Traces in oriented homology theories, II of algebraic varieties

Traces in oriented homology theories, II of algebraic varieties K. Pimenov 02.02.2005 1 General part 1.1 Introduction The present preprint is the second one in a short series of preprints. The main aim of this series is to give a proof of the only non-trivial result on algebraic cobordism stated by V.Voevodsky in [V, Th.3.22]. In this preprint we introduce the notion of an oriented homology theory on algebraic varieties following the outline of [PS], [P1] and [Pa]. The main result of the preprint is Theorem 5.1.4. It states that for a given oriented homology theory there exists a family of pull-back operators, one operator for each projective morphism of smooth varieties. This family of operators satisfies a list of properties and is called a trace structure on the theory. The operators are called trace operators. It is proved that the trace structure is unique provided that it is normalized in a natural way. So in this preprint 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 homology theory we mean a covariant functor A from this category to the category of abelian groups endowed with a functor transformation : A(X, U) A(U) and satisfying the localization, Nisnevich excision and homotopy invariance properties (1.3.1). We consider four structures a homology theory A can be equipped with: an orientation on A, a Thom structure on A, a Chern structure on A and a trace structure. The first part of the preprint has already published as [Pim]. It concerns the relation of the first three structures to each other. Namely we prove that there are natural bijections between the first three of the structures. A heart of these section is Projective Bundle Theorem 2.2.1. We give a rather geometric proof for it. The main theorem of this section is theorem 2.4.3 which uses a blow up technique developed in the proof of Theorem 2.2.1. In contrast with the paper [Ne] we work just with a homology theory A rather than with a pair (A, A ) of (co)homology. We use an extra structure on A called Chern structure (see 2.1.1) and do not use a (co)homological multiplication at all. A Chern structure on a multiplicative pair in a sense of [Ne] gives obviously a Chern structure in our sense on the homology theory A : e Z pim (L) = c Nen(L) : A Z (X) AZ (X). the author is supported by grant PD02-1.1-368 of Russian Government 1

A construction of the trace structure is given in the second part of the preprint. Section 3.1.2 led on thoroughly a work [P1]. Subsection 4.1 delights some new aspects which are appeared in homological context. In this section we prove that for an oriented theory homology groups are modules over the Lazard ring. We do not prove in this text that there is a bijection between trace structures and orientations since our definition of orientation uses supports but there are no supports when we discuss trace structure. However we prove uniqueness theorem in subsection 5.3. In Appendix we show a backward relation between Trace structure and Chern prestructure. Besides we extend uniqueness theorem to a pretheory context. Given an oriented commutative ring T -spectrum E one can form as a cohomology theory E so a homology theory E. For a smooth projective variety X of dimension d with a structural morphism f : X pt the class [X] := f! (1) E 2d,d (X) is called the fundamental class of X. It is proved in [PY, Thm.2.2] that the cap product with this class [X] : E p,q (X) E 2d p,d q (X) is an isomorphism (the Poincare duality isomorphism). Our theorem 6.1.3 allows us to deduce some functorial properties of this fundamental class to complete the proof of Voevodsky Theorem. The first part of the preprint was written during the visit of the author to the Universität Bielefeld. The author is very grateful to H.Abels for shown hospitality. The author thanks I.Panin very much for useful discussion on the subject of the present preprint. 1.2 Terminology and notation Let k be a field. The term variety is used in this text to mean a reduced quasi-projective 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 the reduced scheme, so Z is considered as a closed subvariety of X. We fix the following notation: Ab - the category of abelian groups; Sm - the category of smooth varieties; SmOp - the category of pairs (X, U) with smooth X and open U in X. Morphisms are morphisms 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 L X (D) (see [Har, Ch.II, 6, 6.13]). P(V ) = P roj(s (V )) - the space of lines in a finite dimensional k-vector space V ; L V = O V ( 1) - the tautological line bundle over P(V ); 1 X - the trivial rank one bundle over X, often we will write 1 for 1 X ; 2

P(E) - the space of lines in a vector bundle E; L E = O E ( 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; 1.3 Homology theories 1.3.1 Definition. A homology theory is a covariant functor SmOp A Ab together with a functor morphism : A (X, U) A (U) satisfying the following properties 1. Localization: the sequence A (U) j A(X) i A (X, U) P A (U) j A(X) 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; 3. Homotopy invariance: the operator A (X A 1 ) A (X) induced by the projection X A 1 X is an isomorphism. The operator P is called the boundary operator and is written usually as. Let us stress that the functor takes values in the category of abelian groups rather than in the category of graded abelian groups. The subscript in A is used only to stress that we work with a covariant functor. A morphism of homology theories ϕ : (A, A ) (B, B ) is a functor transformation ϕ : A B commuting with the boundary morphisms in the sense that for every pair P = (X, U) SmOp one has P B ϕ P = ϕ U P A. We write also A Z (X) for A (X, U), where Z = X U, and call the group A Z (X) homology of X with the support on Z. The operator is called the support restriction operator for the pair (X, U). A (X) i A Z (X) (1) Below in the text we will often denote a support restriction operator appearing in various contexts by β. 1.4 General properties of homology theories We specify here certain properties of an arbitrary homology theory A which are useful below in the text. 3

1.4.1. The localization property implies that A (X) = A (X, X) = 0. Therefore A ( ) = A ( ) = 0. 1.4.2. 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 the level of A then the third of these morphisms induces an isomorphism on the level of A. 1.4.3. Localization sequence for a triple. Let T Y X be closed subsets of a smooth variety X. Let : A T (X) A (X T ) be the boundary map for the pair (X, X T ). Consider the support restriction map e : A (X T ) A Y T (X T ) and set Y,T = e : A T T (X) AY (X T ). We claim that the sequence... A Y (X) α A T (X) A Y T (X T ) β A Y (X) α A T (X)... with the obvious mappings α and β is a complex and moreover it is exact. We call this sequence the localization sequence for the triple (X, X T, X Y ). If Y = X, then this sequence coincides with the localization sequence for the pair (X, X T ). 1.4.4. Mayer-Vietoris sequence. If X = U 1 U 2 is a union of two open subsets U 1 and U 2 and if Y is a closed subset in X, then set T i = Y U i, Y i = Y U i, U 12 = U 1 U 2 Y 12 = U 12 Y. Consider the morphism of the localization sequences for the triples X Y T 1 and U 2 Y 2 T 1 induced by the inclusion of the triples (U 2, U 2 T 1, U 2 Y 2 ) (X, X T 1, X Y ) A Y 12 (U 2 T 2 ) β 2 A Y 2 (U 2 ) A T 1 (U 2 ) β 1 α 2 γ A Y 1 (X T 1) α 1 A Y (X) e A T 1 (X) A Y 12 (U 2 T 2 ) A Y 1 (X T 1). The map γ is an isomorphism by the excision property. Also by excision property we may identify A Y 12 (U 2 T 2 ) with A Y 12 (U 12 ) and A Y 1 (X T 1) with A Y 1 (U 1). Set d = γ 1 e : A Y (X) A Y 12 (U 12 ). We claim that the sequence... A Y (X) d A Y 12 (U 12 ) (β 1, β 2 ) A Y 1 (U 1) A Y 2 (U 2) α 1+α 2 A Y (X)... is exact and call this sequence the Mayer-Vietoris sequence of the open covering X = U 1 U 2. The proof of the exactness is straightforward and we skip it. The Mayer-Vietoris sequence is natural in the following sence.if f : X X is a morphism and X = U 1 U 2 is a Zariski covering of X such that f(u i) U i and if Y is a closed subset in X containing f 1 (Y ), then the mappings f : A Y (X ) A Y (X), f : A Y i (U i ) AY i (U i), f : A Y 12 (U 12 ) AY 12 (U 12 ) form a morphism of the corresponding Mayer-Vietoris sequences. 4

