INVARIANTS, TORSION INDICES AND ORIENTED COHOMOLOGY OF COMPLETE FLAGS

B. CALMÈS, V. PETROV, AND K. ZAINOULLINE

Contents

1. Introduction 1

Part 1. Invariants, torsion indices and formal group laws. 4 2. Formal group rings 4 3. Differential operators Δ_{α} and C_{α} 7 4. Endomorphisms of a formal group ring 10 5. Torsion indices and augmented operators 12 6. Invariants and the characteristic map 15 7. The product structure of the cohomology 17

Part 2. Bott-Samelson resolutions for oriented theories. 19 8. Algebraic cobordism and oriented cohomology theories 19 9. A sequence of split ℙ^{1}-bundles 21 10. An elementary step of the Bott-Samelson resolution 23 11. Bott-Samelson varieties 25 12. Bott-Samelson resolutions 26

Part 3. Algebraic and geometric comparison. Applications. 27 13. Comparison results 27 14. Formulas for push-forwards 30 15. Algorithm for multiplying in h^{∗}(G/B) 32 16. Landweber-Novikov operations 32 17. Examples of computations 33 References 35

1. Introduction

Let H be an algebraic cohomology theory endowed with Chern classes c_{i} such that for any two line bundles L_{1} and L_{2} over a variety X we have

(1)  c_{1}(L_{1} ⊗ L_{2}) = c_{1}(L_{1}) + c_{2}(L_{2}).

The basic example of such a theory is the Chow group CH of algebraic cycles modulo rational equivalence.

Let G be a split semi-simple linear algebraic group over a field k and let T be a split maximal torus inside G contained in a Borel subgroup B. Consider the variety G/B of Borel subgroups of G with respect to T. In two classical papers [7] and [8] Demazure studied the cohomology ring H(G/B; ℤ) and provided an algorithm to compute H(G/B; ℤ) in terms of generators and relations.

The first author was supported by EPSRC grant EP/E01786X/1. The second author is supported by RFBR 08-01-00756, 09-01-00878, 09-01-90304, 09-01-91333. The third author is supported by NSERC Discovery Grant.

The main object of his consideration was the so called characteristic map

(2)  𝔠: S^{∗}(M) → H(G/B; ℤ),

where S^{∗}(M) is the symmetric algebra of the character group M of T. In [7] Demazure interpreted this map from the point of view of invariant theory of the Weyl group W of G by identifying its kernel with the ideal generated by non-constant invariants S^{∗}(M)^{W}. The cohomology ring H(G/B; ℤ) was then replaced by a certain algebra constructed in terms of operators and defined in purely combinatorial terms.

In the present paper, we generalize most of the results of [7] to the case of an arbitrary oriented cohomology theory h, i.e. when (1) is replaced by

c_{1}(L_{1} ⊗ L_{2}) = F (c_{1}(L_{1}), c_{1}(L_{2}))

where F is the formal group law associated to h. Such theories were extensively studied by Levine-Morel [13], Panin-Smirnov [16], Merkurjev [14] and others. Apart from the Chow ring, other examples include algebraic K-theory, étale cohomology H_{et}^{∗}(−, μ_{m}), (m, char(k)) = 1, Morava K-theories, connective K-theory, elliptic cohomology and the universal such theory: algebraic cobordism Ω.

To generalize the characteristic map (2), we first introduce a substitute for the symmetric algebra S^{∗}(M) = CH^{∗}(BT). This new combinatorial object, which we call a formal group ring, is denoted by R[[M]]_{F}, where R = h(pt) is the coefficient ring, and can be viewed as a substitute of the cohomology ring of the classifying space h(BT) of T. As in [7], we introduce a subalgebra D(M)_{F} of the R-linear endomorphisms of R[[M]]_{F} generated by specific differential operators and by taking its R-dual we obtain H(M)_{F}, a combinatorial substitute for the cohomology ring h(G/B; ℤ). The characteristic map (2) then turns into the map

𝔠: R[[M]]_{F} → H(M)_{F} .

The Weyl group W acts naturally on R[[M]]_{F} and the main result of our paper (theorems 6.4 and 6.9) says that:

Theorem. If the torsion index of G is invertible in R and R has no 2-torsion, then the characteristic map is surjective and its kernel is generated by W -invariant elements in the augmentation ideal.

Demazure’s methods to prove this theorem in the special case of the additive formal group law do not work in general, for the following reason: the main objects used in his proofs are operators Δ_{w} for every w ∈ W. They are defined first for simple reflections, and afterwards for any w by decomposing it into simple reflections and composing the corresponding operators. It is then proved that the resulting composition is independent of the decomposition of w. For more general formal group laws, similar operators can still be defined for simple reflections (see Definition 3.5), but independence of the decomposition does not hold, as it was observed in [5]. Geometrically, it can be translated into the fact that the cobordism class of a desingularized Schubert variety depends on the desingularization, and not only on the Schubert variety itself (see Lemma 13.9). We overcome this problem by working with suitable filtrations such that the associated graded structures are covered by the additive case of Demazure (see in particular Proposition 4.4). We therefore encourage the reader unfamiliar with [7] to start by having a quick look at it before reading our sections 3 to 7.

As an immediate application of the developed techniques, we provide an efficient algorithm for computing the cohomology ring H(M)_{F} = h(G/B; ℤ). To do this we generalize the Bott-Samelson approach introduced in [1] and [8]. For oriented topological theories, some algorithms were considered by Bressler-Evens in [4, 5] and for algebraic theories in characteristic 0 by Hornbostel-Kiritchenko in [12]. See remark 15.2 for a comparison.

Note that the theorem also provides another approach to computing the cohomology ring h(G/B; ℤ) by looking at the subring of invariants R[[M]]_{F}^{W}. Observe that in the classical case when h = CH (or K_{0}) and G is simply-connected it is known that R[[M]]_{F}^{W} is a power series ring in basic polynomial invariants (resp. fundamental representations). In general, the structure of R[[M]]_{F}^{W} remains unknown.

Finally, for the reader primarily interested in topology, let us mention that while all our proofs are algebraic and written in the language of algebraic geometry, the results apply as they are to topological cobordism or other complex oriented theories. Indeed, there is a canonical ring morphism

Ω^{∗}(G/B) → MU^{∗}(G/B(ℂ))

(see [13, Ex. 1.2.10]) that is an isomorphism because both are free modules over the Lazard ring with bases corresponding to each other (given by desingularized Schubert cells).

The paper consists of three parts.

In the first part, we generalize the results of [7] by introducing and studying the generalized characteristic map 𝔠: R[[M]]_{F} → H(M)_{F}. In section 2, we introduce the formal group ring R[[M]]_{F} and prove its main properties. In section 3, we define the main operators Δ and C on R[[M]]_{F}. In sections 4 and 5 we study the subalgebra D(M)_{F} of operators generated by Δ (resp. C) and multiplications. In section 6 we define H(M)_{F} and prove the main theorem. In section 7 we introduce a product on H(M)_{F} compatible with the characteristic map 𝔠.

In the second part, we generalize some of the results of [8] to arbitrary oriented cohomology theories. In section 8, we discuss properties of oriented theories. In sections 9 to 12, we carry out the Bott-Samelson desingularization approach.

In the last part, we apply the results of the first and the second parts to obtain information about the ring structure in h(G/B; ℤ). In section 13, we prove that our algebraic replacement H(M)_{F} is isomorphic (as a ring) to the oriented cohomology h^{∗}(G/B; ℤ) and that the characteristic maps 𝔠_{G/B} and 𝔠 correspond to each other via this isomorphism. In section 14, we give an algebraic description of the push-forward to the point and we prove various formulas. In sections 15 and 16, we explain an algorithm to compute the ring structure of h^{∗}(G/B; ℤ) and the Landweber-Novikov operations on algebraic cobordism. Finally, in section 17, we give multiplication tables for Ω^{∗}(G/B; ℤ) for groups G of rank 2.

Notation. Let k denote a base field of arbitrary characteristic. A variety over k means a reduced irreducible scheme of finite type over k. By X and Y we always mean smooth varieties over k. The base point Spec k is denoted by pt.

A ring always means a commutative ring with a unit and R always denotes a ring. A ring R′ is called an R-algebra if it comes equipped with an injective ring homomorphism R ↪ R′. The letter M always denotes an abelian group.

All formal group laws are assumed to be one-dimensional and commutative. Let F denote a formal group law and let 𝕃 denote the Lazard ring, i.e. the coefficient ring of the universal formal group law U.

Acknowledgments. Our sincere gratitude goes to Michel Demazure for his answers to our questions concerning the paper [7]. We would like to thank the authors of the Macaulay 2 software [10], and, in particular Dan Grayson for his quick and accurate responses on the Macaulay 2 mailing list. We would also like to thank Vladimir Chernousov, Fabien Morel and Burt Totaro for their encouraging attention to our work.

Part 1. Invariants, torsion indices and formal group laws.

2. Formal group rings

Let R be a ring, let M be an abelian group and let F be a formal group law over R. In the present section we introduce and study the formal group ring R[[M]]_{F}. For this, we use several auxiliary facts concerning topological rings and their completions which can be found in [3, III, §2]. The main result here is the decomposition theorem 2.10. At the end we provide some examples of computations of R[[M]]_{F}.

2.1. Definition. Let R be a ring and let S be a set. Let R[x_{S}] := R[x_{s}, s ∈ S] denote the polynomial ring over R with variables indexed by S. Let ϵ : R[x_{S}] → R be the augmentation morphism which maps any x_{s} to 0. Consider the ker(ϵ)-adic topology on R[x_{S}] given by ideals ker(ϵ)^{i}, i ≥ 0, which form a fundamental system of open neighborhoods of 0. Note that a polynomial is in ker(ϵ)^{i} if and only if its valuation is at least i, hence, we have ∩_{i} ker(ϵ)^{i} = {0} and the ker(ϵ)-adic topology is Hausdorff.

We define R[[x_{S}]] to be the ker(ϵ)-adic completion of the polynomial ring R[x_{S}].

2.2. Remark. By definition, an element in R[[x_{S}]] can be written uniquely as a formal sum

∑_{s_{1},...,s_{m}∈S} a_{s_{1},...,s_{m}} · x_{s_{1}} … x_{s_{m}}, a_{s_{1},...,s_{m}} ∈ R,

where for any positive m there is only a finite number of non-zero coefficients a_{s_{1},...,s_{m}}. In particular, when S is a finite set of order n, the ring R[[x_{S}]] is the usual ring of power series in n variables.

2.3. Let F be a formal group law over a ring R (see [13, p.4]). Given an integer m ≥ 0 we use the notation

x +_{F} y = F (x, y), m ·_{F} x = \underbrace{x +_{F} … +_{F} x}_{m times} and (−m) ·_{F} x = −_{F} (m ·_{F} x)

By associativity of F, for any m_{1}, m_{2} ∈ ℤ we have

(m_{1} + m_{2}) ·_{F} x = (m_{1} ·_{F} x) +_{F} (m_{2} ·_{F} x).

Now, here comes the definition playing a central role in the sequel.

2.4. Definition. Let M be an abelian group and let F be a formal group law over a ring R. Consider the ring R[[x_{M}]] and let J_{F} be the closure of the ideal generated by the elements x_{0} and x_{λ+μ} − (x_{λ} +_{F} x_{μ}) for all λ, μ ∈ M. We define the formal group ring R[[M]]_{F} to be the quotient

R[[M]]_{F} = R[[x_{M}]]/J_{F} .

The class of x_{λ} in R[[M]]_{F} will be denoted by the same letter.

2.5. By definition, R[[x_{M}]] is a complete Hausdorff ring with respect to the topology induced by the kernel of the augmentation R[[x_{M}]] → R. Since J_{F} is clearly contained in this kernel, the augmentation map R[[x_{M}]] → R factors through the quotient R[[M]]_{F}. Therefore, R[[M]]_{F} is a complete Hausdorff ring with respect to the I_{F} -adic topology, where I_{F} denotes the kernel of the augmentation R[[M]]_{F} → R.

Let f: R → R′ be a morphism of rings respecting the formal group laws, i.e. sending every coefficient of F to the corresponding coefficient of F′. Then, for every abelian group M, f induces a ring homomorphism f_{⋆}: R[[M]]_{F} → R′[[M]]_{F′} sending x_{λ} ∈ R[[M]]_{F} to x_{λ} ∈ R′[[M]]_{F′} for every λ ∈ M. This morphism sends I_{F} to I_{F′} and, hence, is continuous.

Let now f: M → M′ be a morphism of abelian groups. It induces a continuous ring homomorphism f̂: R[[M]]_{F} → R[[M′]]_{F} sending x_{λ} to x_{f(λ)}. Moreover, if f is surjective, then so is f̂.

Finally, let f: F → F′ be a morphism of formal group laws over R, i.e. a formal power series f ∈ R[[x]] such that f(x +_{F} y) = f(x) +_{F′} f(y). Then, f induces a continuous ring homomorphism f^{⋆}: R[[M]]_{F′} → R[[M]]_{F} sending x_{λ} to f(x_{λ}).

We have therefore proved:

2.6. Lemma. Via the above constructions (−)_{⋆}, ˆ (−) and (−)^{⋆}, the assignment taking (R, M, F) to the topological ring R[[M]]_{F} is covariant with respect to ring morphisms R → R′ and morphisms of abelian groups M → M′, and is contravariant with respect to morphisms of formal group laws F → F′.

2.7. Let 𝕃 be the Lazard ring of coefficients of the universal formal group law U. For any formal group law F over R, there is a unique ring homomorphism ϑ_{F} : 𝕃 → R sending the universal formal group law U on 𝕃 to F. By universality, the morphism

R ⊗_{𝕃} 𝕃[[M]]_{U} → R[[M]]_{F} , r ⊗ z ↦ r · (ϑ_{F})_{⋆}(z)

is an isomorphism.

2.8. Since R[[M]]_{F} is an R-algebra, we may consider the formal group law F as an element of R[[M]]_{F} [[x, y]]. Let a, b ∈ I_{F}. The specialization at x = a and y = b defines a pairing

⊞: I_{F} × I_{F} → I_{F} .

Similarly, we define ⊟: I_{F} → I_{F} using the inverse of F and ⊡: ℤ × I_{F} → I_{F} by applying ⊞ or ⊟ iteratively. From the associativity, commutativity, inverse properties of F and the continuity of the quotient map R[[x_{M}]] → R[[M]]_{F} it follows that x_{λ+μ} = x_{λ} ⊞ x_{μ} and

x_{−λ} = x_{−λ} ⊞ (x_{λ} ⊟ x_{λ}) = (x_{−λ} ⊞ x_{λ}) ⊟ x_{λ} = 0 ⊟ (x_{λ}) = ⊟(x_{λ}).

2.9. A formal group law F on R induces via a ring homomorphism R → R′ a formal group law on R′. In particular, if M and N are abelian groups, then using R → R[[M]]_{F} = R′, we can define R[[M]]_{F} [[N]]_{F}, which is naturally an R[[M]]_{F} - algebra. By functoriality, R[[M ⊕ N]]_{F} is also an R[[M]]_{F} -algebra. We define two morphisms of R[[M]]_{F} -algebras

ϕ: R[[M ⊕ N]]_{F} → R[[M]]_{F} [[N]]_{F} and ψ: R[[M]]_{F} [[N]]_{F} → R[[M ⊕ N]]_{F}

as follows:

The map R[x_{M⊕N}] → R[[x_{M}]][[y_{N}]] sending x_{(λ,γ)} to x_{λ} +_{F} y_{γ} extends to a continuous map f: R[[x_{M⊕N}]] → R[[x_{M}]][[y_{N}]]. Consider the composition

R[[x_{M⊕N}]] →^{f} R[[x_{M}]][[y_{N}]] →^{g} R[[M]]_{F} [[y_{N}]] →^{h} R[[M]]_{F} [[N]]_{F}

where g is induced by R[[x_{M}]] → R[[M]]_{F} and h is the quotient map. We have

h ∘ g ∘ f(x_{(λ,γ)} +_{F} x_{(μ,δ)}) =^{(1)} h ∘ g((x_{λ} +_{F} y_{γ}) +_{F} (x_{μ} +_{F} y_{δ})) =

=^{(2)} h ∘ g((x_{λ} +_{F} x_{μ}) +_{F} (y_{γ} +_{F} y_{δ})) =^{(3)} h((x_{λ} ⊞ x_{μ}) +_{F} (y_{γ} +_{F} y_{δ})) =

=^{(4)} (x_{λ} ⊞ x_{μ}) +_{F} (y_{γ} ⊞ y_{δ}) =^{(5)} x_{λ+μ} +_{F} y_{γ+δ} =^{(6)} h(x_{λ+μ} +_{F} y_{γ+δ}) =

=^{(7)} h ∘ g(x_{λ+μ} +_{F} y_{γ+δ}) =^{(8)} h ∘ g ∘ f(x_{(λ+μ,γ+δ)}),

where (1) and (8) hold by definition and continuity of f, (2) by associativity and commutativity of F, (3) and (7) by definition of g, (4) and (6) by definition of h and (5) by definition of ⊞. It shows that h ∘ g ∘ f factors through R[[M ⊕ N]]_{F} as

h ∘ g ∘ f: R[[x_{M⊕N}]] → R[[M ⊕ N]]_{F} →^{ϕ} R[[M]]_{F} [[N]]_{F} ,

where ϕ is a morphism of R[[M]]_{F} -algebras.

