MOTIVES AND ORIENTED COHOMOLOGY OF A LINEAR ALGEBRAIC GROUP

ALEXANDER NESHITOV

Abstract. For a cellular variety X over a field k of characteristic 0 and an algebraic oriented cohomology theory h of Levine-Morel we construct a filtration on the cohomology ring h(X) such that the associated graded ring is isomorphic to the Chow ring of X. Taking X to be the variety of Borel subgroups of a split semisimple linear algebraic group G over k we apply this filtration to relate the oriented cohomology of G to its Chow ring. As an immediate application we compute the algebraic cobordism ring of a group of type G_{2} and of some other groups of small ranks, hence, extending several results by Yagita.

Using this filtration we also establish the following comparison result between Chow motives and h-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically split variety Y gives rise to a h-motivic decomposition of Y with the same generating function. Moreover, under some conditions on the coefficient ring of h the obtained h-motivic decomposition will be irreducible. We also prove that if Chow motives of two twisted forms of Y coincide, then their h-motives coincide as well.

1. Introduction

We work over the base field k of characteristic 0. For an algebraic oriented cohomology theory h of Levine-Morel [11] and a cellular variety X of dimension N we construct a filtration

h(X) = h^{(0)}(X) ⊇ h^{(1)}(X) ⊇ … ⊇ h^{(N)}(X) ⊇ 0

on the cohomology ring such that the associated graded ring

Gr^{∗} h(X) = ⨁_{i≥0} h^{(i)}(X)/ h^{(i+1)}(X)

is isomorphic (as a graded ring) to the Chow ring CH^{∗}(X, Λ) of algebraic cycles modulo rational equivalence relation with coefficients in a ring Λ. We exploit this filtration and isomorphism in two different contexts:

First, we consider the (cellular) variety X = G/B of Borel subgroups of a split semisimple linear algebraic group G over k. By [7, Prop. 5.1] the cohomology ring h(G) can be identified with a quotient of h(G/B), so there is an induced filtration on h(G). One of our key results (Prop. 4.3) shows that CH^{∗}(G, Λ) covers the associated graded ring Gr^{∗} h(G) and describes the kernel of this surjection. As an immediate application for h = Ω (the algebraic cobordism of Levine-Morel) we compute the cobordism ring for groups G_{2}, SO_{3}, SO_{4}, Spin_{n} for n = 3, 4, 5, 6 and P GL_{n} for n ≥ 2, in terms of generators and relations, hence, extending several previously known results by Yagita [19]; as an application for h = K_{0} (the Grothendieck K_{0}) we construct certain elements in the difference between topological and the Grothendieck γ-filtration on K_{0}(X), hence, extending some of the results by Garibaldi-Zainoulline [6].

The second deals with the study of h-motives of generically cellular varieties. The latter is a natural generalization of the notion of the Chow motives to the case of an arbitrary algebraic oriented cohomology theory of Levine-Morel. It was introduced and studied by Nenashev-Zainoulline in [13] and Vishik-Yagita in [17].

Let Λ denote the coefficient ring of h and let Λ^{i} denote its i-th graded component. We prove the following theorem which relates h-motives of generically cellular varieties to its Chow motives:

Theorem A. Let X be a generically cellular variety over k, i.e. cellular over the function field k(X). Assume that the Chow motive of X with coefficients in Λ^{0} splits as

M^{CH}(X, Λ^{0}) = ⨁_{i≥0} R(i)^{⊕c_{i}}, c_{i} ≥ 0,

for some motive R which splits as a direct sum of twisted Tate motives \overline{R} = ⨁_{j≥0} Λ^{0}(j)^{⊕d_{j}} over its splitting field.

Then the h-motive of X (with coefficients in Λ) splits as

M^{h}(X) = ⨁_{i≥0} R_{h}(i)^{⊕c_{i}}

for some motive R_{h} and over the same splitting field R_{h} splits as a direct sum of twisted h-Tate motives \overline{R_{h}} = ⨁_{j≥0} Λ(j)^{⊕d_{j}}.

This result can also be derived from the arguments of [17] where it is proved that sets of isomorphism classes of objects of category of Chow motives and Ω-motives coincide. However, our approach gives more explicit correspondence between idempotents defining the (Chow) motive R and the h-motive R_{h}. The latter allows us to prove the following result concerning the indecomposability of the h-motive R_{h}:

Theorem B. Assume that Λ^{1} = … Λ^{N} = 0, where N = dim X.

If the Chow motive R is indecomposable (over Λ^{0}), then the h-motive R_{h} is indecomposable (over Λ).

and also the following comparison property:

Theorem C. Suppose that X, Y are generically cellular and Y is a twisted form of X, i.e. Y becomes isomorphic to X over some splitting field.

If M^{CH}(X, Λ^{0}) ≅ M^{CH}(Y, Λ^{0}), then M^{h}(X) ≅ M^{h}(Y).

The paper is organized as follows: In section 2 we recall concepts of an algebraic oriented cohomology theory h of Levine-Morel and the corresponding category of h-motives. In section 3 we introduce the filtration on the cohomology ring h(X) of a cellular variety X which plays a central role in the paper. In section 4 we apply the filtration to obtain several comparison results between CH(G) and h(G). In particular, in section 5 we compute the algebraic cobordism Ω for some groups of small ranks and construct explicit elements in the difference of between the topological and the γ-filtration on K_{0}(G/B). In section 6 we apply the filtration to obtain comparison results between h-motives and Chow-motives of generically split varieties.

Acknowledgments. I am grateful to my PhD supervisor Kirill Zainoulline for useful discussions concerning the subject of this paper. The work has been supported by the Ontario Trillium Graduate Scholarship, RFBR grant 12-01-33057 and by NSERC 396100-2010 and ERA grants of K. Zainoulline.

2. Preliminaries

In the present section we recall notions of an algebraic oriented cohomology theory, a formal group law and of a cellular variety. We recall the definition of the category of h-motives with the inverted Tate object.

Oriented cohomology theories. The notion of an algebraic oriented cohomology theory was introduced by Levine-Morel [11] and Panin-Smirnov [14]. Let Sm_{k} denote the category of smooth varieties over Spec k = pt. An algebraic oriented cohomology theory h^{∗} is a functor from Sm_{k}^{op} to the category of graded rings. We will denote by f^{h} : h^{∗}(Y) → h^{∗}(X) the induced morphism f: X → Y and call it the pullback of f. By definition, the functor h^{∗} is equipped with the pushforward map f_{h} : h^{∗}(X) → h^{dim Y −dim X+∗}(Y) for any projective morphism f: X → Y. These two structures satisfy the axioms of [11, Def. 1.1.2]. We denote its coefficient ring h^{∗}(pt) by Λ^{∗}. As for the Chow groups, we will also use the lower grading for h, i.e. h_{i}(X) = h^{dim X−i}(X) for an irreducible variety X.

Formal group law. For an oriented cohomology theory h^{∗} there is a notion of the first Chern class of a line bundle. For X ∈ Sm_{k} and a line bundle L over X it is defined as c_{1}^{h}(L) = z^{h}z_{h}(1) ∈ h^{1}(X) where z: X → L is a zero section. There is a commutative associative 1-dimensional formal group law F over Λ^{∗} such that for any two line bundles L_{1} , L_{2} over X we have c_{1}^{h}(L_{1} ⊗ L_{2}) = F (c_{1}^{h}(L_{1}), c_{1}^{h}(L_{2})) [11, Lem. 1.1.3]. We will use the notation x +_{F} y for F (x, y). For any x we will denote by −_{F} x the unique element such that x +_{F} (−_{F} x) = 0. For any n ∈ ℤ we will denote by n ·_{F} x the expression x +_{F} … +_{F} x (n times) if n is positive, and (−_{F} x) +_{F} … +_{F} (−_{F} x) (−n times) if n is negative. By [11] there is a natural transformation that commutes with pushforwards:

ν_{X}: Ω^{∗}(X) ⊗_{𝕃^{∗}} Λ^{∗} → h^{∗}(X),

where 𝕃 = Ω(pt) is the Lazard ring and the map 𝕃^{∗} → Λ^{∗} is obtained by specializing the coefficients of the universal formal group law to the coefficients of the corresponding F.

Cellular and generically cellular varieties. A variety Y ∈ Sm_{k} is called cellular if there is a filtration of Y = Y_{0} ⊇ Y_{1} ⊇ … ⊇ Y_{m} ⊇ ∅ such that each Y_{i} \ Y_{i+1} is a disjoint union of affine spaces of the same rank c_{i}: Y_{i} \ Y_{i+1} ≅ 𝔸_{k}^{c_{i}} ∐ … ∐ 𝔸_{k}^{c_{i}}.

We call a variety X generically cellular if X_{k(X)} is a cellular variety over the function field k(X).

2.1. Example. Let G be a split semisimple algebraic group, B its Borel subgroup containing a fixed maximal split torus T and W the corresponding Weyl group. For any w ∈ W let l(w) denote its length. Let w_{0} ∈ W denote the longest element of W and N = l(w). It is well known that the flag variety X = G/B has the cellular structure given by the Schubert cells X_{w}:

X = X_{w_{0}} ⊇ ⋃_{l(w)=N−1}X_{w} ⊇ ⋃_{l(w)=N−2}X_{w} ⊇ … ⊇ X_{e} = pt,

where X_{w} is the closure of BwB/B in X.

