Equivariant Cobordism of Schemes

Amalendu Krishna

Received: October 22, 2010 Revised: October 21, 2011

Communicated by Alexander Merkurjev

Abstract. Let k be a field of characteristic zero. For a linear algebraic group G over k acting on a scheme X, we define the equivariant algebraic cobordism of X and establish its basic properties. We explicitly describe the relation of equivariant cobordism with equivariant Chow groups, K-groups and complex cobordism.

We show that the rational equivariant cobordism of a G-scheme can be expressed as the Weyl group invariants of the equivariant cobordism for the action of a maximal torus of G. As applications, we show that the rational algebraic cobordism of the classifying space of a complex linear algebraic group is isomorphic to its complex cobordism.

2010 Mathematics Subject Classification: Primary 14C25; Secondary 19E15 Keywords and Phrases: Algebraic cobordism, group actions

1. Introduction

Let k be a field of characteristic zero. Based on the construction of the motivic algebraic cobordism spectrum MGL by Voevodsky, Levine and Morel [31] gave a geometric construction of the algebraic cobordism and showed that this is a universal oriented Borel-Moore homology theory in the category of varieties over the field k. Their definition was extended by Deshpande [9] in the equivariant set-up that led to the notion of the equivariant cobordism of smooth varieties acted upon by linear algebraic groups. This in particular allowed one to define the algebraic cobordism of the classifying spaces analogous to their complex cobordism. Apart from its many applications in the equivariant set up which are parallel to the ones in the non-equivariant world, an equivariant cohomology theory often leads to the description of the corresponding non-equivariant cohomology by mixing the geometry of the variety with the representation theory of the underlying groups.

Our aim in this first part of a series of papers is to develop the theory of equivariant cobordism in the category of all k-schemes with action of a linear algebraic group. We establish the fundamental properties of this theory and give applications. In the second part [20] of this series, we shall give many important applications of the results of this paper. Some further applications of the results of this paper to the computation of the non-equivariant cobordism rings appear in [21], [24] and [15]. We now describe some of the main results in of this paper. Let G be a linear algebraic group over k. In this paper, a scheme will mean a quasi-projective k-scheme and all G-actions will be assumed to be linear. If X is a smooth scheme with a G-action, Deshpande defined the equivariant cobordism Ω_{∗}^{G}(X) using the coniveau filtration on the Levine-Morel cobordism of certain smooth mixed spaces. This was based on the construction of the Chow groups of classifying spaces in [39] and the equivariant Chow groups in [10]. Using a niveau filtration on the algebraic cobordism, which is based on the analogous filtration on any Borel-Moore homology theory as described in [2, Section 3], we define the equivariant algebraic cobordism of any k-scheme with G-action in Section 4. This is defined by taking a projective limit over the quotients of the Levine-Morel cobordism of certain mixed spaces by various levels of the niveau filtration. In order to make sense of this construction, one needs to prove various properties of the above niveau filtration which is done in Section 3. These equivariant cobordism groups coincide with the one in [9] for smooth schemes. We also show in Section 5 how one can recover the formula for the cobordism group of certain classifying spaces directly from the above definition, by choosing suitable models for the underlying mixed spaces. In Section 5, we establish the basic properties such as functoriality, homotopy invariance, exterior product, projection formula and existence of Chern classes for equivariant vector bundles in Theorem 5.2. Although we do not have the equivariant version of the localization sequence for the algebraic cobordism, we shall show that the restriction map induced by a G-equivariant open immersion is indeed surjective. In Section 7, we show how the equivariant cobordism is related to other equivariant cohomology theories such as equivariant Chow groups, equivariant K-groups and equivariant complex cobordism. Using some properties of the niveau filtration and known relation between the non-equivariant cobordism and Chow groups, we deduce an explicit formula (cf. Proposition 7.2) which relates the equivariant cobordism and the equivariant Chow groups of k-schemes. Using this and the main results of [17], we give a formula in Theorem 7.4 which relates the equivariant cobordism with the equivariant K-theory of smooth schemes. We also construct a natural transformation from the algebraic to the equivariant version of the complex cobordism for schemes over the field of complex numbers. Our next main result of this paper is Theorem 8.6, where we show that for a connected linear algebraic group G acting on a scheme X, there is a canonical isomorphism Ω_{∗}^{G}(X) ⟶^{≅} (Ω_{∗}^{T} (X))^{W} with rational coefficients, where T is a split maximal torus of a Levi subgroup of G with Weyl group W. This is mainly achieved by the Morita isomorphism of Proposition 5.4 and a detour to the motivic cobordism MGL and its extension MGL′ to singular schemes by Levine [28]. The use of MGL′-theory in our context is motivated by the recent comparison result of Levine [29] which shows that the Levine-Morel cobordism theory is a piece of the more general MGL′-theory. As an easy consequence of Proposition 7.2, we recover Totaro’s cycle class map (cf. [39])