In the opposite direction, we proceed as follows: We extend the functorial morphism R[[M]]_{F} → R[[M ⊕N]]_{F} to a morphism R[[M]]_{F} [y_{N}] → R[[M ⊕N]]_{F} by sending y_{γ} to x_{(0,γ)}. It induces a morphism R[[M]]_{F} [[y_{N}]] → R[[M ⊕ N]]_{F} on completions, which factors through a continuous morphism ψ: R[[M]]_{F} [[N]]_{F} → R[[M ⊕ N]]_{F}.

2.10. Theorem. The morphisms of R[[M]]_{F} -algebras ϕ and ψ defined above are inverses to each other. In other words, we have an isomorphism

R[[M ⊕ N]]_{F} ≃ R[[M]]_{F} [[N]]_{F} .

Proof. We have

ψ ∘ ϕ(x_{(λ,γ)}) = ψ(x_{λ} +_{F} y_{γ}) = ψ(x_{λ}) +_{F} ψ(y_{γ}) = x_{(λ,0)} +_{F} x_{(0,γ)} = x_{(λ,γ)}

where the second equality holds by continuity of ψ. The other composition can be checked on y_{γ}, since we are dealing with morphisms of R[[M]]_{F} -algebras, and we have ϕ ∘ ψ(y_{γ}) = ϕ(x_{(0,γ)}) = x_{0} +_{F} y_{γ} = 0 +_{F} y_{γ} = y_{γ}. □

2.11. Lemma. Let M = ℤ. Then sending x_{m} to m ·_{F} x defines a ring isomorphism

R[[ℤ]]_{F} ≃ R[[x]].

In particular, if M is a free abelian group of rank one, then R[[M]]_{F} is isomorphic to R[[x]].

Proof. The morphism ϕ : R[x_{ℤ}] → R[[x]] sending x_{n} to n·_{F} x extends to a morphism ϕ̂: R[[x_{ℤ}]] → R[[x]]. By continuity, ϕ̂ satisfies ϕ̂(x_{m} +_{F} x_{n}) = ϕ̂(x_{m})+_{F} ϕ̂(x_{n}) as well as ϕ̂(−_{F} (x_{m})) = −_{F} ϕ̂(x_{m}). Thus ϕ̂(x_{m+n}) = ϕ̂(x_{m} +_{F} x_{n}) and ϕ̂ factors through a ϕ̄ : R[[ℤ]]_{F} → R[[x]]. A map ψ̂ in the opposite direction is defined similarly using completeness, by sending x to x_{1}. By continuity, checking that ϕ̄ ∘ ψ̂ = id and ψ̂ ∘ ϕ̄ = id can be done on generators. Namely, ϕ̄ ∘ ψ̂(x) = ϕ̄(x_{1}) = x by definition, and ψ̂ ∘ ϕ̄(x_{n}) = ψ̂(n ·_{F} x) = n ⊡ ψ̂(x) = n ⊡ x_{1} = x_{n}. □

2.12. Corollary. Let ϕ : M → ℤ^{⊕n} be an isomorphism. Then ϕ induces an isomorphism

R[[M]]_{F} ≃ R[[x_{1}, …, x_{n}]].

Proof. It follows from Lemma 2.11 and Theorem 2.10 by induction on n. □

2.13. Remark. Note that the right hand side is independent of F, although the isomorphism depends on F. Also note that if R is an integral domain, so is R[[ℤ^{⊕n}]]_{F}.

2.14. Example. Similarly, one can prove that R[[ℤ/n]]_{F} ≃ R[[x]]/(n·_{F} x) by sending x_{m} to m ·_{F} x. Observe that n ·_{F} x = nx + x^{2}q. In particular, if n is invertible in R, then (n ·_{F} x) = (x) and R[[ℤ/n]]_{F} ≃ R.

Let us now examine what happens at a finite level in R[[M]]_{F}.

2.15. Let R[M]_{F} denote the subring of R[[M]]_{F} defined as the image of the subring R[x_{M}] by means of the composition R[x_{M}] → R[[x_{M}]] → R[[M]]_{F}. Then the ring R[[M]]_{F} is the completion of R[M]_{F} at the ideal I_{F} ∩ R[M]_{F}. By the functoriality of R[[M]]_{F} (see 2.6) the assignment

(R, M, F) ↦ R[M]_{F}

is a covariant functor with respect to morphism of rings R → R′ and morphisms of abelian groups M → M′. Moreover, if M → M′ is surjective, then so is R[M]_{F} → R[M′]_{F}.

2.16. Example. If M = ℤ, the isomorphism of Lemma 2.11 maps R[ℤ]_{F} to polynomials in n ·_{F} x. In particular, unless F and the formal inverse have a finite number of nonzero coefficients only, R[ℤ]_{F} does not map to R[x].

2.17. Remark. The morphism ϕ from 2.9 doesn’t necessarily send R[M ⊕ N]_{F} to R[M]_{F} [N]_{F}. The morphism ψ from 2.9 sends R[M]_{F} [N]_{F} into R[M ⊕ N]_{F} but is not necessarily surjective.

2.18. Example (cf. 8.3). The additive formal group law over R is given by F (x, y) = x + y. In this case we have ring isomorphisms

R[[M]]_{F} ≃ ∏_{i=0}^{∞} S_{R}^{i}(M) and R[M]_{F} ≃ ⨁_{i=0}^{∞} S_{R}^{i}(M)

where S_{R}^{i}(M) is the i-th symmetric power of M over R and the isomorphisms are induced by sending x_{λ} to λ ∈ S_{R}^{1}(M).

2.19. Example (cf. 8.4). The multiplicative periodic formal group law over R is given by F (x, y) = x + y − βxy, where β is an invertible element in R. Consider the group ring

R[M] := {∑_{j} r_{j}e^{λ_{j}} | r_{j} ∈ R, λ_{j} ∈ M} .

Let tr : R[M] → R be the trace map, i.e. a R-linear map sending any e^{λ} to 1. Let R[M]^{∧} be the completion of R[M] at ker(tr). Then we have ring isomorphisms

R[[M]]_{F} ≃ R[M]^{∧} and R[M]_{F} ≃ R[M]

induced by x_{λ} ↦ β^{−1}(1 − e^{λ}) and e^{λ} ↦ (1 − βx_{λ}) = (1 − βx_{−λ})^{−1}.

2.20. Example (cf. 8.5). The multiplicative non-periodic formal group law over R is given by F (x, y) = x + y − vxy, where v is not invertible in R. Specializing R[[M]]_{F} at v = 0 or v = β where β is invertible in R, we obtain the formal group rings of the previous examples.

2.21. Remark. Let h be an oriented cohomology theory as defined in [13, Def. 1.1.2] and let F be the associated formal group law (see Section 8). Let M be the group of characters of a split torus T over k. Then the formal group ring R[[M]]_{F} can be viewed as an algebraic substitute of the completed equivariant cohomology ring h_{T} (pt)^{∧} or the cohomology ring of the classifying space h(BT). Its finite counterpart R[M]_{F} could then play the role of the equivariant cohomology ring h_{T} (pt) itself.

3. Differential operators Δ_{α} and C_{α}

In the present section we introduce two linear operators on R[[M]]_{F}. The first operator Δ_{α}^{F} is a generalized version of the operator Δ_{α} from [7, §3 and §4]. Indeed, one recovers the results of loc. cit. when R = ℤ and F is the additive formal group law from Example 2.18. A version of the second operator C_{α}^{F} was already used in [5, §5] for topological complex cobordism.

3.1. Consider a reduced root system as in [7, §1], i.e. a free ℤ-module M of finite rank, a finite subset of M whose elements are called roots and a map associating a coroot α^{∨} ∈ M^{∨} to a root α, satisfying certain axioms.^{1} The reflection map λ ↦ λ − α^{∨}(λ)α is denoted by s_{α}.

3.2. The Weyl group W associated to a reduced root system is the subgroup of linear automorphisms of M generated by reflections s_{α}. It acts linearly on M and thus by R-algebra automorphisms on R[[M]]_{F} using the functoriality in M of R[[M]]_{F}. We have the following obvious equalities for any u ∈ R[[M]]_{F}:

s_{α}(u) = s_{−α}(u)  and  s_{α}^{2}(u) = u.

3.3. Lemma. For any n ∈ ℤ, the element x − x +_{F} (n ·_{F} y) is uniquely divisible by y in R[[x, y]].

Proof. Since y is a regular element in R[[x, y]], we just need to prove divisibility. Note that for any power series g(x, y), the series g(x, 0) − g(x, y) is divisible by y. Apply it to g(x, y) = x +_{F} (n ·_{F} y). □

3.4. Corollary. Assume that R is integral. For any u ∈ R[[M]]_{F}, the element u − s_{α}(u) is uniquely divisible by x_{α}.

Proof. First note that R[[M]]_{F} is an integral domain by 2.13, so we just need to prove divisibility. Since

s_{α}(x_{λ}) = x_{λ−α^{∨}(λ)α} = x_{λ} ⊞ (−α^{∨}(λ) ⊡ x_{α})

it holds by the previous lemma when u = x_{λ}. Then, by the formula

(3) uv − s_{α}(uv) = (u − s_{α}(u))v + u(v − s_{α}(v)) − (u − s_{α}(u))(v − s_{α}(v))

the result holds by induction on the degree of monomials for any element in R[M]_{F}. Finally, it holds by density on the whole R[[M]]_{F}. □

For a given root α we define a linear operator Δ_{α}^{F} on R[[M]]_{F} as follows:

3.5. Definition. First, using Corollary 3.4 we define a linear operator Δ_{α}^{U} on 𝕃[[M]]_{U}, where U is the universal formal group law over the Lazard ring 𝕃, as

Δ_{α}^{U}(u) = \frac{u − s_{α}(u)}{x_{α}}, where u ∈ 𝕃[[M]]_{U}.

Finally, identifying R[[M]]_{F} with R ⊗_{𝕃} 𝕃[[M]]_{U} via the isomorphism from 2.7 we define the desired operator Δ_{α}^{F} on R[[M]]_{F} as id_{R} ⊗Δ_{α}^{U}. We will simply write Δ_{α}(u) when the formal group law F over R is understood.

3.6. Remark. Observe that if R is integral, then the operator Δ^{F} can be defined directly using the same formula as for Δ^{U}. If R is torsion, then it is not the case. Indeed, take R = ℤ/2, M = ℤ, F to be an additive law and α = 2 to be a root. Then x_{α} = x_{1+1} = x_{1} + x_{1} = 2x_{1} = 0 in R[[M]]_{F}.

3.7. Remark. We could instead have defined Δ_{α} as Δ_{α}(u) = \frac{s_{α}(u)−u}{x_{−α}}. This exchanges Δ_{α} with −Δ_{−α} so it is easy to switch from one convention to the other. Both conventions give the same classical operator when the formal group law is additive.

3.8. Proposition. The following formulas hold for any u, v ∈ R[[M]]_{F}, λ ∈ M and w ∈ W (Compare to [7, §3]^{2}).

(1) Δ_{α}(1) = 0, Δ_{α}(u)x_{α} = u − s_{α}(u),

(2) Δ_{α}^{2}(u)x_{α} = Δ_{α}(u) + Δ_{−α}(u), Δ_{α}(u)x_{α} = Δ_{−α}(u)x_{−α},

(3) s_{α}Δ_{α}(u) = −Δ_{−α}(u), Δ_{α}s_{α}(u) = −Δ_{α}(u), (4) Δ_{α}(uv) = Δ_{α}(u)v + uΔ_{α}(v) − Δ_{α}(u)Δ_{α}(v)x_{α} = Δ_{α}(u)v + s_{α}(u)Δ_{α}(v),

1This is often called a root datum instead of a root system in the literature, but we follow [7].

2 There is a sign mistake in Equation (3) of [7, §3]. One should read Δ_{α} = −Δ_{−α}

(5) wΔ_{α}w^{−1}(u) = Δ_{w(α)}(u).

Proof. All formulas can be proved in 𝕃[[M]]_{U} (which is an integral domain) and then specialized to any other R[[M]]_{F}. Formula (4) follows from equation (3) above. Formula (5) follows from the fact that α^{∨}(w^{−1}(λ)) = w(α)^{∨}(λ). All other formulas follow by definition. □

From now on, we fix a basis of simple roots (α_{1}, …, α_{n}), with associated simple reflections s_{1}, …, s_{n}. This defines a length function on W. Let I denote a sequence (i_{1}, …, i_{l}) of l integers in [1, n] and let w(I) = s_{i_{1}} ⋯ s_{i_{l}} be the corresponding product of simple reflections. The decomposition I is reduced if w has length l. We define the linear operator Δ_{I} as the composition

Δ_{I} = Δ_{α_{i_{1}}} ∘ ⋯ ∘ Δ_{α_{i_{l}}} .

The operators Δ_{I} have the following important property:

3.9. Theorem. Let F be a formal group law of the form F (x, y) = x + y − v · xy for some v ∈ R. Let I and I′ be two reduced decompositions of w in simple reflections. Then Δ_{I} = Δ_{I′}.

Proof. See [8, Theorem 2 p. 86]. The proof assumes v = 1 but it works for any other value. □

When the formal group law is of the above type, the previous theorem justifies the notation Δ_{w} instead of Δ_{I}, but in general, we have to keep the dependence on the decomposition.

3.10. Remark. Theorem 3.9 was proved in [4] (see loc. cit. Thm.3.7) for topological oriented theories. Moreover, the result of [4] says that the formal group law has to be of the above form for such an independence of the decomposition to hold.

For a given root α we define another linear operator C_{α}^{F} on R[[M]]_{F} as follows:

3.11. Definition. Let g(x, y) be the power series defined by

x +_{F} y = x + y − xy · g(x, y)

and let e_{α} be the element g(x_{α}, x_{−α}) in R[[M]]_{F}. We set

C_{α}^{F} (u) = ue_{α} − Δ_{α}^{F} (u), where u ∈ R[[M]]_{F} .

We will simply write C_{α}(u) when the formal group law F over R is understood. For a sequence of integers I in [1, n], we define the operator C_{I} in the same way as Δ_{I}.

3.12. Proposition. The following formulas hold for any u, v ∈ R[[M]]_{F}, λ ∈ M and w ∈ W.

(1) C_{α}(1) = e_{α}, C_{α}(x_{−α}) = 2,

(2) C_{α}(u)x_{α}x_{−α} = ux_{α} + s_{α}(u)x_{−α}, C_{α}(ux_{−α}) = u + s_{α}(u),
(3) C_{α}s_{α}(u) = C_{−α}(u), s_{α}C_{α}(u) = C_{α}(u),
(4) C_{α}(uv) = C_{α}(u)v + s_{α}(u)C_{α}(v) − s_{α}(u)ve_{α} = C_{α}(u)v − s_{α}(u)Δ_{α}(v),
(5) wC_{α}w^{−1}(u) = C_{w(α)}(u),
(6) C_{α}Δ_{α} = Δ_{α}C_{α} = Δ_{α}C_{−α} = 0.

Proof. All these formulas can easily be derived from the ones of Proposition 3.8 using the definition of C_{α} and the fact that e_{α}x_{α}x_{−α} = x_{α} + x_{−α}. □

3.13. Proposition. Let W_{α} be the subgroup of order 2 generated by s_{α} in W. Let R[[M]]_{F}^{W_{α}} denote the subring of fixed elements of R[[M]]_{F} under the action of W.

If the ring R has no 2-torsion, then the operators Δ_{α} and C_{α} are R[[M]]_{F}^{W_{α}}-linear. In particular, they are R[[M]]_{F}^{W} -linear.

Proof. For the operator Δ_{α}, by formula (4) of 3.8 it suffices to show that Δ_{α}(u) = 0 for u ∈ R[[M]]_{F}^{W_{α}}. According to formula (3) of 3.8, it is equivalent to Δ_{α}(u) = −Δ_{α}(u) which holds if R[[M]]_{F} has no 2-torsion.

The same facts for C_{α} then follow from its definition in terms of Δ_{α}. □

3.14. Lemma. Assume R has no 2-torsion. Then for any root α, the element x_{α} is regular in R[[M]]_{F} and the set of elements u such that Δ_{α}(u) = 0 is R[[M]]_{F}^{W_{α}}.

Proof. Let ω_{1}, …, ω_{n} be the fundamental weights corresponding to a choice of simple roots α = α_{1}, …, α_{n} of the root system. Possibly extending M, we can assume that the ω_{i} are a basis of M. By the isomorphism R[[M]]_{F} ≃ R[[x_{ω_{1}}, …, x_{ω_{n}}]] of Corollary 2.12, x_{α} becomes ∑_{i} <α, α_{i} > x_{ω_{i}} + u, u ∈ I^{2}. One easily checks that this element is regular if at least one of the <α, α_{i} > is regular in R. In particular, this is the case if <α_{1}, α_{1} >= 2 is regular in R. By equation (1) of Proposition 3.8, the last part of the claim is then clear. □

3.15. Proposition. Let f : R → R′ be a ring morphism sending a formal group law F over R to a formal group law F′ over R′. Then, the operators Δ and C satisfy

f_{∗}Δ_{α}^{F} = Δ_{α}^{F′}f_{∗}  and  f_{∗}C_{α}^{F} = C_{α}^{F′}f_{∗}

where f_{∗} : R[[M]]_{F} → R′[[M]]_{F′} is the morphism of Lemma 2.6.