1.4.5. Let i r : X r X 1 X 2 be the natural inclusion (r = 1, 2). Let Y r X r be a closed subset for (r = 1, 2) Then the induced map A Y 1 (X 1) A Y 2 (X 2) A Y 1 Y 2 (X 1 X 2 ) is an isomorphism. Proof. This follows from the Mayer-Vietoris property and the fact that A ( ) = 0. 1.4.6. Strong homotopy invarance. 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 natural map p : A S (T ) AZ (X) is an isomorphism. If s : X T is a section then the induced operator s : A Z (X) A S (T ) is an isomorphism as well. Proof. First consider the case Z = X. Then S = T and we have to check that the map p : A (T ) A (X) is an isomorphism. Choose a finite Zariski open covering X = U i such that T i = p 1 (U i ) is isomorphic to the trivial vector bundle over each U i and then use the morphism of the Mayer-Vietoris sequences and the homotopy invariance property of the homology theory A. To prove the general case consider the localization sequences for the pairs (X, X Z) and (T, T S). The morphism p : (T, T S) (X, X Z) form a morphism of these two long exact sequences. The 5-Lemma completes the proof. 1.4.7. Deformation to the normal cone. The deformation to the normal cone is a well-known construction (for example, see [Fu]). Since the construction and its property (3) 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 X t together with a smooth morphism p t : X t A 1 and a closed imbedding i t : Y A 1 X t such that the map p t i t coincides with the projection Y A 1 A 1 and the fiber of p t over 1 A 1 is canonically isomorphic to X and the base change of i t by means of the imbedding 1 A 1 coincides with the imbedding i : Y X; the fiber of p t over 0 A 1 is canonically isomorphic to N and the base change of i t by means of the imbedding 0 A 1 coincides with the zero section Y N. Thus we have the diagram (N, N Y ) i 0 (X t, X t Y A 1 ) i 1 (X, X Y ) (2) Here and everywhere below in the text we identify a variety with its image under the zero section of any vector bundle over this variety. Recall a construction of X t, p t and i t. For that take X t to be the blow-up of X A1 with the center Y {0}. Set X t = X t X where X is the the proper preimage of X {0} under the blow-up map. Let σ : X t X A 1 be the restriction of the blow-up map σ : X t X A1 to X t and set p t to be the composition of σ and the projection X A 1 A 1. The proper preimage of Y A 1 under the blow-up map is mapped isomorphically to Y A 1 under the blow-up map. Thus the inverse isomorphism gives the desired imbedding i t : Y A 1 X t (observe that i t (Y A 1 ) does not cross X). 5

It s not difficult to check that the imbedding i t satisfies the mentioned two properties ( the preimage of X 0 under the map σ consists of two irreducible components: the proper preimage of X and the exceptional divizor P(N 1). Their intersection is P(N) and i t (Y A 1 ) crosses P(N 1) along P(1) = the zero section of the normal bundle N ). We claim that the diagram (2) consists of isomorphisms on the level of A. We will not here give the proof of this theorem because the proof is straightforward analogous to one given in cohomological context in [Pa] (c.f. Theorem 2.2.8). 1.4.8 Theorem. The following diagram consists of isomorphisms A Y (N) i0 A Y A1 (X t ) i1 A Y (X). (3) Moreover for each closed subset Z Y the following diagram consists of isomorphisms as well A Z (N) i0 A Z A1 (X t ) i1 A Z (X). (4) 1.4.9 Corollary. Let j 0 : P(1 N) X t be the imbedding of the exceptional divisor into X t and let j 1 = e t i 1 : X X t, where e t : X t X t is the open inclusion. Then the mapping is an isomorphism. Proof. Consider the commutative diagram A P(1) (P(1 N)) j0 A Y A1 (X t ) (5) A Y (N) e i 0 A Y A1 (X t ) e t A P(1) (P(1 N)) j 0 A Y A1 (X t ) where the vertical arrows are the natural mappings. The vertical arrows are isomorphisms by the excision property. The operator i 0 is an isomorphism by the first item of Theorem 1.4.8. Thus the operator j 0 is an isomorphism. 1.4.10. Let X be a smooth variety and let L be a line bundle over X. Let E = 1 L and let ī L : X = P(L) P(E) be the closed imbedding induced by the direct summand L of E. Let A (P(E)) β A P(1) (P(E)) be the support restriction operator and let ī L : A (P(L)) A (P(E)) be the natural mapping. We claim that the following sequence 0 A (P(L)) īl A (P(E)) β A P(1) (P(E)) 0. (6) is exact. 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 i L : 6

P(L) U is just the zero section of this line bundle, ī L = j i L and the natural operator i L : A (P(L)) A (U) is an isomorphism (the inverse to the one q ). Now consider the pair (P(E), U). By the localization property 1.3.1 the following sequence... A (U) j A (P(E)) β A P(1) (P(E))... is exact. If P(E) p X is the natural projection then the operator i L p : A (P(E)) A (U) splits j. This implies the injectivity of j and the surjectivity of β in the long sequence above. To proof that the sequence (6) is short exact it remains to recall that the operator i L is an isomorpism and ī L = j i L. 1.4.11. We use here the notation from 1.4.7. Let e t : X t X t be the open inclusion and let p : P(1 N) Y be the projection and let s : Y P(1 N) be the section of the projection identifying Y with the subvariety P(1) in P(1 N). The following commutative diagram will be repeatedly used below in the text j 0 P(1 N) X j 1 t X s I i t Y k 0 Y A 1 k 1 Y, where I t = e t i t and j 0 is the inclusion of the exceptional divisor and j 1 = e t i 1 and k 0, k 1 are the closed imbedding given by y (y, 0) and y (y, 1) respectively. 1.4.12 Lemma (Useful lemma). Under the notation of 1.4.7 let j t : V t = X t Y A 1 X be the inclusion. If the support restriction operator A (P(1 N)) A P(1) (P(1 N)) is surjective then Im(j 0 ) + Im(jt ) = A (X t ), in the other words the operator j 0 jt : A (P(1 N)) A (V t ) A (X t ) is an epimorphism. In particular this holds if Y is a divisor on X. Proof. Consider the commutative diagram A (P(1 N)) β j 0 A (X t) β t A P(1) (P(1 N)) j 0 A Y A1 (X t ). where β and β t are the support restriction operators. The bottom operator j 0 is an isomorphism by Corollary 1.4.9. The map β is surjective by the very assumption (if Y is a divisor in X then β is surjective by 1.4.10). Since the composition β t j 0 coincides with the one j 0 β it is surjective as well. The localization sequence for the pair (X t, V t) shows that Im(j t) = Ker(β t). The Lemma follows. 7

