POINCARÉ DUALITY FOR PROJECTIVE VARIETIES

IVAN PANIN SERGE YAGUNOV

In 1895 in his first topological memoir “Analysis Situs” [P] Henri Poincaré established a foundation of Algebraic Topology. Besides defining conceptions of homology and cohomology, he formulated and proved that Betti numbers of a (compact, orientable) manifold are symmetric. This result, together with a bunch of its refinements and consequences, is now called the Poincaré Duality. In a fancier language the theorem of Poincaré may be rewritten as follows.

Let us consider a complex oriented cohomology theory represented by a ringspectrum E. Then, for every complex projective algebraic variety X one defines a fundamental class [X] ∈ E_{2n}(X) (here n stays for the complex dimension of X). The cap-product ⌢[X] : E^{∗}(X) → E_{2n−∗}(X) provides an isomorphism of cohomology and homology groups of X and called the Poincaré Duality isomorphism.

From the modern point of view it looks pretty interesting to obtain an analogue of this result in the context of Algebraic Geometry. In the current preprint we consider the category of smooth varieties Sm and two functors on it (contravariant and covariant) A^{∗} and A_{∗} with values in ℤ/2-graded abelian groups. We postulate certain properties of these functors and derive the Poincaré Duality. We consider only theories which take values in ℤ/2-graded abelian groups, however all the arguments work for ℤ-grading as well.

In the first section we define a notion of pseudo-representable theory which mimics the pair (E^{∗}, E_{∗}) of theories represented by a spectrum E. In order to state the main result we need a whole bunch of products (two cross-products, two slant products, and two inner products) substituting ones appearing from a spectrum multiplication μ: E ∧ E → E. For this need we introduce a technical notion of a “multiplicative pair” which encodes all necessary structures. We also need a good replacement of the notion of orientability. For this end we axiomatize transfer structures in our theory which associate to every projective morphism f: Y → X two maps f_{!} : A^{∗}(Y) → A^{∗}(X) and f^{!} : A_{∗}(X) → A_{∗}(Y) satisfying a natural list of properties and compatible in certain sense (see Axiom B.5). It may be shown that the existence of these structures is related to the “nice” theory of characteristic classes for our functors (see [PS], where the transfer structure called “integration”).

Having all these data in hands, we define a fundamental class of a projective variety X as [X] =^{def} π^{!}(1) ∈ A_{0}(X), where π: X → pt is the structure morphism. Then, our main result claims that the map

⌢ [X]: A^{∗}(X) →^{≃} A_{∗}(X)

is the grade-preserving isomorphism.

In fact, our result holds for a rather wide class of theories represented by oriented multiplicative T -spectra in the sense of Voevodsky [Vo]. This includes, for example, Motivic Cohomology and Algebraic Cobordism (we should indicate here that the transfer structure for homology theories represented by an oriented T -spectra is still under construction). As a further refinement we can expect, for example, that the duality given in [FV] might be seen as a case of Poincaré Duality.

The question raised in the paper was solved during the work of the second author at the Universität Gesamthohschule Essen and the current text was written during his short-time visit to the Universität Bielefeld. The second author is very grateful to H.Esnault, S.Müller–Stach, E.Viehweg, and U.Rehmann for shown hospitality and excellent working possibilities during his visits.

The first author is in debt to A.Merkur’ev for inspirating discussions at the initial stage of the work.

Both authors are grateful to TMR ERB FMRX CT-97-0107, INTAS, and RFFR grant networks for particular financial support during their work.

The first author thanks very much for the particular support for the years 2001–2 from the ” Support Fund of National Science” of the Russian Academy of Sciences. Notation. Throughout the paper we use greek letters to denote elements of

cohomology groups and latin for homological ones; Δ always denotes a diagonal morphism; ℙ(V) = P roj(S^{∗}(V^{∨})) is the space of lines in a finite k-vector space V; Symbol 1 denotes trivial bundle; ℙ(E) = P roj(S^{∗}(E^{∨})) is the space of lines in a vector bundle E; O_{E}(−1) is the tautological line bundle on ℙ(E); O_{E}(1) is the line bundle on ℙ(E) dual to O_{E}(−1); We reserve letter z for the zero-section X →^{z} E and s for the section X →^{s} ℙ(1 ⊕ E) induced by the zero-section of E. Both the sections z and s are called below the zero sections; pt = Spec k; For the convenience of perception we usually move indexes up and down oppositely to the predefined positions of ∗ or !.

1. Initial data

Consider a category Sm/k of smooth algebraic varieties over a field k. Let A^{∗} and A_{∗} be functors (cohomology and homology pretheories) A^{∗}: (Sm/k)^{op} → ℤ/2-Ab and A_{∗}: Sm/k → ℤ/2-Ab taking their values in the category of ℤ/2-graded abelian groups.

Definition 1.1. Let functors A^{∗} and A_{∗} (cotravariant and covariant, respectively), be endowed with a product structure consisting of two cross-products

\underline{×}: A_{p}(X) ⊗ A_{q}(Y) → A_{p+q}(X × Y)
\overline{×}: A^{p}(X) ⊗ A^{q}(Y) → A^{p+q}(X × Y)

and two slant-products

/: A^{p}(X × Y) ⊗ A_{q}(Y) → A^{p−q}(X)
\: A^{p}(X) ⊗ A_{q}(X × Y) → A_{q−p}(Y).

Define two inner products

⌣: A^{p}(X) ⊗ A^{q}(X) → A^{p+q}(X)
⌢: A^{p}(X) ⊗ A_{q}(X) → A_{q−p}(X),

as α ⌣ β = Δ^{∗}(α\overline{×}β) and α ⌢ a = α\Δ_{∗}(a). We say that functors A^{∗} and A_{∗} make a multiplicative pair (A^{∗}, A_{∗}) if the mentioned products satisfy the following five axioms.

(A.1) The cup-product makes the group A^{∗}(X) an associative skew-commutative

ℤ/2-graded unitary ring and this structure is functorial. (A.2) The cap-product makes the group A_{∗}(X) a unital A^{∗}(X)-module (we have

1 ⌢ a = a for every a ∈ A_{∗}(X)) and this structure is functorial in the sense that α ⌢ f_{∗}(a) = f_{∗}(f^{∗}(α) ⌢ a).

(A.3) Associativity relations. For α ∈ A^{∗}(X × Y), β ∈ A^{∗}(Y), γ ∈ A^{∗}(X),

a ∈ A_{∗}(Y), and b ∈ A_{∗}(X), we have: (i) α/(β ⌢ a) = (α ⌣ p_{Y}^{∗} (β))/a (ii) γ ⌣ (α/a) = (p_{X}^{∗}(γ) ⌣ α)/a (iii) (α/a) ⌢ b = p_{∗}^{X}((α ⌢ (a\underline{×}b)), where morphisms p_{X} and p_{Y} are corresponding projections.

(A.4) Functoriality for slant-product: For morphisms f: X → X′, g: Y → Y′, and elements α ∈ A^{∗}(X′ × Y′) and a ∈ A_{∗}(Y), one has: (f × g)^{∗}(α)/a =

f^{∗}(α/g_{∗}(a)).

(A.5) In the homology group of the final object pt we are given an element [pt] ∈

A_{0}(pt) such that for every α ∈ A^{∗}(pt), one has: α/[pt] = α. One can easily verify that the latter property implies the existance of the iso-

morphism ω: A^{∗}(pt) →^{≃} A_{∗}(pt) given as ω(α) = α ⌢ [pt]. Throughout the paper we implicitly use this identification and usually denote [pt] by 1, taking into account that ω(1) = [pt].