2.2. Example. Let ζ ∈ Z^{1}(k, G) be a 1-cocycle with values in G. Then the twisted form _{ζ}(G/B) of X = G/B provides an example of a generically split variety.

h-motives. The notion of h-motives for the algebraic oriented cohomology theory h was studied by Nenashev-Zainoulline in [13], and Vishik-Yagita in [17]. We refer to [17, §2] for definition of the category of effective h-motives. In the present paper we will deal with the category of h-motives M_{h} with the inverted Tate object. It is constructed as follows:

Let SmProj_{k} denote the category of smooth projective varieties over k. Following [5] we consider the category Corr_{h} defined as follows: For X, Y ∈ SmProj_{k} with irreducible X and m ∈ ℤ we set

Corr_{m}(X, Y) = h_{dim X+m}(X × Y).

Objects of Corr_{h} are pairs (X, i) with X ∈ SmProj_{k} and i ∈ ℤ. For X ∈ SmProj_{k} with irreducible components X_{l} define the morphisms

Hom_{Corr}((X, i), (Y, j)) = ∐_{l} Corr_{i−j}(X_{l}, Y).

For α ∈ Hom((X, i), (Y, j)) and β ∈ Hom((Y, j), (Z, k)) the composition is given by the usual correspondence product: α∘β = (p_{XZ})_{h}((p_{Y Z})^{h}(β)·(p_{XY})^{h}(α)), where p_{XY} , p_{Y Z}, p_{XZ} denote the projections from X × Y × Z onto the corresponding summands.

Taking consecutive additive and idempotent completion of Corr_{h} we obtain the category M_{h} of h-motives with inverted Tate object. Objects of this category are (∐_{i}(X_{i}, n_{i}), p) where p is a matrix with entries p_{i,j} ∈ Corr_{n_{i}−n_{j}}(X_{i}, X_{j}) such that p ∘ p = p. Morphisms between objects are given by the set

Hom((∐(X_{i}, n_{i}), p), (∐(Y_{j}, m_{j}), q)) = q ∘ ⨁_{i,j} Corr_{n_{i}−m_{j}}(X_{i}, Y_{j}) ∘ p

considered as a subset of ⨁_{i,j} Corr_{n_{i}−m_{j}}(X_{i}, Y_{j}). This is an additive category where each idempotent splits. There is a natural tensor structure inherited from the category Corr_{h}:

(∐(X_{i}, n_{i}), p) ⊗ (∐(Y_{j}, m_{j}), q) = (∐_{i,j}(X_{i} × Y_{j}, n_{i} + n_{j}), p × q)

where p×q denotes the projector p_{(i_{1},j_{1})(i_{2},j_{2})} = p_{i_{1},i_{2}} ×q_{j_{1},j_{2}} : X_{i_{1}} ×Y_{j_{1}} → X_{i_{2}} ×Y_{j_{2}}.

There is a functor M^{h} : SmProj_{k} → M_{h} that maps a variety X to the motive M^{h}(X) = ((X, 0), id_{X}) and any morphism f: X → Y to the correspondence (Γ_{f})_{h}(1) ∈ h_{dim X}(X × Y) = Corr_{0}(X, Y), where Γ_{f}: X → X × Y is the graph inclusion. We will denote by Δ: X → X × X the diagonal embedding. Then Δ_{h}(1) is the identity in Corr_{0}(X, X).

Denote M^{h}(pt) by Λ and ((pt, 1), id_{pt}) by Λ(1). We call Λ(1) the h-Tate motive. We write Λ(n) for Λ(1)^{⊗n} and M^{h}(X)(n) for M^{h}(X)⊗Λ(n). The motive M^{h}(X)(n) is called the n-th twist of the motive M^{h}(X).

By definition we have

h^{i}(X) = Hom_{M_{h}}(M^{h}(X), Λ(i)) and h_{i}(X) = Hom_{M_{h}}(Λ(i), M^{h}(X)).

2.3. Lemma. For X ∈ SmProj_{k} with the structure morphism π: X → pt any isomorphism M^{h}(X) ≅ ⨁_{i} Λ(α_{i}) corresponds to a choice of two Λ-basis sets

{τ_{i} ∈ h^{α_{i}}(X)}_{i} and {ζ_{i} ∈ h_{α_{i}}(X)}_{i}

such that π_{h}(τ_{i}ζ_{j}) = δ_{i,j} in Λ and ∑_{i} ζ_{i} ⊗ τ_{i} = Δ_{h}(1) in h(X × X).

Proof. In the direct sum decomposition M^{h}(X) ≅ ⨁_{i} Λ(α_{i}) the i-th projection p_{i}: M^{h}(X) → Λ(α_{i}) is defined by an element τ_{i} ∈ h^{α_{j}}(X) and the i-th inclusion ı_{i}: Λ(α_{i}) → h(X) is defined by an element ζ_{i} ∈ h_{α_{i}}(X). Then by definition of the direct sum we obtain

π_{h}(τ_{i}ζ_{j}) = δ_{i,j} and ∑_{i} ζ_{i} ⊗ τ_{i} = Δ_{h}(1).

Let us check that {ζ_{i}}_{i} form a basis of h(X). Indeed, we have h(X) = ⨁_{j} h^{j}(X) = ⨁_{j} Hom_{M_{h}}(M^{h}(X), Λ(j)) ≅ ⨁_{j} Hom_{M_{h}}(⨁_{i} Λ(α_{i}), Λ(j)) = ⨁_{i} Λ_{α_{i}−∗} and ζ_{i} are the images of standard generators. So {ζ_{i}}_{i} form a Λ-basis of h(X). Finally, since {τ_{i}}_{i} are dual to {ζ_{i}}_{i}, {τ_{i}}_{i} is also a basis. □

2.4. Remark. Observe that any isomorphism M^{h}(X) ≅ ⨁ Λ(α_{i}) gives rise (canonically) to an isomorphism h^{∗}(X) ≅ ⨁_{i} Λ^{∗−α_{i}}.

3. Filtration on the cohomology ring

In the present section we construct a filtration on the oriented cohomology h(X) of a cellular variety X which will play an important role in the sequel.

3.1. Proposition. Assume that X is a cellular variety over k with the cellular decomposition X = X_{0} ⊇ X_{1} ⊇ … ⊇ X_{n} ⊇ ∅ where X_{i} \ X_{i+1} = ∐_{c_{i}} 𝔸^{α_{i}}. Then

(1) the h-motive of X splits as M^{h}(X) = ⨁_{i} Λ(α_{i})^{⊕c_{i}};

(2) the Künneth formula holds, i.e. the natural map h(X)⊗_{Λ}h(X) → h(X ×X) is an isomorphism;

(3) the specialization maps ν_{X}: Ω(X)⊗Λ → h(X) and ν_{X×X}: Ω(X ×X)⊗Λ → h(X × X) are isomorphisms.

Proof. By [5, Cor. 66.4] the Chow motive M^{CH}(X) splits, then [17, Cor. 2.9] implies that the motive M^{Ω}(X) splits into a sum of twisted Tate motives M^{Ω}(X) = ⨁_{i∈I} 𝕃(α_{i})^{⊕c_{i}}. By Lemma 2.3 there are elements ζ_{i,j}^{Ω} ∈ h_{α_{i}}(X) and τ_{i,j}^{Ω} ∈ h^{α_{i}}(X), j ∈ {1..c_{i}} such that π_{Ω}(ζ_{i,j}^{Ω}τ_{i′,j′}^{Ω}) = δ_{(i,j),(i′,j′)} and Δ_{Ω}(1) = ∑_{i,j} ζ_{i}^{Ω} ⊗ τ_{i}^{Ω}. Denote ζ_{i,j}^{h} = ν(ζ_{i,j}^{Ω} ⊗ 1) and τ_{i,j}^{h} = ν(τ_{i,j}^{Ω} ⊗ 1). Since ν commutes with pullbacks and pushforwards, π_{h}(ζ_{i,j}^{h}τ_{i′,j′}^{h}) = δ_{(i,j),(i′,j′)} and Δ_{h}(1) = ∑_{i,j} ζ_{i,j}^{h} ⊗τ_{i,j}^{h}. Then by Lemma 2.3 we have M^{h}(X) = ⨁_{i} Λ(α_{i})^{⊕c_{i}}, so (1) holds.

The Künneth map fits into the diagram

[figure: h(X) ⊗_{Λ} h(X); h(X × X); (⨁_{i} Λ^{∗−α_{i}}) ⊗_{Λ} (⨁_{j} Λ^{∗−α_{j}}); ⨁_{i,j} Λ^{∗−α_{i}−α_{j}}]

where the bottom arrow is an isomorphism, so the Künneth formula (2) holds.

Note that the natural map ν_{X} can be factored as follows

ν_{X}: Ω(X)⊗Λ = ⊕_{m}Hom_{M_{Ω⊗Λ}}(⨁ Λ(α_{i}), Λ(m)) → Hom_{M_{h}}(⨁ Λ(α_{i}), Λ(m)) = h(X).

