebraic cobordism theory (Section 2.1.11) it would be convenient to enrich MGL with a symmetric ring structure [13, Section 4]. For that we construct another T -spectrum MGL which is a commutative symmetric ring spectrum by the very construction and which is weekly equivalent to MGL as the T -spectrum. The desired T -spectra MGL is described in [23, 2.5.5]. A ring structure on the algebraic cobordism theory was introduced as well in [10]. 2.5.6. Singular Cohomology of the Real Point Sets Let k = R and let A = Aev Aodd with Aev (X, U ) = H p ((X (R), U (R); Z/2) 0 and Aodd (X, U ) = 0 (see Section 2.1.3). Take as a boundary the usual boundary map for the pair (X (R), U (R)). Clearly is grade-preserving with respect to the grading we choose on A. Now the cup product makes A a Z/2-graded-commutative ring theory.

3. Orientations In this section A is a ring cohomology theory. We introduce three following structures which A can be endowed with: an orientation, a Chern structure and a Thom structure. We show that there is a natural one-to-one correspondence between these structures (see Theorems 3.5, 3.35 and 3.36).

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Let us recall that for a vector bundle E over a variety X we identify X with z(X ), where z: X E is the zero section. DEFINITION 3.1. An orientation on the theory A is a rule assigning to each smooth variety X , to each its closed subset Z and to each vector bundle E/ X an operator thE : AZ (X ) AZ (E ) Z which is a two-sided A(X )-module isomorphism and satisfies the following properties 1. invariance: for each vector bundle isomorphism : E F the diagram commutes - AZ (F ) AZ (X ) -- A
id thE Z

AZ (X ) -- - AZ (E ) 2. base change: for each morphism f : (X ,X - Z ) (X, X - Z) with closed subsets Z X and Z X and for each vector bundle E/ X and for its pullback E over X and for the projection g : E = E âX X E the diagram commutes AZ (X ) --- AZ (E ) A A
f g thE Z

thE Z

AZ (X ) --- AZ (E ) 3. for each vector bundles p : E X and q : F X the following diagram commutes - AZ (X ) -- thF
Z

thE Z

thE Z

AZ (E ) p
thZ

(F )

AZ (F ) -- - AZ (E F) and both compositions coincide with the operator th

thZ

q (E )

E F Z

.

The operators thE are called Thom isomorphisms. The theory A is called oriZ entable if there exists an orientation of A. The theory A is called oriented if an orientation is chosen and fixed. Next we are going to describe a number of data which allow to orient A.

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3.2. CHE RN AND T HOM S T RUCT URE S O N A In this section A is a ring cohomology theory. If X is a smooth variety we write 1X for the trivial rank 1 bundle over X . Often we will just write 1 for 1X if it is clear from a context what the variety X is. DEFINITION 3.2. A Chern structure on A is an assignment L c(L) which associate to each smooth X and each line bundle L/ X a universally central element c(L) A(X ) satisfying the following properties 1. functoriality: c(L1 ) = c(L2 ) for isomorphic line bundles L1 and L2 ; f A (c(L)) = c(f (L)) for each morphism f : Y X ; 2. nondegeneracy: the operator (1, ): A(X ) A(X ) A(X â P1 ) is an isomorphism where = c(O(-1)) and O(-1) is the tautological line bundle on P1 ; 3. vanishing: c(1X ) = 0 A(X ) for any smooth variety X . The element c(L) A(X ) is called Chern class of the line bundle L. (It will be proved below in Lemma 3.29 that the elements c(L) are nilpotent). Let E be a vector bundle over a smooth X and m AX (E ) be an element. We will say below in the text that m is A(X )-central, if for any element a A(X ) one has the relations m a = a m in AX (E ) (we consider elements of A(X ) as elements of A(E ) by means of the pull-back operator induced by the projection E X ). We will say that m is universally A(X )-central if for any morphism f : X X and the vector bundle E = X âX E and its projection F : E E the element F A (m) AX (E ) is A(X )-central. Observe that for a universally A(X )-central element m AX (E ) the element zA (i A (m)) in A(X ) is universally central in the sense of Definition 2.15 (here i A : AX (E ) A(E ) is the support extension operator and z: X E is the zero section of E ). DEFINITION 3.3. One says that A is endowed with a Thom structure if for each smooth variety X and each line bundle L/ X it is chosen and fixed a universally A(X )-central element th(L) AX (L) satisfying the following properties 1. functoriality: A (th(L2 )) = th(L1 ) for each isomorphism : L1 L2 of line bundles; A fL (th(L)) = th(LY ) for each morphism f : Y X and each line bundle L/ X , where LY = L âX Y is the pull-back line bundle over Y and fL : LY L is the projection to L; 2. nondegeneracy: the cup-product th(1): A(X ) AX (X âA1 ) is an isomorphism (here X is identified with X â{0}).

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The element th(L) AX (L) is called the Thom class of the line bundle L.Now we are going to describe a one-to-one correspondence between Chern and Thom structures on A. LEMMA 3.4. Assume A is endowed with a Chern structure L c(L). Let L be a line bundle over a smooth X and let E = 1 L and let p : P(E ) X be the projection. Identify the group AP(1) (P(E )) with a subgroup of A(P(E )) via the support extension operator AP(1)(P(E )) A(P(E )) from the sequence (8). Then the element c(OE (1) p L) A(P(E )) belongs to the subgroup AP(1) (P(E )) of the group A(P(E )). Below we will often write ¯ (L) for c(OE (1) p L). th Proof. The projection to the base X identifies the closed subvariety P(L) with the variety X . The restriction of the line bundle OE (1) to P(L) is coincides with L . Thus the restriction of OE (1) p L to P(L) is the trivial bundle. Now if iL : P(L) P(E ) is the inclusion from (8) then
A iL (c(OE (1) p L)) = c(iL (OE (1) p L)) = 0.

The exactness of the sequence (8) completes the proof. Now we are ready to describe the mentioned one-to-one correspondence. Assuming that A is endowed with a Thom structure L th(L) endow A with a Chern structure as follows. For a line bundle L over a smooth X set c(L) = [zA i A ](th(L)) A(X ), (10) where i A : AX (L) A(L) is the support extension operator (see Definition 2.1) and zA : A(L) A(X ) is the operator induced by the zero section z: X L. Assuming that A is endowed with a Chern structure L c(L) endow A with a Thom structure as follows. For a line bundle L over a smooth X consider the vector bundle E = 1 L, the projection p : P(E ) X , the natural inclusion th e: L P(E ) and the pull-back eA : AP(1)(P(E )) AX (L). The element ¯ (L) = c(OE (1) p L) A(P(E )) belongs to the subgroup AP(1) (P(E )) by lemma above. Now set th(L) = eA (c(OE (1) p L)) A(L , L0 ) = AX (L). (11)

THEOREM 3.5. For any ring cohomology theory A the following assertions hold. 1. If A is endowed with a Thom structure L th(L) then the assignment L c(L) given by (10) endows A with a Chern structure. 2. If A is endowed with a Chern structure L c(L) then the assignment L th(L) given by (11) endows A with a Thom structure. 3. The constructions described in the items 2 and in 1 are inverse of each other: namely if L c(L) is a Chern structure on A and L th(L) is a Thom structure on A, then the relation (10) holds for all line bundles if and only if the relation (11) holds for all line bundle.

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Let L c(L) be a Chern structure on A and let L th(L) be a Thom structure on A. In the case when (11) holds for all line bundles (or (10) holds for all line bundles which is the same) we say that the Chern structure and the Thom structure on A correspond to each other. The item 1 describes the arrow from Section 1. The item 2 describes a unique arrow inverse to the arrow . Proof. We start with the following lemma. LEMMA 3.6. Let L c(L) be a Chern structure on A. Let O(-1) be the tautological line bundle on the projective line P1 and let O(1) be the dual bundle and let = c(O(-1)), and = O(1). Then 2 = 0 = 2 and c(O(1)) = -c(O(-1)). Proof of lemma. Since O(-1)|{0} is the trivial bundle one has |{0} = 0. Thus A{0} (P1 ) and A{} (P1 ). Therefore the element 2 A(P1 ) is in the image of A{0}{} (P1 ) = A (P1 ). This last group vanishes by the property Section 2.2.1 and thus 2 = 0. Similarly 2 = 0. The A(p t )-module A(P1 â P1 ) is a free module with the free bases 1, 1, 1 and by the property of the Chern classes. Consider an element = c(p1 (O(-1) p2 (O(1)))). Write it in the form = a00 1 1 + a10 1 + a01 1 + a11 ,where aij areelementsin A(p t ). Restricting the element to {0}â {0}, to P1 â {0} and to the diagonal (P1 ) one gets the following relations: a00 = 0, a00 + a10 = in A(P1 ) and a00 + (a10 + a01 ) = 0 in A(P1 ). Thus a10 = 1 and a10 + a01 = 0. Therefore a01 = -a10 = -1. The chain of the relations = {0}âP1 = a00 + a01 = - completes the proof of lemma. Now we are ready to prove the assertion (2) of theorem. The functoriality of the assignment L th(L) is obvious. Now to prove that the element th(L) is universally A(X )-central it suffices to prove that the element th(L) is A(X )-central. Consider the element ¯ (L) = c(OE (1) p L)) AP(1) (P(E )). This element is th th A(P(E ))-central because it is the Chern class. Since th(L) = eA (¯ (L)) and the pull-back map A(P(E )) A(L) is surjective the element th(L) is A(X )-central. It remains to prove the non-degeneracy property of the element th(1). For that consider the commutative diagram with exact rows 0
/ AX
â{0}

(X â P1 )
O

/ A(X â P1 ) O
(¯ (1),1) th

/ A(X â A1 ) O
pr
A

/0

¯ (1) th

0

/ A(X )

/ A(X ) A(X )

/ A(X )

/ 0.