Proof. This is clear by construction. □

4. Endomorphisms of a formal group ring

In the present section we introduce and study the subalgebra D(M)_{F} of Rlinear endomorphisms of a formal group ring R[[M]]_{F} generated by the Δ operators of the previous section. The main result (Theorem 4.11) says that operators Δ_{I_{w}}^{F}, where w runs through all elements of the Weyl group and I_{w} is any chosen reduced decomposition of w, form a basis of D(M)_{F} as an R[[M]]_{F} -module.

4.1. Let R[[M]]_{F} be a formal group ring corresponding to a formal group law F over a ring R and an abelian group M (see Def. 2.4). Let I_{F} be the kernel of the augmentation map R[[M]]_{F} → R as in 2.5. By convention, we set I_{F}^{i} = R[[M]]_{F} for any i ≤ 0. We define the associated graded ring

Gr_{R}^{∗}(M, F) = ⨁_{i=0}^{∞} I_{F}^{i} /I_{F}^{i+1}.

4.2. Lemma. The morphism of graded R-algebras

ϕ : S_{R}^{∗}(M) → Gr_{R}^{∗}(M, F)

defined by sending λ to x_{λ} is an isomorphism.

Proof. The map is well-defined since x_{λ+μ} − (x_{λ} + x_{μ}) is in I_{F}^{2} , and it is obviously surjective. Let us define a map in the other direction. Recall that by definition R[[M]]_{F} = R[[x_{M}]]/J_{F}. Let Ĩ denote the kernel of the augmentation map R[[x_{M}]] → R (see 2.5). Then we have

I_{F} = Ĩ/J_{F} and I_{F}^{i} /I_{F}^{i+1} = Ĩ^{i}/(J_{F} ∩ Ĩ^{i} + Ĩ^{i+1}).

Since Ĩ^{i}/Ĩ^{i+1} is given by monomials of degree i, we may define a map of R-modules

ψ_{i}: Ĩ^{i}/Ĩ^{i+1} → S_{R}^{i}(M)

by sending a monomial of degree i in some x_{λ}-s to the symmetric product of the λ-s involved. This map passes to the quotient since (J_{F} ∩ Ĩ^{i} + Ĩ^{i+1})/Ĩ^{i+1} is generated as an R-module by elements of the form ρ.(x_{λ} + x_{μ} − x_{λ+μ}) where ρ is a monomial of degree i − 1. The sum ⊕_{i}ψ_{i} passes through the quotient as well and, hence, gives the desired inverse map. □

4.3. Consider now a reduced root system M as in Section 3 and let R[[M]]_{F} be the associated formal group ring. Consider the operators Δ_{α}^{F} introduced in Def. 3.5. By their very definition, the operators Δ_{α}^{F} send I_{F}^{i} to I_{F}^{i−1}. Hence, they induce R-linear operators of degree −1 on the graded ring Gr_{R}^{∗}(M, F), denoted by GrΔ_{α}^{F}. Similarly, we define graded versions of operators C_{α}^{F} (see Def. 3.11) on Gr_{R}^{∗}(M, F), denoted by GrC_{α}^{F}.

4.4. Proposition. The isomorphism ϕ of Lemma 4.2 exchanges the operator GrΔ_{α}^{F} (resp. GrC_{α}^{F}) on Gr_{R}^{∗}(M, F) with Δ_{α}^{F_{a}} (resp. −Δ_{α}^{F_{a}}) on the symmetric algebra S_{R}^{∗}(M), where F_{a} denotes the additive formal group law as in Example 2.18 and the Δ_{α}^{F_{a}} = −C_{α}^{F_{a}} are the classical operators of [7].

Proof. Induction on the degree using (4) of Prop. 3.8. □

4.5. Definition. Let D(M)_{F} be the subalgebra of R-linear endomorphisms of R[[M]]_{F} generated by the Δ_{α}^{F} for all roots α and by multiplications by elements of R[[M]]_{F}. Note that by formula (1) of Proposition 3.8, D(M)_{F} contains s_{α} and it contains C_{α}^{F} by its definition. Let D(M)_{F}^{(i)} be the sub R[[M]]_{F} -module of D(M)_{F} generated by the uΔ_{α_{1}}^{F} ⋯ Δ_{α_{n}}^{F}, where u ∈ I_{F}^{m} and m − n ≥ i. This defines a filtration on D(M)_{F}

D(M)_{F} ⊇ ⋯ ⊇ D(M)_{F}^{(i)} ⊇ D(M)_{F}^{(i+1)} ⊇ ⋯ ⊇ 0

with the property that D(M)_{F} = ∪_{i}D(M)_{F}^{(i)}. We define the associated graded R[[M]]_{F} -module

Gr^{∗}D(M)_{F} = ⨁_{i=−∞}^{∞} D(M)_{F}^{(i)}/D(M)_{F}^{(i+1)}.

4.6. Proposition. The filtration on D(M)_{F} above has the following properties: (1) For any root α, we have

Δ_{α}^{F} D(M)_{F}^{(i)} ⊆ D(M)_{F}^{(i−1)}, D(M)_{F}^{(i)}Δ_{α}^{F} ⊆ D(M)_{F}^{(i−1)}

and similarly for C_{α}^{F}

(2) For any integer j, we have I_{F}^{j} D(M)_{F}^{(i)} ⊆ D(M)_{F}^{(i+j)}

(3) For any operator D ∈ D(M)_{F}^{(i)}, we have D(I_{F}^{j}) ⊆ I_{F}^{i+j}

(4) ∩_{i}D(M)_{F}^{(i)} = {0}

Proof. The claim (1) follows from equation (4) in Prop. 3.8 (resp. equation (4) in Prop. 3.12). The claim (2) is obvious. The claim (3) follows from the fact that Δ_{α}^{F} (I_{F}^{i}) ⊆ I_{F}^{i−1}. To prove (4) observe that any D in ∩_{i}D(M)_{F}^{(i)} sends R[[M]]_{F} to ∩_{i}I_{F}^{i} = 0 by (3) so is the zero operator. □

4.7. By claim (3) of Prop. 4.6 the graded module Gr^{∗}D(M)_{F} acts by graded endomorphisms on the graded ring Gr_{R}^{∗}(M, F), which is isomorphic to S_{R}^{∗}(M) by Lemma 4.2. Let F_{a} denote the additive formal group law. Then the associated subalgebra D = D(M)_{F_{a}} coincides with one considered in [7, §3]. By Prop. 4.4, the class of Δ_{α}^{F} (resp. C_{α}^{F}) in Gr^{(−1)}D(M)_{F} acts by Δ_{α}^{F_{a}} (resp. −Δ_{α}^{F_{a}}) on S_{R}^{∗}(M) and the class of the multiplication by x_{α} in Gr^{(1)}D(M)_{F} acts by the multiplication by α on S_{R}^{∗}(M).

4.8. Proposition. The graded module Gr^{∗}D(M)_{F} over the graded ring Gr_{R}^{∗}(M, F) ≃ S_{R}^{∗}(M) is isomorphic to D.

Proof. It is a sub-module of the R-linear endomorphisms of Gr^{∗}R[[M]]_{F} generated by the same elements as D. □

4.9. Let A be a ring and let I be an ideal of A, such that A is complete and Hausdorff for the I-adic topology. Let M be an A-module together with a Ifiltration (M^{(i)})_{i∈ℤ}, i.e. IM^{(i)} ⊆ M^{(i+1)} for any i. Consider the associated graded ring Gr^{∗}A and the associated graded Gr^{∗}A-module Gr^{∗}M.

The filtration is called exhaustive (see [2, III, §2, 1]) if any λ ∈ M belongs to some M^{(i)}, i.e. M = ∪_{i}M^{(i)}. For an exhaustive filtration let ν(λ) denote the largest integer such that λ ∈ M^{ν(λ)} and let λ̄ denote the class of λ in M^{(ν(λ))}/M^{(ν(λ)+1)} if ν(λ) exists and 0 if ν(λ) does not exist. Note that λ̄ is nonzero if and only if λ is not in ∩_{i}M^{(i)}.

4.10. Lemma. Let M be an A-module with an exhaustive I-filtration. Let λ_{1}, …, λ_{n} be elements of M. Then

(1) Assume A is complete for the I-adic topology and M is Hausdorff. If λ̄_{1}, …, λ̄_{n} generate Gr^{∗}M as a Gr^{∗}A-module, then λ_{1}, …, λ_{n} generate M as an A-module.

(2) Assume A is Hausdorff. If λ̄_{1}, …, λ̄_{n} are independent elements in the Gr^{∗}A-module Gr^{∗}M, then λ_{1}, …, λ_{n} are independent in the A-module M.

(3) Assume A is Hausdorff complete and M is Hausdorff. If Gr^{∗}M is a finitely generated free Gr^{∗}A-module, then M is a finitely generated free A-module.

Proof. Let A[i] be A itself considered as an A-module with the shifted filtration I^{j+ν(λ_{i})} (by convention, I^{j} = A for j ≤ 0). The claims (1) and (2) then follow from [3, III, §2, 8, Cor. 1 and 2] applied to X = ⨁_{i} A[i], Y = M and ⨁_{i} A[i] → M the map sending (a_{i})_{i} to ∑_{i} a_{i}λ_{i}. The last claim (3) is an immediate consequence of (1) and (2). □

We now come to the main result of this section:

4.11. Theorem. Let D(M)_{F} be the algebra of operators defined above. For each element w ∈ W, choose a reduced decomposition I_{w} in simple reflections. Then the operators Δ_{I_{w}}^{F} (resp. C_{I_{w}}^{F}) form a basis of D(M)_{F} as an R[[M]]_{F} -module.

Proof. When the formal group law is additive, this is proved in [7, §4, Cor. 1] for the Δ_{w}^{F_{a}} (and C_{w}^{F_{a}} = (−1)^{l(w)}Δ_{w}^{F_{a}}). The filtered module D(M)_{F} is Hausdorff by point (4) in Proposition 4.6, and R[[M]]_{F} is complete. We can therefore apply point (3) of Lemma 4.10 to deduce the general case from the corresponding fact on the associated graded objects, which is the additive case by Proposition 4.8. □

5. Torsion indices and augmented operators

In the present section we study the augmentation ϵD(M)_{F} of the algebra of operators D(M)_{F}. The main result is Proposition 5.4 which is what becomes of Theorem 4.11 when the augmentation ϵ is applied. Similarly, we then examine the filtration on ϵD(M)_{F} obtained by applying ϵ to the one on D(M)_{F} introduced in 4.5.

5.1. Consider the torsion index 𝔱 of the root system as defined in [7, §5] and its prime divisors, called torsion primes. It has been computed for simply connected root systems of all types (see [7, §7, Prop. 8] and [18, 19]). The results of these computations are summarized in the following table:

Type  A_{l} B_{l}, l ≥ 3 C_{l} D_{l}, l ≥ 4 G_{2} F_{4} E_{6} E_{7}  E_{8}
Torsion primes ∅  2  ∅  2  2 2, 3 2, 3 2, 3  2, 3, 5
Torsion index  1  2^{e(l)}  1 2^{e(l−1)}  2 2 · 3 2 · 3 2^{2} · 3 2^{6} · 3^{2} · 5

The exponent e(l) is equal to l−⌊log_{2}(\binom{l+1}{2}+1)⌋ except for certain values of l equal or slightly larger than a power of 2 which are explicitly given in [19, Thm. 0.1]. Since 𝔱(R_{1} × R_{2}) = 𝔱(R_{1}) · 𝔱(R_{2}), we may compute torsion indices of semi-simple root systems by reducing to irreducible ones.

5.2. Let M be the character group of a root system of G and let F be a formal group law over a ring R. Let ϵ: R[[M]]_{F} → R be the augmentation map with the kernel I_{F}. Let N be the length of the longest element w_{0} of the Weyl group W of G. Let F_{a} be an additive formal group law.

By definition of the torsion index (see [7, §5]) there exists an element a of the symmetric algebra S_{R}^{N}(M) such that

Δ_{w_{0}}^{F_{a}}(a) = 𝔱.

Let ϕ: S_{R}^{∗}(M) →^{≃} Gr_{R}^{∗}(M, F) be an isomorphism from Lemma 4.2. Let u_{0} ∈ I_{F}^{N} be an element such that ū_{0} = ϕ(a). Therefore, for any reduced decomposition I_{0} of w_{0} we have ϵΔ_{I_{0}}^{F}(u_{0}) = 𝔱 or, in other words, by Prop. 4.4

Δ_{I_{0}}^{F}(u_{0}) = 𝔱 + I_{F} and C_{I_{0}}^{F}(u_{0}) = (−1)^{N}𝔱 + I_{F} in R[[M]]_{F} .

5.3. Lemma. Let I be a sequence of simple reflections and let l(I) be its length. Then

(1) for any i ∈ ℤ, Δ_{I}^{F} (I_{F}^{i}) ⊆ I_{F}^{i−l(I)} and C_{I}^{F} (I_{F}^{i}) ⊆ I_{F}^{i−l(I)};

(2) if l(I) ≤ i and I is not reduced, then

Δ_{I}^{F} (I_{F}^{i}) ⊆ I_{F}^{i−l(I)+1}  and  C_{I}^{F} (I_{F}^{i}) ⊆ I_{F}^{i−l(I)+1};

(3) let u_{0} ∈ I_{F}^{N} be the element chosen above. If l(I) ≤ N then

ϵΔ_{I}^{F} (u_{0}) = (−1)^{N}ϵC_{I}^{F} (u_{0}) = \begin{cases}𝔱 & if I is reduced and l(I) = N \\ 0 & otherwise. \end{cases}

Proof. Claim (1) follows from Prop. 4.6, (1). To check (2), it therefore suffices to check that GrΔ_{I}^{F} is zero. It follows from the corresponding fact in the additive case [7, §4, Prop. 3, (a)] after applying Prop. 4.4. The “otherwise” case of (3) follows from (2). The case where I is reduced and has length N follows from the definition of u_{0}. □

5.4. Proposition. Consider the R-module ϵD(M)_{F} of all R-linear forms ϵD, where D ∈ D(M)_{F}. Assume the torsion index 𝔱 is regular in R. Then for any choice of a collection of reduced decomposition I_{w} for each w ∈ W, the (ϵΔ_{I_{w}})_{w∈W} (resp. the (ϵC_{I_{w}})_{w∈W}) form an R-basis of ϵD(M)_{F}.

Proof. By Theorem 4.11, the ϵΔ_{I_{w}} generate ϵD(M)_{F}. We prove that the ϵΔ_{I_{w}} such that l(I_{w}) ≤ i are independent by induction on i. It is obviously true when i < 0. Let ∑_{l(w)≤i} r_{w}ϵΔ_{I_{w}} = 0. By induction, it suffices to show that the coefficients r_{w} with l(w) = i are zero. For any v and w of length i, the concatenation I_{w} + I_{v^{−1}w_{0}} is a reduced decomposition of w_{0} if and only if v = w. Thus, by Lemma 5.3.(1) and (3) we have ∑_{l(w)≤i} r_{w}ϵΔ_{I_{w}}(Δ_{I_{v^{−1}w_{0}}} (u_{0})) = r_{v}𝔱 which implies that r_{v} = 0 by regularity of 𝔱. The same proof works for the C_{I_{w}}. □

5.5. Lemma. Assume 𝔱 is regular in R.

(1) For any reduced decomposition I_{w} of w and any element u ∈ I_{F}^{i} ∖ I_{F}^{i+1}, the element uΔ_{I_{w}}^{F} is in D(M)_{F}^{(i−l(w))} ∖ D(M)_{F}^{(i−l(w)+1)}.

(2) The group D(M)_{F}^{(i)} is the set of D ∈ D(M)_{F} such that D(I_{F}^{j}) ⊆ I_{F}^{i+j}.

Proof. It suffices to prove (1) after inverting the torsion index. By Lemma 5.3.(1), the operator uΔ_{I_{w}}^{F} is in D(M)_{F}^{(i−l(w))}. Consider the element y = Δ_{I_{w^{−1}w_{0}}}^{F} (u_{0}).

It is in I_{F}^{l(w)}. By Lemma 5.3.(3) the element s = Δ_{I_{w}}^{F}(y) is invertible. Hence, uΔ_{I_{w}}^{F}(y) = us cannot be in I_{F}^{i+1}, and therefore uΔ_{I_{w}}^{F} is not in D(M)_{F}^{(i−l(w)+1)}.

To prove (2), first note that Prop. 4.6.(3) immediately gives one inclusion. Suppose D is such that D(I_{F}^{j}) ⊆ I_{F}^{i+j}. Using Theorem 4.11 we may write it as

D = ∑_{w} u_{w}Δ_{I_{w}}^{F}.

Recall that ν(u_{w}) denotes the largest integer such that u_{w} ∈ I_{F}^{v} (see 4.9). Consider the set S of w ∈ W such that ν(u_{w}) − l(w) is minimal and take an element w_{1} ∈ S such that l(w_{1}) is maximal. To show that D ∈ D(M)_{F}^{(i)}, it suffices to show that ν(u_{w_{1}}) − l(w_{1}) ≥ i by minimality of elements of S.

Let y_{1} = Δ_{I_{w{−1 \atop 1}w_{0}}}^{F} (u_{0}) be the corresponding element in I^{l(w_{1})}. We have

D(y_{1}) = u_{w_{1}}Δ_{I_{w_{1}}}^{F} (y_{1}) + ∑_{w∈S∖{w_{1}}} u_{w}Δ_{I_{w}}(y_{1}) + ∑_{w∉S} u_{w}Δ_{I_{w}}(y_{1}).