Thus ν_{X} is an isomorphism. The same reasoning proves the statement for ν_{X×X}, hence, (3) holds. □

3.2. Definition. Let X be a cellular variety. Consider two basis sets ζ_{i} ∈ h_{α_{i}}(X) and τ_{i} ∈ h^{α_{i}}(X) provided by Proposition 3.1 and Lemma 2.3. We define the filtration h^{(l)}(X) as the Λ-linear span

h^{(l)}(X) = ⨁_{N−α_{i}⩾l} Λζ_{i} = ⨁_{α_{i}⩾l} Λτ_{i}.

We denote h^{(l/l+1)}(X) = h^{(l)}(X)/ h^{(l+1)}(X) and Gr^{∗} h(X) = ⨁_{l} h^{(l/l+1)}(X) to be the associated graded group. Lemma 3.4 implies that the latter is a graded ring.

3.3. Remark. In the case when the theory h is generically constant and satisfies the localization property, the filtration introduced above coincides with the topological filtration on h(X), i.e. with the filtration where the l-th term is generated over Λ by classes [Z → X] of projective morphisms Z → X birational on its image and dim X − dim Z ⩽ l. This fact follows from the generalized degree formula [11, Thm. 4.4.7].

3.4. Lemma. h^{(l_{1})}(X) · h^{(l_{2})}(X) ⊆ h^{(l_{1}+l_{2})}(X).

Proof. We have τ_{i}^{Ω}τ_{j}^{Ω} = ∑_{l} a_{l}ζ_{l}^{Ω} in Ω(X) for some a_{l} ∈ 𝕃. Then α_{i} + α_{j} = deg(a_{l}) + α_{l}. Since deg(a_{l}) ⩽ 0, α_{l} ⩾ α_{i} + α_{j} ⩾ l_{1} + l_{2} for any nontrivial a_{l}. Since ζ_{i}^{h} = ν(ζ_{i}^{Ω} ⊗ 1) we have ζ_{i}^{h}ζ_{j}^{h} = ∑_{l}(a_{l} ⊗ 1)ζ_{l}^{h} with α_{l} ⩾ α_{i} + α_{j} ⩾ l_{1} + l_{2}. So ζ_{i}^{h}ζ_{j}^{h} ∈ h^{(l_{1}+l_{2})}(X). □

3.5. Proposition. For a cellular X there is a graded ring isomorphism:

Ψ: ⨁_{i=0}^{N} h^{(i/i+1)}(X) → CH(X, Λ).

Proof. By Proposition 3.1 it is sufficient to prove the statement for h = Ω. Observe that Ω^{(l/l+1)}(X) is a free 𝕃-module with the basis τ_{i}^{Ω} + h^{(l+1)}(X) with α_{i} = l and CH^{i}(X, 𝕃) is a free 𝕃-module with basis τi^{CH} with α_{i} = l. Thus the 𝕃-module homomorphism Ψ_{l} defined by

Ψ_{l}(τ_{i}^{Ω} + h^{(i+1)}(X)) = τi^{CH}

is an isomorphism.

Let us check that Ψ = ⨁ Ψ_{l} preserves multiplication. For any i, j we have

τ_{i}^{Ω}τ_{j}^{Ω} = ∑_{m} a_{m}τ_{m}^{Ω}  (∗)

for some a_{m} ∈ 𝕃. Then for any m we have deg(a_{m}) + α_{m} = α_{i} + α_{j}. Then in h^{(α_{i}+α_{j}/α_{i}+α_{j}+1)} we have

τ_{i}^{Ω}τ_{j}^{Ω} = ∑_{α_{m}=α_{i}+α_{j}} a_{m}τ_{m}^{Ω} modulo + h^{(α_{i}+α_{j}+1)}(X)

Observe that 𝕃^{0} = ℤ and for all a_{m} ∈ 𝕃 such that deg(a_{m}) < 0 we have that a_{m} ⊗ 1_{ℤ} = 0 in ℤ. Thus tensoring (∗) with 1_{ℤ} we get

τi^{CH}τj^{CH} = ∑_{α_{m}=0}(a_{m} ⊗ 1)τm^{CH}.

So Ψ_{α_{i}+α_{j}}(τ_{i}^{Ω} + h^{(α_{i}+1)}(X) · τ_{j}^{Ω} + h^{(α_{j}+1)}(X)) = τi^{CH} · τj^{CH}. Hence, Ψ is a graded ring isomorphism. □

3.6. Lemma. Ψ(ζ_{i}^{h} + h^{(α_{i}+1)}(X)) = ζi^{CH}.

Proof. It is sufficient to show the statement for h = Ω^{∗}. Consider the expansion ζ_{i}^{Ω} = ∑ a_{j}τ_{j}^{Ω} for some a_{j} ∈ 𝕃 with deg a_{j} + α_{j} = N − α_{i}. Since deg a_{j} ⩽ 0 we have

ζ_{i}^{Ω} = ∑_{deg a_{j}=0} a_{j}τ_{j}^{Ω} mod Ω^{(N−α_{i}+1)}(X).

Therefore, Ψ(ζ_{i}^{Ω} + Ω^{(N−|w|+1)}(X)) = ζi^{CH}. □

4. Oriented cohomology of a group

In the present section, using the filtration introduced in 3.2 we compute algebraic cobordism for some groups of small ranks and for P GL_{n}, n ≥ 2. We also construct nontrivial elements in the difference between the topological and the γ-filtration on K_{0}(G/B).

In this section we assume that the associated to h weak Borel-Moore homology theory satisfies the localization property of [11, Definition 4.4.6]. Examples of such theories include h(−) = Ω(−) ⊗ Λ, or any oriented cohomology theory in the sense of Panin-Smirnov [14].

Consider the variety X = G/B where G is a split semisimple algebraic group. Let π_{G/B}: G → X be the quotient map. According to Example 2.1 X is cellular. For any w ∈ W we fix a minimal decomposition w = s_{i_{1}} … s_{i_{m}} into simple reflections. Denote the corresponding multiindex by I_{w} = (i_{1}, … i_{m}) and consider the Bott-Samelson variety X_{I_{w}}/B [4, §11]. Then p_{I_{w}}: X_{I_{w}}/B → G/B is a desingularisation of the Schubert cell X_{w}. Take

ζ_{w} = (p_{I_{w}})_{h}(1) ∈ h_{l(w)}(X) = Hom_{M_{h}}(Λ(l(w)), M^{h}(X)

to be the embedding in the direct sum decomposition ⨁_{w∈W} Λ(l(w)) ≅ M^{h}(G/B). So, with this choice of isomorphism M^{h}(G/B) ≅ ⨁_{w∈W} Λ(l(w)) the basis given by Lemma 2.3 coincides with the basis ζ_{I_{w}} constructed in [4, §13].

Let Λ[[T^{∗}]]_{F} be the formal group algebra introduced by Calmès-Petrov-Zainoulline in [4, §2], where T^{∗} is the character lattice of T and F is the formal group law of the theory h. There is the characteristic map 𝔠_{F} : Λ[[T^{∗}]]_{F} → h(G/B) such that 𝔠_{F} (x_{λ}) = c_{1}^{h}(L(λ)) for a character λ. By [7, Prop. 5.1] there is a short exact sequence

(1)  0 → 𝔠(I_{F}) → h(G/B) ⟶^{π_{G/B}^{h}} h(G) → 0,

where I_{F} denotes the ideal in Λ[[T^{∗}]]_{F} generated by x_{α} for α ∈ T^{∗}. By [4, Lem. 4.2] there is a graded algebras isomorphism ψ: ⨁_{m=0}^{∞} I_{F}^{m}/I_{F}^{m+1} → S_{Λ}^{∗}(T^{∗}) where S_{Λ}^{∗}(T^{∗}) denotes the symmetric algebra over T^{∗}. Let F_{a} denote the additive formal group law. We will need the following 4.1. Lemma. The following diagram commutes:

\xymatrix{⨁_{m=0}^{∞} I_{F}^{m}/I_{F}^{m+1} \ar[r]^{Gr𝔠_{F}} \ar[d]^{ψ} & ⨁_{m=0}^{∞} h^{(m/m+1)}(G/B) \ar[d]^{Ψ} \\ S_{Λ}^{∗}(T^{∗}) \ar[r]^{𝔠_{Fa}} & CH(G/B, Λ)}

Proof. By definition, it is sufficient to prove that Ψ(pr_{1}(c_{1}^{h}(L_{λ}))) = c1^{CH}(L_{λ}) Consider the expansion c_{1}^{Ω}(L_{λ}) = ∑ r_{w}τ_{w}^{Ω} we have that r_{w} ∈ 𝕃^{0} = ℤ for |w| = 1. Then c_{1}^{h}(L_{λ}) = ∑(r_{w} ⊗ 1_{Λ})τ_{w}^{h} and

Ψ(pr_{1}(c_{1}^{h}(L_{λ}))) = ∑_{|w|=1} r_{w}τw^{CH} = c_{1}^{Ω}(L_{λ}) ⊗ 1_{ℤ} = c1^{CH}(L_{λ}).  □

4.2. Lemma. For the additive group law the induced filtration satisfies

𝔠(I_{F_{a}})CH(X, Λ) ∩ CH^{i}(X, Λ) = ∑ 𝔠(x_{α})CH^{i−1}(X, Λ).