The non-degeneracy property of the Chern class and the relation ¯ (1) = th -c(O(-1)) = - show that the middle vertical arrow is an isomorphism. The right vertical arrow is an isomorphism by the homotopy invariance property. Therefore the left vertical arrow is an isomorphism as well.

549.tex; 27/10/2003; 16:37; p.23


24 Now consider the commutative diagram A(X )

id th(1) ¯ (1) th

I. PANIN

/ AX

â{0}

(X â P1 )

e
A

A(X )

/ AX

â{0}

(X â A1 ).

The map eA is an isomorphism by the excision property. Therefore the bottom arrow is an isomorphism as well. The non-degeneracy property of the class th(1) is proved. To prove the assertion (1) of theorem we need in some preliminaries. NOTATION 3.7. Let M be a line bundle over a smooth variety X and let eA : AP(1) (P(1 M)) AX (M ) be the excision isomorphism induced by the open inclusion e: M P(1 M). For an element AX (M ) set = (eA )-1 ( ) AP(1) (P(1 M )). ¯ Since the support extension map AP(1) (P(1 M)) A(P(1 M)) is injective (8) we will often write for the image of this element in A(P(1 M)). If = th(M ) ¯ is the Thom class of M then we will often write ¯ (M ) for the element . th ¯ The following two observations will be useful for the proof as well · if : X1 X is a morphism of smooth varieties and M1 = (M ) is the line bundle over X1 and : PX1 (1 M1 ) PX (1 M) = P(1 M) is the induced morphism of the projective bundles then for 1 = A ( ) one has the relation ¯ 1 = A (). ¯ · if s : X P(1 M) is the section identifying X with P(1) then one has ¯ s A () = zA (i A ( )),where z is the zero section of M and i A : AX (M ) A(M ) is the support extension operator. Now under Notation 3.7 one has the following lemma. LEMMA 3.8. Let an assignment L th(L) be the Thom structure on A. Let L c(L) be the assignment given by the formula (10). Then for the line bundle th O(1) on P1 one has the relation ¯ (1) = c(O(1)) in A(P1 ). Proof of lemma. Let P2 be a rational point and let : P P2 be the blowup of the projective plane P2 at the point . The linear projection P2 - P1 extends canonically to a morphism p : P P1 . Using this morphism the variety P is naturally identified with the projective bundle P(1 L) over the projective line P1 , where L = O(1). Under this identification the preimage -1 () of the point coincides with the subvariety P(L) of the projective bundle P(1 L). The subvariety P(1) P(1 L) is the image of a section s1 : P1 P(1 L) of the projection p . The image of P1 under the composite map s1 is a projective

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ORIENTED COHOMOLOGY THEORIES OF ALGEBRAIC VARIETIES

25

line l in P2 which avoids the point . Let x P1 be a rational point and let j : P1 = p -1 (x ) P(1 L) be the imbedding of the fiber into the total space. One can summarize these data in the following diagram P
j 1 s
L



/ pt
i 2

P


1 p
0

/ P(1 L)
i



pt

/P

/8 P qqq q p qqq qqq s1 qqq
1

.

¯ th Now set = th(L) AX (L), then = ¯ (L) AP(1)(P(1 L)). In the commutative diagram of the pull-backs Al (P2 ) w -- -
A =u

AP(1) (P(1 L)) v

Al (P2 -) -- - AP(1) (P(1 L) - P(L)), the maps w , t and v are isomorphisms. In fact, w and v are isomorphisms by the excision property and t is isomorphism because the map identifies P(1 L) - P(L) with P2 -. Therefore the fourth arrow u = A is an isomorphism as well. ¯ Therefore there exists an element Al (P2 ) such that A ( ) = . The mappings j, s1 : P1 P2 are two linear imbeddings of the projective line into P2 . Therefore by the property (Section 2.10) one has the relation ( j)A ( ) = ( s1 )A ( ) in A(P1 ). Thus one gets the chain of relations in A(P1 )
A ¯ ¯ j A () = ( j)A ( ) = ( s1 )A ( ) = s1 ().

t

By the two observations mentioned just below Notation 3.7 one gets the relations A ¯ th ¯ th j A () = ¯ (1) and s1 () = c(L) = c(O(1)). Thus ¯ (1) = c(O(1)) and lemma is proved. Now we are ready to prove assertion (1) of theorem. The functoriality of the class L c(L) given by the formula (10) is obvious. To prove that for the trivial line bundle 1 over a smooth variety X one has c(1) = 0 consider a section s : X X â A1 of the trivial bundle 1 which takes a point x X to the point (x , 1). If z is the zero section of the same bundle then the pull-back mappings zA and s A coincides. In fact both are the inverse to the pull-back map p A : A(X ) A(X â A1 ) induced by the projection p : X â A1 X .If i A : AXâ0 (X â A1 ) A(X â A1 ) is the support extension map, then c(1) = (zA i A )(1) = (s A i A )(1) and it remains to show that s A i A = 0.

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26 For that consider a commutative diagram AX
â{0} j

I. PANIN

(X â A1 ) A

--- A(X â A1 ) A
s

i

A

A (X â (A1 -{0})) ---

A(X ),

where the pull-back map j A is induced by the inclusion j : X â (A1 - {0}) X â A1 and the bottom horizontal arrow is the pull-back induced by the inclusion s : (X, ) (X â (A1 - {0}), X â (A1 - {0})). The group A (X â (A1 - {0}) vanishes by the vanishing property (Section 2.2.1). Thus s A i A = 0 which proves the relation c(1) = 0. It remains to prove the non-degeneracy property of the class L c(L). For that consider the assignment L c (L) = c(L ). Clearly the class c is functorial and satisfies the vanishing property. Moreover the map (1,c (O(-1))): A(X ) A(X ) A(X â P1 ) is an isomorphism by the non-degeneracy property of the Thom class L th(L) and the very last lemma. Thus the assignment L c (L) is a Chern structure. Now the previous lemma shows that c (O(-1)) = -c (O(1)). Thus c(O(1)) = -c(O(-1)) and therefore the map (1,c(O(-1))): A(X ) A(X ) A(X â P1 ) is an isomorphism as well. The non-degeneracy property of the class L c(L) is proved and hence the assertion (1) of theorem is proved as well. The third assertion of the theorem is proved just after Section 3.3 because the proof of the third assertion presented in this text uses Theorem 3.9. 3.3. P ROJ E C T I VE B UNDL E T HE ORE M We are going to construct higher Chern classes for a ring cohomology theory A endowed with a Chern structure L c(L). Following the known Grothendieck's method one has to compute cohomology of a projective bundle. THEOREM 3.9 (Projective bundle cohomology). Let A be a ring cohomology theory endowed with a Chern structure L c(L) on A. Let X be a smooth variety and let E/ X be a vector bundle with rkE = n. For E = c(OE (-1)) A(P(E )) we have an isomorphism (1,E ,... ,
n-1 E

): A(X ) A(X ) ··· A(X ) A(P(E )),

where (and elsewhere) we denote the operator of -product with a universally central element by the symbol of the element. n Moreover, for trivial E we have E = 0. In addition, all the assertions hold if the element E = c(OE (1)) A(P(E )) is used instead of E . Proof. This variant of the proof is based on an oral exposition of Suslin. Let {0} = [1 : 0 : ··· : 0] Pn be a point and let An be an affine subspace in Pn defined by the inequality x0 = 0. Let Pn be a hypersurface in Pn defined by xi = 0 i

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ORIENTED COHOMOLOGY THEORIES OF ALGEBRAIC VARIETIES

27

and let An = Pn An . Let pi : An A1 be the projection on the i th axis and let i i ¯ ji : A1 An be the i th axis. Finally let ji : P1 Pn be the closed imbedding extending the imbedding ji . Let resi : APn (Pn ) AAn (An ) be the pull-back map induced by the imbedding i i n A Pn . Let res: A{0} (An ) A{0} (Pn ) be the pull-back map induced by the same imbedding. The element = c(O(-1)) A(P1 ) vanishes being restricted to P1 -{0}. Thus ¯ the element t = belongs to the subgroup A{0} (P1 ) of the group A(P1 ).Set ¯ t = j A (t) A{0} (A1 ), where j A is the pull-back map A{0} (P1 ) A{0} (A1 ).Set
A A A th(n) = p1 (t ) p2 (t ) ··· p1 (t ) A{0} (An ).