By Lemma 5.3.(1) and (2), two sums on the right are in I_{F}^{ν(u_{w_{1}} )+1}. As in the proof of (1), u_{w_{1}}Δ_{I_{w_{1}}}^{F} (y_{1}) is not in I^{ν(u_{w_{1}} )+1} so D(y_{1}) is in I_{F}^{ν(u_{w_{1}} )} ∖ I_{F}^{ν(u_{w_{1}} )+1}. Since

D(y_{1}) is also in I_{F}^{l(w_{1})+i}, we obtain that ν(u_{w_{1}}) − l(w_{1}) ≥ i. □

We now consider the filtration on ϵD(M)_{F} that is the image by ϵ of the filtration on D(M)_{F} defined in 4.5, i.e.

ϵD(M)_{F} ⊇ ⋯ ⊇ ϵD(M)_{F}^{(i)} ⊇ ϵD(M)_{F}^{(i+1)} ⊇ ⋯ ⊇ 0.

5.6. Proposition. Assume 𝔱 is regular in R. This filtration satisfies the following.

(1) For any i ∈ ℤ and ϵD ∈ ϵD(M)_{F}^{(i)}, we have ϵD(I_{F}^{−i+1}) = 0.

(2) Let E be a subset of W of elements of length l, and let the I_{w} be reduced decompositions of each of the w ∈ E. Then any nonzero R-linear combination

f = ∑_{w∈S} r_{w} · ϵΔ_{I_{w}}^{F} (resp. ϵC_{I_{w}}^{F})

is in ϵD(M)_{F}^{−l} ∖ ϵD(M)_{F}^{−l+1}. In particular, for any w

ϵΔ_{I_{w}}^{F} ∈ ϵD(M)_{F}^{−l(w)} ∖ ϵD(M)_{F}^{−l(w)+1} (resp. with C_{I_{w}}^{F}).

(3) For any choice of reduced decompositions I_{w} for every element w ∈ W, the R-module ϵD(M)_{F}^{(−i)} is a free R-module with basis the ϵΔ_{I_{w}} (resp. ϵC_{I_{w}}) with l(w) ≤ i.

(4) For any i ≤ −N, we have ϵD(M)_{F}^{(i)} = ϵD(M)_{F} and for any i > 0, we have ϵD(M)_{F}^{(i)} = 0, i.e. the filtration is of the form

ϵD(M)_{F} = ϵD(M)_{F}^{(−N)} ⊇ ϵD(M)_{F}^{(−N+1)} ⊇ ⋯ ⊇ ϵD(M)_{F}^{(0)} ⊇ 0.

Proof. Claim (1) follows from Lemma 5.5.(2). To prove (2), we first apply f to Δ_{I_{w^{−1}w_{0}}}^{F} (u_{0}). By Lemma 5.3.(3), we obtain 𝔱r_{w}, which has to be zero by (1) for the

element f to be in I_{F}^{l+1}. Therefore, each r_{w} is zero. The last two claims follow. □

6. Invariants and the characteristic map

We now come to the definition of an algebraic replacement H(M)_{F} for the oriented cohomology h^{∗}(G/B) when G is a split semi-simple simply connected algebraic group corresponding to the root system and B is a Borel of G. The identification of H(M)_{F} with h^{∗}(G/B) is the subject of section 13.

6.1. Definition. Let H(M)_{F} be the R-dual of ϵD(M)_{F} and let

𝔠^{F} : R[[M]]_{F} → H(M)_{F}

denote the natural map obtained by duality, i.e. sending u to the evaluation at u. When the formal group law is clear from the context, we write 𝔠 for 𝔠^{F}.

Again, when F = F_{a} is additive, this H(M)_{F} corresponds to the one defined by Demazure in [7, §3].

6.2. Remark. Observe that when R has no 2-torsion, since the operators Δ_{α}^{F} are R[[M]]_{F} -linear by Proposition 3.13, the characteristic map satisfies 𝔠(fu) = ϵ(f)𝔠(u) when f is in R[[M]]_{F}^{W}.

6.3. Theorem. Let (I_{w})_{w∈W} be a choice of reduced decompositions. When the torsion index 𝔱 is regular in R, there is a unique R-basis z_{I_{w}}^{Δ} of H(M)_{F} such that the characteristic map is given by

𝔠(u) = ∑_{w} ϵΔ_{I_{w}}^{F}(u)z_{I_{w}}^{Δ}.

Similarly, there is a unique R-basis z_{I_{w}}^{C} of H(M)_{F} such that the characteristic map is given by

𝔠(u) = ∑_{w} ϵC_{I_{w}}^{F}(u)z_{I_{w}}^{C}.

Proof. We take the bases that are dual to the ones of Proposition 5.4. □

6.4. Theorem. Assume that the torsion index 𝔱 is invertible in R. Then for any choice of a collection of reduced decompositions I_{w} for every w ∈ W, the 𝔠(Δ_{I_{w}}(u_{0})) (resp. the 𝔠(C_{I_{w}}(u_{0}))) form an R-basis of the image of 𝔠. In particular, the characteristic map 𝔠 is surjective.

Proof. We have 𝔠(u_{0}) = 𝔱.z_{I_{0}}^{Δ} and

𝔠(Δ_{I_{w^{−1}w_{0}}} (u_{0})) = 𝔱.z_{w}^{Δ} + ∑_{l(v)>l(w)} ϵΔ_{I_{v}}Δ_{I_{w^{−1}w_{0}}} (u_{0})z_{I_{v}}^{Δ}

by point (3) of Lemma 5.3. All the z_{I_{w}}^{Δ} are thus in the image of the characteristic map by decreasing induction on the length of w. The same proof works when replacing Δ by C. □

Still assuming that 𝔱 is regular in R, let f : R → R′ be a ring morphism sending a formal group law F over R to a formal group law F′ over R′. By restriction (through f), H_{R′}(M)_{F′} is a R-module. Let f_{H} : H(M)_{F} → H_{R′}(M)_{F′} be the R-linear map sending the element z_{I_{w}}^{Δ} in H(M)_{F} to the corresponding one in H_{R′}(M)_{F′}.

6.5. Proposition. We have the following:

(1) The assignment f → f_{H} defines a functor from the category of formal group laws (R, F) (𝔱 regular in R) to the category of pairs (R, M) where M is a module over R, and the morphisms are the obvious ones.

(2) We have f_{H}𝔠^{F} = 𝔠^{F′}f_{∗}.

(3) The map f_{H} defined above is independent of the choice of the reduced decompositions I_{w}.

Proof. Point (1) holds by construction. Point (2) follows from the formula for the characteristic map given in Theorem 6.3. Since f_{∗} is defined independently of the I_{w}, the last item (3) holds if the characteristic map is surjective. The general case follows by embedding H_{R}(M)_{F} into H_{R[1/𝔱]}(M)_{F} where the characteristic map is surjective by Theorem 6.4. □

6.6. Proposition. Assume that 𝔱 is invertible in R. Fix a reduced decomposition I_{w} for each w ∈ W. Then

(1) For any x ∈ R[[M]]_{F}, the system of linear equations in R[[M]]_{F}

Δ_{I_{v}}(x) = ∑_{w∈W} r_{w}Δ_{I_{v}}Δ_{I_{w}}(u_{0})

for all v in W, has a unique solution (r_{w})_{w∈W}.

(2) If (r_{w})_{w∈W} is the solution of (1), then for any D ∈ D(M)_{F}, we have the

equality

D(x) = ∑_{w∈W} r_{w}DΔ_{I_{w}}(u_{0}).

(3) If (r_{w})_{w∈W} is the solution of (1) and if R has no 2-torsion, then all the r_{w}

are in fact in R[[M]]_{F}^{W}. The same is true when replacing Δ by C everywhere in points (1), (2) and (3).

Proof. Let A be a ring and I be an ideal of A contained in its Jacobson radical. A matrix with coefficients in A is invertible if and only if the corresponding matrix with coefficients in A/I is invertible. Since I_{F} is contained in the radical of R[[M]]_{F}, it suffices to show that the matrix of the system in R[[M]]_{F} /I_{F} is invertible. If we order the v’s by increasing length and the w’s by decreasing length, Lemma 5.3 shows that the matrix is lower triangular with 𝔱 on the diagonal, and it is therefore invertible. Point (2) follows from point (1) since the Δ_{I_{w}} (resp. the C_{I_{w}}) form a basis of D(M)_{F} as an R[[M]]_{F} -module by Theorem 4.11. Let us prove point (3). For any simple root α and for any v ∈ W, we have

Δ_{α}Δ_{I_{v}}(x) = ∑_{w∈W} Δ_{α}(r_{w}Δ_{I_{v}}Δ_{I_{w}}(u_{0}))

= ∑_{w∈W} r_{w}Δ_{α}Δ_{I_{v}}Δ_{I_{w}}(u_{0}) + ∑_{w∈W} Δ_{α}(r_{w})s_{α}Δ_{I_{v}}Δ_{I_{w}}(u_{0})

using point (4) of Proposition 3.8. But we also have

Δ_{α}Δ_{I_{v}}(x) = ∑_{w∈W} r_{w}Δ_{α}Δ_{I_{v}}Δ_{I_{w}}(u_{0})

by point (2) with D = Δ_{α}Δ_{I_{v}}. So

∑_{w∈W} Δ_{α}(r_{w})s_{α}Δ_{I_{v}}Δ_{I_{w}}(u_{0}) = 0

for any v ∈ W. Applying the ring automorphism s_{α}, we obtain that the s_{α}Δ_{α}(r_{w}) are solution of the system (1) with x = 0 and are therefore 0 by uniqueness. Thus Δ_{α}(r_{w}) = 0 and r_{w} is fixed by s_{α} for any simple root α by Lemma 3.14 hence by the whole Weyl group W, since it is generated by the simple reflections.

Point (3) for C is proved exactly in the same way (still using Δ_{α}, not C_{α}). □

6.7. Theorem. Assume that the torsion index 𝔱 is invertible in R and that R has no 2-torsion. Choose a reduced decomposition I_{w} for every w ∈ W. The elements Δ_{I_{w}}(u_{0}) (resp. C_{I_{w}}(u_{0})) form a basis of R[[M]]_{F} as an R[[M]]_{F}^{W} -module.

Proof. We need to show that any x can be decomposed in a unique way as x = ∑_{w∈W} r_{w}Δ_{I_{w}}(u_{0}). Note that this is the row v = 1 of the system (1) of Proposition 6.6. By R[[M]]_{F}^{W} -linearity of any Δ_{I_{v}} (see Prop. 3.13), if that row is satisfied with coefficients r_{w} in R[[M]]_{F}^{W}, the rest of the system is satisfied, so this proves uniqueness. Existence of the decomposition then follows from points (1) and (3) of the proposition. The same proof goes through with the C_{I_{w}}(u_{0}). □

6.8. Remark. The previous theorem for the (uncompleted) symmetric algebra, i.e. the additive case, is [7, §6, Théorème 2, (c)], but the proof given there is incorrect: in the notation of loc. cit., the ideal I is only known a priori to be of the right form to apply the graded Nakayama lemma when it is tensored by ℚ, a fact that cannot be assumed, this is the whole point of the theorem. When contacted by one of the authors, Demazure kindly and quickly supplied another proof which we adapted to our setting in Proposition 6.6. There is a slight difference: in the symmetric algebra case, the matrix of the system is upper triangular with diagonal 𝔱 and is therefore invertible, whereas in our case, the lower triangular part does not vanish because our formal group law is not additive, but the strictly upper triangular part is in the radical because our ring is complete.

6.9. Theorem. Assume that the torsion index 𝔱 is invertible in R and that R has no 2-torsion. Then the kernel of the characteristic map 𝔠 : R[[M]]_{F} → H(M)_{F} is the ideal of R[[M]]_{F} generated by elements in I_{F}^{W}.

Proof. By Remark 6.2, the ideal generated by I_{F}^{W} is included in ker 𝔠. Conversely, let x ∈ ker 𝔠, and decompose it as ∑_{w∈W} r_{w}Δ_{I_{w}}(u_{0}) with the r_{w} ∈ R[[M]]_{F}^{W} by the previous theorem. We then have 0 = 𝔠(x) = ∑_{w∈W} ϵ(r_{w})𝔠(Δ_{I_{w}}(u_{0})). But the 𝔠(Δ_{I_{w}}(u_{0})) form an R-basis of the image of 𝔠 by Theorem 6.4, so ϵ(r_{w}) = 0 for all w ∈ W and x is the ideal generated by I_{F}^{W}. □

7. The product structure of the cohomology

In this section, we explain how to define a product on H(M)_{F} in order that the characteristic map be a ring homomorphism. Then, we study the structure of H(M)_{F}.

7.1. Lemma. For any D ∈ D(M)_{F},

(1) there is a finite family (D_{i}, D′_{i}) of elements of D(M)_{F} such that D(uv) = ∑_{i} D_{i}(u)D′_{i}(v) for any elements u, v ∈ R[[M]]_{F}.

(2) Furthermore if D is in D(M)_{F}^{(j)}, each D_{i} and D′_{i} can be chosen in D(M)_{F}^{(m_{i})} and D(M)_{F}^{(m′_{i})} respectively, such that m_{i} + m′_{i} ≥ j.

Proof. To show (1), the elements of D(M)_{F} satisfying a statement form a sub-R[[M]]_{F} -algebra of D(M)_{F}, and it is clear for generators of D(M)_{F} by equation (4) of Prop. 3.8. This same equation also proves (2) for generators by induction. □

Let R[[M]]_{F}^{o∨} be the continuous R-dual of R[[M]]_{F}, i.e. the set of R-linear morphisms f from R[[M]]_{F} to R such that f(I_{F}^{i}) = 0 for some i (depending on f).

7.2. Lemma. The R-module R[[M]]_{F}^{o∨} is flat. When 𝔱 is regular in R, the R-module ϵD(M)_{F} is also flat.

Proof. For any i, the R-module I_{F}^{i} /I_{F}^{i−1} is a free R-module since it is isomorphic to S_{R}^{i}(M). Thus, by induction, for any i, the quotient R[[M]]_{F} /I_{F}^{i} is a finitely generated free R-module, thus flat. Now R[[M]]_{F}^{o∨} is the direct limit of the (R[[M]]_{F} /I^{i})^{∨} (usual R-duals) and it is therefore flat. When 𝔱 is regular, the module ϵD(M)_{F} is a finitely generated free R-module by 5.4, so it is flat. □

7.3. Theorem. Let 𝔱 be regular in R. There is a unique ring structure on H(M)_{F} such that the characteristic map 𝔠 : R[[M]]_{F} → H(M)_{F} is a ring morphism.

Proof. Since R → R[1/𝔱] is an injection and H(M)_{F} is a free R-module, we can check uniqueness after inverting 𝔱, in which case it is obvious since the characteristic map is surjective. Let us now prove the existence of this product. The R-module R[[M]]_{F}^{o∨} has a natural structure of coalgebra induced by the collection of maps (R[[M]]_{F} /I^{i})^{∨} → (R[[M]]_{F} /I^{i})^{∨} ⊗ (R[[M]]_{F} /I^{i})^{∨} dual to the product, and taking direct limits. The inclusion ϵD(M)_{F} ⊆ R[[M]]_{F}^{o∨} induces the diagonal map ϵD(M)_{F} ⊗_{R} ϵD(M)_{F} → R[[M]]_{F}^{o∨} ⊗_{R} R[[M]]_{F}^{o∨} which is injective by Lemma 7.2. To show that ϵD(M)_{F} is a subcoalgebra of R[[M]]_{F}^{o∨}, it therefore suffices to show that the composition ϵD(M)_{F} → R[[M]]_{F}^{o∨} → R[[M]]_{F}^{o∨} ⊗_{R} R[[M]]_{F}^{o∨} factors through the image of ϵD(M)_{F} ⊗_{R} ϵD(M)_{F} in R[[M]]_{F}^{o∨} ⊗_{R} R[[M]]_{F}^{o∨}. This is ensured by Lemma 7.1. Now H(M)_{F} is the dual of a coalgebra, and is therefore an R-algebra. By construction, the characteristic map is a ring homomorphism. □

7.4. We now filter the ring H(M)_{F} by setting that H(M)_{F}^{(i)} is the set of forms on ϵD(M)_{F} that are zero on ϵD(M)_{F}^{(−i+1)}. By Proposition 5.6, we therefore have H(M)_{F}^{(0)} = H(M)_{F} and H(M)_{F}^{(N+1)} = 0. In other words, the filtration has the form

H(M)_{F} = H(M)_{F}^{(0)} ⊇ H(M)_{F}^{(1)} ⊇ ⋯ ⊇ H(M)_{F}^{(N)} ⊇ 0.

We consider the augmentation map ε : H(M)_{F} → R defined as the evaluation at ϵ ∈ ϵD(M)_{F}.

7.5. Proposition. Assume 𝔱 is regular in R. We have:

(1) H(M)_{F}^{(1)} is the kernel of ε.

(2) The filtration is compatible with the product:

H(M)_{F}^{(j)}H(M)_{F}^{(m)} ⊆ H(M)_{F}^{(j+m)}

(3) The characteristic map 𝔠 is a morphism of filtered rings.