1.4.13. Let i : P(V ) P(W ) and j : P(V ) P(W ) be two linear imbeddings that is imbeddings induced by linear imbeddings V into W. If the dimension of V is strictly less than the dimension of W, then i = j : A (P(V )) A (P(W )). In fact, in this case there exists a linear automorphism φ of W which has the determinant 1 and such that j = φ i. Since φ 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 φ = id. Therefore j = φ i = i. 2 Chern and Thom structures on A 2.1 Ghern structure for line bundles In this section A is a homology theory. If X is a smooth variety we write 1 X for the trivial rank one bundle over X. Often we will just write 1 for 1 X if it is clear from a context what the variety X is. 2.1.1 Definition. A Chern structure on A is an assignment which associate to each smooth X, closed subset Z X and each line bundle L/X a homomorphism e Z (L) : A Z (X) A Z (X) satisfying the following properties 1. functoriality: Let ϕ : (X, U ) (X, U) be a morphism of varieties, Z = X U, Z = X U and L be a linear bundle on X. Then the following diagram commutes A Z (X ) ϕ A Z (X) e Z (ϕ L) A Z (X ) ϕ e Z (L) A Z (X). Chern homomorphisms commutes with the boundary operator from long localization sequences. e Z (L 1 ) = e Z (L 2 ) for isomorphic line bundles L 1 and L 2 ; 2. nondegeneracy: the operator (p, p e) : A (X P 1 ) A (X) A (X) is an isomorphism, where O( 1) is the tautological line bundle on P 1, e = e(q O( 1)), q : X P 1 P 1 and p : X P 1 X are the projections; 3. vanishing: e Z (1 X ) vanishes for any pair Z X. The homomorphisms e Z (L) will be called Chern homomorphisms of the line bundle L. (It will be proved below in 2.2.9 that these homomorphisms are nilpotent). if Z = X the superscript in e Z (L) will be omitted. 8

2.1.2 Lemma. Let L be a tautological line bundle O( 1) over X P 1. Then one has the following relations: e(l) 2 = e(l)e(l ) = 0; e(l ) = e(l) Hom(A (X P 1 ), A (X P 1 )). Proof. To prove the first assertion we show that for any two line bundle L 1 and L 2 over P 1 one has e(q L 2 )e(q L 1 ) = 0 where q : X P 1 P 1 is the projection. Fix two points {0}, { } P 1 and consider a commutative diagram A (X P 1 ) β A X {0} γ (X P 1 ) A X {0} (X (P 1 { })) e(q L 1 ) e X {0} (q L 1 ) e X {0} (q L 1 ) A (X P 1 ) β A X {0} (X P 1 ) γ A X {0} (X (P 1 { })) where β is a support restriction and γ is an excision isomorphism. The right vertical arrow vanishes because q L 1 X A 1 is a trivial line bundle. Therefore the middle vertical arrow vanishes. We get β e(q L 1 ) = 0. By 1.4.10 the sequence 0 A (X) s A (X P 1 ) β A X {0} (X P 1 ) 0 is exact. It implies that Im(e(q L 1 )) Im(s ). Since the line bundle s q L 2 is trivial one has e(q L 2 )e(q L 1 )(A (X P 1 )) (e(q L 2 )s )(A (X)) = (s e(s q L 2 ))(A (X)) = 0 and the first part of the lemma is proved. In the proof of the second part we assume for simplicity that X = pt. Let i 1, i 2 : P 1 P 1 P 1 be closed imbeddings, p 1, p 2 : P 1 P 1 P 1 be corresponding projections and : P 1 P 1 P 1 be a diagonal imbedding. Also let s : pt P 1 P 1 be any closed imbedding and P : P 1 P 1 pt be a projection. The projection P 1 pt will be denoted by a small letter p. We claim the following relation holds: = i 1 + i2 s p. (7) By property 2 there is an isomorphism A (P 1 P 1 ) A (pt) A (pt) A (pt) A (pt) given by (P, P e(p 1L), P e(p 2L), P e(p 1L)e(p 2L)). In order to check the relation 7 we must check four equations: P = P (i 1 + i2 s p ) P e(p 1 L) = P e(p 1 L)(i1 + i2 s p ) P e(p 2L) = P e(p 2L)(i 1 + i 2 s p ) P e(p 1 L)e(p 2 L) = P e(p 1 L)e(p 2 L)(i1 + i2 s p ). The first one is obvious. To prove the other three ones it is enough to observe that e(p r L) = e(l); e(p r L)ir = ir e(l); e(p r L)ik = 0 for k r. Then both hands in the second and third equations are equal to p e(l) and both hands in the fourth equation are zero. 9

Now we are ready to complete the proof of this lemma. Consider a line bundle M = p 1L p 2L over P 1 P 1. Then i 1M = L; i 2M = L ; M = 1 P 1. By equation (7) one has P e(m) = P e(m)(i 1 + i2 s p ). Computing left and right hand we get 0 = p (e(l) + e(l )). To prove the relation u = v Hom(A (P 1 ), A (P 1 )) it is sufficient by property 2 in Definition 2.1.1 to prove that p u = p v & p e(l)u = p e(l)v. The first one in our case (u = e(l), v = e(l )) is checked just above. In the second one both sides of the relation vanish because e(l) 2 = 0 and e(l)e(l ) = 0 by the first assertion of this Lemma. 2.1.3 Definition. If one has a Chern structure (L, X, Z) e Z (L) on a homology theory A then the assignment (L, X, Z) (e ) Z (L) = e Z (L ) will be called a dual Chern structure with respect to {e}. We must check that all properties of Chern structure are valid. The only point to prove is the nondegeneracy. It is hold because by Lemma 2.1.2 e (O( 1)) = e(o(1)) = e(o( 1). 2.1.4 Definition. One says that A is endowed with a Thom structure if for each smooth variety X, closed subset Z X and each line bundle L/X it is chosen and fixed a homomorphism th Z (L) : A Z (L) AZ (X) satisfying the following properties 1. functoriality: Let Z X and Z X be closed subsets, L/X be a line bundle over X; ϕ : X X be a morphism such that ϕ 1 (Z) Z. Then the following diagram commutes A Z th Z (L ) (L ) A Z (X ) ϕ L A Z (L) th Z (L) ϕ A Z (X), where L = L X X is a line bundle over X and ϕ L : L L is a morphism of line bundles induced by ϕ. Homomorphisms th Z (L) commutes with from a long localization sequence. 2. If τ : L 1 L 2 is an isomorphism of line bundles over X then for any closed subset Z X one has th Z (L 2 ) τ = th Z (L 1 ) Hom(A Z (L 1), A Z (X)). 3. nondegeneracy: th(1) : A X (X A 1 ) A (X) is an isomorphism ( here X is identified with X {0} ). 2.1.5 Remark. Chern and Thom homomorphisms commutes with from the Mayer- Vietoris sequence. 2.1.6 Lemma. Any homomorphism th Z (L) is an isomorphism. 10