Let ei : (Pn , ) (Pn , Pn - Pn ) and let e: (Pn , ) (Pn , Pn -) be the inclui sions. The pull-back operators eiA : APn (Pn ) A(Pn ) and eA : A{0} (Pn ) A(Pn ) i are just the support extension operators. LEMMA 3.10. Let Y be a smooth variety and let Z Y be a closed subset. Let pr: Y â A1 A1 and p : Y â A1 Y be the projections. Then the composition operator ( prA (t )) p A : AZ (Y ) AZ is an isomorphism. LEMMA 3.11. The map th(n) : A(p t ) A{0} (An ) is an isomorphism. LEMMA 3.12. Let n = c(O(-1)) A(Pn ) and n = c(O(1)) A(Pn ). Then n n n +1 = 0 and n +1 = 0. LEMMA 3.13. The support extension map APn (Pn ) A(Pn ) is injective, the i ¯ ¯ element n = c(O(-1)) coincides with eiA (ti ) for an appropriative element ti A n n ¯ APn (P ) and the relation resi (ti ) = pi (t ) holds in AAn (A ). i i th LEMMA 3.14. The element n A(Pn ) coincides with eA (¯ (n) ) for an appropriative element ¯ (n) A{0} (Pn ). th If the support extension operator eA : A{0} (Pn ) A(Pn ) is injective, then the th relation th(n) = res(¯ (n) ) holds in A{0} (An ). Remark 3.15. The linear projection Pn -{0} Pn Pn -{0} makes Pn -{0} 0 a line bundle over Pn and the subvariety Pn is the zero section of this line bundle. 0 0 By the homotopy invariance property the pull-back operator A(Pn -{0}) A(Pn ) 0 is an isomorphism.
â{0}

(Y â A1 )

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28

I. PANIN

Given these five lemmas complete the proof of theorem as follows. The general case is reduced to the case of the trivial vector bundle E via the Mayer­Vietoris arguments using Proposition 2.18. If E = 1k+1 then P(E ) = X â Pk . To use shorter notation we prove the theorem only for the case of the projective space Pk itself. We proceed the proof by the induction on the integer k .If k = 1, then the theorem holds by the very definition of the Chern structure on A. We will assume below that theorem holds for all integers k < n and prove theorem for k = n. Consider the localization sequence for the pair (Pn , Pn -{0}) ··· A{0} (Pn ) - A(Pn ) - A(Pn -{0}) ···. If i A(Pi ) is the Chern class of the line bundle O(-1) on Pi ,then n |Pn-1 = n-1 j j n-1 and n |Pn-1 = n-1 . By the inductive assumption the elements 1,n-1 ,...,n-1 form a free base of the A(p t )-module A(Pn-1 ). Therefore the map A(Pn ) A(Pn ) is a split surjection. 0 By Remark 3.15 the pull-back operator A(Pn -{0}) A(Pn ) is an isomorph0 ism. Thus the pull-back operator A(Pn ) A(Pn -{0}) is a split surjection as well. Now the localization sequence for the pair (Pn , Pn - {0}) shows that the support extension map A{0} (Pn ) A(Pn ) is an injection. Therefore one gets a short exact sequence 0 A{0} (Pn ) - A(Pn ) - A(Pn ) 0, 0 where is the pull-back map. One more consequence of the injectivity of the support extension operator A{0} (Pn ) A(Pn ) is the relation
A A A th p1 (t ) p2 (t ) ··· p1 (t ) = res(¯ n )

in A{0} (An ) which now holds by Lemma 3.14. An A(p t )-linear map s : A(Pn ) 0 j j A(Pn ) taking the element n-1 to n (j = 0, 1,... ,n - 1) splits the surjection . n n The element n A(Pn ) belongs to the subgroup A{0} (Pn ) because n-1 = 0 in n-1 n A(P ) by Lemma 3.12. It remains to show that the map : A(p t ) A{0} (Pn ) is an isomorphism. For that consider the diagram A(p t )

id th(n) ¯ th
(n)

/ A{0} (Pn ) / A{0} (An ).
res

A(p t )

It commutes by Lemma 3.14. The operator res is an isomorphism by the excision property. The operator th(n) is an isomorphism by Lemma 3.11. Thus the operator ¯ (n) is an isomorphism as well. To prove theorem it remains to prove th Lemmas 3.10­3.14.

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29

Remark 3.16. In certain texts proofs of the projective bundle theorem for some specific cohomology theories contain the following gap. It is verified that there is an isomorphism A(p t ) A{0} (Pn ), and it is missed to check that specifically the = operator n : A(p t ) A{0} (Pn ) is an isomorphism. Proof of Lemma 3.10. Let pr : Y â P1 P1 and p: Y â P1 Y be the ¯ ¯ projections. We will write for short t for the operator (prA (t )) p A and will ¯ write in this proof for the operator ¯ A ( ) p a : AZ (Y ) AZ âP1 (Y â P1 ) pr and write 1 for the operator p A : AZ (Y ) AZ âP1 (Y â P1 ). We begin with ¯ verifying that the operator (1, ): AZ (Y ) AZ (Y ) AZ
âP
1

(Y â P1 )

(12)

is an isomorphism. In fact, by Proposition 2.17 the diagram commutes (here U = Y-Z)
A(Y â P1 )
(1,)

- -- --

A(U â P1 )
(1,)





- -- AZ âP1 (Y â P1 ) - - -- --- A(Y â P1 ) (1,) (1,)

A(Y ) A(Y ) - -- A(U ) A(U ) - -- AZ (Y ) AZ (Y ) - - A(Y ) A(Y ). -- -- ---

Since = c(O(-1)) A(P1 ) the five-lemma proves that the operator (12) is an isomorphism. The next step is to check that the operator : AZ (Y ) AZ
â{0}

(Y â P1 )


(13) (Y â P1 ) - AZ


is an isomorphism. For that consider the localization sequence ··· AZ â{0} (Y â P1 ) - AZ - Z â{0}) - ···
âP
1

âA

1

(Y â P1 -

for the triple (Y â P1 ,Y â P1 - Z â{0},Y â P1 - Z â P1 ). We claim that the operator is always surjective (and thus the operator is always injective and therefore the localization sequence splits in short exact sequences). In fact, if i : Y â A1 Y â P1 - Z â{0} is the open inclusion and q : Y â P1 - Z â {0} is the projection then q i = p : Y â A1 Y and thus i A q A = p A . The pull-back operator i A : AZ âA1 (Y â P1 - Z â {0}) AZ âA1 (Y â A1 ) is an isomorphism by the excision property and the pull-back operator p A : AZ (Y ) AZ âA1 (Y â A1 ) is an isomorphism by the strong homotopy invariance property. Thus q A : AZ (Y ) AZ âA1 (Y â P1 - Z â{0}) is an isomorphism. This proves the surjectivity of and the injectivity of . We are ready to verify that the operator (13) is an isomorphism. For that consider the diagram
0

/ AZ

1 â{0} (Y â P )



O

/ AZ

1 âP1 (Y â P )



O

/ AZ
pr

1 âA1 (Y â P - Z â{0})

O

/0



(,1) in

qA

0

/ AZ (Y )

/ AZ (Y ) AZ (Y )

/ AZ (Y )

/ 0,

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30

I. PANIN

where the operator in is the inclusion to the first summand and the operator pr is the projection on the second summand. The diagram commutes because the cup-product is functorial. The sequence on the top is short exact and the map q A is an isomorphism as was just checked above. The operator (, 1) is an isomorphism as was checked as well above in this proof. Thus the operator (13) is an isomorphism as well. The proof of the lemma is completed as follows. Consider the diagram AZ (Y )

id t

/ AZ

â{0}

(Y â P1 )


AZ (Y )

/ AZ

â{0}

(Y â A1 ),



where is the pull-back operator induced by the inclusion Y â A1 Y â P1 . The operator is an isomorphism by the excision property. Since the operator is an isomorphism the operator t is an isomorphism as well. Lemma 3.10 is proved. Proof of Lemma 3.11. For every integer i let pi,i : Ai A1 be the projection of the affine space Ai to its last coordinate. Using the induction by n it straightforward to check that the cup-product operator th(n) : A(p t ) A{0} (An ) coincides with the composition operator -- -- A(p t ) - A{0} (A1 ) - - ··· - - A{0} (An ). Each arrow in this sequence of arrows is an isomorphism by Lemma 3.10. The lemma follows. Proof of Lemma 3.12. For every integer i = 0, 1,... ,n one has n |Pn -Pn = i 0 because the Chern class of a trivial line bundle vanishes. Thus n belongs to the image of the support extension operator eiA : APn (Pn ) A(Pn ), say n = i n ¯ ¯ eiA (ti ) for appropriative element ti APn (Pn ). Now the element n +1 coincides i ¯ ¯¯ with the image of the cup-product t0 t1 ··· tn under the support extension map n ···Pn (Pn ) A(Pn ). The group APn ···Pn (Pn ) vanishes because n Pn = . AP0 0i n n 0 n n Thus n +1 = 0. Similarly one gets the relation n +1 = 0. The lemma is proved. Proof of Lemma 3.13. The localization sequence for the pair (Pn , Pn - Pn ) cuts i into short exact sequences 0 APn (Pn ) A(Pn ) A(Pn - Pn ) 0 i n because the composite map A(p t ) - A(Pn ) A(Pn - Pn ) is an isomorphism. i This proves the first assertion of the lemma. ¯ ¯ The fact that n = eiA (ti ) for the element ti APn (Pn ) is proved in the proof of i i Lemma 3.12.
p
A

t

p

A 2,2

(t )

p

A n,n

(t )