CH^{∗}(BG) → MU^{∗}(BG) ⊗_{𝕃} ℤ → H^{∗}(BG, ℤ)

for a complex linear algebraic group G. It is conjectured that this map is an isomorphism of rings. This conjecture has been shown to be true by Totaro for some classical groups such as BGL_{n}, O_{n}, Sp_{2n} and SO_{2n+1}. Although, we can not say anything about this conjecture here, we do show as a consequence of Theorem 8.6 that the map CH^{∗}(BG) → MU^{∗}(BG) ⊗_{𝕃} ℤ is indeed an isomorphism of rings with the rational coefficients (see Theorem 8.9 for the full statement). We do this by first showing that there is a natural ring homomorphism Ω^{∗}(BG) → MU^{∗}(BG) (with integer coefficients) which lifts Totaro’s map. We then show that this map is in fact an isomorphism with rational coefficients using Theorem 8.6. We now make a remark on our definition and the notation for the equivariant cobordism groups. In many of the topology texts, the cobordism rings of classifying spaces are expressed as rings which are complete. For example, one often writes MU^{∗}(ℂP^{∞}) as the formal power series 𝕃[[t]] instead of the graded power series ring. This does not allow one to write an expression of the cobordism in each degree. Since our interest is to give an expression of the cobordism groups in each component, we shall express the equivariant cobordism of a smooth scheme as a graded ring. This notation has been used earlier by other authors in the topological context (see [25, Section 2], [27]). We refer the reader to Subsection 6.1 for more about the comparison between the two notations. We end this introduction with the following comment. One of the initial motivations for this article was to find a definition of the equivariant algebraic cobordism which has all the expected properties of an equivariant cohomology theory and which is simultaneously, simple enough to compute. Although much of this objective is achieved, the equivariant cobordism as considered here has two drawbacks. The first one is that there is not always a natural map from the algebraic to the complex equivariant cobordism for complex varieties with group action (cf. Proposition 7.5). A more serious problem is the lack of the localization sequence (cf. Proposition 5.3). One way to take care of these two problems is to consider Voevodsky’s motivic cobordism MGL from the equivariant point of view. It turns out that although this approach does have certain advantages, it becomes computationally much harder. So the challenge is to study and analyze the situations when these two approaches yield the same answer so that one can use either of the two, depending on what one would like to prove. These questions will be studied in further detail in [23].

2. Recollection of algebraic cobordism

In this section, we briefly recall the definition of algebraic cobordism of Levine-Morel. We also recall the other definition of this object as given by Levine-Pandharipande. Since we shall be concerned with the study of schemes with group actions and the associated quotient schemes, and since such quotients often require the original scheme to be quasi-projective, we shall assume throughout this paper that all schemes over k are quasi-projective.

Notations. We shall denote the category of quasi-projective k-schemes by V_{k}. By a scheme, we shall mean an object of V_{k}. The category of smooth quasiprojective schemes will be denoted by V_{k}^{S}. If G is a linear algebraic group over k, we shall denote the category of quasi-projective k-schemes with a G-action and G-equivariant maps by V_{G}. The associated category of smooth G-schemes will be denoted by V_{G}^{S}. All G-actions in this paper will be assumed to be linear. Recall that this means that all G-schemes are assumed to admit G-equivariant ample line bundles. This assumption is always satisfied for normal schemes (cf. [36, Theorem 2.5], [37, 5.7]).

2.1. Algebraic cobordism. Before we define the algebraic cobordism, we recall the Lazard ring 𝕃. It is a polynomial ring over ℤ on infinite but countably many variables and is given by the quotient of the polynomial ring ℤ[A_{ij}|(i, j) ∈ ℕ^{2}] by the relations, which uniquely define the universal formal group law F_{𝕃} of rank one on 𝕃. This formal group law is given by the power series

F_{𝕃}(u, v) = u + v + ∑_{i,j≥1}a_{ij}u^{i}v^{j},

where a_{ij} is the equivalence class of A_{ij} in the ring 𝕃. The Lazard ring is graded by setting the degree of a_{ij} to be 1 − i − j. In particular, one has 𝕃_{0} = ℤ, 𝕃_{−1} = ℤa_{11} and 𝕃_{i} = 0 for i ≥ 1, that is, 𝕃 is non-positively graded. We shall write 𝕃_{∗} for the graded ring such that 𝕃_{∗,i} = 𝕃_{−i} for i ∈ ℤ. We now define the algebraic cobordism of Levine and Morel [31]. Let X be an equi-dimensional k-scheme. A cobordism cycle over X is a family α = [Y ⟶^{f} X, L_{1}, ⋯, L_{r}], where Y is a smooth scheme, the map f is projective, and L_{i}’s are line bundles on Y. Here, one allows the set of line bundles to be empty. The degree of such a cobordism cycle is defined to be deg(α) = dim_{k}(Y) − r and its codimension is defined to be dim(X) − deg(α). Let Z^{∗}(X) be the free abelian group generated by the cobordism cycles of the above type. Note that this group is graded by the codimension of the cycles. In particular, for j ∈ ℤ, Z^{j}(X) is the free abelian group on cobordism cycles α = [Y ⟶^{f} X, L_{1}, ⋯, L_{r}], where Y is smooth and irreducible and codimension of α is j. We impose several relations on Z^{∗}(X) in order to define the algebraic cobordism group. The first among these is the so called dimension axiom: let R_{dim}^{∗}(X) be the graded subgroup of Z^{∗}(X) generated by the cobordism cycles α = [Y ⟶^{f} X, L_{1}, ⋯, L_{r}] such that dim_{k}Y < r. Let