Remark 1.2. In that follows we mostly make a deal with ⌢, ⌣, /, and \underline{×}-products. As well as in topology, it is easy to recover the rest of the structure from these four products (see [Ad, Sw]).

To make more clear the origins of the notion of multiplicative pair, consider a simple example.

Example 1.3. Let E be a ring-spectrum and E^{∗} , E_{∗} denote (co-)homology theories built by E. Then, these functors make a multiplicative pair E = (E^{∗}, E_{∗}) such that all the products are corresponding topological ones. The proof consists of direct checking all the properties and may be found in any book on Algeraic Topology (see, for example, [Ad, Sw]).

Definition 1.4. A pseudo-representable theory A is a multiplicative pair (A^{∗}, A_{∗}) of functors which are defined at the begining of this section and both A^{∗} and A_{∗} are additive and strongly homotopy invariant. Namely, we require:

Additivity: Let i_{r}: X_{r} ↪ X_{1} ⨿ X_{2} be natural inclusions (r = 1, 2). Then the induced maps A^{∗}(X_{1} ⨿ X_{2}) → A^{∗}(X_{1}) ⊕ A^{∗}(X_{2}) and A_{∗}(X_{1}) ⊕ A_{∗}(X_{2}) → A_{∗}(X_{1} ⨿ X_{2}) are isomorphisms.

Strong Homotopy Invariance: For any variety X, a vector bundle E over X, and an E-torsor 𝔈 with the projection p: 𝔈 → X, the natural maps p^{∗}: A^{∗}(X) → A^{∗}(𝔈) and p_{∗}: A_{∗}(𝔈) → A_{∗}(X) are isomorphisms.

Remark 1.5. In the spirit of [Pa, Definition 1.1.1] this makes the functor A^{∗} a cohomology pretheory.

Definition 1.6. We say that a pseudo-representable theory A is oriented if for every projective morphism f: Y → X there are given maps f_{!} : A^{∗}(Y) → A^{∗}(X) and f^{!} : A_{∗}(X) → A_{∗}(Y) such that a triple (A^{∗}, ⌣, f ↦ f_{!}) is an oriented cohomology pretheory (see [Pa, Definition 1.1.7]) and the family of maps f^{!} satisfies the followimg list of properties: (B.1) Functoriality. We have: (f ∘ g)^{!} = g^{!} ∘ f^{!} and id_{X}^{!} = id. (B.2) Base-Change property for transversal squares. For any transversal square

(see [Pa, 1.1.2] or [PY, 1.1] for definitions)

\xymatrix{Y′ \ar@{^{(}->}[r]^{f̄} \ar[d]_{ḡ} & X′ \ar[d]^{g} \\ Y \ar@{^{(}->}[r]^{f} & X}

the diagram

\xymatrix{A_{∗}(Y′) \ar[d]_{ḡ_{∗}} & A_{∗}(X′) \ar[l]_{f̄^{!}} \ar[d]_{g_{∗}} \\ A_{∗}(Y) & A_{∗}(X) \ar[l]_{f^{!}}}

commutes. For an arbitrary morphism f: Y → X we have a commutative diagram

\xymatrix{A_{∗}(Y × ℙ^{n}) \ar[r]^{(f×id)_{∗}} & A_{∗}(X × ℙ^{n}) \\ A_{∗}(Y) \ar[u]_{p_{Y}^{!}} \ar[r]_{f_{∗}} & A_{∗}(X). \ar[u]_{p_{X}^{!}}}

(B.3) Gysin exact sequence (localization) Let i: Y ↪ X be a closed embedding and j: X − Y ↪ X the corresponding open inclusion. Then, there is an exact sequence

A_{∗}(X − Y) →^{j_{∗}} A_{∗}(X) →^{i^{!}} A_{∗}(Y).

Before formulating the next axiom we should introduce the notion of an Euler class. For a line bundle L over X we set e(L) =^{def} z^{∗}z_{!}(1), where z : X → L is the zero-section (see [Pa, 1.1.4] for details). (B.4) Projective Bundle Theorem (PBT). For X ∈ Sm/k and a rank n vector

bundle E →^{p} X over X set ζ = e(O_{E}(1)) ∈ A^{∗}(ℙ(E)). Then, the map

⊕_{i=0}^{n−1}ψ_{i}: A_{∗}(ℙ(E)) →^{≃} ⨁_{i=0}^{r−1} A_{∗}(X).

where ψ_{i} = p_{∗} ∘ (ζ^{⌣i} ⌢ −), is an isomorphism.

Finally, we require both transfer structures to be consistent in the following way: (B.5) Let L be a line bundle over X. Then, for the zero-section s: X → ℙ(1 ⊕ L)

and every a ∈ A_{∗}(ℙ(1 ⊕ L)), we have: s_{∗}s^{!}(a) = s_{!}(1) ⌢ a.

Corollary 1.7. Let u_{1} and u_{2} be natural embeddings of varieties Y_{1} and Y_{2} to Y = Y_{1} ⨿ Y_{2}. For a closed embedding i: Y ↪ X set i_{r} = i ∘ u_{r} : Y_{r} ↪ X. Then,

i^{!} = u_{∗}^{1} ∘ i_{1}^{!} + u_{∗}^{2} ∘ i_{2}^{!} : A_{∗}(X) → A_{∗}(Y).

We do not prove this Corollary in this preprint but rather add it to the list of axioms for oriented pseudo-representable theory. However, similarly to the cohomological case, it can be derived from the other axioms (the proof is parallel to [Pa, 1.7.1]).

Remark 1.8. By the definition the triple (A^{∗}, ⌣, f ↦ f_{!}) is an oriented cohomology pretheory (see [Pa, Definition 1.1.7]). Thus, for a projective morphism f : Y → X the map f_{!} : A^{∗}(Y) → A^{∗}(X) is a two-side A^{∗}(X)-module homomorphism, i.e.

f_{!}(f^{∗}(α) ⌣ β) = α ⌣ f_{!}(β) (1.1)

f_{!}(α ⌣ f^{∗}(β)) = f_{!}(α) ⌣ β.

Remark 1.9. The reader could see that axioms B.1–B.4. are in obvious sense dual to the axioms for oriented cohomology pretheory given in [Pa].

Remark 1.10. Axiom B.5 may be replaced by a more natural one which gives an equivalent condition. For the zero-section z: X → L we require the relation z^{!} ∘ z_{∗} = e(L) ⌢: A_{∗}(X) → A_{∗}(X) to be held.

To complete this section we formulate two simple consequences of the listed axioms which we need below.

Lemma 1.11. For p: X × Y → Y and α ∈ A^{∗}(Y), we have: p^{∗}(α) = 1 × α.

Lemma 1.12. Let L be a line bundle over X. Then, for the zero-section s: X → ℙ(1 ⊕ L) and α ∈ A^{∗}(L) one has: s_{!}s^{∗}(α) = s_{!}(1) ⌣ α.

2. Poincaré Duality Theorem

Definition 2.1. Let A be an oriented pseudo-representable theory and X ∈ Sm/k projective variety with structure morphism π: X → pt. Then, we call an element π^{!}(1) ∈ A_{0}(X) the fundamental class of X and denote it by [X].

Theorem 2.2 (Poincaré Duality). Let A be an oriented pseudo-representable theory. For every projective X ∈ Sm/k, denote by D^{•}: A^{∗}(X) → A_{∗}(X) the map D^{•}(α) = α ⌢ [X] and by D_{•}: A_{∗}(X) → A^{∗}(X) the map D_{•}(a) = Δ_{!}(1)/a, where Δ: X → X ×X is the diagonal morphism. Then, the maps D^{•} and D_{•} are mutually inverse isomorphisms.