Proof. Note that 𝔠(I_{a})CH(X, Λ) is generated by the elements 𝔠(x_{α}) ∈ CH^{1}(X, R). For any element z = ∑ 𝔠(x_{α})y_{α} of 𝔠(I_{a})CH(X, Λ) we have that z lies in the CH^{i}(X, Λ) if and only if y_{α} ∈ CH^{i−1}(X, Λ). Therefore 𝔠(I_{a})CH(X, Λ)∩CH^{i}(X, Λ) = ∑ 𝔠(x_{α})CH^{i−1}(X, Λ). □

Let h^{(i)}(G) denote the image π_{G/B}^{h}(h^{(i)}(G/B)) and let h^{(i/i+1)}(G) denote the quotient h^{(i)}(G)/ h^{(i+1)}(G).

4.3. Proposition. For every i there is an exact sequence:

0 → \frac{𝔠(I) h^{(i−1)}(X)}{h^{(i+1)}(X)} → \frac{(𝔠(I) h(X))∩h^{(i)}(X)}{h^{(i+1)}(X)} → CH^{i}(G, Λ) → h^{(i/i+1)}(G) → 0.

Proof. By [3, Prop. 2, §2.4] we obtain from (1) the short exact sequence:

0 → \frac{(𝔠(I) h(X))∩h^{(i)}(X)}{h^{(i+1)}(X)} → \frac{h^{(i)}(X)}{h^{(i+1)}(X)} → h^{(i/i+1)}(G) → 0.

By Lemma 4.2 applied to the case of additive formal group law, the above sequence turns into

0 → ∑ 𝔠(x_{α})CH^{i−1}(X, R) → CH^{i}(X, R) → CH^{i}(G, R) → 0.

Observe that for isomorphism (Ψ^{i})^{−1} we have

(Ψ^{i})^{−1} (∑ 𝔠(x_{α})CH^{i−1}(X, Λ)) = (\frac{𝔠(I) h^{(i−1)}(X)}{h^{(i+1)(X)}}) ⊆ (\frac{𝔠(I) h^{(} X)∩h^{(i)}(X)}{h^{(i+1)(X)}}) (∗)

Then we get the following diagram with exact rows:

[figure: 0; ∑ 𝔠(x_{α})CH^{i−1}(X, Λ); CH^{i}(X, Λ); (Ψ^{i})^{−1}; CH^{i}(G, Λ); 0; 0; \frac{𝔠(I) h(X)∩h^{(i)}(X)}{h^{(i+1)(X)}}; h^{(i/i+1)}(X); h^{(i/i+1)}(G); 0]

The latter sequence and (∗) gives rise to the exact sequence

0 → \frac{𝔠(I) h^{(i−1)}(X)}{h^{(i+1)}(X)} → \frac{(𝔠(I) h(X))∩h^{(i)}(X)}{h^{(i+1)}(X)} → CH^{i}(G, Λ) → h^{(i/i+1)}(G) → 0. □

4.4. Corollary. Assume that pullbacks pG/B^{CH}(X_{w_{i}}) of Schubert cells generate CH(G) as ℤ-algebra for some elements w_{1}, … w_{m} in W. Then the pullbacks of Schubert cells p_{G/B}^{h}(ζ_{w_{i}}) generate h(G) as Λ-algebra.

Proof. By 4.3 classes of p_{G/B}^{h}(ζ_{w_{i}}) generate the associated graded ring ⨁_{i} h^{(i/i+1)}(G). Then p_{G/B}^{h}(ζ_{w_{i}}) generate h(G), since the filtration is finite. □

The following observations will be useful for computations

4.5. Lemma. If CH^{i}(G) = 0 then \frac{𝔠(I) h^{(i−1)}(G/B)}{h^{(i+1)}(G/B)} = \frac{h^{(i)}(G/B)}{h^{(i+1)}(G/B)}

4.6. Lemma. Assume that CH^{1}(G) = 0. Then for i ⩾ 2

𝔠(I) h(G/B) ∩ h^{(i)}(G/B) = 𝔠(I) h^{(1)}(G/B) ∩ h^{(i)}(G/B)

Proof. Note that ideal 𝔠(I) h(G/B) is generated by 𝔠(x_{α}) where α runs over the basis of the character lattice. □

5. Examples of computations

The results of the previous section allow us to obtain some information concerning the ring h(G) from CH(G). Moreover, in some cases it allows us to compute h(G).

We follow the notation of the previous section. We denote ⨁ h^{(i/i+1)}(G) by Gr^{∗} h(G). For a ∈ h^{i}(G) let \overline{a} ∈ h^{(i/i+1)}(G) denote its residue class. Let α be the projection CH^{∗}(G, Λ) → ⊕Gr^{∗} h(G).

Algebraic cobordism of G_{2}. According to [12] we have

CH^{∗}(G_{2}, ℤ) = ℤ[x^{3}]/(x_{3}^{2}, 2x_{3})

where x_{3} = π^{CH}(ζ_{212}), π: G → G/B is the projection and ζ_{212} is the Schubert cell corresponding to the word w = s_{2}s_{1}s_{2}. Let y_{3} denote the pullback π^{Ω}(ζ_{212}^{Ω}) of the corresponding Schubert cell in the ring Ω(G/B) (see Theorem 13.12 of [4]). Observe that α^{3}(x_{3}) = \overline{y_{3}}. Since CH(G_{2}, 𝕃) is generated by 1 and x_{3}, Gr^{∗}Ω(G_{2}) is generated by 1 and \overline{y_{3}}. Then by[3, §2.8] Ω(G_{2}) is generated by 1 and y_{3} ∈ Ω^{(3)}(G_{2}). Since 2x_{3} = 0, then \overline{2y_{3}} so 2y_{3} ∈ Ω^{(4)}(G_{2}) which is zero since CH^{i}(G_{2}) = 0 for i ⩾ 4. Thus 2y_{3} = 0 and y_{3}^{2} ∈ Ω^{(6)}(G_{2}) = 0.

Let us now compute Ω^{(3)}(G_{2}). Proposition 4.3 gives us the exact sequence

0 → \frac{𝔠(I)Ω^{(2)}(G_{2}/B)}{Ω^{(4)}(G_{2}/B)} → \frac{𝔠(I)Ω(G_{2}/B)∩Ω^{(3)}(G_{2}/B)}{Ω^{(4)}(G_{2}/B)} → 𝕃/2 · x_{3} → Ω^{(3)}(G_{2}).

Note that since 𝔠(I)Ω(G/B) is generated by 𝔠(x_{1}) and 𝔠(x_{2}). By Lemma 4.5 we have

\frac{𝔠(I)Ω(G_{2}/B)}{Ω^{(4)}(G_{2}/B)} = \frac{⟨ζ_{12121},ζ_{21212}⟩Ω(G_{2}/B)}{Ω^{(4)}(G_{2}/B)}

Then 𝔠(I) ∩ Ω^{(3)}(G_{2}/B)/Ω^{(4)}(G_{2}/B) equals to the set

{x = aζ_{12121} + bζ_{21212} | x ∈ Ω^{(3)}}.

It is enough to consider only a, b in Ω^{(1)}(G_{2}/B) \ Ω^{(2)}(G_{2}/B) since for a, b ∈ Ω^{(0)}(G_{2}/B) \ Ω^{(1)}(G_{2}/B) we have x ∉ Ω^{(2)}(G_{2}/B). So we consider

a = r_{1}ζ_{12121} + r_{2}ζ_{21212} and b = s_{1}ζ_{12121} + s_{2}ζ_{21212} for r_{1}, r_{2}, s_{1}, s_{2} ∈ 𝕃.

Using the multiplication table for G_{2}/B from [4] we obtain that x equals

(r_{2} + s_{1} + s_{2})ζ_{1212} + (3r_{1} + r_{2} + s_{1})ζ_{2121} + (r_{2} + s_{1} + 3r_{1})a_{1}ζ_{121} + (r_{2} + s_{1})a_{1}ζ_{212}

modulo Ω^{(4)}(G_{2}/B). Then x ∈ Ω^{(3)}(G_{2}/B)/Ω^{(4)}(G_{2}/B) iff r_{2} + s_{1} + s_{2} = 0 and 3r_{1} + r_{2} + s_{1} = 0. Therefore, r_{2} + s_{1} = −3r_{1} and s_{2} = 3r_{1}. So

x + Ω^{(4)}(G_{2}/B) = −3r_{1}a_{1}ζ_{212} + Ω^{(4)}(G_{2}/B).

Hence, the kernel of 𝕃 · 2x_{3} → Ω^{(3)}(G_{2}) is generated by 3a_{1}x_{3}. Then

Ω^{(3)}(G_{2}) = 𝕃/(2, 3a_{1}) · y_{3} = 𝕃/(2, a_{1}) · y_{3}

and we obtain that

(2)  Ω(G_{2}) = 𝕃[y^{3}]/(y_{3}^{2}, 2y_{3}, a_{1}y_{3}).