Z_{dim}^{∗}(X) = \frac{Z^{∗}(X)}{R_{dim}^{∗}(X)}.

For a line bundle L on X and cobordism cycle α as above, we define the Chern class operator on Z_{dim}^{∗}(X) by letting c_{1}(L)(α) = [Y ⟶^{f} X, L_{1}, ⋯, L_{r}, f^{∗}(L)]. Next, we impose the so called section axiom. Let R_{sec}^{∗}(X) be the graded subgroup of Z_{dim}^{∗}(X) generated by cobordism cycles of the form [Y → X, L] − [Z → X], where Y ⟶^{s} L is a section of the line bundle L on Y which is transverse to the zero-section, and Z ↪ Y is the closed subvariety of Y defined by the zeros of s. The transversality of s ensures that Z is a smooth variety. In particular, [Z → X] is a well-defined cobordism cycle on X. Define

\underline{Ω}^{∗}(X) = \frac{Z_{dim}^{∗}(X)}{R_{sec}^{∗}(X)}.

The assignment X ↦ \underline{Ω}^{∗}(X) is called the pre-cobordism theory. Finally, we impose the formal group law on the cobordism using the following relation. For X as above, let R_{FGL}^{∗}(X) ⊂ 𝕃⊗_{ℤ}\underline{Ω}^{∗}(X) be the graded 𝕃submodule generated by elements of the form

\{F_{𝕃} (c_{1}(L), c_{1}(M)) (x) − c_{1}(L ⊗ M)(x)|x ∈ \underline{Ω}^{∗}(X), L, M ∈ Pic(X)\} .

We define the algebraic cobordism group of X by

(2.1)  Ω^{∗}(X) = \frac{𝕃⊗_{ℤ}\underline{Ω}^{∗}(X)}{R_{FGL}^{∗}(X)}.

If X is not necessarily equi-dimensional, we define Z_{∗}(X) to be same as Z^{∗}(X) except that Z_{∗}(X) is now graded by the degree of the cobordism cycles. In particular, Z_{i}(X) is the free abelian group on cobordism cycles [Y ⟶^{f} X, L_{1}, ⋯, L_{r}] such that f is projective and Y is smooth and irreducible such that dim(Y) − r = i. One then defines Ω_{∗}(X) to be the quotient of 𝕃_{∗} ⊗_{ℤ} \underline{Ω}_{∗}(X) in the same way as above. Note that for X equi-dimensional of dimension d and i ∈ ℤ, one has Ω^{i}(X) ≅ Ω_{d−i}(X). Observe that Ω^{∗}(X) is a graded 𝕃-module such that Ω^{j}(X) = 0 for j > dim(X) and Ω^{j}(X) can be non-zero for any given −∞ < j ≤ dim(X). Similarly, Ω_{∗}(X) is a graded 𝕃_{∗}-module which has no component in the negative degrees and it can be non-zero in arbitrarily large positive degree. The following is the main result of Levine and Morel from which most of their other results on algebraic cobordism are deduced. We refer to loc. cit. for more properties.

Theorem 2.1. The functor X ↦ Ω_{∗}(X) is the universal Borel-Moore homology on the category V_{k}. In other words, it is universal among the homology theories on V_{k} which have functorial push-forward for projective morphism, pull-back for smooth morphism (any morphism of smooth schemes), Chern classes for line bundles, and which satisfy Projective bundle formula, homotopy invariance, the above dimension, section and formal group law axioms. Moreover, for a k-scheme X and closed subscheme Z of X with open complement U, there is a localization exact sequence

Ω_{∗}(Z) → Ω_{∗}(X) → Ω_{∗}(U) → 0.