One can extract the following nice consequence of the Poincaré Duality theorem, which enables us to interpret transfer maps in a way topologists like to.

Corollary 2.3. For projective X, Y ∈ Sm/k and a morphism f: X → Y, one has:

f_{!} = D_{•}^{Y} f_{∗}D_{X}^{•}
f^{!} = D_{X}^{•}f^{∗}D_{•}^{Y} ,

where D_{X} and D_{Y} are introduced above duality operators for varieties X and Y, respectively.

The proof of Theorem 2.2 is based on two projection formulae for cap- and slantproducts. In this section we formulate these assertions and derive the theorem from them. The rest of the paper is devoted to the proofs of the assertions.

Theorem 2.4. For X, Y ∈ Sm/k, an arbitrary projective morphism f : Y → X, and any elements α ∈ A^{∗}(Y) and a ∈ A_{∗}(X), the relation

(2.1)  f_{∗}(α ⌢ f^{!}(a)) = f_{!}(α) ⌢ a

holds in the group A_{∗}(X).

We need several corollaries of this theorem.

Corollary 2.5. Let τ : X × X → X × X be the transposition morphism. Then, for any elements α ∈ A^{∗}(X), β ∈ A^{∗}(X × X), and a ∈ A_{∗}(X × X), we have:

a) Δ_{!}(α) ⌢ a = Δ_{!}(α) ⌢ τ_{∗}(a)
b) Δ_{!}(α) ⌣ β = Δ_{!}(α) ⌣ τ^{∗}(β)

in A_{∗}(X × X) (A^{∗}(X × X), respectively).

Proof. First of all, we show that Δ^{!}∘τ_{∗} = Δ^{!}. In fact, this follows from transversality of the diagram

(2.2)  \xymatrix{X \ar[r]^{Δ} \ar[d]_{id} & X × X \ar[d]^{τ} \\ X \ar[r]^{Δ} & X × X.}

(Since the map τ is flat we may apply the transversality criterion given in [Fu].) The base change property for transfers in A_{∗} tells us that Δ^{!} ∘ τ_{∗} = id_{∗} ∘Δ^{!}. Since id_{∗} = id, one gets the desired result.

To complete the proof of a), it remains to observe that by Theorem 2.4 one has the chain of relations

Δ_{!}(α) ⌢ a = Δ_{∗}(α ⌢ Δ^{!}(a)) = Δ_{∗}(α ⌢ Δ^{!}(τ_{∗}(a))) = Δ_{!}(α) ⌢ τ_{∗}(a).

The proof of case b) goes similarly, using cohomological projection formula (1.1):

Δ_{!}(α) ⌣ β = Δ_{!}(α ⌣ Δ^{∗}(β)) = Δ_{!}(α ⌣ Δ^{∗}(τ^{∗}(β))) = Δ_{!}(α) ⌣ τ^{∗}(β).

□

Corollary 2.6. For every projective X ∈ Sm/k and its structure morphism π: X → pt, one has the following relation in A(pt):

(2.3)  π_{∗}([X]) = π_{!}(1).

Proof. Apply Theorem 2.4 to the morphism π in the case: α = 1 ∈ A^{∗}(X) and a = 1 ∈ A_{∗}(pt) = A^{∗}(pt). □

Corollary 2.7. Let V be a vector bundle over Y ∈ Sm/k, a morphism p: ℙ(1 ⊕ V) → Y be the projection and let s : Y → ℙ(1 ⊕ V) be the zero-section. Then, for every elements α ∈ A^{∗}(Y) and a ∈ A_{∗}(ℙ(1 ⊕ V)), one has:

a)  s^{!}(a) = p_{∗}(s_{!}(1) ⌢ a) and
b)  s_{!}(α) = p^{∗}(α) ⌣ s_{!}(1).

Proof. For a), we have:

(2.4)  p_{∗}(s_{!}(1) ⌢ a) = p_{∗}s_{∗}(1 ⌢ s^{!}(a)) = s^{!}(a),

since p_{∗}s_{∗} = id. As well, for b):

(2.5)  s_{!}(α) = s_{!}(s^{∗}p^{∗}(α) ⌣ 1) = p^{∗}(α) ⌣ s_{!}(1).

□

Let us state here the second important theorem concerning the interaction of the slant-product and transfers. First, we need to introduce a piece of notation which will be used pretty often throughout the paper.

Notation 2.8. Dealing with varieties X, Y, T, and a morphism Y →^{f} X we denote the morphism F = (id ×f): T × Y → T × Y by a corresponding capital letter.

Theorem 2.9. Let f : Y → X be a projective morphism of smooth varieties. Let, also T ∈ Sm/k. Then, for every elements α ∈ A^{∗}(T × Y) and a ∈ A_{∗}(X) one has the following equality in A^{∗}(T):

α/f^{!}(a) = F_{!}(α)/a.

Corollary 2.10. Let Y, T ∈ Sm/k with projective Y and let q: Y → pt be the structure map of Y. Then, for every α ∈ A^{∗}(T × Y), one has the equality:

α/[Y] = Q_{!}(α).

Proof. By Theorem 2.9 one has in A^{∗}(T)

(2.6)  α/[Y] = α/q^{!}(1) = Q_{!}(α)/1 = Q_{!}(α),

which proves the Corollary. □

Corollary 2.11. Let X be a smooth projective variety. Then, in A^{∗}(X), we have:

(2.7)  Δ_{!}(1)/[X] = 1.

Proof. Denoting by p : X × X → X the projection morphism, we have a chain of equalities:

(2.8)  Δ_{!}(1)/[X] = p_{!}(Δ_{!}(1)) = (p ∘ Δ)_{!}(1) = id_{!}(1) = 1,

which derives this Corollary from Corollary 2.10. □

Now we are ready to prove the main result.

Proof of Theorem 2.2. This is an easy consequence of Corollaries 2.11 and 2.5.

Let p_{1}, p_{2} : X × X → X denote corresponding projections. Observe, that for every γ ∈ A^{∗}(X × X) one has the relation Δ_{!}(1) ⌣ γ = γ ⌣ Δ_{!}(1). In fact,

(2.9)  Δ_{!}(1) ⌣ γ = Δ_{!}(1 ⌣ Δ^{∗}(γ)) = Δ_{!}(Δ^{∗}(γ) ⌣ 1) = γ ⌣ Δ_{!}(1).

Applying Corollary 2.5.b, we have:

Δ_{!}(1)/(α ⌢ [X]) = (Δ_{!}(1) ⌣ p_{2}^{∗}(α))/[X] = (Δ_{!}(1) ⌣ p_{1}^{∗}(α))/[X]
(2.10)  = (p_{1}^{∗}(α) ⌣ Δ_{!}(1))/[X] = α ⌣ (Δ_{!}(1)/[X])

By 2.11, we have: Δ_{!}(1)/[X] = 1. Thus, Δ_{!}(1)/(α ⌢ [X]) = α.

On the other hand, using 2.5.a, one has:

(Δ_{!}(1)/a) ⌢ [X] = p_{∗}(Δ_{!}(1) ⌢ (a\underline{×}[X])) = p_{∗}(Δ_{!}(1) ⌢ ([X]\underline{×}a))
(2.11)  = (Δ_{!}(1)/[X]) ⌢ a.

Since Δ_{!}(1)/[X] = 1, one gets (Δ_{!}(1)/a) ⌢ [X] = a. □

Thus, to prove Theorem 2.2 it is sufficient to prove Theorems 2.4 and 2.9. We start proving Theorem 2.4.