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31

¯ To prove the relation resi (ti ) = piA (t ) consider the commutative diagram A(Pn ) O
A ei

¯ jiA

/ A(P1 ) O
i
A

APn (Pn ) i
res
i

¯ jiA

/ A{0} (P1 )
j
A

AAn (An ) i



jiA

/ A{0} (A1 ),

where the maps eiA and i A are the support extension operators. The relation ¯ ¯¯ ¯ jiA (n ) = in A(P1 ) and the injectivity of the map i A prove the relation jiA (ti ) = t 1 A¯ 1 in the group A{0} (P ). Since t = j (t) in A{0} (A ) hence one gets the relation ¯ jiA (resi (ti )) = t in A{0} (A1 ). The pull-back homomorphisms jiA : AAn (An ) i 1 A{0} (A ) and piA : A{0} (A1 ) AAn (An ) are inverse to each other isomorphisms by i ¯ the homotopy invariance property. This proves the desired relation resi (ti ) = piA (t ) n in AAn (A ). i Proof of Lemma 3.14. Lemma 3.12 (applied to Pn-1 ) shows that the element n n |Pn vanishes. By Remark 3.15 the pull-back operator A(Pn - {0}) A(Pn ) 0 0 n n is an isomorphism. Thus the element n |Pn -{0} vanishes as well. Therefore n = th th eA (¯ (n) ) for an appropriative element ¯ (n) A{0} (Pn ). This proves the first assertion of the lemma. To prove the last assertion of the lemma consider the commutative diagram
n i =1

A(Pn )
O



/ A(Pn ) O
e
A

A ei

n i =1

APn (Pn ) i



/ A{0} (Pn )

resi

res

n i =1

AAn (An ) i





/ A{0} (An ),

where the maps eiA and eA are the cup-products. The the injectivity of the map group A{0} (Pn ). Now the ¯ resi (ti ) = piA (t ) prove the in the group A{0} (An ).

are the support extension maps and the horizontal arrows commutativity of the upper square of this diagram and ¯ ¯ t th eA prove the relation t1 t2 ··· ¯n = ¯ (n) in the commutativity of the bottom square and the relations A A A desired relation p1 (t ) p2 (t ) ··· pn (t ) = res(¯ (n) ) th

Now Lemmas 3.10­3.14 are proved. The proof of Theorem 3.9 is completed.

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I. PANIN

COROLLARY 3.17 (Projective bundle cohomology with supports). Under the hypotheses of Theorem 3.9 the map (1,E ,... ,
n-1 E

) : AZ (X ) AZ (X ) ··· AZ (X ) AP

(EZ )

(P(E ))

is an isomorphism where EZ = E |Z is the restriction of the vector bundle E to Z . The short exact sequence (14) written-down below is useful as well. Namely, let X be a smooth variety and let M and N be two vector bundles over X . Let ¯ ¯ iM : P(M ) P(M N) and iN : P(N ) P(M N) be the closed imbeddings induced by the direct summands M and N , respectively. Let p : P(M N) X be A the projection. Let jM : AP(M ) (P(M N)) A(P(M N)) be the support exten¯A sion operator and let iN : A(P(M N)) A(P(N )) be the pull-back operator. COROLLARY 3.18. With these notation under the hypotheses of Theorem 3.9 the sequence 0 AP
(M )

(P(M N)) - A(P(M N)) - A(P(N )) 0

A jM

¯A iE

(14)

is short exact. To prove this consider U = P(M N) - P(M ) with the open inclusion j : U P(M N) and observe that U becomes a vector bundle over X by means of the linear projection q : U P(N ). The obvious inclusion iN : P(N ) U is ¯ just the zero section of this vector bundle, iN = j iN and the pull-back operator A iN : A(U ) A(P(N )) is an isomorphism (the inverse to the one q A ). Now consider the pair (P(M N), U ). By the localization property (Definition 2.1) the following sequence ··· AP
(M )

(P(M N)) - A(P(M N)) - A(U ) ···

A jM

j

A

is exact. We claim that this sequence splits in short exact sequences with the A surjective j A and the injective jM . To prove this claim observe that one has the relation ¯A N = iN (
M N

)

which holds because the restriction of the line bundle OM N (-1) to P(N ) is ¯A ON (-1). Thus N iN (A(P(M N ))) and by the projective bundle theorem ¯A (Theorem 3.9) the operator iN : A(P(M N)) A(P(N )) is surjective. The A ¯ operator iN is an isomorphism, iN = j iN and thus j A : A(P(M N)) A(U ) A is surjective and the support extension operator jM is injective. Now the sequence A ¯ (14) is short exact because the operator iN is an isomorphism and iN = j iN .The corollary is proved. The last corollary and Lemma 2.4 prove the following corollary.

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33

COROLLARY 3.19. Under the hypotheses of Theorem 3.9 and the notation of Lemma 2.4 let Vt = Xt - it (Y â A1 ) and let jt : Vt Xt be the open inclusion. Then
A Ker(j0 ) Ker(jtA ) = (0). A In the other words the operator (j0 ,jtA ): A(Xt ) A(P(1 N)) A(Vt ) is a monomorphism.

3.4. E N D O F T HE P ROOF OF T H E ORE M 3. 5 The third assertion of Theorem 3.5 is proved in this section. Assume we are given with a Chern structure L c(L) on A and let L th(L) be the Thom structure given by (11). We will now check that for each line bundle L over a smooth variety X one has zA (i A (th(L))) = c(L). For that consider the commutative diagram AP(1)(P(1 L))

e
A

¯ iA

/ A(P(1 L)) / A(L)
z
A

AX (L)

i

A



e

A

A(X ) The chain of relations (here z = e z) ¯ ¯ zA (i A (th(L))) = zA (eA (c(O(1) p (L)))) = c(z (O(1) L)) = c(L) proves the desired relation. In the rest of the proof Notation 3.7 are used. Now suppose we are given with a Thom structure L th(L) on A and let L c(L) be the Chern structure on A given by the formula (10). For a line bundle L over a smooth X consider the vector bundle E = 1 L, the projection p : P(E ) X , the natural inclusion e: L P(E ) and the pull-back eA : AP(1) (P(E )) AX (L). We have to check the relation (11). Since the operator eA is an isomorphism it suffices to check the relation in A(P(1 L)) ¯ (L) = c(O(1) q (L)) th To do this we need in some preliminary lemmas. LEMMA 3.20. The elements ¯ (L) th Moreover both elements belongs to Proof. Let j : P(E ) - P(1) projection p : P(E ) X to P(E ) - and c(OE (1) p (L)) are central in A(P(E )). the ideal AP(1) (P(E )). P(E) be the inclusion. The restriction of the P(1) makes the last variety in a line bundle over (15)

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34

I. PANIN

X . The inclusion sL : X P(E ) - P(1) identifying X with the subvariety P(L) in P(E ) is the zero section of the mentioned line bundle. By the strong homotopy A property of the pretheory A the pull-back operator sL : A(P(E ) - P(1)) A(X ) is an isomorphism. The line bundle sL (OE (1) p (L)) coincides with the line bundle L L and therefore it is the trivial line bundle. Thus the Chern class A c(sL (OE (1) p (L)) vanishes and the element sL (c(OE (1) p (L))) vanishes as well. Therefore j A (c(OE (1) p (L))) = 0in A(P(E ) - P(1)), which proves the inclusion c(OE (1) p (L)) AP(1) (P(E )). The class ¯ (L) is in the subgroup AP(1)(P(E )) by the very definition of the th class ¯ (L). th The element c(OE (1) p (L)) is central in A(P(E )) because it is a Chern class. To prove that the element ¯ (L) is central recall that for every smooth variety th X and every line bundle L over X the element th(L) AX (L) is A(X )-central. Now for every element a A(P(E )) one has a chain of relations in AX (L) th th eA (¯ (L) a) = th(L) eA (a ) = eA (a ) th(L) = eA (a ¯ (L)). Since the support extension operator eA : one gets the relation ¯ (L) a = a th ¯ (L) A(P(E )) is central. The lemma th AP(1) (P(E )) AX (L) is an isomorphism ¯ (L) in AP(1)(P(E )). Thus the element th is proved.

LEMMA 3.21. The operator ( ¯ (L)): A(X ) AP(1) (P(E )) is an isomorphism. th Proof. Consider the following commutative diagram A(X ) --- AP(1)(P(E )) A
id e ¯ (L) th

A(X ) ---

th(L)

AX (L).