Observe that taking the latter equality modulo 2 we obtain the result established by Yagita in [19].

Algebraic cobordism of groups SO_{n}, Spin_{m} for n = 3, 4 and m = 3, 4, 5, 6. According to [12]

CH(Spin_{i}) = ℤ for i = 3, 4, 5, 6.

Then by Proposition 4.3 we obtain

(3)  Ω(Spin_{i}) = 𝕃 for i = 3, 4, 5, 6.

We have CH(SO_{3}) = ℤ[x_{1}]/(2x_{1}, x_{1}^{2}) where x_{1} = π^{CH}(ζ_{w_{0}s_{1}}). Since CH^{i}(SO_{3}) = 0 for i ⩾ 2, Ω^{(2)}(SO_{3}) = 0. For i = 1 two left terms of exact sequence of Proposition 4.3 coincide, so there is an isomorphism CH^{1}(SO_{3}) → Ω^{(1)}(SO_{3}). Hence, we obtain

(4)  Ω(SO_{3}) = 𝕃[y^{1}]/(2y_{1}, y_{1}^{2}), where y_{1} = π^{Ω}(ζ_{w_{0}s_{1}}).

Since CH(SO_{4}) = ℤ[x_{1}]/(2x_{1}, x_{1}^{2}) the same reasoning proves that

(5)  Ω(SO_{4}) = 𝕃[y^{1}]/(2y_{1}, y_{1}^{2}).

Oriented cohomology of P GL_{n}.

5.1. Lemma. For any oriented cohomology theory h with the coefficient ring Λ and the formal group law F we have

h(P GL_{n}) = Λ[x]/(x^{n}, nx^{n−1}, … \binom{n}{d}x^{d}, …, nx, n ·_{F} x).

Proof. Consider the variety of complete flags X = SL_{n}/B. Let F_{i} denote the tautological vector bundle of dimension i over X. Let L_{1} = F_{1} and L_{i} = F_{i}/F_{i−1} for i = 2, … n. Then, by [9, Thm. 2.6] we have

h(X) ≅ Λ[x_{1}, … x_{n}]/S(x_{1}, … x_{n})  (∗)

where S(x_{1}, … x_{n}) denotes the ideal generated by positive degree symmetric polynomials in variables x_{1}, … x_{n}, and the isomorphism sends x_{i} to the Chern class c_{1}^{h}(L_{i}). The maximal split torus T ⊆ SL_{n} consists of diagonal matrices with trivial determinant. Let χ_{i} ∈ T̂ denote the character that sends the diagonal matrix to its i-th entry. So, the character lattice equals to M = ℤχ_{1} ⊕ … ⊕ ℤχ_{n}/(χ_{1} + … + χ_{n}). Observe that L_{i} coincides with the line bundle L(χ_{i}), so by definition we have that x_{i} = 𝔠(χ_{i}), where 𝔠: Λ[[M]]_{F} → h(X) is the characteristic map. Note that the roots of P GL_{n} = P SL_{n}/μ_{n} are equal nχ_{1}, χ_{2} − χ_{1}, …, χ_{n} − χ_{1}.

According to [7, 5.1] we have

h(P GL_{n}) = h(X)/(𝔠(nχ_{1}), 𝔠(χ_{2} − χ_{1}), …, 𝔠(χ_{n} − χ_{1})).

Then in the quotient we have

\overline{𝔠(χ_{i})} = \overline{𝔠(χ_{1} + χ_{i} − χ_{1})} = \overline{𝔠(χ_{1}) +_{F} 𝔠(χ_{i} − χ_{1})} = \overline{𝔠(χ_{1})}.

Taking x = \overline{𝔠(χ_{1})} by (∗) we get

h(P GL_{n}) = Λ[x]/(S(x, …, x), n ·_{F} x).

According to [9] S(x_{1}, … x_{n}) is generated by polynomials f_{n}(x_{n}), f_{n−1}(x_{n}, x_{n−1}), …, f_{1}(x_{n}, …, x_{i}) where f_{i}(x_{n}, …, x_{i}) denotes the sum of all degree i monomials in x_{n}, …, x_{i}. Note that \binom{n}{d} equals to the number of degree d monomials in n − d + 1 variables. Then substituting x_{1} = … x_{n} = x we obtain that x^{n}, nx^{n−1}, \binom{n}{d}x^{n−1}, …, nx generate the ideal S(x, …, x). □

5.2. Example. For a prime number p and 0 < d < p the coefficient \binom{p}{d} is divisible by p. By [8, Rem. 5.4.8] over Λ/pΛ we have p ·_{F} x = pβ_{0}(x) + β_{1}(x^{p}). Thus, the ideal I = (x^{p}, px^{p−1}, … \binom{p}{d}x^{d}, …, px, p ·_{F} x) is generated by x^{p}, px. So for any prime p we have

h(P GL_{p}) = Λ[x]/(px, x^{p}).

In the case h = K_{0} this agrees with [20, 3.6].

Topological and the γ-filtration. Proposition 4.3 allows to estimate the difference between the topological and the Grothendieck γ-filtration on K_{0}(G/B) for a split linear algebraic group G. Namely, consider two filtrations on K_{0}(G/B):

γ-filtration: γ^{i}(G/B) = ⟨c_{1}(L(λ)) | λ ∈ T^{∗}⟩ [20, Definition 4.2], topological filtration: τ^{i}(G/B) = ⟨[O_{V}] | codim (V) ⩾ i⟩.

5.3. Proposition. Let G be a split semisimple simply connected linear algebraic group such that CH^{i}(G) = 0 for 1 ⩽ i ⩽ n − 1 and CH^{n}(G) ≠ 0. Let ζ_{w} be a Schubert cell such that π^{CH}(ζ) is nontrivial in CH^{n}(G).

Then γ^{i}(G/B) + τ^{i+1}(G/B) = τ^{i}(G/B) for i < n and the class of ζ_{w}^{K_{0}} is nontrivial in τ^{n}(G/B)/γ^{n}(G/B).

Proof. As shown in [15] K_{0}(G) = ℤ for a simply connected group G. Then characteristic map 𝔠 is surjective [6, §1B]. We have K_{0}^{(1)}(G/B) = τ^{1} = γ^{1}. Note that K_{0}^{(i)}(G/B) = τ^{i}. Then γ^{1}τ^{0} ∩τ^{i} = τ^{i} and the Proposition 4.3 gives us a short exact sequence for all i ⩾ 1:

0 → \frac{γ^{1}τ^{i−1}}{τ^{i+1}} → \frac{τ^{i}}{τ^{i+1}} → CH^{i}(G, ℤ[β, β^{−1}]) → 0.

Then for any 1 ⩽ i < n we have γ^{1}τ^{i−1}/τ^{i+1} = τ^{i}/τ^{i+1}. By induction we get τ^{i} = γ^{i} + τ^{i+1} for i < n and for i = n we get By induction we get τ^{i} = γ^{i} + τ^{i+1} for i < n and for i = n we get

0 → \frac{γ^{n}}{τ^{n+1}} → \frac{τ^{n}}{τ^{n+1}} → CH^{n}(G, ℤ[β, β^{−1}]) → 0.

So for any nontrivial element of of CH^{n}(G) the class of its preimage is nontrivial in τ^{n}/γ^{n}. □

6. Applications to h-motivic decompositions

Throughout this section we consider a generically cellular variety X of dimension N and an oriented cohomology theory h^{∗} that is generically constant and is associated with weak Borel-Moore homology h_{∗} which satisfies the localization property. These assumptions imply that the generalized degree formula of Levine-Morel [11, Theorem 4.4.7] holds. The aim of this section is to prove theorems A, B and C of the introduction which provide a comparison between the Chow motive M(X) and the h-motive M^{h}(X) of X.

Let L be the splitting field of X and \overline{X} = X ×_{Spec k} Spec L. Let p denote the projection p: \overline{X} × \overline{X} → X × X. Since \overline{X} is cellular, we may consider a filtration on h(\overline{X}) introduced in 3.2. It gives rise to a filtration on h(\overline{X} × \overline{X}) = h(\overline{X}) ⊗_{Λ} h(\overline{X}). Namely, we set

h^{(l)}(\overline{X} × \overline{X}) = ∑_{i+j=l} h^{(i)}(\overline{X}) ⊗_{Λ} h^{(j)}(\overline{X}).

On h(X × X) we consider the induced filtration

h^{(l)}(X × X) = (p^{h})^{−1}(h^{(l)}(\overline{X} × \overline{X})).

Denote the quotient h^{(l)}(\overline{X} × \overline{X})/ h^{(l+1)}(\overline{X} × \overline{X}) by h^{(l/l+1)}(\overline{X} × \overline{X}) and denote by pr_{l}: h^{(l)}(\overline{X} × \overline{X}) → h^{(l/l+1)}(\overline{X} × \overline{X}) the usual projection. Denote

h_{2N−i}^{(i)}(X × X) = h^{(i)}(X × X) ∩ h_{2N−i}(X × X) and
h_{2N−i}^{(i)}(\overline{X} × \overline{X}) = h^{(i)}(\overline{X} × \overline{X}) ∩ h_{2N−i}(\overline{X} × \overline{X}).