For this end, it is convinient to introduce a class 𝔙 of projective morphisms f : Y → X for which the relation

(2.12)  f_{∗}(α ⌢ f^{!}(a)) = f_{!}(α) ⌢ a

holds in A_{∗}(X) for every elements α ∈ A^{∗}(Y) and a ∈ A_{∗}(X).

We prove Theorem 2.4 showing consequently that the following classes of morphisms are contained in the class 𝔙.

• Zero-section morphisms of line bundles: s: Y ↪ ℙ(1 ⊕ L);

• Closed embeddings i: D ↪ X of smooth divisors;

• Zero-sections of finite sum of line bundles:

s: Y ↪ ℙ(1 ⊕ L_{1} ⊕ L_{2} ⊕ ⋯ ⊕ L_{n}));

• Zero-sections of arbitrary vector bundles: s: Y ↪ ℙ(1 ⊕ V);

• Closed embeddings i: Y ↪ X;

• Projections p: X × ℙ^{n} → X;

• Arbitrary projective morphisms.

3. Auxillary Facts

In this section we collect several useful lemmas, which are utilized repeatedly in proofs of Theorems 2.4 and 2.9 in consequent sections.

Lemma 3.1. Suppose, we are given a transversal square

\xymatrix{X′ \ar[d]_{ḡ} & Y′ \ar[d]^{g} \ar[l]_{f̄} \\ X & Y \ar[l]^{f}}

with projective morphism f. Let α ∈ A^{∗}(Y) and a ∈ A_{∗}(X′). Then, the following relations hold:

(1) f_{∗}(α ⌢ f^{!}ḡ_{∗}(a)) = ḡ_{∗}f̄_{∗}(g^{∗}(α) ⌢ f̄^{!}(a))

(2) f_{!}(α) ⌢ ḡ_{∗}(a) = ḡ_{∗}(f̄_{!}g^{∗}(α) ⌢ a)

Moreover, for a variety T ∈ Sm/k and an element β ∈ A^{∗}(T ×Y), we have:

(3) β/f^{!}ḡ_{∗}(a) = G^{∗}(β)/f̄^{!}(a)

(4) F_{!}(β)/ḡ_{∗}(a) = F̄_{!}G^{∗}(β)/a.

Proof. All these relations may be easily obtained using the base-change property and natural formulae from the first section. We illustrate it proving the first relation:

f_{∗}(α ⌢ f^{!}ḡ_{∗}(a)) = f_{∗}(α ⌢ g_{∗}f̄^{!}(a))
(3.1)  = f_{∗}g_{∗}(g^{∗}(α) ⌢ f̄^{!}(a)) = ḡ_{∗}f̄_{∗}(g^{∗}(α) ⌢ f̄^{!}(a)).
□

Remark 3.2. The reader could see that all the cases of the previous lemma may be considered as consequences of a general “lifting principle” formulation and proof of which is left to the reader.

The second fact that we need is a useful lemma, which is actually a “dualization” of “Useful Lemma 1.4.2” from [Pa].

To state this lemma, consider the following deformation diagram, in which B denotes the blowup of X × 𝔸^{1} at Y × {0}. This diagram has transversal squares.

(3.2)  \xymatrix{ & B − Y × 𝔸^{1} \ar[d]_{k_{B}} \\ ℙ(1 ⊕ N) \ar@{^{(}->}[r]_{k_{0}} \ar[d]^{p} & B \ar[r]^{σ} & X \ar@{^{(}->}[l]^{k_{1}} \\ Y \ar@{^{(}->}[r]_{j_{0}} \ar@{^{(}->}[u]^{i_{0}} & Y × 𝔸^{1} \ar@{^{(}->}[u]^{i_{t}} & Y \ar@{^{(}->}[l]^{j_{1}} \ar@{^{(}->}[u]^{i_{1}}}

The morphism σ here is the natural retraction B = (X × 𝔸^{1})_{Y ×{0}} →^{σ} X.

Lemma 3.3 (Homological useful lemma). Im k_{∗}^{B} + Im k_{∗}^{0} = A_{∗}(B)

Proof. Let us take b ∈ A_{∗}(B). Since the map j_{∗}^{0} is an isomorphism and i_{0}^{!}p^{!} = id, we can, using the transversal base-change property, lift b up to b̄ = p^{!}(j_{∗}^{0})^{−1}i_{t}^{!}(b) ∈ A_{∗}(ℙ(1 ⊕ N)), such that i_{t}^{!}(k_{∗}^{0}(b̄)) = i_{t}^{!}(b). Then, the Gysin exact sequence implies that k_{∗}^{0}(b̄) − b ∈ Im k_{∗}^{B}. □

This lemma plays an important role in the proofs of Theorems 2.4 and 2.9 because of the following proposition.

Proposition 3.4. Let i: Y ↪ X be a closed embedding of smooth varieties and N be the corresponding normal bundle. If the zero-section morphism s: Y ↪ ℙ(1⊕N) belongs to 𝔙, then i itself belongs to 𝔙.

Proof. Let us pay attention to Diagram 3.2. We take the morphism i as i_{1} in this diagram and s as i_{0}. First of all, using the proposition assumption, we shall show that the morphism i_{t} in Diagram 3.2 belongs to the class 𝔙. Namely, we prove that for every elements α_{t} ∈ A^{∗}(Y × 𝔸^{1}) and a_{t} ∈ A_{∗}(B), we have:

(3.3)  i_{∗}^{t}(α_{t} ⌢ i_{t}^{!}(a_{t})) = i_{!}^{t}(α_{t}) ⌢ a_{t}.

Using Lemma 3.3 we can rewrite a_{t} as a sum k_{∗}^{B}(a_{B})+k_{∗}^{0}(a_{0}), where a_{0} ∈ A_{∗}(ℙ(1⊕ N)) and a_{B} ∈ A_{∗}(B − Y × 𝔸^{1}). From the Gysin exact sequence, we have:

(3.4)  i_{t}^{!}k_{∗}^{B} = 0 and
(3.5)  k_{B}^{∗}i_{!}^{t} = 0.

Therefore, i_{∗}^{t}(α_{t} ⌢ i_{t}^{!}k_{∗}^{B}(a_{B})) = 0 and i_{!}^{t}(α_{t}) ⌢ k_{∗}^{B}(a_{B}) = 0. In fact, to get the second relation, we use 3.5 as follows:

(3.6)  i_{!}^{t}(α_{t}) ⌢ k_{∗}^{B}(a_{B}) = k_{∗}^{B}(k_{B}^{∗}i_{!}^{t}(α_{t}) ⌢ a_{t}) = 0.

This implies that

(3.7)  i_{∗}^{t}(α_{t} ⌢ i_{t}^{!}(a_{t})) = i_{∗}^{t}(α_{t} ⌢ i_{t}^{!}k_{∗}^{0}(a_{0})).

Applying Lemma 3.1 to the left-hand-side square of Diagram 3.2 and denoting j_{0}^{∗}(α_{t}) by α_{0}, one has:

(3.8)  i_{∗}^{t}(α_{t} ⌢ i_{t}^{!}k_{∗}^{0}(a_{0})) = k_{∗}^{0}i_{∗}^{0} (j_{0}^{∗}(α_{t}) ⌢ i_{0}^{!}(a_{0})) = k_{∗}^{0}i_{∗}^{0} (α_{0} ⌢ i_{0}^{!}(a_{0})) .

Similarly, one has:

(3.9)  i_{!}^{t}(α_{t}) ⌢ a_{t} = k_{∗}^{0} (i_{!}^{0}(α_{0}) ⌢ a_{0}) .