Proof. The usual arguments using Mayer-Vietoris and long exact sequence for a triple. Now we are going to describe a one-to-one correspondence between Chern and Thom structures on A. 2.1.7 Lemma. Suppose that a homology theory theory A is equipped with a Chern structure. Then the following assertions hold 1. Let e = e(o P n( 1)) and β : A (P n ) A Pn 1 (P n ) be a support restriction. Then there exists a unique homomorphism ê : A Pn 1 (P n ) A (P n ) such that ê β = e. 2. Let L/X be a line bundle, Z X be a closed subset. Let p : P(1 L) X be a projection, s : X P(1 L) be a closed imbedding identifying X with P(1), s L : X P(1 L) be a closed imbedding identifying X with P(L) and β : A p 1 (Z) (P(1 L)) A s (Z) (P(1 L)) be the support restriction. Consider the line bundle M = O 1 L (1) p L over P(1 L). Then s M = L, s L M = 1 X; there exists a unique homomorphism α Z (L) : A s (Z) (P(1 L)) A Z (X) such that α Z (L) β = p e(m). Proof. To prove the first statement consider a long localization sequence for pair (P n, P n P n 1 ). As A (P n P n 1 ) = A (A n ) = A (pt) by homotopic invariance each third arrow in this sequence j : A (P n P n 1 ) A (P n ) is mono. Therefore the long sequence splits into short exact sequences 0 A (A n ) j A (P n ) β A Pn 1 (P n ) 0. (8) As j O P n( 1) is trivial then e j = j e(j O P n( 1)) = 0. Hence e factors through the factorgroup A Pn 1 (P n ) as required. The proof of the second statement is similar the first one. First of all we compute s M and s L M. One has s O 1 L ( 1) = 1 X and s L O 1 L( 1) = L. Therefore s M = 1 X s p L = L and s L M = L L = 1 X. Consider a long localization sequence for triple P(1 L) p 1 (Z) s (Z):... A p 1 (Z) s (Z) (P(1 L) P(1)) j A p 1 (Z) (P(1 L)) β A s (Z) (P(1 L))... where A p 1 (Z) s (Z) (P(1 L) s (Z)) = A p 1 (Z) s (Z) (P(1 L) P(1)) by the excision property. Since P(1 L) P(1) is a line bundle over P(L) (isomorphic to L ) then s L : A Z (X) (P(1 L) P(1)) and (p P(1 L) P(1) ) are inverse to each other isomorphisms. Hence j in the long sequences has splitting (s L ) 1 p and this sequence splits into short exact ones: A p 1 (Z) s (Z) 0 A Z (X) sl A p 1 (Z) (P(1 L)) β A s (Z) (P(1 L)) 0. (9) 11

Since s L M is trivial then the homomorpfism ep 1 (Z) (M) factors through β: e p 1 (Z) (M) = ê p 1 (Z) (M) β. We can set α Z (L) = p êp 1 (Z) (M). (10) 2.1.8 Remark. Notation used in the previous lemma will be used below in this section without any further explanation. 2.1.9 Lemma. Let γ : A Z (L) = As (Z) (P(1 L) P(L)) A s (Z) (P(1 L)) be the excision isomorphism. Suppose that A is endowed with a Chern structure. Let α Z (L) = α Z (L) γ where a homomorphism α Z (L) is constructed in Lemma 2.1.7. Then homomorphisms α Z (L) define a Thom structure on a homology theory A. Proof. The functiorial properties of such constructed α Z (L) are obvious consequences from the analogous properties of Chern homomorphisms. The only item we need to check is nondegeneracy. By the nondegeneracy property of Chern structure we have a commutative diagram: 0 A (X A 1 ) j A (X P 1 ) β A X {0} (P 1 ) 0 0 A (X) id i A (X P 1 ) p ζ α(1) A (X) 0 where ζ = e(o(1)) : A (X P 1 ) A (X P 1 ) is the Chern homomorphism. The top row is exact in the same way as sequence (8). The bottom row is exact by the nondegeneracy of the dual Chern structure (c.f. Definition 2.1.3). As two of vertical arrows are isomorphisms the third one is an isomorphism as well. 2.1.10 Remark. The bar-notation introduced above will be used below without any further explanation. 2.1.11 Lemma. Let {th Z (L)} be a Thom structure on a homology theory A. Define a homomorphism c Z (L) as the composition [ ] A Z s (X) A p 1 (Z) (P(1 L)) β A s (Z) (P(1 L)) thz (L) A Z (X). Then the assignment (L, X, Z) c Z (L) is a Chern structure on A. Moreover, if we begin with a Chern structure {e Z (L)} and take a Thom structure to be the one corresponding to the Chern structure {e Z (L)} by the construction from lemma 2.1.9 then for any (L, X, Z) one has c Z (L) = e Z (L). Proof. Functorial properties are obvious. 12

To proof the nondegeneracy property for a just defined Chern structure it is sufficient to prove nondegeneracy of the dual Chern structure. The last will follow from the fact that the sequence 0 A (X) A (X P 1 ) p c(o(1)) A (X) 0 is exact. For simplicity of notation we proceed only the case X = pt. Consider P 1 P 2 and a point { } P 2 P 1. Then P 2 { } is turned out to be a line bundle L over P 1 which is isomorphic to O(1). Then P(1 L) is isomorphic to the blow up P 2 { } of the P 2 at the point { }. The exceptional divisor D is identified with P(L) under this isomorphism. We will write P 2 for P 2 { } and σ : P 2 P 2 for the blowing up. By the excision property one has A P(1) (P(1 L) D) = A P(1) (P(1 L)) and A P1 (P2 { }) = A P1 (P2 ). Since σ : P(1 L) D P 2 { } is an isomorphism then the operator A P(1) (P(1 L)) σ A P1 (P 2 ) is an isomorphism. Take a projective line l on P 2 crossing the point. Let {0} = P 1 l and l P 2 be the strict transform of l. It is easy to see that l is just a fibre of projective bundle P(1 L) over a point {0}. Then closed imbedding i : l P(1 L) can be regarded as a morphism of the projective bundles l/{0} and P(1 L)/P 1. Let i 0 : P 1 P 2, τ : P 1 l be arbitrary linear embeddings. By the functoriality of the Thom isomorphisms we have a commutative diagram A (P 1 τ ) β 0 {0} id A (l) A (l) A {0} th(1) (l) A ({0}) id σ i σ i i i A (P 1 ) i 0 A (P 2 ) β A P1 (P2 ) σ 1 A P(1) (P(1 L)) th(l) A (P 1 ). The left hand side square commutes by 1.4.13. The bottom row composition map is the homomorphism c(l) by the very definition of c(l). Since the diagram commutes one has the relation p c(l) = β 0 th(1). Replacing in the short exact sequence of type 8. 0 A (pt) A (P 1 ) β A {0} (P 1 ) 0 the term A {0} (P 1 ) by the one A (pt) we get the short exact sequence 0 A (pt) A (P 1 ) th(1) β A (pt) 0 Since p c(l) = th(1) β the nondegeneracy property of the c(l) follows. It remains to prove the relation c(l) = e(l) for any line bundle L over X. 13