The operator eA is an isomorphism by the excision property. The operator th(L) is an isomorphism because th(L) is the Thom class. The lemma follows. LEMMA 3.22. The diagram commutes - A(X ) AP(1)(P(E )) -- c( ¯ (L) th A(X ) -- - A(X ),
id s
A

L)

th and s A (¯ (L)) = s A (c(OE (1) p (L))). Proof. Clearly zA eA = s A : AP(1) (P(E )) A(X ), where z: X L is the th zero section of L. Thus s A (¯ (L)) = zA (th(L)) = c(L) by the very definition of c(L). The commutativity of the diagram is checked. It remains to check that

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c(L) = s A (c(OE (1) p (L))). This is obvious because the line bundle s (OE (1)) is trivial and the line bundle s (p (L)) coincides with the line bundle L. CLAIM 3.23. For any variety X and any line bundle L over X there exists a finitedimensional vector space V and a diagram of the form - X X - P(V )
p f

(16)

in which X is a torsor under a vector bundle over X and the morphism f is such that the line bundles p (L) and f (OV (-1)) are isomorphic. Proof of the claim. To construct the diagram (16) recall that by the Jouanalou trick [14] there is a torsor X /X under a vector bundle over X such that X is an affine variety. Now take the projection p : X X and consider the pull-back p (L) of the line bundle L. Since the variety X is affine the line bundle p (L) can be induced from a projective space via a morphism f : X P(V ). The claim is proved. Proof of the relation (15). Take X = P and L = OP (1) and consider the commutative diagram from the last lemma. By the projective bundle theorem (Theorem 3.9) the ring A(p t )[[t ]] of formal power series in one variable is identified with the ring A(P ) identifying the variable t with the Chern class c(L). Thus the operator c(L): A(X ) A(X ) is injective. The operator ¯ (L) is an th isomorphism by Lemma 3.21. Hence the operator s A : AP(1)(P(E )) A(X ) is th injective. Now the relation c(OE (1) p (L)) = ¯ (L) in A(X ) holds by the last lemma. The relation (15) is proved in the considered case. Clearly this implies the relation (15) in the case X = P(V ) and L = OV (1) for any finite-dimensional k -vector space V . The general case of the relation (15) will be reduced now to this particular case. Let X be a variety and let L be a line bundle over X . By Claim 3.23 there exists a diagram of the form (16) such that the pull-back operator p A : A(X ) A(X ) is an isomorphism and the line bundles L = p (L) and f (OV (1)) are isomorphic. Set E = 1L and EV = 1OV (1) and let p : P(E ) X and pV : P(EV ) P(V ) be the projections. A choice of a line bundle isomorphism L f (OV (1)) gives rise to the following Cartesian diagram -- P(E ) -- - P(EV ) P(E ) - pV p
p P F

X - -- X -- - P(V ). Clearly one has relations c(OE (1) (p ) (L )) = F A (c(OEV (1) (pV ) (LV ))), and c(OE (1) (p ) (L )) = P A (c(OE (1) p (L))).

p

f

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As we already know c(OEV (1) (pV ) (LV )) = ¯ (OV (1)) in A(P(EV )). Now one th has the chain of relations in P(E ) P A (c(OE (1) p (L))) = c(OE (1) (p ) (L )) = F A (c(OEV (1) (pV ) (LV ))) th th th th = F A (¯ (OV (1))) = ¯ (f (OV (1))) = ¯ (L ) = P A (¯ (L)). Since the pull-back operator P The theorem is proved. 3.5. SPLITTIN G PR IN C I PLE Let A be a ring cohomology theory endowed with a Chern structure L c(L) on A. Here a variant of splitting principle is given which will be used in the text below. It will be convenient to fix certain notation. Let p : Y X and f : X X be morphisms. Then we will write Y for the scheme X âX Y and write p for the projection X âX Y X and f for the projection X âX Y Y . LEMMA 3.24. Let E be a rank n vector bundle over a smooth variety X . Then there exists a smooth morphism r : T X such that the vector bundle r (E ) is a direct sum of line bundles and for each closed subset Z of X and for S = r -1 (Z ) the pull-back map r A : AZ (X ) AS (T ) is a split injection and moreover for a smooth variety X and any morphism f : X X the pull-back map (r )A : A(X ) A(T ) is a split injection. Proof. Let E be a rank n vector bundle over a smooth variety X and let p : P(E ) X be the associated projective bundle over X and let OE (-1). Then there is the canonical short exact sequences of vector bundles on P(E ) with the rank n - 1 vector bundle E 0 OE (-1) p (E ) E 0, and the pull-back map p A : AZ (X ) Ap-1 (Z ) (P(E )) is a split injection by the projective bundle theorem. Repeating this construction several times one gets a smooth variety Y , a morphism q : Y X and a filtration (0) E1 E2 ··· En = q (E ) of the vector bundle q (E ) such that all the quotients Ei /Ei -1 are line bundles. Moreover the pull-back map q A : AZ (X ) Aq -1 (Z )(Y ) is a split injection and for a smooth variety X and any morphism f : X X the pull-back map (q )A : A(X ) A(Y ) is a split injection as well. CLAIM 3.25. Let S be a smooth variety and let F1 ,F2 be two vector bundles over S and let : F1 F2 be a vector bundle epimorphism and let K = ker( ). Then there is an affine bundle g : T S such that the epimorphism g ( ) splits. Assuming for a moment this claim complete the proof of lemma as follows. Take S = Y and consider the filtration (0) E1 E2 ··· En = q (E ) on the vecA

is an isomorphism the desired relation follows.

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tor bundle q (E ). Applying several times claim one gets an affine bundle g : T S such that one has a direct sum decomposition g (q (E )) = n=1 (g (Ei /Ei -1 )) of i the vector bundle (q g) (E ). Show that the morphism r = q g : T X have the desired property. For each smooth variety S an each morphism S S the pull-back map (g )A : A(S ) A(T ) is an isomorphism because T is an affine bundle over S . Now if X is a smooth variety and f : X X is a morphism, then Y is smooth over X and therefore S = Y is smooth as well. The pull-back map (q )A : A(X ) A(Y ) is a split injection by the projective bundle cohomology. Thus the composite map (q g ): A(X ) A(T ) is a split injection as well. Proof of claim. Let Hom(F2 ,F1 ) be the scheme representing the sheaf Hom(F2 ,F1 ) and let : Hom(F2 ,F1 ) Hom(F2 ,F2 ) be the morphism corresponding to the morphism Hom(F2 ,F1 ) Hom(F2 ,F2 ) induced by the vector bundle map : F1 F2 .Let id: S Hom(F2 ,F2 ) be the section of the projection Hom(F2 ,F2 ) S corresponding to the identity map F2 F2 . Let Sect( ) = -1 (id(S )) be a closed subscheme of the scheme Hom(F2 ,F1 ) and let g : T = Sect( ) S be the projection. The scheme T represents the sheaf of sections of the sheaf epimorphism . Thus there exists a canonical section s : g (F2 ) g (F1 ) of the epimorphism g ( ): g (F1 ) g (F2 ). This section gives rise by a standard way to a vector bundle isomorphism g (F1 ) g (F2 ) g (K ). = To prove claim it remains to observe that the variety T = Sect( ) is a torsor under the vector bundle Hom(F2 ,K ). The claim is proved. 3.6. C H E R N C L A SSES Let A be a ring cohomology theory. DEFINITION 3.26. A Chern classes theory on A is an assignment which associate to each smooth variety X and each vector bundle E on X certain elements ci (E ) A(X ) (i = 0, 1,... ) which are universally central and satisfy the following properties 1. c0 (E ) = 1: the restriction of the assignment L c1 (L) to line bundles is a Chern structure on A. 2. functoriality: ci (E ) = ci (E ) for isomorphic vector bundles E and E ; f A (ci (E )) = ci (f (E )) for each morphism f : Y X . 3. Cartan formula: cr (E ) = c0 (E1 ) cr (E2 ) + ··· + cr (E1 ) c0 (E2 ) for each short exact sequence 0 E1 E E2 0 of vector bundles. 4. Vanishing property: cm (E ) = 0for m > rk(E ).

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THEOREM 3.27. Let A be endowed with a Chern structure L c(L). Then there exists a unique Chern classes theory on A such that for each line bundle L one has c1 (L) = c(L). Moreover the Chern classes ci (E ) are nilpotent for i > 0. Proof. First prove the uniqueness assertion. If there are two assignments E/ X ci (E ) and E/ X ci (E ) satisfying the required properties. Then they coincide on line bundles by the properties 1 and 4. Therefore they coincide on direct sums of line bundles by the Cartan formula 3. Thus they coincide on all vector bundles by the splitting principle (Lemma 3.24). It remains to construct a Chern classes theory. We follow here the well-known construction of Grothendieck [12]. Let X be a smooth variety and E/ X beavector bundle with rkE = n. Set = c(OE (-1)). By Theorem 3.9 there are unique elements ci (E ) A(X ) such that n - c1 (E )
n-1

+ ··· + (-1)n cn (E ) = 0.