By the proposition assumption, we have the relation i_{∗}^{0}(α_{0} ⌢ i_{0}^{!}(a_{0})) = i_{!}^{0}(α_{0}) ⌢ a_{0}. Combining this with equalities (3.7), (3.8), and (3.9) finishes the proof of (3.3).

We now go further and move the desired relation one more step to the right in Diagram 3.2. Namely, we prove that for every elements α_{1} ∈ A^{∗}(Y) and a_{1} ∈ A_{∗}(X), we have:

(3.10)  i_{∗}^{1}(α_{1} ⌢ i_{1}^{!}(a_{1})) = i_{!}^{1}(α_{1}) ⌢ a_{1}.

For this end we observe that k_{∗}^{1} is a monomorphism, therefore it is sufficient to check that

(3.11)  k_{∗}^{1}i_{∗}^{1}(α_{1} ⌢ i_{1}^{!}(a_{1})) = k_{∗}^{1}(i_{!}^{1}(α_{1}) ⌢ a_{1}).

Using that the map j_{1}^{∗} is an isomorphism we choose α_{t} ∈ A^{∗}(Y × 𝔸^{1}) as α_{t} = (j_{1}^{∗})^{−1}(α_{1}). We also set: a_{t} = k_{∗}^{1}(a_{1}) ∈ A_{∗}(B). Applying Lemma 3.1 to the right-hand-side square of Diagram 3.2, we have:

(3.12)  k_{∗}^{1}i_{∗}^{1}(α_{1} ⌢ i_{1}^{!}(a_{1})) = k_{∗}^{1}i_{∗}^{1}(j_{1}^{∗}(α_{t}) ⌢ i_{1}^{!}(a_{1})) = i_{∗}^{t}(α_{t} ⌢ i_{t}^{!}(a_{t})).

By the same arguments, we get:

(3.13)  k_{∗}^{1}(i_{!}^{1}(α_{1}) ⌢ a_{1}) = i_{!}^{t}(α_{t}) ⌢ a_{t}.

Taking these two relations together with (3.3) finishes the proof of relation (3.10) and of the proposition as well. □

4. Proof of Theorem 2.4

We begin with the following very simple case of the Theorem.

Lemma 4.1. Let L be a line bundle over a smooth variety Y. Then the zero-section s: Y ↪ ℙ(1 ⊕ L) belongs to 𝔙.

Proof. Since the map s is a section of the projection map p, we can write α = s^{∗}p^{∗}(α). Then, we have:

(4.1)  s_{∗}(α ⌢ s^{!}(a)) = s_{∗}(s^{∗}p^{∗}(α) ⌢ s^{!}(a)) = p^{∗}(α) ⌢ s_{∗}s^{!}(a).

Using the normalization property, we can rewrite the latter expression as:

(4.2) p^{∗}(α) ⌢ s_{∗}s^{!}(a) = p^{∗}(α) ⌢ (s_{!}(1) ⌢ a) = (p^{∗}(α) ⌣ s_{!}(1)) ⌢ a = s_{!}(α) ⌢ a.
□

Corollary 4.2. Let i: D ↪ X be a smooth divisor. Then, i ∈ 𝔙.

Proof. Since, in this case, the corresponding normal bundle is a line bundle, the Corollary follows from Lemma 4.1 and Proposition 3.4. □

Proposition 4.3. Let W = L_{1} ⊕⋯⊕L_{n} be an n-dimensional vector bundle over a variety Y which splits in the sum of line bundles. Then, the zero-section morphism s: Y ↪ ℙ(1 ⊕ W) belongs to the class 𝔙.

Proof. The map s can be decomposed in the following obvious way:

(4.3)  Y ↪^{i_{1}} ℙ(1 ⊕ L_{1}) ↪^{i_{2}} ⋯ ↪^{i_{n}} ℙ(1 ⊕ W)

where the map i_{j} the zero-section of L_{j}. Now we can rewrite the desired formula

s_{∗}(α ⌢ s^{!}(a)) = i_{∗}^{n}i_{∗}^{n−1} … i_{∗}^{2}i_{∗}^{1}(α ⌢ i_{1}^{!}i_{2}^{!} … i_{n−1}^{!}i_{n}^{!}(a))
= i_{∗}^{n}i_{∗}^{n−1} … i_{∗}^{2}(i_{!}^{1}(α) ⌢ i_{2}^{!} … i_{n−1}^{!}i_{n}^{!}(a))
(4.4)  = ⋯ = i_{!}^{n}i_{!}^{n−1} … i_{!}^{2}i_{!}^{1}(α) ⌢ a = s_{!}(α) ⌢ a

by iterative application of Corollary 4.2. □

In order to proceed with the case of an arbitrary vector-bundle, we need the homological analogue of the splitting principle. To state it, consider a vector bundle E → Y of constant rank n over a smooth variety Y. Let GL_{n} be the corresponding principal GL_{n}-bundle over Y, let T_{n} ⊂ GL_{n} be the diagonal tori, and let Y′ = GL_{n}/T_{n} be the orbit variety with the projection morphism p : Y′ → Y. Finally, we denote by E′ = E ×_{Y} Y′ the pull-back of the vector bundle E.

Proposition 4.4. The bundle E′ splits in a direct sum of line bundles and the map p_{∗}: A_{∗}(Y′) → A_{∗}(Y) is a universal splitting epimorphism (which means for any base-change Z → Y the induced map A_{∗}(Z ×_{Y} Y′) → A_{∗}(Z) is a splitting epimorphism).

Proof. The projection GL_{n} → Y′ and the natural T_{n}-action on GL_{n} makes it a principal T_{n}-bundle over Y′. Moreover, if GL′_{n} = GL_{n} ×_{Y} Y′ is the bull-back of GL_{n}, there is a natural isomorphism of principal GL_{n}-bundles

(4.5)  GL_{n} ×_{T_{n}} GL_{n} → GL′_{n}

over Y′. The bundle E′ over Y′ corresponds exactly to the principal GL_{n}-bundle GL′_{n}. Thus, the mentioned isomorphism of principal GL_{n}-bundles over Y′ shows that the bundle E′ splits in a direct sum of line bundles (say corresponding to the fundamental characters χ_{1}, χ_{2}, …, χ_{n} of the tori T_{n}). This proves the first assertion of the proposition.

To prove the second one, consider a Borel subgroup B_{n} in GL_{n} (say the subgroup of all upper triangle matrises) and let U_{n} be the maximal unipotent subgroup of B_{n} (the group of upper triangle matrises with 1’s on the diagonal). Let F = GL_{n}/B_{n} (this is just the flag bundle over Y assosiated to E). The bundle F comes equipped with projections q : F → Y and r : Y′ → F, where the projection r is induced by the inclusion T_{n} ⊂ B_{n}. Using the natural U_{n}-action on GL_{n}, it is easy to check that there is a tower of morphisms:

(4.6)  GL_{n} = S_{m} → S_{m−1} → ⋯ → S_{1} = F,

which has on each level a principal G_{a}-bundle (so, every level is a torser over the trivial rank one vector bundle). By the strong homotopy invariance property, the induced map on homology r_{∗} : A_{∗}(Y′) → A_{∗}(F) is an isomorphism.

As it was already mentioned, F is a full flag bundle over Y associated to the bundle E. Thus, there is a tower of morphisms

(4.7)  F = Z_{s} → Z_{s−1} → ⋯ → Z_{1} = Y

in which each level is a projective bundle associated to a vector bundle. Thus, by PBT, we have a split epimorphism in homology induced on each floor. Thus, the map q_{∗}: A_{∗}(F) → A_{∗}(Y) is a split epimorphism as well.

These proves that the map p_{∗} : A_{∗}(Y′) → A_{∗}(Y) is also an epimorphism.