It was also shown in loc. cit. that the natural composite map

Φ : 𝕃 → 𝕃 ⊗_{ℤ} \underline{Ω}^{∗}(k) ↠ Ω^{∗}(k)
a ↦ [a]

is an isomorphism of commutative graded rings. As an immediate corollary of Theorem 2.1, we see that for a smooth k-scheme X and an embedding σ : k → ℂ, there is a natural morphism of graded rings

(2.2)  Φ_{X}^{top} : Ω^{∗}(X) → MU^{2∗}(X_{σ}(ℂ)),

where MU^{∗}(X_{σ}(ℂ)) is the complex cobordism ring of the complex manifold X_{σ}(ℂ) given by the complex points of X ×_{k} ℂ. This map is an isomorphism for X = Spec(k). In particular, there are isomorphisms of graded rings

(2.3)  𝕃 ⟶^{≅} Ω^{∗}(k) ⟶^{≅} MU^{2∗} ⟶^{≅} MU^{∗},

where MU^{∗} is the complex cobordism ring of a point. As a corollary, we see that for any field extension k ↪ K, the natural map Ω^{∗}(k) → Ω^{∗}(K) is an isomorphism.

2.2. Cobordism via double point degeneration. To enforce the formal group law on the algebraic cobordism in order to make it an oriented cohomology theory on the category of smooth varieties, Levine and Morel artificially imposed this condition by tensoring their pre-cobordism theory with the Lazard ring. Although they were able to show that the resulting map Z_{∗}(X) → Ω_{∗}(X) is still surjective, they were unable to describe the explicit geometric relations in Z_{∗}(X) that define Ω_{∗}(X). This was subsequently accomplished by Levine-Pandharipande [32]. We conclude our introduction to the algebraic cobordism by briefly discussing the construction of Levine-Pandharipande. For n ≥ 1, let □^{n} denote the space (ℙ_{k}^{1} − {1})^{n}.

Definition 2.2. A morphism Y ⟶^{π} □^{1} is called a double point degeneration, if Y is a smooth scheme and π^{−1}(0) is scheme-theoretically given as the union A ∪ B, where A and B are smooth divisors on Y which intersect transversely. The intersection D = A ∩ B is called the double point locus of π. Here, A, B and D are allowed to be disconnected or, even empty.

For a double point degeneration as above, notice that the scheme D is also smooth and O_{D}(A+B) is trivial. In particular, one sees that N_{A/D} ⊗_{D} N_{B/D} ≅ O_{D}. This is turn implies that the projective bundles ℙ(O_{D} ⊕ N_{A/D}) → D and ℙ(O_{D} ⊕ N_{B/D}) → D are isomorphic, where N_{A/D} and N_{B/D} are the normal bundles of D in A and B respectively. Let ℙ(π) → D denote any of these two projective bundles.

Let X be a k-scheme and let Y ⟶^{f} X × □^{1} be a projective morphism from a smooth scheme Y. Assume that the composite map π : Y → X × □^{1} → □^{1} is a double point degeneration such that Y_{∞} = π^{−1}(∞) is smooth. We define the cobordism cycle on X associated to the morphism f to be the cycle

(2.4)  C(f) = [Y_{∞} → X] − [A → X] − [B → X] + [ℙ(π) → X].

Let M_{∗}(X) be the free abelian group on the isomorphism classes of the morphisms [Y ⟶^{f} X], where Y is smooth and irreducible and f is projective. Then M_{∗}(X) is a graded abelian group, where the grading is by the dimension of Y. Let R_{∗}(X) be the subgroup of M_{∗}(X) generated by all cobordism cycles C(f), where C(f) is as in (2.4). Note that R_{∗}(X) is a graded subgroup of M_{∗}(X). Define

(2.5)  ω_{∗}(X) = \frac{M_{∗}(X)}{R_{∗}(X)}.

Theorem 2.3 ([32]). There is a canonical isomorphism

(2.6)  ω_{∗}(X) ⟶^{≅} Ω_{∗}(X)

of oriented Borel-Moore homology theories on V.

3. Niveau filtration on algebraic cobordism

In this section, we introduce the niveau filtration on the algebraic cobordism which plays an important role in the definition of the equivariant algebraic cobordism. Our main result here is a refined localization sequence for the cobordism which preserves the niveau filtration. This new localization sequence will have interesting consequences in the study of the equivariant cobordism. Let X be a k-scheme of dimension d. For j ∈ ℤ, let Z_{j} be the set of all closed subschemes Z ⊂ X such that dim_{k}(Z) ≤ j (we assume dim(∅) = −∞). The set Z_{j} is then ordered by the inclusion. For i ≥ 0, we define

Ω_{i}(Z_{j}) = \underrightarrow{lim}_{Z∈Z_{j}}Ω_{i}(Z) and put

Ω_{∗}(Z_{j}) = ⨁_{i≥0} Ω_{i}(Z_{j}).