(17)

Set c0 (E ) = 1and cm (E ) = 0 if m > n. CLAIM 3.28. Classes ci (E ) satisfy the theorem. The rest of the proof is devoted to the proof of this claim. The property c0 (E ) = 1 holds by the very definition. To prove the property c1 (L) = c(L) for a line bundle L observe that P(L) = X and OL (-1) = L over X . Thus = c(L) in A(X ) and the relation (17) shows that c1 (L) = c(L). LEMMA 3.29. For each line bundle L over a smooth X the class c(L) A(X ) is nilpotent. To prove this lemma recall that by Claim 3.23 one can find a diagram of the form (16) with a torsor under a vector bundle p : X X and a morphism f : X P(V ) such that the line bundles L = p (L) and f (OV (1)) over X are isomorphic. The class c(OV (1)) A(P(V )) is nilpotent by Lemma 3.12. Thus the class c(L ) = f A (c(OV (1))) is nilpotent as well. The pull-back map p A : A(X ) A(X ) is an isomorphism by the strong homotopy invariance (Section 2.2.6). Therefore the class c(L) A(X ) is nilpotent as well. The lemma is proved. Now prove the functoriality of the classes ci . A vector bundle isomorphism : E E induces an isomorphism : P(E ) P(E ) of the projective bundles and a line bundle isomorphism (OE (-1)) OE (-1) over P(E ). Therefore A (c(OE (-1))) = c(OE (-1)) in A(P(E )). Now the relations ci (E ) = ci (E ) follows immediately from the projective bundle cohomology and the relation (17). The property f A (ci (E )) = ci (f (E )) is proved similarly. For the rest of the proof we need the following claim.

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CLAIM 3.30. For a rank r vector bundle F set ct (F ) = 1 + c1 (F )t + ··· + cn (F )t r . Let T be a smooth variety and let F = r=1 Li for certain line bundles Li over T . i Then one has
r

ct (F ) =
i =1

ct (Li ).

In particular the elements ci (F ) are universally central and nilpotent. (The nilpotence of the class c1 (L) is proved just above.) Assuming for a moment Claim 3.30 complete the proof of Claim 3.28 as follows. By the splitting principle (Lemma 3.24) there exists a smooth variety T and a morphism r : T X such that each the vector bundle r (Ei ) is a sum of line bundles and the pull-back map r A : A(X ) A(T ) is injective. The Claim 3.30 and the injectivity of the map r A show the Cartan formula ct (E ) = ct (E1 )ct (E2 ). Furthermore the Claim 3.30 shows that the elements r A (ci (E )) A(T ) are universally central. In particular for a smooth variety X and a morphism f : X X and for T = X âX T the element (f )A (r A (ci (E ))) is central in A(T ). By the same splitting principle the pull-back map (r )A : A(X ) A(T ) is injective. Now the relation (f )A (r A (ci (E ))) = (r )A (f A (ci (E ))) and the injectivity of the map (r )A : A(X ) A(T ) show that the element f A (ci (E )) is central in A(X ). Thus the elements ci (E ) are universally central. Finally the Claim 3.30 shows that the elements r A (ci (E )) are nilpotent. The injectivity of the map r A proves the nilpotence of the elements ci (E ) A(X ). It remains to prove the Claim 3.30. If = c(OF (-1)), where OF (-1) is the tautological line bundle on P(F ) then it suffices to prove the relation ( - c1 (Li )) = 0 in A(P(F )). To prove the very ¯ last relation set F i = L1 ··· Li ··· Ln , where the bar means that the corresponding summand has to be omitted. Since OF (-1)|P(Li ) = Li over X the element - c1 (Li ) vanishes being restricted to P(Li ). Therefore - c1 (Li ) belongs to the subgroup AP(F i ) (P(F )) of the group A(P(F )) (see (14)). Thus the cup-product n i =1 ( - c1 (Li )) belongs to the subgroup AP(F i ) (P(F )) of the group A(P(F )). Since the intersection n=1 P(F i ) is empty the group AP(F i ) (P(F )) vanishes by i the vanishing property. Hence indeed the relation n=1 ( - c1 (Li )) = 0 holds in i A(P(F )) and n=1 (1 + c1 (Li )t ) = ct (F ) in A(X ). i Finally since each of the elements c1 (Li ) is universally central and nilpotent hence each of the elements cj (E ) is universally central and nilpotent as well. Claim 3.30 is proved. PROPOSITION 3.31. Let X be a smooth variety and let E be a vector bundle over X of the constant rank n. Let p : P(E ) X be the projection. Then the element cn (OE (1) p (E )) A(P(E )) vanishes. Proof. Define a rank n - 1 vector bundle Q over P(E ) by the short exact sequence 0 OE (-1) p (E ) Q 0. Tensoring this short exact sequence with the line bundle OE (1) one gets a short exact sequence 0 O

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OE (1) p (E ) OE (1) Q 0. Now the Cartan formula for the Chern classes gives the relation cn (OE (1) p (E )) = c1 (O)cn-1 (OE (1) Q). Thus cn (OE (1) p (E )) = 0. 3.7. ORI E N T I NG A T HE ORY In this section A is a ring cohomology theory. Two theorems in this section shows how one can construct an orientation using a Chern structure (or a Thom structure) on A and how one can construct a Chern structure (or a Thom structure) using an orientation. Before to state theorems it is convenient to fix a notion of Thom classes theory, which is equivalent to the notion of orientation but it is defined in terms of elements rather than in terms of homomorphisms. The definition of A(X )-central elements in AX (E ) (for a vector bundle E over a smooth variety X ) is given just below the definition of a Chern structure. DEFINITION 3.32. A Thom classes theory on A is an assignment which associate to each smooth variety X and to each vector bundle E over X an element th(E ) AX (E ) satisfying the following properties (1) th(E ) is A(X )-central; (2) A (th(F )) = th(E ) for each vector bundle isomorphism : E F ; (3) f A (th(E )) = th(f (E )) for each morphism f : Y X with a smooth variety Y; (4) the operator A(X ) AX (E ), a th(E ) a is an isomorphism; (5) multiplicativity property: for the projections qi : E1 E2 Ei (i = 1, 2) one has (18) q1 th(E1 ) q2 th(E2 ) = th(E1 E2 ) AX (E1 E2 ). The element th(E ) is called the Thom class of the vector bundle E . LEMMA 3.33. If is an orientation on the theory A then the assignment E thE (1) AX (E ) is a Thom classes theory on A. We write thX (E ) for the eleX ment thE (1) AX (E ). X If an assignment E/ X th(E ) AX (E ) is a Thom classes theory on A, then the family of homomorphisms th(E ): AZ (X ) AZ (E ) form an orientation on A. The two mentioned correspondences between orientations and Thom classes theories are inverse to each other. Proof. It is obvious. LEMMA 3.34. If an assignment E/ X AX (E ) is a Thom classes theory on A, then its restriction to line bundles is a Thom structure on A. If two Thom classes theories coincide on each line bundle then they coincide. Proof. The first assertion is obvious. To prove the second assertion consider two Thom classes theories E th(E ) AX (E ) and E th (E ) AX (E )

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which coincide on line bundles. To prove that for a vector bundle E one has the relation th(E ) = th (E ) one may assume by the splitting principle (Lemma 3.24) that E = Li is a direct sum of line bundles. Let qi : E Li be the projection to the i th summund. Now the chain of relations th(E ) = qiA (th(Li )) = qiA (th (Li )) = th (E ) completes the proof of the assertion. THEOREM 3.35. Given a Chern structure L c(L) on A (or the corresponding by Theorem 3.5 Thom structure L th(L) on A) there exists an orientation (X,Z ,E ) thE on A such that the following properties hold Z 1. for each smooth variety X and each line bundle L/ X one 2. for each smooth X and each line bundle L/ X one has c(L) a where a A(X ) is any element, i A : AX (L) extension operator for the pair (L , L - X), z: X L is has th(L) = thL (1); X zA i A thL (a ) = X A(L) is the support the zero section.

Moreover the required orientation is uniquely determined both by the property (1) and by the property (2). This theorem describes the arrow and the composition from Section 1. THEOREM 3.36. If (X,Z ,E ) thE is an orientation on A then the assignment Z L zA i A thL (1) is a Chern structure on A, the assignment L thL (1) is X X a Thom structure on A and so constructed Chern and Thom structures correspond to each other. Moreover the construction of an orientation by means of a Chern (or a Thom) structure given by Theorem 3.35 and the construction of a Chern and a Thom structure by means of an orientation are inverse of each other. This theorem describes the arrow and the composition from Section 1. Moreover it states that the arrow and the composition are inverse to each other, and it states that the composition and the arrow are inverse to each other. Proof of Theorem 3.35. To construct an orientation on A it suffices (see Lemma 3.33) to construct a Thom classes theory E th(E ) AX (E ). Let E/ X be a rank n vector bundle and let F = E 1. Let p : P(F ) X be the projection. The support extension operator AP(1) (P(F )) A(P(F )) is injective because the sequence (14) is exact. The same exact sequence and Proposition 3.31 show that the element cn (OF (1) p E) A(P(F )) belongs to the subgroup AP(1) (P(F )).Set ¯ (E ) = cn (OF (1) p E) AP(1) (P(F )), th and define the element th(E ) AX (E ) as follows th th(E ) = eA (¯ (E )) = eA (cn (OF (1) p E)) AX (E ). (20) (19)

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To show that the assignment E th(E ) AX (E ) is a Thom classes theory it remains to check the properties (1)­(5) from Definition 3.32. The second and the third property follows immediately from the functoriality of the Chern classes (Theorem 3.27). The element ¯ (L) A(P(F )) is central because it is a Chern class. Since the th pull-back map eA : A(P(F )) A(E ) is surjective the element th(L) = eA (¯ (L)) th is A(X )-central. For a smooth variety Y and a morphism f : Y X one has th(f (E )) = f A (th(E )) in AY (f (E )). Thus the element th(E ) is universally A(X )-central. This proves the property (1). To prove the fourth property consider the commutative diagram AP(1)(P(F ))
O
¯ (E ) th e
A

/ AX (E ) O
th(E )

A(X )

id

/ A(X ).