Proof. Point (1) is clear since ϵ generates D(M)_{F}^{(0)}. Let h_{1} ∈ H(M)_{F}^{(j)} and h_{2} ∈ H(M)_{F}^{(m)}. The product h = h_{1}.h_{2} is defined as h(ϵD) = ∑_{i} h_{1}(ϵD_{i})h_{2}(ϵD′_{i}) where the (D_{i}, D′_{i}) are as in Lemma 7.1. By the same lemma, if D ∈ D(M)_{F}^{(j+m+1)}, they can be chosen such that D_{i} ∈ D(M)_{F}^{j_{i}} and D′_{i} ∈ D(M)_{F}^{m_{i}} with j_{i} + m_{i} ≥ j + m + 1. Thus, for any i, either j_{i} ≥ j +1 or m_{i} ≥ m+1, so every term in the sum is zero and h ∈ H(M)_{F}^{(i+j)}. For point (3), take an element u in I_{F}^{i}. By definition, 𝔠(u) is the evaluation at u in ϵD(M)_{F}^{∨} = H(M)_{F} so by Prop. 5.6.(1), 𝔠(u) is in H(M)_{F}^{(i)}. □

7.6. Proposition. Let I_{w} be choices of reduced decompositions for every w ∈ W. Assume the torsion index 𝔱 is regular in R. Then

(1) For any i, the elements z_{I_{w}}^{Δ} (resp. z_{I_{w}}^{C}) with l(w) ≥ i generate H(M)_{F}^{(i)}.

(2) The elements z_{I_{w_{0}}}^{Δ} and z_{I_{w_{0}}}^{C} of Theorem 6.3 do not depend on the choices of decompositions I_{w} and are equal up to the sign (−1)^{N}. The element z_{I_{w_{0}}}^{C} is denoted by z_{0} and for any u_{0} as in section 5, we have 𝔠(u_{0}) = (−1)^{N}𝔱z_{0}.

Proof. Point (1) follows from point (3) of Proposition 5.6 since the z_{I_{w}}^{Δ} (resp. z_{I_{w}}^{C}) are the dual basis to the ϵΔ_{I_{w}} (resp. ϵC_{I_{w}}). By Theorem 6.3, we have 𝔠(u_{0}) = 𝔱z_{I_{w_{0}}}^{Δ} . But the left hand side is independent of the choices of the I_{w} and the right hand side is independent of the choice of u_{0}, which proves the claim. We also have 𝔠(u_{0}) = (−1)^{N}𝔱z_{0}. □

7.7. We now consider operators on H(M)_{F}. For any root α, multiplication on the right by C_{α} (resp. Δ_{α})) defines an endomorphism on ϵD(M)_{F}, and therefore one on H(M)_{F} by duality. This operator is denoted by A_{α} (resp. B_{α}). In other words, for any h ∈ H(M)_{F} and ϵD ∈ ϵD(M)_{F}, we have

A_{α}(h)(ϵD) = h(ϵDC_{α})  and  B_{α}(h)(ϵD) = h(ϵDΔ_{α})

As for the operators Δ and C, we use the notation A_{I} and B_{I} for a sequence I of simple reflections.

7.8. Proposition. When 𝔱 is regular in R, these operators satisfy:

(1) A_{α} ∘ 𝔠(u) = 𝔠 ∘ C_{α}(u) and B_{α} ∘ 𝔠(u) = 𝔠 ∘ Δ_{α}(u).

(2) A_{α}H(M)_{F}^{(i)} ⊆ H(M)_{F}^{(i−1)} and A_{α}H(M)_{F}^{(i)} ⊆ H(M)_{F}^{(i−1)}

(3) For any reduced decomposition I_{0} of w_{0}, the element A_{I_{0}}(z_{0}) (resp. B_{I_{0}}(z_{0})) is invertible in H(M)_{F}.

(4) If I is nonreduced and of length N, then the element A_{I}(z_{0}) (resp. B_{I}(z_{0})) is in H(M)_{F}^{(1)} = ker ε.

Proof. Point (1) follows directly from the definitions of the characteristic map and of the operators. Point (2) follows from point (1) of Proposition 4.6. For point (3), first note that by Proposition 7.5, an element h ∈ H(M)_{F} is invertible if and only if ε(h) is invertible in R. Now

𝔱 εA_{I_{0}}(z_{0}) = εA_{I_{0}}(𝔱z_{0}) = εA_{I_{0}}𝔠(u_{0}) = ε𝔠(C_{I_{0}}(u_{0})) = ϵC_{I_{0}}(u_{0}) = 𝔱.

Therefore, εA_{I_{0}}(z_{0}) = 1. The same proof works for εB_{I_{0}}(z_{0}). For point (4), the same series of equalities is used, except that the last term is zero by Prop. 5.3. □

7.9. Proposition. Let I_{w} be a choice of reduced decompositions for all w ∈ W. Assume 𝔱 is regular in R. Then the elements A_{I_{w}}(z_{0}) (resp. B_{I_{w}}(z_{0})) with l(w) ≤ N − i form an R-basis of H(M)_{F}^{(i)}.

Proof. We prove it for the A_{I_{w}}(z_{0}), the proof for the B_{I_{w}}(z_{0}) is similar. First note that by Proposition 7.6, z_{0} is the unique element of H(M)_{F} such that z_{0}(ϵC_{I_{w}}) = δ_{w,w_{0}}. Thus, if l(w)+l(v) ≤ N, we have A_{I_{w}}(z_{0})(ϵC_{I_{v}}) = δ_{w,v^{−1}w_{0}}. In other words, if we decompose A_{I_{w}} on the basis of the z_{I_{v}}, the coordinate on z_{I_{v}} is δ_{w,v^{−1}w_{0}} if l(v) + l(w) ≤ N. Thus, the A_{I_{w}}(z_{0}) with l(w) ≤ N − i can be expressed as linear combinations of the z_{I_{v}} with l(v) ≥ i and the matrix of their expressions is unitriangular up to a correct ordering of rows and columns and is therefore invertible. □

7.10. Proposition. Assume 𝔱 is regular in R and R′, respectively endowed with formal group laws F and F′. Let f : R → R′ be a ring morphism sending F to F′. Then the morphism f_{H} : H_{R}(M)_{F} → H_{R′}(M)_{F′} introduced before Proposition 6.5 is actually a morphism of filtered rings. It satisfies

A_{I}^{F′}f_{H} = f_{H}A_{I}^{F}  and  B_{I}^{F′}f_{H} = f_{H}B_{I}^{F}

for any sequence I.

Proof. Both facts are clear by Point 2 of 6.5 when the characteristic map is surjective, and therefore when 𝔱 is regular by extending scalars to R [1/t]. □

Part 2. Bott-Samelson resolutions for oriented theories.

8. Algebraic cobordism and oriented cohomology theories

In the present section we recall the notion of oriented cohomology theory and the notion of algebraic cobordism Ω following the book of Levine and Morel [13]. As main examples of oriented theories we consider the Chow ring CH, Grothendieck’s K^{0} and connective K-theory k.

8.1. According to [13, §1.1] an oriented cohomology theory h is a contravariant functor from the category of smooth varieties over a field k of an arbitrary characteristic to the category of graded rings satisfying the standard cohomological axioms: localization, homotopy invariance and push-forwards for projective morphisms (see [13, Def.1.1.2]).

A universal theory Ω satisfying these properties was constructed in [13, §2] assuming the base field k has characteristic 0. It is called algebraic cobordism. An element of codimension i in Ω(X), i.e. in Ω^{i}(X), is additively generated by classes [Y → X] of projective maps of codimension i from smooth schemes Y.

For any map f: X_{1} → X_{2} the induced functorial map f^{∗}: Ω^{i}(X_{2}) → Ω^{i}(X_{1}) is called the pull-back. For a projective map f: X_{1} → X_{2}, there is a corresponding push-forward map f_{∗}: Ω(X_{1}) → Ω(X_{2}) shifting the cohomological degree by dim X_{2} − dim X_{1} when X_{1} and X_{2} are equidimensional and given by

[Y_{1} → X_{1}] ↦ [Y_{1} → X_{1} ⟶^{f} X_{2}].

8.2. An oriented cohomology theory h comes with a formal group law F over the coefficient ring h(pt) such that

F (c_{1}^{h}(L_{1}), c_{2}^{h}(L_{2})) = c_{1}^{h}(L_{1} ⊗ L_{2}),

where L_{1} and L_{2} are lines bundles on X and c_{1}^{h} denotes the first Chern class in the cohomology theory h (see [13, Cor.4.1.8]). For Ω, the associated formal group law U

U(x, y) = x + y + ∑_{i,j≥1} a_{ij}x^{i}y^{j}, where a_{ij} ∈ Ω(pt).

turns out to be universal. The ring of coefficients Ω(pt) is generated by the coefficients a_{ij} and coincides with the classical Lazard ring 𝕃.

There is a canonical map pr_{h} : Ω → h sending the coefficients of U to the corresponding coefficients of F, and hence inducing a morphism of formal group laws.

8.3. Example. Consider the Chow ring CH(X) of algebraic cycles on X modulo rational equivalence. According to [13, Thm. 4.5.1] the canonical map pr_{CH} : Ω → CH induces an isomorphism Ω ⊗_{𝕃} ℤ →^{≃} CH of oriented cohomology theories. In particular, pr_{CH} : Ω → CH is surjective and its kernel is generated by 𝕃_{>0}, the subgroup of elements of positive dimension in the Lazard ring. Observe that pr_{CH} restricted to the coefficient rings 𝕃 → ℤ = CH(pt) coincides with the augmentation map. The associated formal group law, denoted by F_{0}, is called the additive formal group law and is given by F_{0}(x, y) = x + y.

8.4. Example. Consider the oriented cohomology theory K(X) = K^{0}(X)[β, β^{−1}], where K^{0}(X) is the Grothendieck K^{0} of X. Observe that K(pt) = ℤ[β, β^{−1}]. According to [13, Cor.4.2.12] the canonical map pr_{K} : Ω → K sending [ℙ^{1}] to β induces an isomorphism Ω ⊗_{𝕃} ℤ[β, β^{−1}] → K of oriented cohomology theories. The associated formal group law, denoted by F_{β}, is called a multiplicative periodic formal group law and is given by F_{β}(x, y) = x + y − βxy.

8.5. Example. Consider a ring homomorphism 𝕃 → ℤ[v] given by [ℙ^{1}] = −a_{11} ↦ v and a_{ij} ↦ 0 for (i, j) ≠ (1, 1). We define a new cohomology theory, called a connective K-theory, by k = Ω ⊗_{𝕃} ℤ[v] (see [13, §4.3.3]). Its formal group law is denoted by F_{v} and is given by F_{v}(x, y) = x + y − vxy. Observe that contrary to Example 8.4 the element v is non-invertible in the coefficient ring k(pt) = ℤ[v].

8.6. Remark. Avoiding the localization axiom one obtains a more general notion of oriented cohomology theory studied by Merkurjev in [14]. In particular, he showed that taking the so-called ‘tilde operation’ of CH one obtains a theory C̃H, defined over a field of any characteristic, which serves as a good replacement for algebraic cobordism in all questions related with cohomological operations. Note that the Lazard ring 𝕃 injects into the coefficient ring C̃H(pt).

We come now to the following important property of an algebraic cohomology theory:

8.7. Definition. We say that a cohomology theory is weakly birationally invariant if for any proper birational morphism f: Y → X the push-forward of the fundamental class f_{∗}(1_{Y}) is invertible.

8.8. Example. Any birationally invariant theory, i.e. such that f_{∗}(1_{Y}) = 1_{X} for any proper birational f: Y → X, is weakly birationally invariant. The Chow ring CH considered over an arbitrary field and the K-theory K considered over a field of characteristic 0 provide examples of birationally invariant theories. According to [13, Thm.4.3.9] the connective K-theory k defined over a field of characteristic 0 is universal among all birationally invariant theories. In particular, the kernel of the canonical map 𝕃 → k(pt) is generated by differences of classes [X] − [X′], where X and X′ are birationally equivalent.

8.9. Lemma. For any smooth variety X over a field of characteristic 0, the kernel of the natural map Ω^{n}(X) → CH^{n}(X) consists of nilpotent elements if n ≥ 0 and is trivial if n = dim X.

Proof. According to [13, Rem. 4.5.6] the kernel is additively generated by products of the form ab, a ∈ 𝕃^{−i}, b ∈ Ω^{n+i}(X), i > 0. Now the claim follows from the fact that Ω^{j}(X) = 0 when j > dim X. □

8.10. Corollary. Over a field of characteristic 0, any oriented cohomology theory in the sense of Levine-Morel is weakly birationally invariant.

Proof. Consider the algebraic cobordism Ω. Let f: Y → X be a proper birational map. The element f_{∗}^{Ω}(1_{Y}) − 1_{X} is in the kernel of the map Ω^{0}(X) → CH^{0}(X). By Lemma 8.9 the difference f_{∗}^{Ω}(1_{Y}) − 1_{X} is nilpotent, therefore, f_{∗}^{Ω}(1_{Y}) is invertible. The statement for an arbitrary theory follows by the universality of Ω. □

9. A sequence of split ℙ^{1}-bundles

In the present section we compute the oriented cohomology h of a variety obtained as a sequence of split ℙ^{1}-bundles. The main tool is the projective bundle theorem for h. Observe that all formulas are given in terms of pull-backs and pushforwards of fundamental classes. This invariant description will play an important role in the sequel.

9.1. Lemma. Let X be a smooth projective variety over a field k. Let p: ℙ_{X}(E) → X be the projective bundle of a vector bundle E of rank 2 over X. Assume that p has a section σ. Then there is a ring isomorphism

h(ℙ_{X}(E)) ≃ h(X)[ξ]/(ξ^{2} − yξ), where ξ = σ_{∗}(1_{X}) and y = p^{∗}σ^{∗}ξ.

Proof. Consider the canonical embedding O_{E}(−1) → p^{∗}E, where O_{E}(−1) is a tautological line bundle over ℙ_{X}(E) (see [9, B.5.5]). Let L denote the quotient E/σ^{∗}O_{E}(−1). By Cartan formula (see [16, Def. 3.26.(3)]) we have

c_{1}^{h}(E) = c_{1}^{h}(σ^{∗}O_{E}(−1)) + c_{1}^{h}(L)  and  c_{2}^{h}(E) = c_{1}^{h}(σ^{∗}O_{E}(−1)) · c_{1}^{h}(L).

According to the projective bundle theorem (see [16, Thm. 3.9 and formula (17)]) applied to p, there is a ring isomorphism

h(ℙ_{X}(E)) ≃ h(X)[t]/(t − a)(t − b),

where a = p^{∗}c_{1}^{h}(σ^{∗}O_{E}(−1)), b = p^{∗}c_{1}^{h}(L) and t = c_{1}^{h}(O_{E}(−1)).

Consider the elements ξ = b −_{F} t and y = b −_{F} a, where F is the formal group law corresponding to h. Observe that by [15, 2.2.9] ξ = c_{1}^{h}(O_{E}(1) ⊗ p^{∗}L) = σ_{∗}(1_{X}) and by the very definition y = c_{1}^{h}(p^{∗}σ^{∗}O_{E}(1) ⊗ p^{∗}L) = p^{∗}σ^{∗}ξ.

Since changing t to ξ induces an automorphism of h(ℙ_{X}(E)), we only need to prove the relation ξ^{2} − yξ = 0. Note that for any power series f, we may write f(x) − f(y) = (x − y)f′(x, y) where f′(x, y) is again a power series. Hence, we have b−_{F} t = (b−_{F} t)−(t−_{F} t) = (t−b)f′_{1}(b, t) and (b−_{F} t)−(b−_{F} a) = (t−a)f′_{2}(a, b, t) for some power series f′_{1} and f′_{2}. Therefore

ξ(ξ − y) = (b −_{F} t)((b −_{F} t) − (b −_{F} a)) = (t − b)(t − a)f′_{1}(b, t)f′_{2}(a, b, t) = 0

and the proof is finished. □

9.2. Corollary. In the notation of Lemma 9.1 we have the following formula for the push-forward of the fundamental class

p_{∗}(1_{ℙ_{X}(E)}) = σ^{∗}(ξ^{−1} + (−_{F} ξ)^{−1}).

Proof. By [20, Thm. 5.30] we have

p_{∗}(1_{ℙ_{X}(E)}) = (σ^{∗}b −_{F} σ^{∗}a)^{−1} + (σ^{∗}a −_{F} σ^{∗}b)^{−1},

where σ^{∗}a and σ^{∗}b are the roots of the bundle E as in the proof of Lemma 9.1. Since y = b −_{F} a = p^{∗}σ^{∗}ξ we obtain the desired formula. □

9.3. Theorem. More generally, let X be a variety obtained by means of a sequence of split ℙ^{1}-bundles, i.e. there is a sequence of varieties X_{i}, 0 ≤ i ≤ N, starting from a point X_{0} = pt and finishing at X_{N} = X such that for each 1 ≤ i ≤ N

p_{i}: X_{i} ≃ ℙ_{X_{i−1}}(E_{i}) → X_{i−1}

is a projective bundle with a section σ_{i}, where E_{i} is some vector bundle of rank 2 over X_{i−1}. Then there is a ring isomorphism

h(X_{N}) ≃ h(pt)[ξ_{1}, …, ξ_{N}]/I,

where I is an ideal generated by elements {ξ_{i}^{2} − y_{i}ξ_{i}}_{i=1...N}, ξ_{i} = p^{∗}σ_{i∗}(1_{X_{i−1}}), y_{i} = p^{∗}σ_{i}^{∗}σ_{i∗}(1_{X_{i−1}}) and p^{∗} denotes the pull-back on X_{N}.