It is immediate that Ω_{∗}(Z_{j}) is a graded 𝕃_{∗}-module and there is a graded 𝕃_{∗}- linear map Ω_{∗}(Z_{j}) → Ω_{∗}(X). Following [2, Section 3], we let Z_{j}/Z_{j−1} denote the ordered set of pairs (Z, Z′) ∈ Z_{j} × Z_{j−1} such that Z′ ⊂ Z with the ordering

(Z, Z′) ≥ (Z_{1}, Z′_{1}) if Z_{1} ⊆ Z and Z′_{1} ⊆ Z′.

If (Z, Z′) ≥ (Z_{1}, Z′_{1}), then the functoriality of the push-forward maps and the localization sequence yield a map Ω_{i}(Z_{1} − Z′_{1}) → Ω_{i}(Z − Z′) (cf. (3.1)). We let

Ω_{i} (Z_{j}/Z_{j−1}(X)) := \underrightarrow{lim}_{(Z,Z′)∈Z_{j}/Z_{j−1}}Ω_{i}(Z − Z′).

Lemma 3.1. For f : X′ → X projective, the push-forward map Ω_{∗}(X′) ⟶^{f_{∗}} Ω_{∗}(X) induces a push-forward map Ω_{∗} (Z_{j}/Z_{j−1}(X′)) → Ω_{∗} (Z_{j}/Z_{j−1}(X)).

Proof. Let (Z, Z′) ∈ Z_{j}/Z_{j−1}(X′). Then (W, W′) = (Im(Z), Im(Z′)) ∈ Z_{j}/Z_{j−1}(X). It suffices now to show that f_{∗} induces a natural map Ω_{∗}(Z − Z′) → Ω_{∗}(W − W′). However, this follows directly from the localization exact sequences

(3.1)  \xymatrix{Ω_{∗}(Z′) \ar[r] \ar[d]_{f_{∗}} & Ω_{∗}(Z) \ar[r] \ar[d]^{f_{∗}} & Ω_{∗}(Z − Z′) \ar[r] \ar@{.>}[d] & 0 \\ Ω_{∗}(W′) \ar[r] & Ω_{∗}(W) \ar[r] & Ω_{∗}(W − W′) \ar[r] & 0}

and the fact that the square on the left is commutative. □

For x ∈ Z_{j}, let

(3.2)  { \atop Ω_{∗}(k(x))} = \underrightarrow{lim}_{U⊆\overline{\{x\}}}Ω_{∗}(U),

where the limit is taken over all non-empty open subsets of \overline{\{x\}}. Taking the limit over the localization sequences

Ω_{∗}(Z′) → Ω_{∗}(Z) → Ω_{∗}(Z − Z′) → 0

for (Z, Z′) ∈ Z_{j}/Z_{j−1}, one now gets an exact sequence

(3.3)  Ω_{∗}(Z_{j−1}) → Ω_{∗}(Z_{j}) → ⨁_{x∈(Z_{j}−Z_{j−1})}{ \atop Ω_{∗}(k(x))} → 0.

Definition 3.2. We define F_{j}Ω_{∗}(X) to be the image of the natural 𝕃_{∗}-linear map Ω_{∗}(Z_{j}) → Ω_{∗}(X). In other words, F_{j}Ω_{∗}(X) is the image of all Ω_{∗}(W) → Ω_{∗}(X), where W → X is a projective map such that dim(Image(W)) ≤ j. Using the localization sequence, this is same as saying that F_{j}Ω_{∗}(X) is the set of all elements s ∈ Ω_{∗}(X) such that i^{∗}(s) = 0 for some open subset i : U ↪ X, whose complement has dimension at most j.

One checks at once that there is a canonical niveau filtration

(3.4) 0 = F_{−1}Ω_{∗}(X) ⊆ F_{0}Ω_{∗}(X) ⊆ ⋯ ⊆ F_{d−1}Ω_{∗}(X) ⊆ F_{d}Ω_{∗}(X) = Ω_{∗}(X).

Lemma 3.3. If f : X′ → X is a projective morphism, then f_{∗} (F_{j}Ω_{∗}(X′)) ⊆ F_{j}Ω_{∗}(X). If g : X′ → X is a smooth morphism of relative dimension r, then g^{∗} (F_{j}Ω_{∗}(X)) ⊆ F_{j+r}Ω_{∗}(X′).

Proof. The first assertion is obvious from the definition. In fact, the pushforward map preserves the niveau filtration at the level of the free abelian groups of cobordism cycles. The second assertion also follows immediately using the fact that for a cobordism cycle [Y → X], one has g^{∗} ([Y → X]) = [Y ×_{X} X′ → X′]. This in turn implies that g^{∗} ∘ f_{∗} = f′_{∗} ∘ g^{′∗} for a Cartesian square

\xymatrix{W′ \ar[r]^{f′} \ar[d]_{g′} & X′ \ar[d]^{g} \\ W \ar[r]_{f} & X}

such that f is projective and g is smooth. □

Proposition 3.4. Let X be a k-scheme and let Z be a closed subscheme of X with the complement U. Then for every j ≥ dim(Z), there is an exact sequence

Ω_{∗}(Z) → F_{j}Ω_{∗}(X) → F_{j}Ω_{∗}(U) → 0.