As in the proof of the second part of Lemma 2.1.7 consider the commutative diagram A Z (X) e Z (L) A Z (X) s p A 1 (Z) (P(1 L)) A s (Z) (P(1 L)) e p 1 (Z) (M) α Z (L) s p A 1 (Z) (P(1 L)) β p A Z (X). The right hand side square commutes by the very definition of the operator α Z (L) from Lemma 2.1.7. The left hand side square commutes by the functoriality of the Chern classes and the relation s M = L. Then c Z (L) = α Z (L) β s = p s e Z (L) = e Z (L). 2.2 Projective bundle theorem. The nearest aim is to compute the homology of a projective bundle. 2.2.1 Theorem (Projective Bundle Homology). Let A be a homology theory equipped with a Chern structure (L, X, Z) e Z (L) on A. Let X be a smooth variety and let E/X be a vector bundle with rk(e) = n + 1. For the endomorphism e = e(o E ( 1)) : A (P(E)) A (P(E)) we have an isomorphism (p, p e,..., p e n ) : A (P(E)) A (X) A (X) A (X) where p : P(E) X is a projection. Moreover, for the trivial rank n + 1 bundle E we have e n+1 = 0. In addition, all the assertions hold if the endomorphism e is replaced by the one e(o E (1)) : A (P(E)) A (P(E)). Proof. This variant of the proof is based on an unpublished notes of I.Panin. Let P n 1 P n be a hyperplane and 0 P n P n 1 be a point. Recall that the projection P n {0} P n 1 with the center 0 makes P n {0} into a line bundle over P n 1 which is isomorphic to O P n 1(1). Let us denote this bundle by L below. Projective bundle P(1 L) is isomorphic to the blow up P n {0} of the P n at the point {0}. We will write P n for the P n {0} and σ : P n P n for the blowing up. If P(1 L) is identified with P n then σ(p(1)) = P n 1 P n. Let us regard our homology theory A to be equipped with Thom structure corresponding given Chern structure by Lemma 2.1.9. We will write below in the proof e k for e(o P k( 1)). 2.2.2 Lemma. Let L 1 and L 2 be two line bundle over P(1 L), and i : P 1 P(1 L) be a closed imbedding of any fiber. Then the following three conditions are equivalent: 1. L 1 = L2 ; 2. s L 1 = s L 2 and i L 1 = i L 2 ; 14

3. s L L 1 = s L L 2 and i L 1 = i L 2. Proof. This lemma is an easy consequence of a general projective bundle Picar group computation. Recall that the sequence 0 P ic(x) p P ic(p(e)) i P ic(p n ) = Z 0 is exact for any projective bundle p : E X. Here i : P n X is any fibre of this bundle. If E = P(1 L) then s and s L be various splitting of this sequence. The lemma follows from this remark directly. 2.2.3 Lemma. Let M = O 1 L (1) p L be a line bundle over P(1 L). Then the diagram below commutes. A (P(1 L)) β A P(1) (P(1 L)) be(m ) A (P(1 L)) σ A (P n ) β cσ σ A Pn 1 (P n ) be(opn ( 1)) A (P n ) Proof. The left square commutes by natural reason. Homomorphisms β and β are taken from sequences (8) and (9). Therefore they both are onto. This implies that one has to check the relation σ ê(m) β = ê n βσ in order to prove that right square commutes. Last relation is equivalent to σ e(m ) = e n σ. It follows from the functoriality of Chern homomorphism and the fact that σ O P n( 1) = O 1 L ( 1) p L. Last equality is easy to prove using lemma 2.2.2: s L σ O P n( 1) = σ O P n( 1) { } is trivial and s L M is trivial by Lemma 2.1.7; restricting both side of the desired relation onto a fibre we see that in both cases we get the line bundle L = O P n 1( 1). So two bundles over P(1 L)) are isomorphic to each other and lemma follows. Before the general case we prove Theorem 2.2.1 for the trivial bundle E = X A n+1. For simplicity of notation we procceed only the case X = pt. In this case P(E) = P n and we proceed the proof by the induction on the integer n. By induction assumption there is an isomorphism γ n 1 = ((p n 1 ), (p n 1 ) e n 1,..., (p n 1 ) e n 1 n 1 ) : A (P n 1 ) A (pt) A (pt) and e n n 1 = 0. Let u : A (P n ) A (P n 1 ) be a composition map [ ] A (P n ) (p,p en,...,p en 1 n ) A (pt) A (pt) γ 1 n 1 A (P n 1 ). Since i e n 1 = e n i Hom(A (P n 1 ), A (P n )) where i : P n 1 P n is a linear embedding then ui = id A (P n 1 ). Under this notation if one prove that is an isomorphism then the theorem follows. (p, u e n ) : A (P n ) A (pt) A (P n 1 ) (11) 15

2.2.4 Lemma. Under induction hypothesis there is a following short exact sequence: 0 A (P n 1 ) i A (P n ) β A { } (P n ) 0. Proof. Given a point { } not lying on P n 1 consider a long exact sequence for the pair (P n, P n { }):... A (P n { }) j A (P n ) β A { } (P n )... Since P n { } can be regarded as a linear bundle on P n 1 then zero section natural map z : A (P n 1 ) A (P n { }) is an isomorphism by strong homotopic invariance 1.4.6. Moreover, since j z = i then the homomorphism A (P n ) z u A (P n { }) is the splitting for j. Hence, j is mono and the long exact sequence splits into short exact sequences: 0 A (P n { }) j A (P n ) β A { } (P n ) 0. We can write A (P n 1 ) for A (P n { }) and so we get a desired sequence. 2.2.5 Lemma. Under induction hypothesis one has: 1. The image of the operator e n : A (P n ) A (P n ) lies in the image of i : A (P n 1 ) A (P n ) and e n+1 n = 0. 2. The image of an operator e(m) : A (P(1 L)) A (P(1 L)) lies in the image of s : A (P n 1 ) A (P(1 L)) where s : P n 1 P(1 L) is a closed imbedding identifying P n 1 with P(1) as in Lemma 2.1.7. 3. The following relations hold where P : P(1 L) P n 1 is a projection. Proof. Consider a diagram s P e(m) = e(m); i ue n = e n (12) A (P n ) β A { } (P n ) j A { } (A n ) A (P n 1 ) e n i A (P n ) β e { } n j e { } n A { } (P n ) A { } (A n ) It is commutative from functoriality of Chern homomorphism. Show that the right vertical arrow vanishes. Since the bundle O P n( 1) becomes trivial being restricted to A n = P n P n 1 j P n. A { } (A n ) e{ } (j O P n ( 1)) A { } (A n ) then the right vertical arrow vanishes by property 3 in definition 2.1.1. The middle vertical arrow is null as well because j is an excision isomorphism. Therefore the diagram shows that β e n = e { } n β = 0. Since Ker(β) = Im(i ) by Lemma 2.2.4 then Im(e n ) Ker(β) = Im(i ) as required. 16