Now let Z → Y be a morphism (with smooth variety Z) and Z′ = Z ×_{Y} Y′. Consider a variety F_{Z} = Z×_{Y} Y′ and projections p′ : Z′ → Z, r′ : Z′ → F_{Z}, and q′ : F_{Z} → Z, which are base-changes of projections p, r, and q, respectively. Clearly, r′ may be decomposed as a tower of principal G_{a}-bundles and thus it induces an isomorphism on the homology groups A_{∗} by the strong homotopy invariance. The morphism q′ is decomposable as a tower of projective bundles associated to certain vector bundles. By PBT, it induces a split epimorphism on the homology groups A_{∗}. Thus the morphism p′ induces a split epimorphism on the homology groups A_{∗}. The proof of the Proposition is completed. □

Proposition 4.5. Let s: Y ↪ ℙ(1⊕V) be the zero-section of the finite-dimensional vector bundle V. Then, s ∈ 𝔙.

Proof. Let us take the variety Y′ from Proposition 4.4 and the pull-back V′ of the bundle V with respect to the morphism p. By 4.4, the bundle V′ splits in a direct sum of line bundles. Consider the pull-back diagram

(4.8)  \xymatrix{ℙ(1 ⊕ V′) \ar[r]^{f̄} \ar[d]_{p̄} & Y′ \ar[d]^{p} \\ ℙ(1 ⊕ V) \ar[r]^{f} & Y.}

The map

(4.9)  p̄_{∗}: A_{∗}(ℙ(1 ⊕ V′)) → A_{∗}(ℙ(1 ⊕ V))

is an epimorphism by Proposition 4.4. Let s : Y → ℙ(1 ⊕ V) and s̄: Y′ → ℙ(1 ⊕ V′) be sections of projections f and f̄ induced by zero-sections of the corresponding vector bundles. Then, the diagram

(4.10)  \xymatrix{ℙ(1 ⊕ V′) \ar[d]_{p̄} & Y′ \ar[d]^{p} \ar[l]_{s̄} \\ ℙ(1 ⊕ V) & Y \ar[l]_{s}}

is transversal.

For every α ∈ A^{∗}(Y) and a ∈ A_{∗}(ℙ(1 ⊕ V)), since the map p̄_{∗}: A_{∗}(ℙ(1 ⊕ V′)) → A_{∗}(ℙ(1 ⊕ V)) is an epimorphism, we can choose some b ∈ A_{∗}(ℙ(1 ⊕ V′)) such that a = p̄_{∗}(b), and then apply Lemma 3.1:

(4.11)  s_{∗}(α ⌢ s^{!}(a)) = s_{∗}(α ⌢ s^{!}p̄_{∗}(b)) = p̄_{∗}s̄_{∗}(p^{∗}(α) ⌢ s̄^{!}(b)).

On the other hand, by the same reason:

(4.12)  s_{!}(α) ⌢ a = s_{!}(α) ⌢ p̄_{∗}(b) = p̄_{∗}(s̄_{!}p^{∗}(α) ⌢ b)

and two expressions on the right-hand-side coincide by Proposition 4.3. □

Proposition 4.6. Let i: Y ↪ X be a closed embedding. Then, i ∈ 𝔙.

Proof. Applying Proposition 3.4 we reduce the question to the case of the zerosection morphism s: Y ↪ ℙ(1 ⊕ N) of the normal bundle N = N_{X/Y}. The morphism s belongs to 𝔙 by Proposition 4.5. □

In order to check that for every integer n ≥ 0 the projection morphism p : X × ℙ^{n} → X belongs to 𝔙 we need the following lemma.

Notation 4.7. We denote, from now on, for a projective morphism f the map f_{∗}f^{!} by f^{⋄} and f_{!}f^{∗} by f_{⋄}

Fix now a variety X ∈ Sm/k and take the n-dimensional projective space ℙ_{X}^{n} over X. Consider the diagonal morphism Δ: ℙ_{X}^{n} → ℙ_{X}^{n} × ℙ_{X}^{n}. (Up to the end of this section the scheme product is implicitly taken over X.) Due to the PBT, the element Δ_{!}(1) may be decomposed as

(4.13)  Δ_{!}(1) = 1 ⊠ ζ^{n} + ζ^{n} ⊠ 1 + ∑_{i,j=1}^{n} a_{ij}ζ^{i} ⊠ ζ^{j},

where a_{ij} ∈ A^{∗}(X) and ζ is the canonical generator of A^{∗}(ℙ_{X}^{n}) as an A^{∗}(X)-algebra.

Lemma 4.8. For the projection morphism p_{n}: ℙ_{X}^{n} → X, we have:

a) p_{n}^{⋄} = − ∑_{j=1}^{n} a_{nj}p_{n−j}^{⋄};

b) p_{⋄}^{n} = − ∑_{j=1}^{n} a_{nj}p_{⋄}^{n−j}.

Proof. Let us prove case a). For n = 0 we, trivially, have p_{0}^{⋄} = id.

Starting from the mentioned decomposition for the diagonal morphism and taking into account the relation s_{ij}^{⋄}(x) = (ζ^{i} ⊠ ζ^{j}) ⌢ x, where s_{ij}: ℙ_{X}^{n−i} × ℙ_{X}^{n−j} ↪ ℙ_{X}^{n} × ℙ_{X}^{n} is the standard embedding, we can rewrite the cap-product with Δ_{!}(1) operator in the form:

(4.14)  (Δ_{!}(1) ⌢) = s_{0n}^{⋄} + s_{n0}^{⋄} + ∑_{i,j=1}^{n} a_{ij}s_{ij}^{⋄}.

This enables us to rewrite the map p_{∗}^{n} as well:

(4.15)  p_{∗}^{n} = p_{∗}^{n}(p_{1,n}Δ)^{⋄} = p_{∗}^{n} ∑_{i,j=0}^{n} a_{ij}(p_{1,n}s_{ij})^{⋄},

where the notation is taken from the diagram below.

(4.16)  \xymatrix{ & ℙ_{X}^{n} × \{0\} \ar[dl]_{p_{n}} &  & ℙ_{X}^{n−i} × \{0\} \ar@{^{(}->}[ll]_{s_{i}} \\ X & ℙ_{X}^{n} × ℙ_{X}^{n} \ar[u]_{p_{1,n}} & ℙ_{X}^{n} × ℙ_{X}^{n−j} \ar@{^{(}->}[l]_{s_{0j}} \ar[ul]_{p_{1,n−j}} \ar[dl]^{p_{n}×id} & ℙ_{X}^{n−i} × ℙ_{X}^{n−j} \ar[u]_{id ×p_{n−j}} \ar@{^{(}->}[l] \ar@{^{(}->}[ll]^{s_{ij}} \\  & \{0\} × ℙ_{X}^{n−j} \ar[ul]^{p_{n−j}}}

Using the commutativity of this diagram and transversaltiy of the corresponding squares, we rewrite a typical summand in Formula 4.15 in the following way: (4.17) p_{∗}^{n}p_{∗}^{1,n}s_{∗}^{ij}s_{ij}^{!}p_{1,n}^{!} = p_{∗}^{n}p_{∗}^{1,n}s_{∗}^{ij}(id ×p_{n−j})^{!}s_{i}^{!} = p_{∗}^{n}p_{∗}^{1,n}s_{∗}^{0j}p_{1,n−j}^{!}s_{i}^{⋄} = p_{∗}^{n}p_{1,n−j}^{⋄}(ζ^{i} ⌢).