Proof. Since j ≥ dim(Z), we see that the image of the map Ω_{∗}(Z) → Ω_{∗}(X) lies in F_{j}Ω_{∗}(X). Using the localization sequence of the algebraic cobordism (cf. Theorem 2.1), we only need to show that the map F_{j}Ω_{∗}(X) → F_{j}Ω_{∗}(U) is surjective. Let F_{j}Z_{∗}(X) be the free abelian group on cobordism cycles [Y ⟶^{f} X] such that Y is irreducible and dim(f(Y)) ≤ j. Note that f(Y) is a closed and irreducible subscheme of X since Y is irreducible and f is projective. It is then clear that F_{j}Z_{∗}(X) ⊂ Z_{∗}(X) and F_{j}Z_{∗}(X) ↠ F_{j}Ω_{∗}(X).

Let [Y ⟶^{f} U] be a cobordism cycle on U such that Y is smooth and irreducible, f is projective and dim(f(Y)) ≤ j. We have a factorization Y ↪ ℙ_{k}^{n} × U → U where the first inclusion is a closed immersion. Let \overline{Y} denote a resolution of singularities of the closure of Y in ℙ_{k}^{n} × X and let \overline{Y} ⟶^{f̄} X be the projection

map. It is then easy to verify that [\overline{Y} ⟶^{f̄} X] is a cobordism cycle on X which restricts to [Y ⟶^{f} U] in Ω_{∗}(U) and dim(f̄(\overline{Y})) = dim(f(Y)) ≤ j. This proves the required surjection. □

Theorem 3.5. Let X be a k-scheme and let Z be a closed subscheme of X with the complement U. Then for every j ∈ ℤ, there is an exact sequence

(3.5)  \frac{Ω_{∗}(Z)}{F_{j}Ω_{∗}(Z)} → \frac{Ω_{∗}(X)}{F_{j}Ω_{∗}(X)} → \frac{Ω_{∗}(U)}{F_{j}Ω_{∗}(U)} → 0.

Proof. Let f : U → X and g : Z → X be the inclusion maps. The surjectivity of the second map in (3.5) follows from the localization sequence of algebraic cobordism (cf. Theorem 2.1). It suffices thus to show that

(3.6)  f^{∗}^{(−1)} (F_{j}Ω_{∗}(U)) ⊆ Image (Ω_{∗}(Z) ⊕ F_{j}Ω_{∗}(X) → Ω_{∗}(X))

in order to prove the theorem. So let α ∈ Ω_{∗}(X) be such that β = f^{∗}(α) ∈ F_{j}Ω_{∗}(U). We can find a closed subscheme q : W ↪ U of dimension at most j and a cobordism cycle β′ ∈

Ω_{∗}(W) such that β = q_{∗}(β′). Let Y be the closure of W in X and let W ⟶^{f′} Y ⟶^{p} X be the open and the closed immersions.

Using Theorem 2.1 (the localization sequence), we can find α′ ∈ Ω_{∗}(Y) such that β′ = f′^{∗}(α′). We conclude from this that f^{∗} (α − p_{∗}(α′)) = 0 in Ω_{∗}(U). Using Theorem 2.1 (the localization sequence) again, we see that α = g_{∗}(γ) + p_{∗}(α′) for some γ ∈ Ω_{∗}(Z). Since dim(Y) = dim(W) ≤ j, it also follows that p_{∗}(α′) ∈ F_{j}Ω_{∗}(X). This proves (3.6) and hence the theorem. □

The following is an immediate consequence of Theorem 3.5.

Corollary 3.6. Let X be a k-scheme. Then for any j ≥ 0 and any closed subscheme Z ⊂ X of dimension at most j, the natural map Ω_{∗}(X) → Ω_{∗}(X−Z) induces an isomorphism

\frac{Ω_{∗}(X)}{F_{j}Ω_{∗}(X)} ⟶^{≅} \frac{Ω_{∗}(X − Z)}{F_{j}Ω_{∗}(X − Z)}.

Lemma 3.7. For a k-scheme X and i ≥ 0, the natural map Ω_{i}(X) → CH_{i}(X) has the factorization

Ω_{i}(X) → \frac{Ω_{i}(X)}{F_{i−1}Ω_{i}(X)} → CH_{i}(X).