This fact implies that e n+1 n = 0. Indeed, let e(a) = i (b) for some b A (P n 1 ). Since e n n 1 = 0 then e n+1 n (a) = e n n e(a) = e n n i (b) = i e n n 1(b) = 0. In order to proof the second part of the lemma we consider the commutative diagram: A (P(1 L)) β A P(L) (P(1 L)) j A P(L) (P(1 L) P(1)) A (P(1)) i e(m) A (P(1 L)) β e P(L) (M) j e P(L) (M) A P(L) (P(1 L)) A P(L) (P(1 L) P(1)). As above j is an excision isomorphism. The projection P : P(1 L) P(1)X is a homotopic equivalence with inverse s L. Since s L M = 1 X by lemma 2.1.7 then j M = P s L M is a trivial line bundle. Hence, ep(l) (M) = 0. From exact sequence (9) one has Ker(β) = Im(s ). Since β e(m) = e P(L) (M) β = 0 then Im(e(M)) Ker(β) = Im(s ). 2.2.6 Remark. One can write M for M and write ê(m ) for e(m) in the second and the third assertion of the lemma because their images coincide. We are ready now to complete the proof of Projective Bundle Theorem 2.2.1. Denote by α the Thom isomorphism th (L) in the dual Thom structure, i.e. α = P ê(m ) = P ê(o 1 L ( 1) p L ) Hom(A P(1) (P(1 L)), A (P n 1 )) (see formula (10)). Here hat-notation is borrowed from Lemma 2.1.7. By the short exact sequence (8) 0 A (A n ) j A (P n ) β A Pn 1 (P n ) 0 one has an isomorphism A (P n ) (p, β ) A (pt) A Pn 1 (P n ). The isomorphism of the second summand A Pn 1 (P n ) with A (P n 1 ) is given by formula α σ 1 where σ : A P(1) (P(1 L)) A Pn 1 (P n ) is an isomorphism. By equation (11) it is sufficient to prove that α σ 1 β = u e n. Applying i to both sides of desired equality one has to prove i α σ 1 β = i u e n. (13) Recall that α = P ê(m ). Therefore i α = σ s P ê(m ) = σ ê(m ) by Lemma 2.2.5. Since σ ê(m ) σ 1 = ê n by Lemma 2.2.3 then the left hand of equality (13) is equal to ê n β = e n. But the right hand of this equality is equal to e n by relation (12). Now we have finished the proof for the case of the trivial vector bundle E. The general case can be proceeded in the usual way by the Mayer-Vietoris argument. 17

2.2.7 Lemma. Let Z be a closed subset in X, L = q O P n( 1) be a line bundle over X P n. Then there is an isomorphism ( p, p e Z Pn (L),..., p (e Z Pn (L)) n) : A Z Pn (X P n ) A Z (X) A Z (X). Proof. The proof is a straightforward consequence from Theorem 2.2.1, the long localization sequence for the pair (X P n, (X Z) P n ) and five lemma. 2.2.8 Corollary. Let E = M N be vector bundles of constant rank. Then there is an exact sequence 0 A (P(N)) i A (P(E)) β A P(M) (P(E)) 0. Proof. Consider a long exact sequence for the pair (P(E), P(E) P(M)). As P(E) P(M) can be regarded as vector bundle over P(N) we may replace A (P(E) P(M)) by A (P(N)): A (P(N)) i A (P(E)) β A P(M) (P(E))... By projective bundle theorem i : A (P(N)) A (P(E)) has an obvious splitting. Indeed, let rkn = n, ξ E = e(o E ( 1)) and ξ N = e(o N ( 1)). Since i ξ N = ξ E i then the homomorphism (p E, p E ξ E,..., p E ξ n 1 E ) : A (P(E)) A (pt) A (pt) = A (P(N)) is a desired splitting. 2.2.9 Lemma. For each line bundle L over a smooth X and closed subset Z X the endomorphism e Z (L) of A Z (X) is nilpotent. Proof. Recall that the assertion holds in the case L = O P n(1) by Lemma 2.2.5 applied to the dual Chern structure. To prove the general case recall that by Claim 3.4.4 from [Pa] one can find for any smooth variety X a diagram of the form X p X f P(V ) (14) 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 (O V (1)) over X are isomorphic. By the strong homotopy invariance we can replace X by X and regard that L = f O V (1). The homomorphism e(o V (1)) A(P(V )) is nilpotent as just mentioned above. Thus the homomorphism e X = e(o X P n(1)) is nilpotent as well. (id, f) Let g : X X P n. Then L = g O X P n(1) and the following diagram commutes: 18

A (X) g A (X P n ) e(l) A (X) e X g A (X P n ). Since pg = id X the natural homomorphism g is injective. Therefore it is sufficient to check that g e(l) n+1 = 0. As g e(l) n+1 = e n+1 X g = 0 the Lemma is proved. 2.2.10 Remark. Define a group A Z P (X P ) as an injective limit of the groups A Z Pn (X P n ). Then by Remark to Lemma 2.2.5 and Lemma 2.2.7 one has that e Z Pn n (A Z Pn (X P n )) = i n 1,n (A Z Pn 1 (X P n 1 )) where e n = e(o X P n( 1)) and i n 1,n : X P n 1 X P n is a linear imbedding. Particularly e Z P (q L) is a surjection where L is a tautological line bundle over P and q : X P P is a projection. 2.2.11 Lemma. Let A be a homology theory equipped with the Thom structure {th Z (L)} and let {c Z (L)} be the Chern structure corresponding to this Thom structure as define in Lemma 2.1.11. Let us take the Thom structure {α Z (L)} corresponding to the Chern structure {c Z (L)} by the construction from lemma 2.1.9. Then for any (L, X, Z) one has α Z (L) = th Z (L). Proof. In the proof of the lemma we will use the notation from lemma 2.1.7. Since α Z (L) is defined by formula (10) one has to check a relation p ĉp 1 (Z) (M) = th Z (L) (15) First of all to prove a relation α Z (L) = th Z (L) we assume the homomorphism c Z (L) to be a surjection. Consider a diagram: A Z (X) c Z (L) A Z (X). A s (Z) β s id (P(1 L)) thz (L) A Z (X) where β : A p 1 (Z) (P(1 L)) A s (Z) (P(1 L)) is a support restriction. The diagram commutes by the very construction of the homomorphism c Z (L) from Lemma 2.1.11. Since c Z (L) is a surjection by assumption and th Z (L) is an isomorphism then β s is a surjection. Therefore equality (15) follows from p ĉp 1 (Z) (M) β s = th Z (L) β s. Since ĉ p 1 (Z) (M)β = c p 1 (Z) (M) and s M = L by Lemma 2.1.7 then the left hand is equal to p c p 1 (Z) (M)s = p s c Z (s M) = c Z (L). Right hand is equal to c Z (L) from the diagram above. Lemma 2.2.10 and following remark imply that the lemma holds in the (universal) case: for the line bundle L = O( 1) over X P. 19