Proof. Follows by induction using Lemma 9.1. □

9.4. Remark. Observe that the element y_{i} appearing in the relations is not necessarily in h(pt). To obtain a complete answer in terms of generators ξ_{1}, …, ξ_{N} and relations with coefficients in h(pt) one has to express the elements y_{i} in terms of the ‘previous’ generators ξ_{1}, …, ξ_{i−1}. In the next section, we show how to obtain such an expression for Bott-Samelson varieties using the characteristic map.

9.5. Example. Let X be as in Lemma 9.1. By [9, 3.2.11] we have the following formula for the total Chern class of the tangent bundle of ℙ_{X}(E):

c^{h}(T_{ℙ_{X}(E)}) = c^{h}(p^{∗}T_{X}) · c^{h}(O_{E}(1) ⊗ p^{∗}E).

Since there is an exact sequence

0 → O_{E}(1) ⊗ p^{∗}σ^{∗}O_{E}(−1) → O_{E}(1) ⊗ p^{∗}E → O_{E}(1) ⊗ p^{∗}L → 0,

by Cartan formula we obtain

c^{h}(O_{E}(1) ⊗ p^{∗}E) = c^{h}(O_{E}(1) ⊗ p^{∗}L) · c^{h}(O_{E}(1) ⊗ p^{∗}σ^{∗}O_{E}(−1)).

Since O_{E}(1) ⊗ p^{∗}σ^{∗}O_{E}(−1) = (O_{E}(1) ⊗ p^{∗}L) ⊗ (p^{∗}σ^{∗}O_{E}(−1) ⊗ p^{∗}L^{∨}), we obtain that

c^{h}(T_{ℙ_{X}(E)}) = c^{h}(p^{∗}T_{X}) · (1 + ξ) · (1 + (ξ −_{F} y)).

More generally, for X_{N} from Theorem 9.3 we obtain by induction

(4)  c^{h}(T_{X_{N}}) = ∏_{i=1}^{N}(1 + ξ_{i})(1 + (ξ_{i} −_{F} y_{i})).

10. An elementary step of the Bott-Samelson resolution

We apply the results of the previous section to compute an oriented cohomology of a split P/B = ℙ^{1}-bundle, where P is a semi-direct product of a split reductive linear algebraic group of semisimple rank one and a connected unipotent group, and B is a Borel subgroup of P. This example appears as an elementary step in the construction of a Bott-Samelson variety.

10.1. Consider a split connected solvable algebraic group B over a field k. Let T denote its split maximal torus. Let X be a scheme on which B acts on the right such that the quotient X/B exists and X → X/B is a principal B-bundle. Let M denote the group of characters of T. Each character λ ∈ M extends to a one-dimensional representation V_{λ} of B and, hence, defines a line bundle L(λ) over X/B whose total space is the fiber product X ×^{B} V_{λ}, i.e. the quotient (X ×_{k} V_{λ})/B by means of the right B-action (x, v)b = (xb, b^{−1}v) (see [1, Ch.1 §3]).

10.2. Definition. Let h be an oriented cohomology theory and let F be the corresponding formal group law over the coefficient ring R = h(pt). We define a ring homomorphism 𝔠: R[[M]]_{F} → h(X/B) from the formal group ring R[[M]]_{F} to the cohomology ring h(X/B) by sending a generator x_{λ} to the first Chern class c_{1}^{h}(L(λ)) of the line bundle L(λ). This map is well-defined since all Chern classes are nilpotent and satisfy c_{1}^{h}(L(λ+μ)) = c_{1}^{h}(L(λ)⊗L(μ)) = c_{1}^{h}(L(λ))+_{F} c_{1}^{h}(L(μ)). It is called the characteristic map.

10.3. We follow the notation of [7, §2]: Let P be a semi-direct product of a connected unipotent group U and a reductive split group L of semi-simple rank 1. Let T be a maximal split torus of L and let M denote its group of characters. Let α be the one of two roots of L with respect to T, let U_{α} be the corresponding unipotent subgroup and let B = T · U_{α} · U be the Borel subgroup of P containing T. Let s_{α}(λ) = λ − α^{∨}(λ) · α denote the reflection corresponding to the root α.

Consider the fibered product X′ = X ×^{B} P, i.e. the quotient (X ×_{k} P)/B by means of the B-action (x, h)b = (xb, b^{−1}h). By definition, X′ is a principal P -bundle over X/B, all fibers of the canonical projection p: X′/B → X/B are isomorphic to P/B ≃ ℙ^{1} and there is an obvious section σ: X/B → X′/B given by x ↦ (x, 1). According to [11, Exrc. 7.10.(c)] there exists a vector bundle E of rank 2 over X/B such that X′/B can be identified with the projective bundle ℙ_{X/B}(E).

10.4. Remark. Observe that in [7, §2] instead of the quotient X′/B the author considers the quotient of X′ modulo the opposite Borel subgroup B′. This doesn’t change much, since there is an obvious isomorphism i: X′/B → X′/B′ induced by an automorphism of X′ sending the class of (x, p) to the class of (x, pn_{α}), where n_{α} is an element of P representing the reflection s_{α}. In particular, for any character λ we have i^{∗}L′(λ) = L′(s_{α}(λ)), where L′(λ) (resp. L′(s_{α}(λ))) is the line bundle over X′/B′ (resp. over X′/B).

10.5. Applying Lemma 9.1 to the projective bundle p: ℙ_{X/B}(E) → X/B with the section σ we obtain an isomorphism

h(X′/B) ≃ h(X/B)[ξ]/(ξ^{2} − yξ), where ξ = σ_{∗}(1_{X/B}) and y = p^{∗}σ^{∗}ξ.

10.6. Lemma. We have σ^{∗}ξ = c_{1}^{h}(L(−α)) = 𝔠(x_{−α}).

Proof. By definition of the push-forward we have ξ = c_{1}^{h}(O(D)), where O(D) is a line bundle corresponding to the divisor D = σ(X/B). The first equality then follows from [8, §2.5 Lem. 3]. The second one follows from the definition of the characteristic map 𝔠. □

10.7. Corollary. We have y = p^{∗}σ^{∗}ξ = p^{∗}𝔠(x_{−α}).

10.8. Lemma. Consider the characteristic map 𝔠′: R[[M]]_{F} → h(X′/B). For any u ∈ R[[M]]_{F}, we have

(1) σ^{∗}𝔠′(u) = 𝔠(u) and (2) 𝔠′(u) = p^{∗}𝔠(s_{α}(u)) + p^{∗}𝔠(Δ_{−α}(u)) · ξ

Proof. By [8, §2.5 Lem. 1 and Prop.1], for any character λ ∈ M, we have two equalities

σ^{∗}L′(λ) = L(λ)  and  L′(s_{α}(λ)) = p^{∗}(L(λ)) ⊗ O(D)^{⊗α^{∨}(λ)},

where L′(λ) is the corresponding line bundle over X′/B and O(D) is a line bundle corresponding to the divisor D = σ(X/B).

By the first equality, formula (1) holds for u = x_{λ}. It therefore holds for all u ∈ R[M]_{F} and then for all u ∈ R[[M]]_{F} by continuity.

By the second equality, we have for u = x_{λ}

𝔠′(x_{s_{α}(λ)}) = p^{∗}𝔠(x_{λ}) +_{F} (α^{∨}(λ)) ·_{F} ξ.

Note that for any power series g(ξ) in a variable ξ we may write g(ξ) = g(0)+ξ·g′(ξ), where g′(ξ) = (g(ξ) − g(0))/ξ is again a power series. Since, ξ^{2} = yξ, we have g(ξ) = g(0) + ξ · g′(y). Applying this to g(ξ) = p^{∗}𝔠(x_{λ}) +_{F} α^{∨}(λ) ·_{F} ξ and observing that g(0) = p^{∗}𝔠(x_{λ}) we obtain

𝔠′(x_{s_{α(λ)}}) = p^{∗}𝔠(x_{λ}) + ξ · g′(y),

where g(y) = p^{∗}𝔠(x_{λ}+_{F} α^{∨}(λ)·_{F} x_{−α}) = p^{∗}𝔠(x_{s_{α}(λ)}) by Cor. 10.7. Then by definition of the operator Δ_{−α} we obtain g′(y) = p^{∗}(−Δ_{−α}(x_{λ})) and, hence, 𝔠′(x_{s_{α(λ)}}) = p^{∗}𝔠(x_{λ}) − p^{∗}𝔠(Δ_{−α}(x_{λ}))ξ. Since Δ_{−α}s_{α} = −Δ_{−α}, applying s_{α} to u we obtain

𝔠′(x_{λ}) = p^{∗}𝔠(s_{α}(x_{λ})) + p^{∗}𝔠(Δ_{−α}(x_{λ})) · ξ.

For general u formula (2) then follows by induction on the degree of monomials and by continuity, using Prop. 3.8.(4) and ξ^{2} = yξ. □

10.9. Remark. Following Remark 3.7, the second formula would look simpler with the other definition of Δ_{α}.

10.10. Proposition. The following formulas hold for any u ∈ R[[M]]_{F}

\begin{matrix}(1) p_{∗}(1_{X′/B}) = 𝔠(C_{α}(1)), & (3) p^{∗}(𝔠(C_{α}(u))) = 𝔠′(C_{α}(u)), \\ (2) p_{∗}(𝔠′(u)) = 𝔠(C_{α}(u)), & (4) p^{∗}p_{∗}(𝔠′(u)) = 𝔠′(C_{α}(u)). \end{matrix}

Proof. The first formula of the proposition follows from Corollary 9.2 applied to the projective bundle p: X′/B → X/B where we identify σ^{∗}ξ with 𝔠(x_{−α}) according to Lemma 10.6.

By point (2) of Lemma 10.8 and the projection formula we have

p_{∗}(𝔠′(u)) = 𝔠(s_{α}(u)) · p_{∗}(1_{X′/B}) + 𝔠(Δ_{−α}(u)) · p_{∗}(ξ) =

By the first formula and the fact that p_{∗}(ξ) = p_{∗}(σ_{∗}(1_{X/B})) = 1_{X/B} it can be rewritten as

𝔠(s_{α}(u)) · 𝔠(C_{α}(1)) + 𝔠(Δ_{−α}(u)) = 𝔠(s_{α}(u)C_{α}(1) + Δ_{−α}(u))

Using the definition of C_{α} and point (3) of Proposition 3.12 we finally obtain

= 𝔠(s_{α}(u) · e_{α} + Δ_{−α}(u)) = 𝔠(C_{−α}(s_{−α}(u))) = 𝔠(C_{α}(u)).

This proves the second formula.

The third formula follows from point (2) of Lemma 10.8 and the fact that C_{−α}Δ_{−α} = 0 (see (6) of Prop. 3.12). The last formula is obtained by the composite of the second and the third one. □

11. Bott-Samelson varieties

In the present section we compute an oriented cohomology of Bott-Samelson varieties. For definition and basic properties of Bott-Samelson varieties, we refer to papers [1], [17] and [5].

11.1. Let G be a split semisimple linear algebraic group over a field k, let T be a split maximal torus of G and let T ⊂ B be a Borel subgroup. Let {α_{1}, α_{2}, …, α_{n}}, where n is the rank of G, be a set of simple roots of G. Let P_{i} be a minimal parabolic subgroup corresponding to a simple root α_{i}, i.e. the subgroup generated by unipotent subgroups U_{α_{j}}, 1 ≤ j ≤ n and U_{−α_{i}}.

11.2. Definition. For an l-tuple of integers I = (i_{1}, i_{2}, …, i_{l}) with 1 ≤ i_{j} ≤ n, we define a variety X_{I} to be the fiber product

X_{I} = P_{i_{1}} ×^{B} P_{i_{2}} ×^{B} … ×^{B} P_{i_{l}}.

If I = ∅, then we set X_{∅} = pt.

Observe that there is a natural right action of B on X_{I}, the quotient X_{I}/B exists and X_{I} → X_{I}/B is a principal B-bundle. The variety X_{I}/B is called a Bott-Samelson variety corresponding to I.

11.3. Let J = (i_{1}, i_{2}, …, i_{l−1}). Then X_{I} = X_{J} ×^{B} P_{i_{l}}, and we are in the situation of 10.3 for P = P_{i_{l}}, α = α_{i_{l}}, X′ = X_{I} and X = X_{J}. In particular, the projection map

p_{I}: X_{I}/B → X_{J}/B

has a structure of a ℙ^{1}-bundle with a section denoted by σ_{I}.

Observe that the structure map π_{I}: X_{I}/B → pt can be written as a composite of projection maps

X_{I}/B → X_{(i_{1},...,i_{l−1})}/B → … → X_{(i_{1})}/B → pt.

Hence, the variety X_{I}/B is obtained by means of a sequence of split ℙ^{1}-bundles.

According to 10.1 for an oriented cohomology theory h with a formal group law F there is a well-defined characteristic map

𝔠_{I}: R[[M]]_{F} → h(X_{I}/B),

where R[[M]]_{F} is the formal group ring of the group of characters M of T. Observe that 𝔠_{∅} = ϵ is the augmentation map. Applying Theorem 9.3 and Corollary 10.7 we obtain an isomorphism

(5)  h(X_{I}/B) ≃ h(pt)[ξ_{1}, ξ_{2}, …, ξ_{l}]/({ξ_{j}^{2} − y_{j}ξ_{j}}_{j=1...l}),

where y_{j} = p^{∗}𝔠_{(i_{1},...,i_{j−1})}(x_{−α_{i_{j}}}) and p^{∗} denotes the pull-back to h(X_{I}/B).

11.4. Theorem. Let X_{I}/B be the Bott-Samelson variety corresponding to an l-tuple I = (i_{1}, …, i_{l}). For any subset K of [1, l] we define

ξ_{K} = ∏_{j∈K} ξ_{j} and Θ_{K} = Θ_{1} ⋯ Θ_{l}, where Θ_{j} = \begin{cases}Δ_{−α_{i_{j}}} & if j ∈ K, \\ s_{α_{i_{j}}} & otherwise. \end{cases}

Then the elements ξ_{K}, where K runs through all subsets of [1, l], form a basis of the free h(pt)-module h(X_{I}/B). As a ring h(X_{I}/B) is a quotient of the polynomial ring h(pt)[ξ_{1}, …, ξ_{l}] modulo the relations

ξ_{j}^{2} = ∑_{K⊆[1,j−1]} ϵΘ_{K}(x_{−α_{i_{j}}})ξ_{K}ξ_{j}.

Proof. The fact that h(X_{I}/B) is a free h(pt)-module with a basis ξ_{K} follows by the projective bundle theorem, since X_{I}/B is obtained by means of a sequence of split ℙ^{1}-bundles. To obtain the relations, we first use Lemma 10.8.(2) to prove by induction that

(6)  𝔠_{(i_{1},...,i_{j})}(u) = ∑_{K⊆[1,j]} ϵΘ_{K}(u)ξ_{K}.

Using this we then compute y_{j} and plug the result into the formula (5). □

The following lemma provides a formula for the push-forward of the structure map π_{I}: X_{I}/B → pt:

11.5. Lemma. Consider the variety X_{I}/B for a tuple I = (i_{1}, …, i_{l}). For any u ∈ R[[M]]_{F} we have

π_{I∗}(𝔠_{I}(u)) = ϵC_{I}(u),

where C_{I} = C_{α_{1}} … C_{α_{l}} is the composite of operators defined in 3.11.

Proof. Decomposing the structure map as π_{I}: X_{I}/B →^{p_{I}} X_{J}/B →^{π_{J}} pt, where J = (i_{1}, …, i_{l−1}) and applying Prop. 10.10.(2) we obtain

π_{I∗}(𝔠_{I}(u)) = π_{J∗}p_{I∗}(𝔠_{I}(u)) = π_{J∗}(𝔠_{J}(C_{α_{l}}(u))).

Repeating this recursively for the structure map π_{J}: X_{J}/B → pt and so on, we obtain the desired formula. □

12. Bott-Samelson resolutions

In the present section we discuss the relations between Bott-Samelson varieties and Schubert varieties on G/B.

12.1. According to Bruhat decomposition the variety of complete flags G/B is a finite disjoint union of affine spaces G/B = ∐_{w∈W} BwB/B, where W is the Weyl group of G. The closure of a cell BwB/B is denoted by X_{w} and is called a Schubert variety. Observe that X_{w} is not smooth in general.

Let w = s_{i_{1}}s_{i_{2}} … s_{i_{l}} be a (reduced) decomposition into a product of simple reflections (here s_{i} denotes s_{α_{i}}) with l = l(w) being the length of w. The multiplication map induces a morphism

q_{I}: X_{I}/B → G/B, where I = (i_{1}, i_{2}, …, i_{l}).

By the results of [1, Ch. 3] (see also [17, §1.7] and [5, §4]) this map factors as q_{I}: X_{I}/B → X_{w} ↪ G/B, where the first map is surjective birational and the second is a closed embedding. Furthermore, when J is I with the last entry removed, we have q_{I} ∘ σ_{I} = q_{J}.