The map eA is an isomorphism by the excision property. Thus the right vertical arrow is an isomorphism if the cup-product with the class ¯ (E ) is an isomorphism. th For that consider the commutative diagram with exact rows 0
/ AP
(1)

(P(F ))
O

/ A(P(F )) O
( ¯ (E ), ) th

/ A(P(E )) O
n

/0 / 0,

¯ (E ) th

0

/ A(X )

/ A(X ) A(X )

/ A(X )

n

n where = (1, E ,... , E-1 ), E = c(OE (1)) A(P(E )) and = n-1 (1, F ,... , F ), F = c(OF (1)) P(F ). The map is an isomorphism by the projective bundle theorem. Thus to prove that the left vertical arrow is an isomorphism it suffices to check that the map (¯ (E ), ) is an isomorphism. Using the Mayer­Vietoris property and Proposith tion 2.18 one may assume that the bundle E is the trivial rank n bundle. In this n th case one has ¯ (E ) = cn (OF (1)n ) = F . Thus the map (¯ (E ), ) coincides in this th n case with the map (1, F ,..., F ) and it is an isomorphism by the projective bundle theorem. The property (4) is proved. Basically the property (5) follows from the Cartan formula for Chern classes. But to give a detailed prove one needs certain preliminaries. Let E = E1 E2 be a vector bundle over a smooth variety X and let Fr = Er 1 (r = 1, 2) and let F = E 1 and let p : P(F ) X be the projection . Let qr : E Ei be the projection and let ir : Er E be the imbedding. We will identify E with the open subset P(F ) - P(E ) of P(F ) and identify Ei with the open subset P(Fr ) - P(Ei ) of P(Fr ).Let P(Fi ) be the subvariety in P(F ) defined by the ¯ direct summund Fr of F . Let the closed imbedding ir : P(Fi ) P(F ) be the one extending the imbedding ir . We will write p : P(F ) X for the projection to X . Let resr : AP(Fr ) (P(F )) AEr (E ) be the pull-back map induced by the imbedding E P(F ). Let res: AP(1) (P(F )) AX (E ) be the pull-back map induced

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by the same imbedding. The support extension operators AP
(Fr )

(P(F )) A(P(F ))

are injective for (r = 1, 2) because they are the operators from the short exact sequences of the form (14). The same exact sequences and Proposition 3.31 show that the elements cn1 (OF (1) p (E1 )) A(P(F )); cn2 (OF (1) p (E2 )) A(P(F )) belongs to the subgroups AP(F2 ) (P(F )) and AP(F1 ) (P(F )), respectively. Consider elements x1 = cn2 (OF (1) p (E2 )) AP(F1 ) (P(F )), x2 = cn1 (OF (1) p (E1 )) AP(F2 ) (P(F )). We claim that they satisfy the following relations
A A q1 (th(E1 )) = res1 (x2 ); q2 (th(E2 )) = res2 (x1 ).

(21)

In fact, the commutative diagram AP(1)(P(F2 )) o
A e2

¯A j2

AP

(F1 )

(P(F ))

res2

AX (E2 ) o



A j2

AE1 (E ),

A ¯A and the relation j2 (x1 ) = ¯ (E2 ) in AP(1) (P(F2 )) show that j2 (res2 (x1 )) = th(E2 ). th A A A Now the relation q2 j2 = id proves the relation res2 (x1 ) = q2 (th(E2 )). The second of the two relations (21) is proved. The first one is proved similarly. So the relations (21) are proved. Consider one more commutative diagram

A(P(F )) â A(P(F )) O
A A I1 âI2



/ A(P(F )) O
I
A

AP

(F1 )

(P(F )) â AP
res1 âres

(F2 )

(P(F ))



/ AP

(1)

(P(F ))

2

res

AE1 (E ) â AE2 (E )





/ AX (E ).

The commutativity of the upper square of this diagram proves the relation x1 x2 = ¯ (E ) in AP(1)(P(E )) because th cn2 (OF (1) p (E2 )) cn1 (OF (1) p (E1 )) = cn (OF (1) p (E ))

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44 in A(P(F )). Now the chain of relations th(E ) = res(¯ (E )) = res(x1 x2 ) = res1 (x1 ) res2 (x2 ) th A A = q1 (th(E1 )) q2 (th(E2 ))

I. PANIN

prove the property (5). Thus the assignment E th(E ) AX (E ) is indeed a Thom classes theory on A. We still have to check that the orientation corresponding to this Thom classes theory by Lemma 3.33 satisfies the requirements 1 and 2 of Theorem 3.35. The property th(L) = thL (1) holds because the map thL is defined as the cupX X product with the class th(L). The requirement 2 is satisfied by the following reasons. The composite map zA i A thL : A(X ) A(X ) is a two-sided A(X )-module map. It takes the unit 1 X to the class c(L) by Theorem 3.5. The requirement is checked. To complete the proof of theorem it remains to prove the uniqueness of the orientation. To prove the uniqueness of the orientation satisfying the property 1 take two orientations and on A satisfying the property 1. Then the assignE ments E th(E ) = thE (1) and E th (E ) = thX AX (E ) are two Thom X classes theories on A by Lemma 3.33. To check that they coincide it suffices by Lemma 3.34 to check that their restrictions to line bundles coincide. This is the case by the requirement 1. Thus = . Now prove the uniqueness of the orientation satisfying the requirement 2. Let L c(L) be a Chern structure and let and be two orientations satisfying the requirement 2. We will show that = . It suffices to check that the corresponding Thom classes theories E th (E ) and E th (E ) coincide (see Lemma 3.33). By Lemma 3.34 the restriction of these Thom classes theories to line bundles are Thom structures on A. By the same lemma the Thom classes theories coincide if the mentioned Thom structures on A coincide. To prove that the two Thom structures on A coincide it suffices by Theorem 3.5 to check that the two corresponding Chern structures L c (L) and L c (L) coincide. This holds because c (L) = c(L) and c (L) = c(L) by the requirement 2. The proof of the relation = is completed. Proof of Theorem 3.36. The assignment E thE (1) is a Thom classes theX ory by Lemma 3.33. Its restriction to line bundles is a Thom structure on A by Lemma 3.34. The first assertion is proved. The assignment L c(L) = zA (i A (thL (1))) = zA (i A (th(L))) is a Chern X structure by Theorem 3.5. The Chern structure L c(L) and the Thom structure L th(L) correspond to each other by the same Theorem 3.5. The first part of theorem is proved. Now verify that the correspondences between orientations and Thom structures in the two theorems are inverse to each other. Now let L th(L) be a Thom structure and let be the corresponding by Theorem 3.35 orientation and let L thL (1) = th (L) be the Thom structure X

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corresponding to by Theorem 3.36. We have to check that for each line bundle L one has th (L) = th(L). If L c(L) is the Chern structure corresponding to the Thom structure L th(L),then th (L) = c1 (O(1) p (L)) by the very construction of the orientation . From the other side the assignment L c1 (O(1)) p (L) is exactly the Thom structure corresponding to the Chern structure L c(L). Thus th (L) = th(L) by Theorem 3.5. Let be an orientation and let L th (L) be the corresponding Thom structure and let be the orientation corresponding to the Thom structure L th (L). We have to check that = . It was proved just above that the Thom structure L th (L) corresponding to coincide with the Thom structure L th (L). Now by Lemma 3.34 the Thom classes theory E th (E ) corresponding to coincides with the Thom classes theory E th (E ) corresponding to . Thus = by Lemma 3.33. It is verified simultaneously that the correspondences between orientations and Chern structures on A described in the two theorems are inverse to each other. The theorem is proved. 3.8. EX A M PLES 3.8.1. Let A be the algebraic K-theory (Section 2.1.8). The rule L [1]-[L ] endows A with a Chern structure (the property (4) follows from [25, Section 8, Theorem 2.1]) and thus orients A. It is interesting to observe that the corresponding Chern class cn of a rank n vector bundle E is exactly the known class -1 (E ) = [1] - [E ] + [2 E ] + ··· + (-1)n [n E ]. 3.8.2. q Let A be the etale cohomology theory A (X ) = + HZ (X, µm ), where m is q =- Z an integer prime to char(k ). Consider the short exact sequence of the etale sheaves âm 0 µm G G 0 and denote by : H 1 (X, Gm ) H 2 (X, µm ) the boundary map. For a line bundle L over a smooth variety X let [L] H 1 (X, Gm ) be its isomorphism class. It is known [17] that the rule L ([L]) endows A with a Chern structure. Thus A is oriented. 3.8.3. p p Let A be the motivic cohomology [28]: AZ (X ) = 0 HZ (X, Z(q)). Recall that q= 2 HM(X, Z(1)) = CH 1 (X ) for a smooth X [28]. For a line bundle L over a smooth variety X let D(L) CH 1 (X ) be the associated class divisor. The rule L D(L) endows A with a Chern structure in the characteristic zero [28, Corollary 4.12.1] (now it is known in any characteristic). Thus A is oriented.