The general case can be be reduced to the universal one as follows. By Jouanolou trick we can find a diagram of the form (14). Therefore one can regard X to be an affine variety. Let f : X P be a morphism such that L = f O P ( 1). Set g : X (id,f) X P. Denoting O X P ( 1) by L one has L = g L. Let G : P(1 L) P(1 L ) be an induced morphism of projective bundles. For a closed subset Z X denote Z P X P by Z. Recall that M = O 1 L ( 1) p L is a line bundle over P(1 L) and M = O 1 L ( 1) P L is a line bundle over P(1 L ). Then one has a commutative diagram: A p 1 (Z) (P(1 L)) thz (L) β A Z (X) s A p 1 (Z) (P(1 L)) p A Z (X). G A P 1 (Z ) (P(1 L )) thz (L ) β A Z g (X P ) G S A P 1 (Z ) (P(1 L )) g P A Z (X P ) One has to check the relation th Z (L) β = p c p 1 (Z) (M). Since g is mono then it is sufficient to check a relation g th Z (L) β = g p c p 1 (Z) (M). The right hand is equal to P G c p 1 (Z) (M) = P c P 1 (Z ) (M )G because G M = M. Since for the universal case the lemma holds then P c P 1 (Z ) (M ) = th Z (L ) β. Therefore the right hand is equal to th Z (L ) β G. Since the diagram commutes the last is equal to the left hand. The lemma follows. 2.2.12 Definition. The Chern structure on a homology theory A is said to be COM- MUTATIVE if for any smooth variety X, closed subset Z X and line bundles L 1 /X and L 2 /X the Chern homomorphisms e Z (L 1 ) and e Z (L 2 ) commute with each other. 2.3 Chern classes 2.3.1 Definition. Let A be a homology theory. We say that A is endowed with a Chern class theory if for any vector bundle E/X and any closed subset Z X there are given homomorphisms c Z i (E) : A Z (X) A Z (X) such that 1. c Z i depends only on an isomorphism class of E; For any morphism f : X X and vector bundle E/X the following diagram is commutative: A (X ) cz i (E ) A (X ) f f A (X) cz i (E) A (X) where E = E X X is a pullback vector bundle over X ; Z X is a closed subset and Z = f 1 (Z); 20

2. c Z 0 (E) = id A Z (X); the restriction of the assignment (L, X, Z) c Z 1 (L) to line bundles is a Chern structure on A ; 3. Any two homomorphisms c Z i (E) and cz j (F ) commute with each other; 4. Cartan formula: For each short exact sequence of vector bundles 0 E 1 E E 2 0 we have c Z r (E) = cz r (E 1)c Z 0 (E 2) + c Z r 1 (E 1)c Z 1 (E 2) +... + c Z 0 (E 1)c Z r (E 2); 5. Vanishing property: c Z i (E) = 0 for i > rke. 2.3.2 Theorem. Let A be endowed with a commutative Chern structure (L, X, Z) e Z (L). Then there exists a unique Chern class theory on A such that for each line bundle L one has c Z 1 (L) = e Z (L). Moreover the Chern class homomorphisms c Z i (E) are nilpotent for i > 0. Proof. For simplicity of notation we proceed only the absolute case. First of all we prove the uniqueness assertion. If there are two assignements E/X c i(e) and E/X c i (E) satisfying the required properties. Then they coincide on line bundles by the properties 2 and 5. Therefore they coincide on direct sums of line bundles by the Cartan formula (4). Thus they coincide on all vector bundles by the splitting principle [PY, Prop.3.5]. It remains to construct a Chern classes theory. Let X be a smooth variety and E/X be a vector bundle with rke = n. Set ξ = e(o E ( 1)) Hom(A (P(E), A (P(E))). Let γ = (p, p ξ,..., p ξ n 1 ) : A (P(E)) A (X) is an isomorphism from n Theorem 2.2.1. Then the homomorphism p ξ n γ : A (X) A (X) can be written uniquely as a row (( 1) n 1 c n, ( 1) n 2 c n 1,..., c 1 ) where c i End(A (X)). By definition the homomorphism c k : A (X) A (X) is an k-th Chern class homomorphism for the vector bundle E. One can say the same in a slightly different form: c 1,..., c n are uniquely determined homomorphism such that p ξ n = c 1 p ξ n 1 c 2 p ξ n 2 + + ( 1) n 1 c n p. (16) In order to define Chern class homomorphism with supports one has to apply Projective Bundle Theorem with supports 2.2.7. 2.3.3 Claim. Homomorphisms c i (E) satisfy the theorem. The rest of the proof is devoted to the proof of this Claim. The property c 0 (E) = 1 holds by the very definition. To prove the property c 1 (L) = e(l) for a line bundle L observe that P(L) = X and O L ( 1) = L over X. Thus ξ = e(l) Hom(A (X), A (X)) and the relation (16) shows that c 1 (L) = e(l). Now prove the property 1 of Definition 2.3.1. A vector bundle isomorphism ϕ : E E induces an isomorphism Φ : P(E) P(E ) of the projective bundles and a line bundle isomorphism Φ (O E ( 1)) O E ( 1) over P(E). Therefore Φ ξ = ξ Φ ; p = p Φ. (17) 21

By the formula (16) one has equation p ξ n = n ( 1) k 1 c k (E) p ξ n k k=1. Using relations (17) one can derive. p (ξ ) n Φ = n ( 1) k 1 c k (E) p (ξ ) n k Φ k=1 Since Φ is an isomorphism then one gets p (ξ ) n = n ( 1) k 1 c k (E) p (ξ ) n k. Therefore by very definition of Chern classes given by (16) c k (E ) = c k (E). The proof of the other funktoriality assertion is essentialy the same. Let f : X X be a morfism, E/X be a vector bundle and E = f E be the induced vector bundle over X. Denote by F : P(E ) P(E). The following relations hold: f p = p F ; ξ F = F ξ ; f p = p F. (18) One need to show that f c k (E ) = c k (E) f. One has the following chain of relations f p (ξ ) n = p p ξ n F = ( = k=1 n ( 1) k c k (E) p ξ n k )F = k=1 n ( 1) k c k (E)f p (ξ ) n k. k=1 From the one hand the homomorphism f p (ξ ) n (γ ) 1 : n : A (X ) A (X) can be given by the row (( 1) n f c n (E ), ( 1) n 1 f c n 1 (E ),..., f c 1 (E )). For the rest of the proof we need in the following 2.3.4 Claim. For a rank r vector bundle E set c t (E) = id + c 1 (E)t + + c r (E)t r End(A (X))[t]. Let T be a smooth variety and let E = r i=1 L i for certain line bundles L i over T. Then one has r c t (E) = c t (L i ). Proof. Let ξ E = e(o E ( 1)) where O E ( 1) is the tautological line bundle on P(E). First of all we shall prove the relation i=1 r (ξ e(p E L i)) = 0 (19) i=1 where p E : P(E) T is a projection. Since all first Chern class homomorphisms commute the formula above has sense. 22