Since p_{∗}^{n}p_{1,n−j}^{⋄} = p_{∗}^{n−j}(p_{n} × id)_{∗}p_{1,n−j}^{!} = p_{n−j}^{⋄}p_{∗}^{n}, we can rewrite the expression (4.15) as follows:

(4.18) p_{∗}^{n} = p_{0}^{⋄}p_{∗}^{n} + p_{n}^{⋄}p_{∗}^{n}(ζ^{n} ⌢) + ∑_{j=1}^{n} a_{nj}p_{n−j}^{⋄}p_{∗}^{n}(ζ^{n} ⌢) + ∑_{i,j=1}^{n−1}(⋯)p_{∗}^{n}(ζ^{i} ⌢).

Operators p_{∗}^{n}(ζ^{i} ⌢) coincide to ones ψ_{i}, which appear at the homology PBT. So that, we have:

(4.19)  p_{∗}^{n} = ψ_{0} = ψ_{0} + (p_{n}^{⋄} + ∑_{j=1}^{n} a_{nj}p_{n−j}^{⋄}) ψ_{n} + ∑_{i,j=1}^{n−1}(⋯)ψ_{i}.

Thus, one gets the relation:

(4.20)  (p_{n}^{⋄} + ∑_{j=1}^{n} a_{nj}p_{n−j}^{⋄}) ψ_{n} = − ∑_{i,j=1}^{n−1}(⋯)ψ_{i}.

By PBT, for any x ∈ A_{∗}(X) we can choose an element φ(x) ∈ A_{∗}(ℙ_{X}^{n}) such that

ψ_{i}(φ(x)) = \{ \begin{matrix}0, & i < n \\ x, & i = n. \end{matrix}

Applying operators on both sides of (4.20) to φ(x), we get:

(4.21)  0 = p_{n}^{⋄} + ∑_{j=1}^{n} a_{nj}p_{n−j}^{⋄}

This finishes the proof of case a). The cohomological relation b) may be proved by dualization of these arguments or found in [Pa]. □

We need the following consequence of Lemma 4.8.

Proposition 4.9. Let p_{n} denote, as before, the projection morphism p_{n}: ℙ_{X}^{n} → X. Then, for every element a ∈ A_{∗}(X), one has:

p_{∗}^{n}(p_{n}^{!}(a)) = p_{!}^{n}(1) ⌢ a.

Proof. Recall that p_{n}^{⋄}(a) = p_{∗}^{n}(p_{n}^{!}(a)) and p_{!}^{n}(1) = p_{!}^{n}(p_{n}^{∗}(1)) = p_{⋄}^{n}(1). Thus, we should verify the relation p_{n}^{⋄}(a) = p_{⋄}^{n}(1) ⌢ a. We proceed by induction on n. The case n = 0 obviously holds. Let the proposition hold for n < N. Then, for p_{N} we have by Lemma 4.8:

(4.22)  p_{N}^{⋄}(a) = − ∑_{j=1}^{N} a_{Nj}p_{N−j}^{⋄}(a)

and

(4.23)  p_{⋄}^{N}(1) ⌢ a = − ∑_{j=1}^{N} a_{Nj}p_{⋄}^{N−j}(1) ⌢ a

By the induction hypothesis the expressions on the right-hand-side coincide. The induction goes. □

Proposition 4.10. For every integer n ≥ 0 the projection map p: ℙ_{X}^{n} → X belongs to the class 𝔙.

Proof. Given elements α ∈ A^{∗}(ℙ_{X}^{n}) and a ∈ A_{∗}(X) we should verify that in A_{∗}(X)

(4.24)  p_{∗}(α ⌢ p^{!}(a)) = p_{!}(α) ⌢ a.

Clearly, both sides of (4.24) are A^{∗}(X)-linear. By PBT, the group A^{∗}(ℙ_{X}^{n}) is generated as A^{∗}(X)-module by elements ζ^{j}. Thus, it suffices to check the Proposition just for these elements. From [Pa], we have a relation ζ^{j} = i_{!}(1) in A(ℙ_{X}^{n}), where i: ℙ_{X}^{n−j} ↪ ℙ_{X}^{n} is the standard embedding map and the element ζ^{j} ∈ A^{∗}(ℙ^{n}) is considered here as lying in A^{∗}(ℙ_{X}^{n}) via the pull-back operator for the projection ℙ_{X}^{n} → ℙ^{n}. Denote by p_{j} the projection map ℙ_{X}^{n−j} → X. Since p ∘ i = p_{j}, we have by Proposition 4.6:

(4.25)  p_{∗}(ζ^{j} ⌢ p^{!}(a)) = p_{∗}(i_{!}(1) ⌢ p^{!}(a)) = p_{∗}i_{∗}(1 ⌢ i^{!}p^{!}(a)) = p_{∗}^{j}p_{j}^{!}(a).

Using Proposition 4.9, we get:

(4.26)  p_{∗}^{j}p_{j}^{!}(a) = p_{!}^{j}(1) ⌢ a = p_{!}i_{!}(1) ⌢ a = p_{!}(ζ^{j}) ⌢ a.

The desired relation p_{∗}(ζ^{j} ⌢ p^{!}(a)) = p_{!}(ζ^{j}) ⌢ a follows. □

Since any projective morphism f can be written as a composition of a closed embedding and a projection map (f = p ∘ i), we have by Propositions 4.6 and 4.10:

f_{∗}(α ⌢ f^{!}(a)) = p_{∗}i_{∗}(α ⌢ i^{!}p^{!}(a)) = p_{∗}(i_{!}(α) ⌢ p^{!}(a))
(4.27)  = p_{!}i_{!}(α) ⌢ a = f_{!}(α) ⌢ a.

This completes the proof of Theorem 2.4.

5. Proof of Theorem 2.9

The strategy of the proof of Theorem 2.9 is very similar to one used in the previous section. It is again convenient to introduce a class 𝔚 of projective morphisms f : Y → X for which the relation

(5.1)  F_{!}(α)/a = α/f^{!}(a)

holds. We show that the following classes of morphisms lie in 𝔚.

• Zero-sections of vector bundles: s: Y ↪ ℙ(1 ⊕ V);

• Closed embeddings i: Y ↪ X;

• Projections p: X × ℙ^{n} → X;

• All projective morphisms. We start with the following lemma.

Lemma 5.1. For any projective morphism f: Y → X and a variety T, one has the relation

p_{X}^{∗}f_{!}(1) = 1 × f_{!}(1) = F_{!}(1)

in A^{∗}(T × X). (We recall that here and below we use the notation, proclaimed in 2.8. So that, F in our case denotes the morphism (id ×f): T × X → T × Y.)

Proof. Considering a transversal square

(5.2)  \xymatrix{T × Y \ar[r]^{F} \ar[d]_{p_{Y}} & T × X \ar[d]^{p_{X}} \\ Y \ar[r]^{f} & X}

and using the base-change property, we have: p_{X}^{∗}(f_{!}(1)) = F_{!}(p_{Y}^{∗} (1)) = F_{!}(1). □

Lemma 5.2. Let V be a vector bundle over a smooth variety Y and let s: Y ↪ ℙ(1 ⊕ V) be the zero-section of the projection p: ℙ(1 ⊕ V) → Y Then, the morphism s belongs to the class 𝔚.

Proof. Given a smooth variety T and elements α ∈ A^{∗}(T ×Y) and a ∈ A_{∗}(ℙ(1⊕V)) we should verify that in A^{∗}(T)

(5.3)  α/s^{!}(a) = S_{!}(α)/a.

Using Corollary 2.7, and Lemma 5.1 one gets a chain of relations:

α/s^{!}(a) = α/p_{∗}(s_{!}(1) ⌢ a) = P^{∗}(α)/(s_{!}(1) ⌢ a)
(5.4)  = P^{∗}(α) ⌣ (1 × s_{!}(1))/a = P^{∗}(α) ⌣ S_{!}(1)/a = S_{!}(α)/a

which proves the required one. □

Now we proceed to the case of general closed embedding.