12.2. Lemma. Consider the characteristic map 𝔠_{G/B}: R[[M]]_{F} → h(G/B). For any u ∈ R[[M]]_{F} we have

q_{I}^{∗}(𝔠_{G/B}(u)) = 𝔠_{I}(u).

Proof. By definition of the map q_{I} and the bundle L(λ) we have q_{I}^{∗}(L(λ)) = L_{I}(λ), where L(λ) (resp. L_{I}(λ)) is the line bundle over G/B (resp. over X_{I}/B) corresponding to a character λ. Hence, q_{I}^{∗}(𝔠_{G/B}(x_{λ})) = 𝔠_{I}(x_{λ}). For a general u it follows by continuity. □

Let w_{0} be the element of maximal length in W and let X = X_{I_{0}}/B be the Bott-Samelson variety where I_{0} is a reduced decomposition of w_{0}.

12.3. Lemma. Assume that h is weakly birationally invariant. Then the pull-back q_{I_{0}}^{∗} is injective.

Proof. Since our theory is weakly birationally invariant, the push-forward of the fundamental class q_{I_{0}∗}(1) is invertible. The lemma now follows by projection formula. □

12.4. Theorem (cf. [5, Prop. 3]). Assume that h is weakly birationally invariant. Consider the quotient map p_{i}: G/B → G/P_{i}, where 1 ≤ i ≤ n. Then for any u ∈ R[[M]]_{F} we have the equality

p_{i}^{∗}p_{i∗}(𝔠_{G/B}(u)) = 𝔠_{G/B}(C_{α_{i}}(u))

Proof. We choose a reduced decomposition I_{0} of w_{0} in simple reflections such that the last reflection is s_{i}. Let J be the same tuple without the last entry. Consider the Cartesian diagram

(7)  \xymatrix{X_{I_{0}}/B \ar[r]^{p_{I_{0}}} \ar[d]_{q_{I_{0}}} & X_{J}/B \ar[d]^{p_{i}∘q_{J}} \\ G/B \ar[r]^{p_{i}} & G/P_{i}}

Since the horizontal map p_{i} is smooth, the diagram is transversal. Since q_{I_{0}}^{∗} is injective by Lemma 12.3, it is enough to prove the equality after applying q_{I_{0}}^{∗} to both sides. By base change and by transversality of the diagram we have

q_{I_{0}}^{∗}p_{i}^{∗}p_{i∗}(𝔠_{G/B}(u)) = p_{I_{0}}^{∗}(p_{i} ∘ q_{J})^{∗}p_{i∗}(𝔠_{G/B}(u)) = p_{I_{0}}^{∗}p_{I_{0}∗}q_{I_{0}}^{∗}(𝔠_{G/B}(u))

By Lemma 12.2 and by Prop. 10.10.(4) we obtain

= p_{I_{0}}^{∗}p_{I_{0}∗}(𝔠_{I_{0}}(u)) = 𝔠_{I_{0}}(C_{α_{i}}(u)) = q_{I_{0}}^{∗}(𝔠_{G/B}(C_{α_{i}}(u)))

and the theorem is proved. □

Part 3. Algebraic and geometric comparison. Applications.

13. Comparison results

In the present section we explain why the ring H(M)_{F} is naturally isomorphic to the ring h^{∗}(G/B) (see 13.12) for most weakly birationally invariant theories.

13.1. Let I be a sequence of simple reflections. Recall that w(I) = s_{i_{1}} ⋯ s_{i_{l}} is the corresponding product of simple reflections. Let

ζ_{I} = (q_{I})_{∗}(1) ∈ h^{∗}(G/B)

denote the push-forward of the fundamental class of the Bott-Samelson variety corresponding to I.

13.2. Assumption. For each element w ∈ W, let I_{w} be a chosen reduced decomposition of w. The elements ζ_{I_{w}}, where w runs through all elements of the Weyl group W, form an R-basis of the cohomology h^{∗}(G/B).

13.3. Lemma. Assumption 13.2 holds for

(1) any oriented cohomology theory h_{1}^{∗} = h_{0}^{∗} ⊗_{R_{0}}R_{1} obtained from another one h_{0}^{∗} for which the assumption already holds;

(2) Chow groups and Chow groups mod n (in arbitrary characteristic);

(3) algebraic cobordism, and therefore all oriented cohomology theories over a base field of characteristic zero;

(4) K-theory (and K-theory mod n) in arbitrary characteristic.

Proof. Point (1) is obvious since a free module stays free by base change. For Chow groups and K-theory, this is classical and for example proved in [8]. For algebraic cobordism, since G/B is a cellular space and classes of Schubert varieties form a ℤ-basis of CH(G/B), the lemma follows from [21, Cor. 2.9]. □

We consider the characteristic map 𝔠_{G/B}: R[[M]]_{F} → h^{∗}(G/B).

13.4. Lemma. Let π: G/B → pt be the structure map. For any sequence I and u ∈ R[[M]]_{F} we have

π_{∗}(ζ_{I} · 𝔠_{G/B}(u)) = ϵC_{I}(u).

Proof. The projection formula and Lemma 11.5 imply that we have

π_{∗}(ζ_{I} · 𝔠_{G/B}(u)) = π_{∗}(q_{I∗}(1) · 𝔠_{G/B}(u)) = π_{∗}q_{I∗}q_{I}^{∗}(𝔠_{G/B}(u)) = π_{I∗}(𝔠_{I}(u)) = ϵC_{I}(u).

□

Recall from Definition 6.1 that we also have an algebraic characteristic map 𝔠 : R[[M]]_{F} → H(M)_{F}.

13.5. Lemma. Assume the torsion index 𝔱 is invertible in R and that Assumption 13.2 is satisfied. Then, there is a unique morphism θ : H(M)_{F} → h^{∗}(G/B) of R-algebras such that 𝔠_{G/B} = θ ∘ 𝔠.

Proof. We fix a reduced decomposition I_{w} for each w ∈ W. By 6.4, the map 𝔠 is surjective. We can therefore find a pre-image u_{w} of each element z_{I_{w}}^{C} of Theorem 6.3. By the form of the characteristic map given there, these elements satisfy ϵC_{I_{w}}(u_{v}) = δ_{w,v}. Let us set τ_{I_{w}} = 𝔠_{G/B}(u_{w}). By the previous lemma, (τ_{I_{w}})_{w∈W} is a basis of h^{∗}(G/B) that is dual basis to the basis (ζ_{I_{w}})_{w∈W} with respect to the bilinear form (ζ, τ) ↦ π_{∗}(ζ.τ), and this form is therefore nondegenerate. We now define θ as the R-linear morphism sending z_{I_{w}}^{C} to τ_{I_{w}}. By the formula π_{∗}(ζ_{I_{w}}.𝔠_{G/B}(u)) = ϵC_{I_{w}}(u), we must have

(8)  𝔠_{G/B}(u) = ∑_{w∈W} ϵC_{I_{w}}(u)τ_{I_{w}}

and the isomorphism θ therefore satisfies 𝔠_{G/B} = θ ∘𝔠 since the z_{I_{w}}^{C} satisfy the same formula with 𝔠. The uniqueness of θ is immediate by surjectivity of 𝔠. □

13.6. Corollary. When 𝔱 is invertible in R and Assumption 13.2 holds, the characteristic map 𝔠_{G/B} is surjective.

Proof. Use Theorem 6.4 and the previous lemma. □

13.7. Corollary. When 𝔱 is invertible in R which has no 2-torsion and Assumption 13.2 holds, the kernel of the characteristic map 𝔠_{G/B} is the ideal of R[[M]]_{F} generated by elements in I fixed by W.

Proof. Use Theorem 6.9 and the previous lemma. □

Let u_{0} ∈ I^{N} be chosen as in section 5.

13.8. Lemma. Let I_{w} be a reduced decomposition of an element w. Then

ζ_{I_{w}} · 𝔠_{G/B}(u_{0}) = \{ \begin{matrix}0 & if w ≠ w_{0} \\ 𝔠_{G/B}(u_{0}) & if w = w_{0} \end{matrix}

Proof. We have ζ_{I_{w}} · 𝔠_{G/B}(u_{0}) = q_{I_{w}∗}(𝔠_{I_{w}}(u_{0})) by the projection formula and the fact that q_{I_{w}}^{∗}𝔠_{G/B} = 𝔠_{I_{w}}. By formula (6), 𝔠_{I_{w}}(u_{0}) = 0 if l(I_{w}) < N i.e. if w ≠ w_{0}. This proves the first part of the formula. The other case will follow if we show that ζ_{I_{w_{0}}} = 1 − ∑_{w≠w_{0}} r_{w}ζ_{I_{w}} for some coefficients r_{w} in R. For this, it suffices to show that the coefficient r_{w_{0}} in front of ζ_{I_{w_{0}}} in the decomposition of 1 on the basis of the ζ_{I_{w}} is 1. We have the commutative diagram

[figure: U; □; i_{U′}; X_{I_{0}}/B; q_{I_{0}}; X_{I′}/B; σ; q_{I′}; U; i_{U}; G/B]

where I_{0} is any reduced decomposition of w_{0}, I′ is I_{0} with its last entry removed, σ is the section of p_{I_{0}}, U is the big open cell in G/B and U′ is its pre-image in X_{I_{0}}/B. By [8, §3.10 and 3.11], U′ is indeed isomorphic to U and it is included in the complement of σX_{I′}/B. Thus, i_{U′}^{∗}σ_{∗} = 0, using that X_{I_{0}}/B → X_{I′}/B is a projective bundle. By base change on the cartesian diagram, we obtain i_{U}^{∗}q_{I′∗} = 0 and i_{U}^{∗}q_{I_{0}∗}(1) = 1. Since any I_{w} can be completed to a decomposition of w_{0}, this proves that i_{U}^{∗}(ζ_{I_{w}}) = 0 if w ≠ w_{0}. Pulling back 1 = ∑_{w} r_{w}ζ_{I_{w}} by i_{U}, we get 1 = r_{w_{0}} (by homotopy invariance, h^{∗}(U) ≃ R). □

13.9. Lemma. We have

𝔱 ζ_{∅} = 𝔠_{G/B}(u_{0})  and  𝔱 ζ_{I} = 𝔠_{G/B}(C_{I^{rev}}(u_{0})).

Proof. Let I_{w_{0}} = (i_{1}, …, i_{N}). With the notation of Theorem 11.4, we have

𝔱 ζ_{∅} = 𝔱 q_{∅∗}(1) = 𝔱 q_{I_{w_{0}} ∗}σ_{I_{w_{0}} ∗} ⋯ σ_{(i_{1})∗}(1) = q_{I_{w_{0}} ∗}(𝔱 ξ_{[1,N]}) = q_{I_{w_{0}} ∗}(𝔠_{I_{w_{0}}} (u_{0}))
= q_{I_{w_{0}} ∗}(1)𝔠_{G/B}(u_{0}) = 𝔠_{G/B}(u_{0})

where the two last equalities follow from the projection formula together with q_{I_{w}}^{∗}𝔠_{G/B} = 𝔠_{I_{w}} and then Lemma 13.8. For the second formula, which coincides with the first one when I is empty, we use induction on the length of I. If I = (i_{1}, …, i_{l}) and J = (i_{1}, …, i_{l−1}), then by base change via diagram (7) we obtain

𝔱 · ζ_{I} = 𝔱 · q_{I∗}(1) = 𝔱 · q_{I∗}p_{I}^{∗}(1) = 𝔱 · p_{i_{l}}^{∗}(p_{i_{l}} ∘ q_{J})_{∗}(1) = p_{i_{l}}^{∗}p_{i_{l}∗}(𝔱 · ζ_{J}),

By the induction step and Theorem 12.4 we obtain

= p_{i_{l}}^{∗}p_{i_{l}∗}(𝔠_{G/B}(C_{J^{rev}}(u_{0}))) = 𝔠_{G/B}(C_{α_{i_{l}}} ∘ C_{J^{rev}}(u_{0})) = 𝔠_{G/B}(C_{I^{rev}}(u_{0}))

and the proof is finished. □

13.10. Lemma. Under the assumptions of Proposition 13.5, we have ζ_{∅} = τ_{I_{w_{0}}} . In particular, τ_{I_{w_{0}}} does not depend on the choices of the reduced decompositions I_{w}.

Proof. By definition of the dual basis (τ_{I_{w}})_{w∈W}, it suffices to show that we have π_{∗}(ζ_{I_{w}} · ζ_{∅}) = δ_{w_{0},w}. By lemmas 13.9 and 13.4, we have

π_{∗}(ζ_{I_{w}} · ζ_{∅}) = π_{∗}(ζ_{I_{w}} · 𝔱^{−1}𝔠_{G/B}(u_{0})) = 𝔱^{−1}ϵC_{I_{w}}(u_{0}) = δ_{w,w_{0}}. □

Let Ã_{i} denote the operator p_{i}^{∗}(p_{i})_{∗} on h^{∗}(G/B) and let Ã_{I} be as usual for a sequence I.

13.11. Lemma. The isomorphism θ of Lemma 13.5 satisfies θA_{I} = Ã_{I}θ and therefore sends the A_{I_{w}}(z_{0}) to the Ã_{I_{w}}(ζ_{∅}).

Proof. It follows from the surjectivity of the characteristic map, the fact that θ∘𝔠 = 𝔠_{G/B}, Theorem 12.4 and Proposition 7.8 point (1). □

We can now show that the isomorphism θ of Lemma 13.5 descends to the case where 𝔱 is regular but not necessarily invertible.

13.12. Theorem. Let h^{∗} be a weakly birationally invariant theory satisfying Assumption 13.2 and such that 𝔱 is regular in R = h^{∗}(pt). Then there is a unique isomorphism of R-algebras θ : H(M)_{F} → h^{∗}(G/B) such that the characteristic map 𝔠_{G/B} is given by θ ∘ 𝔠. It satisfies θ ∘ A_{I} = Ã_{I} ∘ θ.

Proof. We know from 7.9 that the A_{I_{w}}(z_{0}) are a basis of H(M)_{F} and from Assumption 13.2 that the ζ_{I_{w}} are a basis of h^{∗}(G/B). We define θ as the isomorphism sending A_{I_{w}}(z_{0}) to ζ_{I_{w}}. By Lemma 13.11, this definition coincides with the one of Lemma 13.5 when 𝔱 is invertible in R. We therefore have a commutative diagram

\xymatrix{H(M)_{F} ⊗_{R} R[𝔱^{−1}] \ar[r]^{θ⊗id} & h^{∗}(G/B) ⊗_{R} R[𝔱^{−1}] \\ H(M)_{F} \ar[u] \ar[r]^{θ} & h^{∗}(G/B) \ar[u]}

where θ⊗id is the θ of Lemma 13.5 and the vertical maps are injective since H(M)_{F} and h^{∗}(G/B) are free R-modules and R injects in R[𝔱^{−1}]. Since all other maps are ring morphisms, θ is one too, and the required equalities follow from the same ones in the case where 𝔱 is invertible. The morphism θ is unique since it is unique after inverting 𝔱. □

13.13. Remark. Note that the morphism θ is independent of the choices of the I_{w} used in its construction. This is obvious by surjectivity of the characteristic map after inverting 𝔱.

This completes the identification of H(M)_{F}, the algebraic model for the cohomology ring introduced in section 5, with the actual cohomology h^{∗}(G/B).

14. Formulas for push-forwards

In the present section we assume that 𝔱 and 2 are regular in R. Note that given a formal group law F, both R[[M]]_{F} and H(M)_{F} can be defined without any reference to a cohomology theory. The goal of this section is to define an “algebraic push-forward” π_{H}: H(M)_{F} → R and to prove formulas related to it. Of course, when the formal group law comes from a cohomology theory, we prove that the morphism π_{H} corresponds to its geometric counterpart π_{∗}: h(G/B) → R through the isomorphism θ: H(M)_{F} →^{≃} h^{∗}(G/B) from Theorem 13.12. The general idea of the proofs is to use that there is a universal formal group law U over the Lazard ring 𝕃, and therefore that such formulas can be proved in H_{𝕃}(M)_{U}, where they hold for geometric reasons, using algebraic cobordism. Then, they hold in any H(M)_{F} by specialization (even if it has no geometric origin).

14.1. Recall that since (U, 𝕃) is the universal formal group law, for any given formal group law F over a ring R, there is a unique morphism f : 𝕃 → R sending U to F. By Lemma 2.6 and Propositions 6.5 and 7.10, there is a commutative diagram

\xymatrix{𝕃[[M]]_{U} \ar[d]^{f_{∗}} \ar[r]^{𝔠^{U}} & H_{𝕃}(M)_{U} \ar[d]^{f_{H}} \\ R[[M]]_{F} \ar[r]^{𝔠^{F}} & H_{R}(M)_{F}}

Furthermore, the morphism f_{H} commutes with the operators A_{I} and B_{I} and sends z_{0}^{U} to z_{0}^{R} by definition. Therefore, it sends A_{I}^{U}(z_{0}^{U}) to A_{I}^{F} (z_{0}^{R}) for any sequence I.

Let (I_{w})_{w∈W} be a choice of reduced decompositions. Then, by Prop. 7.9, the elements (A_{I_{w}}(z_{0}))_{w∈W} form an R-basis of H(M)_{F}.

14.2. Definition. Let π_{H} : H(M)_{F} → R be the R-linear morphism defined by π_{H}(A_{I_{w}}(z_{0})) = ϵC_{I_{w}}(1).

14.3. Proposition. We have

(1) for any morphism f : R → R′ of rings sending a formal group law F to F′, we have π_{H}^{F} = π_{H}^{F′} ∘ f_{H}.