To prove the very last relation set F = L 1 L r 1. The canonical projection P(E) P(F ) P(L r ) makes P(E) P(F ) into a vector bundle over X = P(L r ) with the zero section P(L r ). Therefore the natural mapping A (P(L r )) A (P(E) P(F )) is an isomorphism. Let j : P(E) P(F ) P(E) be an open inclusion. Since O E ( 1) P(Lr) = L r then the bundles j O E ( 1) and j p E L r coincide. One has a commutative diagram 0 A (P(F )) i A (P(E)) β A P(Lr) (P(E)) 0 0 A (P(F )) ξ F e(p F Lr) i A (P(E)) ξ E e(p E Lr) ξ P(Lr) E e P(Lr) (p E Lr) β A P(Lr) (P(E)) 0 where strings are the short exact sequences of type 2.2.8. The right vertical arrow is null by following reasons. A P(Lr) (P(E)) = A P(Lr) (P(E) P(F )) by exscision property; but the bundles O( 1) and p L r coincide being restricted to P(E) P(F ). So we get that the image of ξ E e(p E L r) lies in the image of i. Since ( r 1 i=1 ξ F e(p F L i)) = 0 by induction hypothesis then r 1 (ξ E e(p EL i ))) i = i (ξ F e(p F L i )) = 0 r 1 ( i=1 and the relation (19) is proved. We need to check that c k (E) is just a symmetric polynomial σ k (e(l 1 ),..., e(l r )). Expanding brackets in equation (19) one has ξ n σ 1 ξn 1 + + ( 1) n σ n = 0 where σ k = σ k(e(p L 1 ),..., e(p L r )). Since p σ k = σ kp then p ξ n = σ 1 p ξ n 1 σ 2 p ξ n 2 + + ( 1) n+1 p. Comparing last formula with the formula (16) one completes the proof. i=1 Claim 2.3.4 and splitting principle [PY, Prop.3.5] allow us to complete the proof of Claim 2.3.3. Let E and F be a vector bundles over X. Prove that the homomorfhisms c k (E) and c l (F ) commute. Indeed, by the splitting principle [PY, Prop.3.5] there exists a smooth variety T and a morphism r : T X such that each the vector bundle r E and r F is a sum of line bundles and r : A (T ) A (X) is a split surjection. The homomorphisms c k (r E) and c l (r F ) commute as symmetric polynomes of line bundles chern classes, which are commute because Chern structure is commutative. Then c k (E)c l (F )r = r c k (r E)c l (r F ) = r c l (r F )c k (r E) = c l (F )c k (E)r. Last implies commutativity of higher chern classes because r is onto. Cartan formula and nilpotence are easily can be proved essentially in the same way. Below we will need Proposition 2.3.5 Proposition. Namely, let Y be a smooth variety and let E be a vector bundles over Y of constant rank n. Let p : P(E) Y and q : P(1 E) Y be the projections and s : Y P(1 E) be the closed imbedding identifying Y with P(1). Then 23

1. the homomorphism c n (O E (1) p E) is null; 2. s q c n (O E (1) q E) = c n (O E (1) q E). Proof. Define a rank n 1 vector bundle Q via the short exact sequence 0 O E ( 1) p E Q 0. Tensoring with line bundle O E (1) and applying Cartan formula we get c n (O E (1) p E) = c 1 (O)c n 1 (Q) = 0. The second part of this proposition is a straightforward extension of the third assertion of Lemma 2.2.5. 2.4 Orienting a theory In this subsection A is a homology theory. Two theorems in this subsection shows how one can construct an orientation using a commutative Chern structure (or a Thom structure) on A and how one can construct a commutative Chern structure (or a Thom structure) using an orientation. 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. 2.4.1 Definition. 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 isomorphism which satisfies the following properties th Z (E) : A Z (E) A Z (X) 1. invariance: for each vector bundle isomorphism ϕ : E F the diagram commutes A Z (E) ϕ th Z (E) A Z (X) id A Z (F ) th Z (F ) A Z (X) 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 pull-back E over X and for the projection g : E = E X X E the diagram commutes A Z (E ) g th Z (E ) A Z (X ) f A Z (E) th Z (E) A Z (X) 24

3. for each vector bundles p : E X and q : F X the following diagram commutes A Z (E F ) th Z (p F ) A Z (E) th Z (q E) th Z (E) A Z (F ) th Z (F ) A Z (X) and both compositions coincide with the operator th Z (E F ). The operators th Z (E) are called Thom isomorphisms. The theory A is called orientable if there exists an orientation of A. The theory A is called oriented if an orientation is chosen and fixed. 2.4.2 Lemma. If an assignment (E, X, Z) th Z (E) is an orientation on A, then its restriction to line bundles is a Thom structure on A. If two orientations coincide on each line bundle then they coincide. Proof. The first assertion is obvious. To prove the second assertion consider two orientations th( ) and th ( ) which coincide on line bundles. To prove that for a vector bundle E one has the relation th Z (E) = (th ) Z (E) one may assume by the splitting principle [PY, Prop.3.5] that E = L i is a direct sum of line bundles. Since for each line bundle L one has th Z (L) = (th ) Z (L) the property 3 in Definition 2.4.1 shows that th Z (E) = (th ) Z (E). 2.4.3 Theorem. Given a commutative Chern structure (L, X, Z) e Z (L) on A and corresponding Thom structure (L, X, Z) α Z (L) there exists an orientation (X, Z, E) th Z (E) on A such that the following properties hold 1. for each smooth variety X, each line bundle L/X and each closed subset Z X one has α Z (L) = th Z (L); 2. for each smooth X, closed subset Z X and each line bundle L/X one has e Z (L) = th Z (L) z where z : (X, X Z) (L, L Z) is the zero section. Moreover the required orientation is uniquely determined both by the property 1 and by the property 2. 2.4.4 Theorem. If (X, Z, E) th Z (E) is an orientation on A then the assignment (L, X, Z) th Z (L) z is a COMMUTATIVE Chern structure on A, the assignment (L, X, Z) th Z (L) is 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 2.4.3 and the construction of a Chern and a Thom structure by means of an orientation are inverse of each other. Proof of Theorem 2.4.3. Let E/X be a rank n vector bundle, Z X be a closed subset and let F = E 1. Recall that the sequence 2.2.8 is exact: 0 A p 1 E (Z) (P(E)) i A p 1 F (Z) (P(F )) β A s (Z) (P(F )) 0. 25