6.1. Lemma. There is a graded ring isomorphism

Φ: ⨁_{i=0}^{2N} h^{(i/i+1)}(\overline{X} × \overline{X}) → CH^{∗}(\overline{X} × \overline{X}, Λ).

Proof. Since

⨁_{i=0}^{2N} h^{(i/i+1)}(\overline{X} × \overline{X}) = ⨁_{i=0}^{N} h^{(i/i+1)}(\overline{X}) ⊗_{Λ} ⨁_{i=0}^{N} h^{(i/i+1)}(\overline{X})

and CH(\overline{X} × \overline{X}, Λ) = CH(\overline{X}, Λ) ⊗_{Λ} CH(\overline{X}, Λ) take Φ = Ψ ⊗ Ψ, where Ψ is defined in 3.5. □

6.2. Remark. The restriction of Φ_{i} gives an isomorphism Φ_{i}: h_{2N−i}^{(i/i+1)}(\overline{X} × \overline{X}) → CH^{i}(\overline{X} × \overline{X}, Λ^{0}).

The following lemma provides an h-version of the Rost Nilpotence Theorem:

6.3. Lemma. The kernel of the pullback map p^{h} : End(M^{h}(X)) → End(M^{h}(\overline{X})) consists of nilpotents.

Proof. Consider a diagram

\xymatrix{End(M^{Ω}(X)) \ar[d] \ar[r]^{p^{Ω}} & End(M^{Ω}(\overline{X})) \ar[d] \\ End(M^{CH}(X)) \ar[r] & End(M^{CH}(\overline{X}))}

where vertical arrows are ring homomorphisms that arise from the canonical map Ω(−) → CH(−). By [17, Prop. 2.7] they are surjective with kernels consisting of nilpotents. The kernel of the bottom arrow consists of nilpotents by [18, Prop 3.1]. Then the kernel of the upper arrow consists of nilpotents as well.

Tensoring the upper arrow with Λ we obtain ker(p^{Ω})⊗Λ → Ω_{N}(X ×X)⊗Λ ⟶^{p^{Ω}⊗id} Ω(\overline{X} ×\overline{X})⊗Λ, so ker(p^{Ω})⊗Λ covers the kernel of p^{Ω} ⊗id, thus ker(p^{Ω} ⊗id) consists of nilpotents. Now the specialization maps fit into the commutative diagram.

\xymatrix{Ω_{N}(X × X) ⊗_{𝕃} Λ \ar[d]^{ν_{X×X}} \ar[r]^{p^{Ω}⊗id} & Ω(\overline{X} × \overline{X}) ⊗_{𝕃} Λ \ar[d]^{≅} \\ h(X × X) \ar[r]^{p^{h}} & h(\overline{X} × \overline{X})}

where the right arrow is an isomorphism by 2.4 and 3.1, and the map ν_{X×X} is surjective. So the kernel of the bottom map consists of nilpotents. □

6.4. Lemma. We have h^{(N+i)}(\overline{X} × \overline{X}) ∘ h^{(N+j)}(\overline{X} × \overline{X}) ⊆ h^{(N+i+j)}(\overline{X} × \overline{X}).

Proof. Consider a generator ζ_{m} ⊗ τ_{n} ∈ h^{(N+i)}(\overline{X} × \overline{X}) where N − α_{m} + α_{n} ⩾ N + i and ζ_{m′} ⊗ τ_{n′} ∈ h^{(N+j)}(\overline{X} × \overline{X}) where N − α_{m′} + α_{n′} ⩾ N + j. The composition

(ζ_{m} ⊗ τ_{n}) ∘ (ζ_{m′} ⊗ τ_{n′}) = deg(τ_{n}ζ_{m′})(ζ_{m} ⊗ τ_{n′}) = δ_{n,m′} · (ζ_{m} ⊗ τ_{n′})

is nonzero iff n = m′. In this case N − m + n′ = (N − m + n) + (N − m′ + n′) − N ⩾ N + i + j. Thus ζ_{m} ⊗ τ_{n′′} lies in h^{(N+i+j)}(\overline{X} × \overline{X}). □

6.5. Remark. Indeed, the lemma implies that h^{(N)}(\overline{X} × \overline{X}) is a ring with respect to the composition product, and h^{(N+1)}(\overline{X} × \overline{X}) is its two-sided ideal. Since the composition of homogeneous elements is homogeneous, h_{N}^{(N)}(\overline{X} × \overline{X}) is also a ring with respect to the composition.

6.6. Lemma. The isomorphism Φ_{N}: h^{(N/N+1)}(\overline{X} × \overline{X}) → CH^{N}(\overline{X} × \overline{X}, Λ) is a ring homomorphism with respect to the composition product.

Proof. This immediately follows from the fact that Φ maps residue classes of ζ_{w}^{h}⊗τ_{v}^{h}

to ζw^{CH} ⊗ τv^{CH}. □

6.7. Lemma. Let Y be a twisted form of X, i.e. Y_{L} ≅ X_{L} = \overline{X}. For every codimension m consider the diagram, where p: \overline{X} ×\overline{X} → X ×Y denotes the projection.

\xymatrix{h_{2N−m}^{(m)}(X × Y) \ar[r]^{pr_{m}∘p^{h}} & h_{2N−m}^{(m/m+1)}(\overline{X} × \overline{X}) \\ CH^{m}(X × Y, Λ^{0}) \ar[r]^{p^{CH}} & CH^{m}(\overline{X} × \overline{X}, Λ^{0}) \ar[u]_{Φ^{m}}}

Then im(Φ^{m} ∘ p^{CH}) ⊆ im pr_{m} ∘ p^{h}.

Proof. Note that CH^{m}(X × Y, Λ^{0}) is generated over Λ^{0} by classes i_{CH}(1) where i: Z̃ → Z ↪ X × Y, Z is a closed integral subscheme of codimension m, Z̃ ∈ Sm_{k} and Z̃ → Z is projective birational. Consider the Cartesian diagram

\xymatrix{Z̃ \ar[d]^{i} & Z̃_{L} \ar[d]^{j} \ar[l]_{q} \\ X ×_{k} Y & \overline{X} ×_{L} \overline{X} \ar[l]_{p}}

Since this diagram is transverse, then

j_{h} ∘ q^{h} = p^{h} ∘ i_{h} and j_{CH} ∘ q^{CH} = p^{CH} ∘ i_{CH}.

By lemma 6.8 we have Φ^{m} ∘ j_{CH}(1) = pr_{m}(j_{h}(1)). Then Φ^{m} ∘ p^{CH}(i_{CH}(1)) = Φ^{m} ∘ j_{CH}(1) = pr_{m}(j_{h}(1)) = pr_{m}(p^{h} ∘ i_{h}(1)) ∈ im pr_{m} ∘ p_{m}^{h}. □

6.8. Lemma. Consider a morphism j: Z̃ → \overline{X} ×_{L} \overline{X}, where Z̃ is a smooth irreducible scheme and j is projective of relative dimension −m. It induces two pushforward maps j_{h} : h(Z̃) → h(\overline{X} × \overline{X}) and j_{CH} : CH(Z̃, Λ) → CH(\overline{X} × \overline{X}, Λ).

Then j_{h}(1) ∈ h_{2N−m}^{(m)}(\overline{X} × \overline{X}) and Φ^{m}(j_{CH}(1)) = pr_{m}(j_{h}(1)).

Proof. Observe that

j_{h}(1) = j_{Ω}(1) ⊗_{𝕃} 1_{Λ} and j_{CH}(1) = j_{Ω}(1) ⊗_{𝕃} 1_{ℤ}.

Expanding in the basis we obtain

j_{Ω}(1) = ∑_{i_{1},i_{2}} r_{i_{1},i_{2}}τ_{i_{1}}^{Ω} ⊗ τ_{i_{2}}^{Ω} for some r_{i_{1},i_{2}} ∈ 𝕃.  (∗)

Since j_{Ω}(1) is homogeneous of degree m, we have

r_{i_{1},i_{2}} ∈ 𝕃^{m−α_{i_{1}} −α_{i_{2}}} .  (∗∗)

Then for every nonzero r_{i_{1},i_{2}} we have α_{i_{1}} +α_{i_{2}} ⩾ m. So each τ_{i_{1}}^{Ω}⊗τ_{i_{2}}^{Ω} ∈ Ω^{(m)}(\overline{X}×\overline{X}) and, thus, j_{Ω}(1) ∈ Ω_{2N−m}^{(m)}(\overline{X} × \overline{X}). Taking (∗) modulo Ω^{(m+1)}(\overline{X} × \overline{X}) we obtain

j_{Ω}(1) + Ω^{(m+1)}(\overline{X} × \overline{X}) = ∑_{α_{i_{1}} +α_{i_{2}} =m} r_{i_{1},i_{2}}τ_{i_{1}}^{Ω} ⊗ τ_{i_{2}}^{Ω} + Ω^{(m+1)}(\overline{X} × \overline{X}).