Proof. By Theorem 2.3, Ω_{∗}(X) is generated by the cobordism cycles [Y → X], where Y is smooth and f is projective. It follows from the definition of the niveau filtration that F_{j}Ω_{∗}(X) is generated by the cobordism cycles of the form

i_{∗} ([Y → Z]), where Z ↪^{ϕ} X is a closed subscheme of X of dimension at most j. Since Ω_{∗} → CH_{∗} is a natural transformation of oriented Borel-Moore homology theories, we get a commutative diagram

\xymatrix{Ω_{i}(Z) \ar[r] \ar[d]_{ϕ_{∗}} & CH_{i}(Z) \ar[d]^{ϕ_{∗}} \\ Ω_{i}(X) \ar[r] & CH_{i}(X).}

The lemma now follows from the fact that CH_{i}(Z) = 0 if j ≤ i − 1. □

Lemma 3.8. For any s ∈ F_{j}Ω_{∗}(X), there are elements a_{i} ∈ 𝕃_{∗} and s_{i} ∈ Ω_{≤j}(X) such that s = ∑_{i} a_{i}s_{i}.

Proof. It is a simple variant of the generalized degree formula [30, Theorem 4.7]. We can assume that s is a homogeneous element of Ω_{∗}(X). We have seen in the proof of Proposition 3.4 that F_{j}Z_{∗}(X) a free abelian subgroup of Z_{∗}(X) such

that F_{j}Z_{∗}(X) ↠ F_{j}Ω_{∗}(X). We can thus assume that s = [Y ⟶^{f} X], where Y is smooth and irreducible and f is projective such that dim(f(Y)) ≤ j. Let ι : f(Y) = W ↪ X denote the inclusion of the closed subset and let U be the complement of W in X. Then the image of s dies in Ω_{∗}(U) under the restriction map. It follows from the localization sequence of the algebraic cobordism (cf. Theorem 2.1) and [30, Theorem 4.7] that we can write s = ι_{∗} (a[W̃ → W] + ∑_{i} a_{i}[Z̃_{i} → Z_{i}]), where W̃ → W is a resolution of singu-

larities of W, Z̃_{i} → Z_{i} is a resolution of singularities of a closed subscheme Z_{i} of W of dimension strictly less than that of W and a, a_{i} ∈ 𝕃_{∗}. It follows from this expression that s = ∑_{i} a_{i}s_{i} such that s_{i} ∈ Ω_{≤j}(X) and a_{i} ∈ 𝕃_{∗} □

Proposition 3.9. Let E ⟶^{f} X be a vector bundle of rank r. Then the pull-back map f^{∗} : Ω_{∗}(X) → Ω_{∗}(E) induces an isomorphism

F_{j}Ω_{∗}(X) ⟶^{≅} F_{j+r}Ω_{∗}(E)

for all j ∈ ℤ. In particular, F_{<r}Ω_{∗}(E) = 0.

Remark 3.10. The reader should be warned that the map f^{∗} shifts the degree of the grading by r.

Proof. Using Lemma 3.8, this can be proved in the same way as [9, Lemma 3.3], where a similar result is proven for smooth varieties and coniveau filtration. We sketch the proof in the singular case. The homotopy invariance of the algebraic cobordism tells us that the natural map Ω_{∗}(X) ⟶^{f^{∗}} Ω_{∗}(E) is an isomorphism. So we only need to show that this map is surjective at each level of the niveau filtration. So let e ∈ F_{j+r}Ω_{∗}(E). We can assume that e ∈ Ω_{i}(E) is a homogeneous element. By Lemma 3.8, we can write e = ∑_{p} a_{p}s_{p}, where each s_{p} is a homogeneous element of Ω_{≤j+r}(E) and a_{p} ∈ 𝕃_{∗}. Since f^{∗} is an isomorphism of graded abelian groups which shifts the degree by r, we can write s_{p} = f^{∗}(x_{p}) such that x_{p} ∈ Ω_{≤j}(X). Letting x = ∑_{p} a_{p}x_{p}, we see that x ∈ F_{j}Ω_{∗}(X) and s = f^{∗}(x). This proves the proposition. □

4. Equivariant algebraic cobordism

In this text, G will denote a linear algebraic group of dimension g over k. All representations of G will be finite dimensional. The definition of equivariant cobordism needs one to consider certain kind of mixed spaces which in general may not be a scheme even if the original space is a scheme. The following well known (cf. [10, Proposition 23]) lemma shows that this problem does not occur in our context and all the mixed spaces in this paper are schemes with ample line bundles.

Lemma 4.1. Let H be a linear algebraic group acting freely and linearly on a k-scheme U such that the quotient U/H exists as a quasi-projective variety. Let

X be a k-scheme with a linear action of H. Then the mixed quotient X ×^{H} U exists for the diagonal action of H on X ×U and is quasi-projective. Moreover, this quotient is smooth if both U and X are so. In particular, if H is a closed subgroup of a linear algebraic group G and X is a k-scheme with a linear action

of H, then the quotient G ×^{H} X is a quasi-projective scheme.