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46

I. PANIN

3.8.4. p p Let A be the K-cohomology [25, Section 7, 5.8]: AZ (X ) = 0 HZ (X, Kq ), q= where K is the sheaf of K-groups. Recall that the sheaf K1 coincides with the sheaf O of invertible functions. For a line bundle L over a smooth variety X let [L] H 1 (X, K1 ) = H 1 (X, O ) be the isomorphism class of L. The rule L [L] endows A with a Chern structure [11, Theorem 8.10] and thus orients A. 3.8.5. Let k = R and A be the Z/2-graded-commutative ring theory from Section 2.5.6. For a line bundle L consider the real line bundle L(R) over the topological space X(R) and set c1 (L) = w1 (L(R)) H 1 (X (R); Z/2Z) Aev (X ) (the first Stiefel­ Whitney class). Since Pn (R) = RP n is the real projective space the rule L c1 (L) endows A with a Chern structure and thus orients A. 3.8.6. Semi-topological Complex and Real K-theories [6] If the ground field k is the field R of reals then the semi-topological K-theory of real algebraic varieties K Rsemi defined in [6] is an oriented theory as it is proved in [6]. For a real variety X it interpolates between the algebraic K-theory of X and Atiyah's real K-theory of the associated real space of complex points, X(C). 3.8.7. Orienting the Algebraic Cobordism Theory In this example the notation of Section 2.5.5 are used. The identity morphism MGL1 to itself gives rise in the standard manner to an element [id1 ] MGL2,1 (MGL1 ). By the very definition MGL1 = Th(T (1)) and T (1) is the tautological line bundle O(-1) over the space G(1) = P(V ) = P . Now set
21 th = [id1 ] MGL2,1 (MGL1 ) = MGLP, (O(-1)).

Consider the fiber A1 The restriction of coincides with the T unite 1 MGL0,0 (p t MGL due to (??).

of T (1) over the point g1 P(V ). the element th to the Thom space Th(A1 ) = A1 /(A1 - {0}) 2, -suspension MGL2,1 (Th(A1 )) = MGL{01 (A1 ) of the } ). Thus the element th orients the algebraic cobordism theory

3.9. T H E F ORMAL GROUP L AW F Let be an orientation of A. Thus A is endowed with the Chern structure which correspond to (see Theorems 3.35 and 3.36). Following Novikov, Mischenko [19] and Quillen [24] we associate a formal group law F with . This formal ¯ group law is defined over the ring Auc := A(p t )uc and gives an expression of the first Chern class of L1 L2 in terms of the first Chern classes of line bundles L1 ,L2 .

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¯ Using Theorem 3.9 identify the formal power series in one variable A[[u]] with the ring A(P ) identifying u with c1 (O(1)) A(P ). The two `projections' pi : P â P P induce two pull-back maps piA : A(P ) A(P â P ). ¯ Using Theorem 3.9 again identify A(P â P ) with A[[u1 ,u2 ]], where ui = pi (u) = c1 (pi (O(1)). Set
¯ F (u1 ,u2 ) = c1 (p1 (O(1) p2 (O(1))) A[[u1 ,u2 ]].

(22)

Since the first Chern class is a universally central the element F belongs to ¯ Auc [[u1 ,u2 ]]. PROPOSITION 3.37. For any X Sm and line bundles L1 /X , L2 /X one has the following relation in A(X ) c1 (L1 L2 ) = F (c1 (L1 ), c1 (L2 )). Here the right-hand side is well defined since the first Chern classes are universally central and nilpotent (Theorem 3.27). Proof. Using the Jouanolou device one may assume (compare with the proof of Lemma 3.29) that Li = fi (O(1)) for a maps fi : X PN .Let f = (f1 ,f2 ): X PN â PN . The chain of relations
c1 (L1 L2 ) = f A (c1 (p1 (O(1)) p2 (O(1)))) = f A (F (u1 ,u2 )) = F (c1 (L1 ), c1 (L2 ))

completes the proof. ¯ PROPOSITION 3.38. The formal power series F Auc [[u1 ,u2 ]] is a commutative formal group law [8] with the `inverse element' I (u) = c1 (O(-1)) ¯ Auc (P ) = Auc [[u]]. ¯ Proof. One has to verify that the formal power series F Auc [[u1 ,u2 ]] satisfies the following properties · · · · normalization: F (u1 ,u2 ) u1 + u2 modulo the degree 2; associativity: F (F (u1 ,u2 ), u3 ) = F (u1 ,F (u2 ,u3 )); commutativity: F (u1 ,u2 ) = F (u2 ,u1 ); `inverse element': F (u, I (u)) = 0 for I (u) = c1 (O(-1)) Auc (P ) = ¯ Auc [[u]].

Proposition 3.37 shows that the associativity and the commutativity follows immediately from the corresponding properties of the tensor products of line bundles. To prove the normalization property consider the element = c1 (p1 (O(1)) 1 1 1 1 p2 (O(1))) A(P â P ).The A(p t )-module A(P â P ) is a free module with the free bases 1, 1, 1 and by the projective bundle theorem. Write the element in the form = a00 1 1 + a10 1 + a01 1 + a11 , where aij are elements in A(p t ). By the projective bundle theorem A(P1 â P1 ) = A[[u1 ,u2 ]]/(u2 ,u2 ) and thus it suffices to prove that a00 = 0 and a10 = 1 = a01 . 1 2

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48 Restricting the element to {0} â {0}, to P1 â {0} and the relation |{0} = 0 one gets the relations: a00 = 0, a00 + a00 + a01 = in A(P1 ). Thus a10 = 1and a01 = 1 and the is proved. The relation F (u, I (u)) = 0 follows from Proposition bundles L1 = O(1) and L2 = O(-1) on P .

I. PANIN

to {0} â P1 and using a10 = in A(P1 ) and normalization property 3.37 applied to the line

DEFINITION 3.39. The formal group law F is called the formal group law associated with A endowed with the orientation . Its the `inverse element' is the series I .
- ¯ Let E : A[[u]] A(P ) be an isomorphism taking the variable u to the A element A = c1 (O(-1)) A(P ). The formal power series - - - F (u1 ,u2 ) = (E E )-1 [c1 (p1 (O(-1)) p2 (O(-1)))] A(p t )uc [[u1 ,u2 ]]

(23)

satisfies exactly the same property as the series F (u1 ,u2 ) above. Namely, for any X Sm and line bundles L1 /X , L2 /X one has the following relation in A(X )
- c1 (L1 L2 ) = F (c1 (L1 ), c1 (L2 )).

Taking X = P â P So there is group law definitions putations it



and line bundles Li = pi (O(-1)) for i = 1, 2 one gets

- F (u1 ,u2 ) = F (u1 ,u2 ).

no difference which line bundle is used O(-1) or O(1). The formal F (u1 ,u2 ) is the same in both cases. We usually for the purposes of will use the tautological line bundle O(-1). However for certain comwill be convenient to use the line bundle O(1).

3.9.1. Examples
H · If A = HM(-, Z()) with the first Chern class c1 then one has the relation H H H c1 (L1 L2 ) = c1 (L1 ) + c1 (L2 ). K · If A = K -theory with the first Chern class defined by c1 (L) = [1]-[L ] then K K K K K one has the relation c1 (L1 L2 ) = c1 (L1 ) + c1 (L2 ) - c1 (L1 ) · c1 (L2 ). · Let k = C and A = the complex cobordism theory equipped with the Chern structure L cf (L) given by the Conner­Floyed class cf [5, pp. 48­52], then the formal group law F is the universal commutative formal group law by [24]. Its ring of coefficients (p t ) is canonically isomorphic to the Lazard ring L [24].

Acknowledgements The subject of Sections (3.3) and (3.9) was inspired by Morel's lectures (Muenster, June, 1999). The author thanks A. Suslin, E. Friedlander, A. Merkurjev and S. Yagunov for their stimulating interest to the subject.

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The author thanks very much for the support the TMR Network ERB FMRX CT-97-0107, the grants of the years 2001­2003 of the `Support Fund of National Science' at the Russian Academy of Science, the Program of the year 2003 `Research in the fundamental domains of the modern mathematic' at the Presidium of RAS, the RFFI-grant 03-01-00633a and the grant INTAS-99-00817. References
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