If α_{i_{1}} + α_{i_{2}} = m then r_{i_{1},i_{2}} ∈ 𝕃^{0} = ℤ by (∗∗). Thus taking j_{h}(1) = j_{Ω}(1) ⊗_{𝕃} 1_{Λ} and j_{CH}(1) = j_{Ω}(1) ⊗_{𝕃} 1_{ℤ} we get

pr_{m}(j_{h}(1)) = pr_{m}( ∑_{α_{i_{1}} +α_{i_{2}} =m} r_{i_{1},i_{2}}τ_{i_{1}}^{h} ⊗ τ_{i_{2}}^{h})

and

j_{CH}(1) = j_{Ω}(1) ⊗_{𝕃} 1_{ℤ} ⊗_{ℤ} 1_{Λ} = ∑_{α_{i_{1}} +α_{i_{2}} =m} r_{i_{1},i_{2}}τi_{1}^{CH} ⊗ τi_{2}^{CH}.

Then Φ^{m}(j_{CH}(1)) = pr_{m}(j_{h}(1)), since Φ^{m}(τi_{1}^{CH} ⊗ τi_{2}^{CH}) = pr_{m}(τ_{i_{1}}^{h} ⊗ τ_{i_{2}}^{h}). □

6.9. Lemma. The kernel of the composition homomorphism

pr_{N} ∘ p^{h} : h_{N}^{(N)}(X × X) → h_{N}^{(N)}(\overline{X} × \overline{X}) → h_{N}^{(N/N+1)}(\overline{X} × \overline{X})

consists of nilpotents.

Proof. This follows from Rost nilpotence and the fact that h^{(N+1)}(\overline{X} × \overline{X}) is nilpotent by Lemma 6.4. □

6.10. Lemma. Let C be an additive category, A, B ∈ Ob(C). Let f ∈ Hom_{C}(A, B) and g ∈ Hom_{C}(B, A)such that f ∘ g − id_{B} is nilpotent in the ring End_{C}(B) and g ∘ f − id_{A} is nilpotent in the ring End_{C}(A). Then A is isomorphic to B.

Proof. Denote α = id_{A} − gf and β = id_{B} − fg. Take natural n such that α^{n+1} = 0 and β^{n+1} = 0. Then gf = id_{A} − α is invertible and (gf)^{−1} = id_{A} + α + … + α^{n}. Analogously (fg)^{−1} = id_{B} + β + … + β^{n}. So we have

gf(id_{A} + (id_{A} − gf) + … + (id_{A} − gf)^{n}) = id_{A}

Since (id_{A} − gf)^{n} = ∑_{i=0}^{n}(−1)^{i}\binom{n}{i}(gf)^{i}, we have

g ∑_{m=0}^{n} (∑_{i=0}^{m}(−1)^{i}\binom{m}{i}(fg)^{i}f) = id_{A}  (∗)

and

fg(id_{B} + (id_{B} − fg) + … + (id_{B} − fg)^{n}) = id_{B}

implies

∑_{m=0}^{n} (∑_{i=0}^{m}(−1)^{i}\binom{m}{i}f(gf)^{i}) g = id_{B}.  (∗∗)

Then take

f_{1} = ∑_{m=0}^{n} (∑_{i=0}^{m}(−1)^{i}\binom{m}{i}(fg)^{i}f) = ∑_{m=0}^{n} (∑_{i=0}^{m}(−1)^{i}\binom{m}{i}f(gf)^{i}) .

Then (∗) implies gf_{1} = id_{A} and (∗∗) implies f_{1}g = id_{B}. So f_{1} and g establish inverse isomorphisms between A and B. □

6.11. Corollary. Suppose p_{1} and p_{2} are two idempotents in End(M^{h}(\overline{X})) such that p_{1} − p_{2} is nilpotent. Then the motives (\overline{X}, p_{1}) and (\overline{X}, p_{2}) are isomorphic.

Proof. Take

f = p_{2} ∘p_{1} ∈ Hom_{M_{h}}((\overline{X}, p_{1}), (\overline{X}, p_{2})) and g = p_{1} ∘p_{2} ∈ Hom_{M_{h}}((\overline{X}, p_{2}), (\overline{X}, p_{1})).

Let us check that f ∘ g − id_{(X,p_{2})} = p_{2}p_{1}p_{2} − p_{2} = p_{2}(p_{1} − p_{2})p_{2} is nilpotent.

It is sufficient to check that (p_{2}(p_{1} − p_{2})p_{2})^{m} = p_{2}(p_{1} − p_{2})^{m}p_{2} for any m. Note that if x ∈ ker p_{2} ∩ im p_{1} then (p_{1} − p_{2})(x) = x. Since p_{1} − p_{2} is nilpotent, x = 0. Thus, ker p_{2} ∩ im p_{1} = 0. Since p_{2} is idempotent, im p_{2} ∩ ker p_{2} = 0. Then endomorphism p_{1} − p_{2} of M(\overline{X}) = ker p_{2} ⊕ im p_{2} can be represented as the matrix

p_{1} − p_{2} = (\begin{matrix}E_{1} & E_{2} \\ 0 & 0 \end{matrix})

where E_{1} is a homomorphism from im p_{2} to im p_{2} and E_{2} is a homomorphism from ker p_{2} to im p_{2}. We have

p_{2}(p_{1} − p_{2})^{m}p_{2} = p_{2} ∘ (\begin{matrix}E_{1}^{m} & E_{1}^{m−1}E_{2} \\ 0 & 0 \end{matrix}) ∘ p_{2} = (\begin{matrix}E_{1}^{m} & 0 \\ 0 & 0 \end{matrix}) = (p_{2}(p_{1} − p_{2})p_{2})^{m}.

Then f ∘ g − id_{(X,p_{2})} = p_{2}(p_{1} − p_{2})p_{2} is nilpotent. Symmetrically, g ∘ f − id_{(X,p_{1})} is nilpotent. So (X, p_{1}) and (X, p_{2}) are isomorphic by Lemma 6.10. □

We are now ready to prove theorems A, B and C of the introduction:

Theorem A. Suppose X is generically cellular. Assume that there is a decomposition of Chow motive with coefficients in Λ^{0}

M^{CH}(X, Λ^{0}) = ⨁_{i=0}^{n} R(α_{i})  (∗)

such that over the splitting field L the motive R equals to the sum of twisted Tate motives: \overline{R} = ⨁_{j=0}^{m} Λ^{0}(β_{j}).

Then there is a h-motive R_{h} such that

M^{h}(X) = ⨁_{i=0}^{n} R_{h}(α_{i})

such that over the splitting field R_{h} splits into the h-Tate motives \overline{R_{h}} = ⨁_{j=0}^{m} Λ(β_{j}).

Proof. We may assume that α_{0} = 0 in (∗). Then each summand R(α_{i}) equals to (X, p_{i}) for some idempotent p_{i} and there are mutually inverse isomorphisms ϕ_{i} and ψ_{i} of degree α_{i} between (X, p_{0}) and (X, p_{i}). So we have

• idempotents p_{i} ∈ CH^{N}(X × X), ∑ p_{i} = Δ_{h}^{X}(1)

• isomorphisms ϕ_{i} ∈ p_{0}∘CH^{N+α_{i}}(X×X)∘p_{i} and ψ_{i} ∈ p_{i}∘CH^{N−α_{i}}(X×X)∘p_{0}

• such that ϕ_{i} ∘ ψ_{i} = p_{0} and ψ_{i} ∘ ϕ_{i} = p_{i}

Consider the diagram of Lemma 6.7

\xymatrix{h_{2N−m}^{(m)}(X × Y) \ar[r]^{pr_{m}∘p^{h}} & h_{2N−m}^{(m/m+1)}(\overline{X} × \overline{X}) \\ CH^{m}(X × Y, Λ^{0}) \ar[r]^{p^{CH}} & CH^{m}(\overline{X} × \overline{X}, Λ^{0}) \ar[u]_{Φ^{m}}} .

By 6.7 the elements Φ^{N} ∘ p^{CH}(p_{i}) and Φ^{N+α_{i}} ∘ p^{CH}(ϕ_{i}) and Φ^{N−α_{i}} ∘ p^{CH}(ψ_{i}) lie in im pr_{N} ∘ p^{h}, im pr_{N−α_{i}} ∘ p^{h} and im pr_{N+α_{i}} ∘ p^{h} respectively.

By Lemma 6.9 the kernel of pr_{N} ∘ p^{h} : h_{N}^{(N)}(X × X) → h_{N}^{(N/N+1)}(\overline{X} × \overline{X}) is nilpotent. Then by [1, Prop. 27.4] there is a decomposition r_{i} such that pr_{N} ∘ p^{h}(r_{i}) = p_{i}.