Proposition 5.3. Any closed embedding morphism i : Y ↪ X of smooth varieties belongs to the class 𝔚.

Proof. Denote by ℙ(1 ⊕ N) the projectiovization corresponding to the normal bundle N = N_{X/Y}. It is endowed with a projection morphism p : ℙ(1 ⊕ N) → Y and the zero-section s: Y ↪ ℙ(1 ⊕ N).

As well as in the proof of Theorem 2.4 the proof settles on the deformation diagram, which actually obtained from (3.2) by multiplication on a variety T ∈ Sm/k. For convinience, we reproduce this diagram here.

(5.5)  \xymatrix{ & T × B − T × Y × 𝔸^{1} \ar[d]_{K_{B}} \\ T × ℙ(1 ⊕ N) \ar@{^{(}->}[r]_{K_{0}} & T × B & T × X \ar@{^{(}->}[l]^{K_{1}} \\ T × Y \ar@{^{(}->}[r]_{J_{0}} \ar@{^{(}->}[u]^{I_{0}} & T × Y × 𝔸^{1} \ar@{^{(}->}[u]^{I_{t}} & T × Y \ar@{^{(}->}[l]^{J_{1}} \ar@{^{(}->}[u]^{I_{1}}}

First of all, we show that I_{t} ∈ 𝔚. Namely, we should prove that for any elements α_{t} ∈ A^{∗}(T × Y × 𝔸^{1}) and a_{t} ∈ A_{∗}(B) the relation

(5.6)  α_{t}/i_{t}^{!}(a_{t}) = I_{!}^{t}(α_{t})/a_{t}.

holds in A^{∗}(T).

By Lemma 3.3, we can rewrite a_{t} as a sum k_{∗}^{B}(a_{B})+k_{∗}^{0}(a_{0}), where a_{0} ∈ A_{∗}(ℙ(1⊕ N)) and a_{B} ∈ A_{∗}(B − Y × 𝔸^{1}). Exactly in the same way as before (in section 3), we may use Gysin exact sequences (in homology and cohomology) to obtain the equalities:

(5.7)  i_{t}^{!}k_{∗}^{B} = 0 and
(5.8)  K_{B}^{∗}I_{!}^{t} = 0.

Therefore, α_{t}/i_{t}^{!}k_{∗}^{B}(a_{B}) = 0 and I_{!}^{t}(α_{t})/k_{∗}^{B}(a_{B}) = K_{B}^{∗}I_{!}^{t}(α_{t})/a_{B} = 0. Now, applying Lemma 3.1, we get:

(5.9)  α_{t}/i_{t}^{!}(a_{t}) = α_{t}/i_{t}^{!}k_{∗}^{0}(a_{0}) = α_{0}/i_{0}^{!}(a_{0}),

where α_{0} = J_{0}^{∗}(α_{t}).

Similarly, we obtain the relation:

(5.10)  I_{!}^{t}(α_{t})/a_{t} = I_{!}^{0}(α_{0})/a_{0}.

By Lemma 5.2, we have: α_{0}/i_{0}^{!}(a_{0}) = I_{!}^{0}(α_{0})/a_{0}, which proves Formula 5.6.

We end up the proof by the same way as in 2.4. Since the map J_{1}^{∗} is an isomorphism, we can set α_{t} = (J_{1}^{∗})^{−1}(α_{1}) ∈ A^{∗}(T × Y × 𝔸^{1}) and a_{t} = k_{∗}^{1}(a_{1}) ∈ A_{∗}(B). Applying Lemma 3.1 again, we get:

(5.11)  α_{1}/i_{1}^{!}(a_{1}) = J_{1}^{∗}(α_{t})/i_{1}^{!}(a_{1}) = α_{t}/i_{t}^{!}(a_{t}) and
(5.12)  I_{!}^{1}(α_{1})/a_{1} = I_{!}^{1}J_{1}^{∗}(α_{t})/a_{1} = I_{!}^{t}(α_{t})/a_{t}.

Combining these equalities with relation (5.6), we get: α_{1}/i_{1}^{!}(a_{1}) = I_{!}^{1}(α_{1})/a_{1}, which proves the proposition. □

Consider now the case of projection.

Proposition 5.4. Let X, T ∈ Sm/k, let p: X×ℙ^{n} → X be the projection morphism and P = id×p: T ×X ×ℙ^{n} → T ×X. Then, for every elements α ∈ A^{∗}(T ×X ×ℙ^{n}) and a ∈ A_{∗}(X), one has a relation

α/p^{!}(a) = P_{!}(α)/a

in A^{∗}(T).

Proof. Consider the following commutative diagram with transversal square:

(5.13)  \xymatrix{X × ℙ^{n} \ar[dr]_{p=p_{0}} & X × ℙ^{n−r} \ar[l]_{i} \ar[d]^{p_{r}} & T × X × ℙ^{n−r} \ar[l]_{q̄} \ar[d]^{P_{r}} \\  & X & T × X \ar[l]_{q}}

Clearly, both sides of the required relation are A^{∗}(T)-linear. So, we may assume that α = ζ_{T ×X}^{r}. (Here and below we denote by β_{Y} ∈ A^{∗}(Y ×ℙ^{∗}) the pull-back of the element β ∈ A^{∗}(ℙ^{∗}) with respect to the projection morphism Y × ℙ^{∗} → ℙ^{∗}.) From the relation ζ_{X}^{r} = i_{!}(1) in A^{∗}(X×ℙ^{n}), we have: ζ_{T ×X}^{r} = I_{!}(1_{T ×X}) ∈ A^{∗}(T ×X×ℙ^{n}). This gives us the following chain of equalities:

ζ_{T ×X}^{r}/p^{!}(a) = I_{!}(1_{T ×X})/p^{!}(a) = 1_{T ×X}/i^{!}p^{!}(a)
(5.14)  = 1/p_{r}^{!}(a) = P_{r}^{∗}(1)/p_{r}^{!}(a) = 1/p_{∗}^{r}p_{r}^{!}(a).

By Proposition 4.9:

(5.15)  1/p_{∗}^{r}p_{r}^{!}(a) = 1/(p_{!}^{r}(1_{X}) ⌢ a) = q^{∗}p_{!}^{r}(1)/a.

Applying the base-change property to the square in the diagram above, we get:

(5.16)  q^{∗}p_{!}^{r}(1_{X}) = P_{!}^{r}q̄^{∗}(1) = P_{!}^{r}(1_{T ×X}) = P_{!}(I_{!}(1)) = P_{!}(ζ_{T ×X}^{r}),

which proves the relation ζ_{T ×X}^{r}/p^{!}(a) = P_{!}(ζ_{T ×X}^{r})/a and the proposition as well. □

Finally, for an arbitrary projective morphism f : Y → X with a decomposition:

(5.17)  f: Y ↪^{i} X × ℙ^{n} →^{p} X

one has:

(5.18)  α/f^{!}(a) = α/i^{!}p^{!}(a) = I_{!}(α)/p^{!}(a) = P_{!}I_{!}(α)/a = F_{!}(α)/a,

which proves the relation α/f^{!}(a) = F_{!}(α)/a, and completes the proof of Theorem 2.9.

References

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Ivan Panin  panin@pdmi.ras.ru
Serge Yagunov  yagunov@pdmi.ras.ru

Steklov Mathematical Institute (St.Petersburg) Fontanka 27, 191011 St.Petersburg, Russia.