(2) Assume that h is an oriented cohomology theory satisfying the assumptions of Theorem 13.12, in which the isomorphism θ: H(M)_{F} →^{≃} h^{∗}(G/B) is defined. Then the morphism π_{H} satisfies π_{H} ∘ θ = θ ∘ π_{∗}. In other words, the morphism π_{H} is an algebraic replacement for the push-forward π_{∗}.

Proof. Since f_{H}(A_{I_{w}}^{F}(z_{0}^{F})) = A_{I_{w}}^{F′}(z_{0}^{F′}), (1) follows from the definition of π_{H}. Since the basis of h(G/B) formed by the ζ_{I_{w}} and the basis of H(M)_{F} formed by the A_{I_{w}}(z_{0}) corresponds to each other through the isomorphism θ, (2) follows from Lemma 13.4. □

14.4. Corollary. For any sequence I the morphism π_{H} of the proposition satisfies

π_{H}(A_{I^{rev}}(z_{0})𝔠(u)) = ϵC_{I}(u).

In particular, it is independent of the choice (I_{w})_{w∈W} of reduced decompositions.

Proof. By Proposition 14.3.(2) and by Lemma 13.4, the formula holds for H_{𝕃}(M)_{U}. In general, it reduces to the case where 𝔱 is invertible. When v ∈ R[[M]]_{F}^{W}, we have ϵC_{I}(vu) = ϵ(v)ϵC_{I}(u) and 𝔠(vu) = ϵ(v)𝔠(u) so by decomposing u on the C_{I_{w}}(u_{0}), which form an R[[M]]_{F}^{W} -basis of R[[M]]_{F}, it suffices to prove the formula for u = C_{I_{w}}(u_{0}). For those, it follows by specialization using f_{H} from H_{𝕃}(M)_{U}. □

14.5. Remark. Note that the elements π_{H}(A_{I^{rev}}(z_{0})) are particularly important because they represent the images of desingularized Schubert varieties in the cohomology of the point.

14.6. Proposition. The morphism π_{H} satisfies

π_{H} (A_{I_{v}^{rev}}(z_{0}).z_{I_{w}}^{C}) = δ_{v,w}.

In other words, the bilinear form π_{H}(−, −) is nondegenerate and the bases (z_{I_{w}}^{C})_{w∈W} and (A_{I_{w}^{rev}}(z_{0}))_{w∈W} are dual to each other.

Proof. The formula can be computed after extending scalars to R [𝔱^{−1}]. Then, since the characteristic map 𝔠 is surjective, the elements z_{I_{w}}^{C} have preimages u_{w}, such that ϵC_{I_{v}}(u_{w}) = δ_{v,w}, by the expression of 𝔠 given in Theorem 6.3. The result then follows from formula (14.4). □

Let I_{0} be a reduced decomposition of the longest element w_{0}. When 𝔱 is invertible, the element y_{0} = C_{I_{0}^{rev}}(u_{0})/𝔱 is invertible (by Lemma 5.3), and 𝔠(y_{0}) = A_{I_{0}^{rev}}(z_{0}) by Proposition 7.8.

14.7. Lemma. When 𝔱 is invertible in R, the morphism π_{H} satisfies

π_{H}(𝔠(u)) = ϵC_{I_{0}} (uy_{0}^{−1}) .

Proof. We have

π_{H}(𝔠(u)) = π_{H}(𝔠(uy_{0}^{−1})𝔠(y_{0})) = π_{H}(𝔠(uy_{0}^{−1}A_{I_{0}^{rev}}(u_{0}))) = ϵC_{I_{0}}(uy_{0}^{−1}). □

14.8. Proposition. The operators C satisfy the formula

ϵC_{I}(uC_{J}(u_{0})) = ϵC_{J^{rev}}(uC_{I^{rev}}(u_{0}))

for any pair of sequences I and J. In particular (u = 1, J = ∅), for any sequence I, we have ϵC_{I}(u_{0}) = ϵC_{I^{rev}}(u_{0}).

Proof. The formula can be proved after extension to R [𝔱^{−1}]. It then follows from the computation of π_{H} in formula (14.4) applied to both ends of

A_{I^{rev}}(z_{0})𝔠(uC_{J}(u_{0})) = 𝔱𝔠(C_{I^{rev}}(u_{0})uC_{J}(u_{0})) = 𝔠(uC_{I^{rev}}(u_{0}))A_{J}(z_{0}). □

14.9. Remark. The formulas given in this section are purely algebraic, but their proofs use geometric properties of cobordism. It would be interesting to have algebraic proofs derived directly from the various formulas in section 3.

15. Algorithm for multiplying in h^{∗}(G/B)

Fixing a choice of reduced decompositions I_{w} for all w ∈ W, we define τ_{I_{w}} as θ(z_{I_{w}}^{C}).

15.1. Proposition. Under the assumptions of Theorem 13.12, the bilinear form (b, a) ↦ π_{∗}(b · a) is non degenerate on h^{∗}(G/B) and for any choice of reduced decompositions I_{w}, the basis (ζ_{I_{w}}) and (τ_{I_{w}}) are dual to each other.

Proof. The form can be computed after scalar extension to R[𝔱^{−1}], in which case it follows from the proof of Lemma 13.5. □

Now the multiplication algorithm goes as follows: Substituting u = C_{I_{w}^{rev}}(u_{0}) in (8) and using that 𝔱 ζ_{I_{w}} = 𝔠_{G/B}(C_{I_{w}^{rev}}(u_{0})) we obtain the transition matrix from the basis (ζ_{I_{w}}) to the basis (τ_{I_{w}}). Substituting u = C_{I_{w}^{rev}}(u_{0})C_{I_{w′}^{rev}}(u_{0}) in (8) we obtain the decomposition of the product ζ_{I_{w}}ζ_{I_{w′}} on the basis (τ_{I_{w}}), and we rewrite it in terms of the ζ_{I_{w}} by using the transition matrix. This also explains how to decompose any ζ_{I} (I not necessarily reduced) on a given basis of ζ_{I_{w}}. Again, it suffices to substitute u = C_{I}(u_{0}) in (8) and use the transition matrix.

15.2. Remark. Note that this algorithm is an improvement over the one in [5] or [12] in the following sense. In both of these articles, it is explained how to decompose a product of two generators ζ_{I} and ζ_{J} as a linear combination of other such generators. But starting from a basis (i.e. the ζ_{I_{w}} for a choice of a reduced decomposition for each w ∈ W), it is not explained how to obtain a linear combination containing only generators ζ_{I} with I among the I_{w} and an algorithm for redecomposing any given ζ_{I} on a chosen basis is not given either. This is crucial to compute multiplication tables.

16. Landweber-Novikov operations

In this section we provide an algorithm for computing the Landweber-Novikov operations S^{LN} on Ω^{∗}(G/B).

16.1. Let us recall briefly the definition of S^{LN} (details can be found in [13, § 4.1.9]). Consider a graded polynomial ring ℤ[t] = ℤ[t_{1}, t_{2}, …, t_{k}, …] in infinite number of variables; for a multi-index I = (i_{1}, i_{2}, …, i_{k}) we set

t^{I} = t_{1}^{i_{1}}t_{2}^{i_{2}} … t_{k}^{i_{k}}.

Let λ_{(t)} denote the formal power series

λ_{(t)}(x) = x + ∑_{i=1}^{∞} t_{i}x^{i+1}.

Consider a twisted theory Ω̃ of Ω[t] = Ω ⊗_{ℤ} ℤ[t] (see [14, §4]). By definition Ω̃(X) = Ω(X)[t] for any X, its Chern class is given by the formula

(9)  cΩ̃ 1(L) = λ_{(t)}(c_{1}^{Ω}(L)),

and its formal group law is given by

F (x, y) = λ_{(t)}(U(λ_{(t)}^{−1}(x), λ_{(t)}^{−1}(y))).

By the universality of Ω there is a natural transformation Ω → Ω̃ given by

a ↦ ∑_{I} S_{I}^{LN}(a)t^{I},  a ∈ Ω(X)

where the components S_{I}^{LN} are called Landweber-Novikov operations on Ω(X).

16.2. By functoriality of R[[M]]_{F} in R (see 2.6) the map 𝕃 ≃ Ω(pt) → Ω̃(pt) ≃ 𝕃[t] induces a homomorphism

𝕃[[M]]_{U} → 𝕃[t][[M]]_{F} ,

while functoriality with respect to formal group laws induces a homomorphism

𝕃[t][[M]]_{F} → 𝕃[t][[M]]_{U_{Ω[t]}}
x_{μ} ↦ λ_{(t)}(x_{μ}).

By (9) we have the following commutative diagram:

\xymatrix{𝕃[[M]]_{U} \ar[r] \ar[d]_{𝔠^{Ω}} & 𝕃[t][[M]]_{F} \ar[r] \ar[d]_{𝔠Ω̃} & 𝕃[t][[M]]_{U_{Ω[t]}} \ar[d]_{𝔠^{Ω[t]}} \\ Ω(G/B) \ar[r] & Ω̃(G/B) \ar[r] & Ω(G/B)[t].}

16.3. An action of the Landweber-Novikov operation S_{I}^{LN} on a basis element ζ_{I_{w}} can be computed as follows: First, we compute the image of u = C_{I_{w}^{rev}}(u_{0}) under the composition of top horizontal arrows. Second, we extract the coefficient at the monomial t^{I} of this image. Finally, we apply the characteristic map 𝔠^{Ω} to that coefficient. The result will give S_{I}^{LN}(ζ_{I_{w}}).

Indeed, by definition S_{I}^{LN}(ζ_{I_{w}}) is equal to the coefficient at t^{I} of the image of ζ_{I_{w}} under the composition of bottom horizontal arrows. Since 𝔱 ζ_{I} = 𝔠_{G/B}(C_{I^{rev}}(u_{0})) and by commutativity of the diagram we are done.

17. Examples of computations

In the present section we list the multiplication tables for rings Ω^{∗}(G/B), where G has rank 2. The results are obtained by means of the algorithm described in Section 15 and realized in Macaulay 2 packages [6]. The answers for the other oriented cohomology theories are easily derived by a specialization of the coefficients of the universal formal group law. For instance, the answer for connective K-theory is obtained by specializing the coefficient a_{1} to v and all others to zero.

17.1. We use the presentation of the Lazard ring

𝕃 = ℤ[a_{1}, a_{2}, …],

where the first generators a_{i} are the following linear combinations of the coefficients a_{ij} of the universal formal group law U:

a_{1} = a_{11};
a_{2} = a_{12};
a_{3} = a_{22} − a_{13};
a_{4} = a_{14};
a_{5} = −9a_{15} + a_{24} + 2a_{33}.

Note that the a_{i} (resp. the a_{ij}) are of cohomological degree −i (resp. a_{1−i−j}). For root systems of rank 2, the longest element is of length at most N = 6 (in the G_{2} case), and we therefore need to compute the universal formal group law up to order 7, which thus only involves a_{1}, …, a_{6}. In fact, a_{6} does not appear in the formulas: it is not difficult to show that a_{N} will not appear in the multiplication formulas for a root system with longest element of length N.

17.2. We use the upper case letter ζ_{I} for the element ζ_{I} = q_{I∗}(1). For brevity ζ_{I} with I = (i_{1}, …, i_{l}) is denoted just by ζ_{i_{1}...i_{l}}, and when I is the empty sequence, ζ_{I} is denoted by pt. Note that when l(w) + l(w′) ≥ N = dim G/B one has

ζ_{I_{w}}ζ_{I_{w′}} = δ_{w,w_{0}w′}pt

so we list the remaining cases only.

17.3 (A_{2} case).

ζ_{121} = 1 + a_{2}ζ_{1};
ζ_{12}^{2} = ζ_{2};
ζ_{21}^{2} = ζ_{1};
ζ_{12}ζ_{21} = ζ_{1} + ζ_{2} + a_{1}pt.

This agrees with the computations of Hornbostel-Kiritchenko in [12].

17.4 (B_{2} case).

ζ_{1212} = 1 + 2a_{2}ζ_{12} + (a_{3} − a_{1}a_{2})ζ_{2};
ζ_{121}^{2} = ζ_{21};
ζ_{212}^{2} = 2ζ_{12} + a_{1}ζ_{2};
ζ_{121}ζ_{212} = ζ_{12} + ζ_{21} + a_{1}ζ_{1} + a_{1}ζ_{2} + (2a_{2} + a_{1}^{2})pt;
ζ_{121}ζ_{12} = ζ_{1} + ζ_{2} + a_{1}pt;
ζ_{121}ζ_{21} = ζ_{1};
ζ_{212}ζ_{12} = ζ_{2};
ζ_{212}ζ_{21} = 2ζ_{1} + ζ_{2} + 2a_{1}pt.

17.5 (G_{2} case).

ζ_{121212} = 1 + 4a_{2}ζ_{1212} + (10a_{3} − 10a_{1}a_{2})ζ_{212}
− (4a_{4} + 9a_{1}a_{3} + 3a_{2}^{2} − 9a_{1}^{2}a_{2})ζ_{12}

− (54a_{5} − 459a_{1}a_{4} − 1188a_{2}a_{3} − 108a_{1}^{2}a_{3} + 1080a_{1}a_{2}^{2} + 108a_{1}^{3}a_{2})ζ_{2};

ζ_{12121}^{2} = 3ζ_{2121} + 3a_{1}ζ_{121} + (13a_{2} + 2a_{1}^{2})ζ_{21} + (2a_{3} + 7a_{1}a_{2} + a_{1}^{3})ζ_{1};
ζ_{21212}^{2} = ζ_{1212} + 5a_{2}ζ_{12} + (6a_{3} − 5a_{1}a_{2})ζ_{2};
ζ_{12121}ζ_{21212} = ζ_{1212} + ζ_{2121} + a_{1}ζ_{121} + a_{1}ζ_{212} + (8a_{2} + a_{1}^{2})ζ_{12} + (8a_{2} + a_{1}^{2})ζ_{21}
+ (4a_{3} + 8a_{1}a_{2} + a_{1}^{3})ζ_{1} + (10a_{3} + 6a_{1}a_{2} + a_{1}^{3})ζ_{2}
+ (−4a_{4} + a_{1}a_{3} + 13a_{2}^{2} + 15a_{1}^{2}a_{2} + a_{1}^{4})pt;
ζ_{12121}ζ_{1212} = ζ_{121} + 3ζ_{212} + 4a_{1}ζ_{12} + 3a_{1}ζ_{21} + (8a_{2} + 4a_{1}^{2})ζ_{1} + (13a_{2} + 5a_{1}^{2})ζ_{2}
+ (a_{3} + 16a_{1}a_{2} + 5a_{1}^{3})pt;
ζ_{12121}ζ_{2121} = 2ζ_{121} + 2a_{1}ζ_{21} + (4a_{2} + a_{1}^{2})ζ_{1};
ζ_{21212}ζ_{1212} = 2ζ_{212} + a_{1}ζ_{12} + 4a_{2}ζ_{2};
ζ_{21212}ζ_{2121} = ζ_{121} + ζ_{212} + a_{1}ζ_{12} + a_{1}ζ_{21} + (5a_{2} + a_{1}^{2})ζ_{1} + (8a_{2} + a_{1}^{2})ζ_{2}
+ (3a_{3} + 6a_{1}a_{2} + a_{1}^{3})pt;
ζ_{12121}ζ_{121} = 3ζ_{21} + 2a_{1}ζ_{1};

ζ_{12121}ζ_{212} = 2ζ_{12} + ζ_{21} + 2a_{1}ζ_{1} + 3a_{1}ζ_{2} + (4a_{2} + 3a_{1}^{2})pt;
ζ_{21212}ζ_{121} = ζ_{12} + 2ζ_{21} + 2a_{1}ζ_{1} + 2a_{1}ζ_{2} + (4a_{2} + 2a_{1}^{2})pt;
ζ_{21212}ζ_{212} = ζ_{12};
ζ_{1212}^{2} = 2ζ_{12} + a_{1}ζ_{2};
ζ_{2121}^{2} = 2ζ_{21} + a_{1}ζ_{1};
ζ_{1212}ζ_{2121} = 2ζ_{12} + 2ζ_{21} + 3a_{1}ζ_{1} + 4a_{1}ζ_{2} + (4a_{2} + 4a_{1}^{2})pt;
ζ_{12121}ζ_{12} = ζ_{1} + 3ζ_{2} + 3a_{1}pt;
ζ_{12121}ζ_{21} = ζ_{1};
ζ_{21212}ζ_{12} = ζ_{2};
ζ_{21212}ζ_{21} = ζ_{1} + ζ_{2} + a_{1}pt;
ζ_{1212}ζ_{121} = 2ζ_{1} + 3ζ_{2} + 4a_{1}pt;
ζ_{1212}ζ_{212} = ζ_{2};
ζ_{2121}ζ_{121} = ζ_{1};
ζ_{2121}ζ_{212} = ζ_{1} + 2ζ_{2} + 2a_{1}pt.

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Université d’Artois, Laboratoire de Mathématiques de Lens, France

E-mail address: baptiste.calmes@univ-artois.fr

Max-Plank-Institut für Mathematik, Bonn, Germany

E-mail address: victorapetrov@googlemail.com

Department of Mathematics and Statistics, University of Ottawa, Canada

E-mail address: kirill@uottawa.ca