Let us construct the isomorphisms between r_{i} and r_{0}. Let ϕ′_{i} and ψ′_{i} be some preimages of Φ^{N+i} ∘ p^{CH}(ϕ_{i}) and Φ^{N−i} ∘ p^{CH}(ψ_{i}). Then [16, Lem. 2.5] implies that there are elements ϕ′′_{i} ∈ r_{0} h_{N}^{(N)}(X × X)r_{i} and ψ′′_{i} ∈ r_{i} h_{N}^{(N)}(X × X)r_{0}, such that ϕ_{i,j}ψ_{i,j} = r_{0,1} and ψ_{i,j}ϕ_{i,j} = r_{i,j}. So the h-motives (X, r_{i}) and (X, r_{0})(α_{i}) are isomorphic. Taking R_{h} = (X, r_{0}) we have

M^{h}(X) = ⨁_{i=0}^{n}(X, r_{i}) = ⨁_{i=0}^{n}(X, r_{0})(α_{i}) = ⨁_{i=0}^{n} R_{h}(α_{i}).

Over the splitting field the motive \overline{R_{h}} becomes isomorphic to (\overline{X}, p^{h}(r_{0})) and pr_{N} ∘ p^{h}(r_{0}) = Φ^{N}(p^{CH}(p_{0})). Since the Chow motive (\overline{X}, p^{CH}(p_{0,1})) splits into ⨁_{j} Λ^{0}(β_{j}) we have p^{CH}(p_{0,1}) = ∑_{j} f_{j} ⊗ g_{j} with f_{j} ∈ CH^{α_{j}}(\overline{X}), g_{j} ∈ CH_{α_{j}}(\overline{X})

and π_{CH}(f_{j}g_{l}) = δ_{j,l}. Take φ_{j} and γ_{j} to be the liftings of f_{j} and g_{j} in h_{N−α_{j}}^{(α_{j})}(\overline{X})

and h_{j}^{(N−α_{j})}(\overline{X}) respectively.

Note that φ_{j}γ_{l}+h^{N+1}(\overline{X}) = Ψ^{N}(f_{j}g_{l}). Since h^{(N+1)}(\overline{X}) = 0, we have π_{h}(φ_{j}γ_{l}) = π_{CH}(f_{j}g_{l}) = δ_{j,l}. Then the element ∑ φ_{j} ⊗ γ_{j} is an idempotent in Corr_{0}(\overline{X} × \overline{X}).

Since

pr_{N}(p^{h}(r_{0})) = Φ^{N}(p^{CH}(p_{0})) = pr_{N}(∑_{j} φ_{j} ⊗ γ_{j}),

p^{h}(r_{0}) − ∑ φ_{j} ⊗ γ_{j} lies in h^{N+1}(\overline{X} × \overline{X}), so is nilpotent. Then by Corollary 6.11 we obtain

\overline{R_{h}} = (\overline{X}, p^{h}(r_{0})) ≅ (\overline{X}, ∑_{j} φ_{j} ⊗ γ_{j}) = ⨁ Λ(β_{j}).  □

6.12. Lemma. Assume that Λ^{1} = … Λ^{N} = 0. Then h_{N}(\overline{X} × \overline{X}) ⊆ h^{(N)}(\overline{X} × \overline{X}) and in the diagram of Lemma 6.7

\xymatrix{h_{N}^{(N)}(X × Y) \ar[r]^{pr_{N} ∘p^{h}} & h_{N}^{(N/N+1)}(\overline{X} × \overline{X}) \\ CH^{N}(X × Y, Λ^{0}) \ar[r]^{p^{CH}} & CH^{N} \overline{X} × \overline{X}, Λ^{0}) \ar[u]_{Φ^{N}}}

the inverse inclusion holds: im pr_{N} ∘ p^{h} ⊆ im Φ^{N} ∘ p^{CH}.

Proof. By the degree formula [11, Thm 4.4.7] h(X × X) is generated as Λ-module by pushforwards i_{h}(1), where i: Z → X × X is projective, Z ∈ Sm_{k} and i: Z → i(Z) is birational. Following [11] we will denote such classes by [Z → X × X]_{h}. Then h_{N}(X × X) is additively generated by elements λ[Z → X × X]_{h}, where λ is homogeneous such that deg λ + codim Z = N. Since Λ^{1} = … Λ^{N} = 0, we have codim Z ⩾ N. Then in Ω(\overline{X} × \overline{X}) we have

[Z_{L} → \overline{X} × \overline{X}]_{Ω} = ∑ ω_{i,j}ζ_{i} ⊗ τ_{j} for some ω_{i,j} ∈ 𝕃.

Since all elements of the Lazard ring have negative degrees and [Z_{L} → \overline{X} × \overline{X}]_{Ω} has degree N, each ζ_{i} ⊗ τ_{j} in the expansion is contained in Ω^{(n)}(\overline{X} × \overline{X}). Then

[Z_{L} → \overline{X} × \overline{X}]_{h} = ν_{\overline{X}×\overline{X}}[Z_{L} → \overline{X} × \overline{X}]_{Ω} ∈ h^{(N)}(\overline{X} × \overline{X}) and

[Z → X × X]_{h} ∈ h^{(N)}(X × X). By the same reasons [Y → X × X] belongs to h^{(N+1)}(X × X) if codim Y > N. Then im pr_{N} ∘ p^{h} is generated over Λ^{0} by classes of [Z_{L} → \overline{X} × \overline{X}]_{h}, where Z → X × X has codimension N.

By Lemma 6.8 for any Z → X × X of codimension N we have

pr_{N} ∘ p^{h}([Z → X × X]_{h}) = Φ^{N} ∘ p^{CH}([Z → X × X]).

Then im pr_{N} ∘ p^{h} ⊆ Φ^{N} ∘ p^{CH} and the theorem is proven. □

Theorem B. Let h be oriented cohomology theory with coefficient ring Λ. Assume that the Chow motive R is indecomposable over Λ^{0} and Λ^{1} = … = Λ^{N} = 0. Then the h-motive R_{h} from theorem A is indecomposable.

Proof. By definition, R_{h} = (X, r_{0}) where r_{0} is an idempotent in h_{N}^{(N)}(X × X). If R_{h} is decomposable, then r_{0} = r_{1} + r_{2} for some idempotents in r_{1}, r_{2} ∈ h_{N}(X × X) Then by Lemma 6.12 r_{1}, r_{2} ∈ h_{N}^{(N)}(X × X) and p_{1} = (Φ^{N})^{−1} ∘ pr_{N} ∘ p^{h}(r_{1}) and p_{2} = (Φ^{N})^{−1} ∘ pr_{N} ∘ p^{h}(r_{2}) are rational idempotents and p^{CH}(p_{0}) = p_{1} + p_{2}. These idempotents are nontrivial, since ker(Φ^{N})^{−1} ∘ pr_{N} ∘ p^{h} is nilpotent. Hence, the Chow motive R = (X, p_{0}) is decomposable, a contradiction. □

6.13. Example. If h = Ω or connective K-theory, all the elements in the coefficient ring have negative degree. Then Theorems A and B prove that h-motivic irreducible decomposition coincides with integral Chow-motivic decomposition. This gives another proof of the result by Vishik-Yagita [17, Cor. 2.8].

6.14. Example. Take h to be Morava K-theory h = K(n)^{∗}. The coefficient ring is 𝔽_{p}[v_{n}, v_{n}^{−1}], where deg(v_{n}) = −2(p^{n} − 1). In the case n > log_{p}(\frac{N}{2} + 1) Theorems A and B prove that M^{K(n)}(X) has the same irreducible decomposition as Chow motive modulo p.

Theorem C. Suppose that X, Y are generically cellular and Y is a twisted form of X, i.e. \overline{Y} ≅ \overline{X}.

If M^{CH}(X, Λ^{0}) ≅ M^{CH}(Y, Λ^{0}), then M^{h}(X) ≅ M^{h}(Y).

Proof. Let f ∈ CH^{N}(X × Y) and g ∈ CH^{N}(Y × X) be correspondences, that give mutually inverse isomorphisms between M^{CH}(X) and M^{CH}(Y). Consider the diagram

\xymatrix{h_{N}^{(N)}(X × Y) \ar[r]^{pr_{N} ∘p^{h}} & h_{N}^{(N/N+1)}(\overline{X} × \overline{X}) \\ CH^{N}(X × Y, Λ^{0}) \ar[r]^{p^{CH}} & CH^{N}(\overline{X} × \overline{X}, Λ^{0}) \ar[u]_{Φ^{N}}}

Then by Lemma 6.7 we can find f_{1} ∈ h_{N}^{(N)}(X × Y) and g_{1} ∈ h_{N}^{(N)}(Y × X) such that pr_{N} ∘ p^{h}(f_{1}) = Φ^{N}(f) and pr_{N} ∘ p^{h}(g_{1}) = Φ^{N}(g). Then g_{1} ∘ f_{1} − Δ_{X} lies in the kernel of the map

h_{N}^{(N)}(X × X) ⟶^{pr_{N} ∘p^{h}} h_{N}^{(N/N+1)}(\overline{X} × \overline{X})

which consists of nilpotents by Lemma 6.9. So g_{1} ∘ f_{1} − Δ_{X} is nilpotent. By the same reasons f_{1} ∘ g_{1} − Δ_{Y} is nilpotent. Then M^{h}(X) and M^{h}(Y) are isomorphic by Lemma 6.10 and the theorem is proven. □

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Department of Mathematics and Statistics, University of Ottawa, Canada

   E-mail address: anesh094@uottawa.ca