Proof. It is already shown in [10, Proposition 23] using [12, Proposition 7.1]

that the quotient X ×^{H} U is a scheme. Moreover, as U/H is quasi-projective, [12,

Proposition 7.1] in fact shows that X ×^{H} U is also quasi-projective. The similar conclusion about G ×^{H} X follows from the first case by taking U = G and by observing that G/H is a smooth quasi-projective scheme (cf. [3, Theorem 6.8]).

The assertion about the smoothness is clear since X×U → X ×^{H} U is a principal H-bundle. □

For any integer j ≥ 0, let V_{j} be an l-dimensional representation of G and let U_{j} be a G-invariant open subset of V_{j} such that the codimension of the complement (V_{j} − U_{j}) in V_{j} is at least j and G acts freely on U_{j} such that the quotient U_{j}/G is a quasi-projective scheme. Such a pair (V_{j}, U_{j}) will be called a good pair for the G-action corresponding to j (cf. [17, Section 2]). It is easy to see that a good pair always exists (cf. [10, Lemma 9]). Let X_{G} denote the

mixed quotient X ×^{G} U_{j} of the product X × U_{j} by the diagonal action of G, which is free. Let X be a k-scheme of dimension d with a G-action. Fix j ≥ 0 and let (V_{j}, U_{j}) be an l-dimensional good pair corresponding to j. For i ∈ ℤ, set

(4.1)  Ω_{i}^{G}(X)_{j} = \frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j})}.

Lemma 4.2. For a fixed j ≥ 0, the group Ω_{i}^{G}(X)_{j} is independent (in a canonical way) of the choice of the good pair (V_{j}, U_{j}).

Proof. Let (V_{j}, U_{j}) and (V′_{j}, U′_{j}) be two good pairs of dimensions and l and l′ respectively corresponding to j. Using the results of Section 3, one can follow the proof of the similar result for the equivariant Chow groups in [10, Proposition 1] to construct a canonical isomorphism

α_{vv′} : \frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j})} ⟶^{≅} \frac{Ω_{i+l′−g} (X ×^{G} U′_{j})}{F_{d+l′−g−j}Ω_{i+l′−g} (X ×^{G} U′_{j})}

as follows. We let V = V_{j} ⊕ V′_{j} and U = (U_{j} ⊕ V′_{j}) ∪ (V_{j} ⊕ U′_{j}). Let G act diagonally on V.

Then it is easy to see that the complement of the open subset X ×^{G} (U_{j} ⊕ V′_{j})

in X ×^{G} U has dimension at most d + l + l′ − g − j. Hence by Corollary 3.6, the map (4.2)

\frac{Ω_{i+l+l′−g} (X ×^{G} U)}{F_{d+l+l′−g−j}Ω_{i+l+l′−g} (X ×^{G} U)} ⟶^{ι_{v}^{∗}} \frac{Ω_{i+l+l′−g} (X ×^{G} (U_{j} ⊕ V′_{j}))}{F_{d+l+l′−g−j}Ω_{i+l+l′−g} (X ×^{G} (U_{j} ⊕ V′_{j}))}

Documenta Mathematica 17 (2012) 95–134

is an isomorphism. On the other hand, the map X ×^{G} (U_{j} ⊕ V′_{j}) ⟶^{ϕ_{v}} X ×^{G} U_{j} is a vector bundle of rank l′ and hence by Proposition 3.9, the map (4.3)

\frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j})} ⟶^{ϕ_{v}^{∗}} \frac{Ω_{i+l+l′−g} (X ×^{G} (U_{j} ⊕ V′_{j}))}{F_{d+l+l′−g−j}Ω_{i+l+l′−g} (X ×^{G} (U_{j} ⊕ V′_{j}))}

is also an isomorphism. Combining the above two isomorphisms, we get the canonical isomorphism

(ι_{v}^{∗})^{−1} ∘ ϕ_{v}^{∗} : \frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j})} ⟶^{≅} \frac{Ω_{i+l+l′−g} (X ×^{G} U)}{F_{d+l+l′−g−j}Ω_{i+l+l′−g} (X ×^{G} U)}.

In the same way, we also get an isomorphism

(ϕ_{v′}^{∗})^{−1} ∘ ι_{v′}^{∗} : \frac{Ω_{i+l+l′−g} (X ×^{G} U)}{F_{d+l+l′−g−j}Ω_{i+l+l′−g} (X ×^{G} U)} ⟶^{≅}

⟶^{≅} \frac{Ω_{i+l′−g} (X ×^{G} U′_{j})}{F_{d+l′−g−j}Ω_{i+l′−g} (X ×^{G} U′_{j})}.

The composite α_{vv′} = ((ϕ_{v′}^{∗})^{−1} ∘ ι_{v′}^{∗}) ∘ ((ι_{v}^{∗})^{−1} ∘ ϕ_{v}^{∗}) is the desired canonical isomorphism (see the proof of [39, Theorem 1.1]). □

Lemma 4.3. For j′ ≥ j ≥ 0, there is a natural surjective map Ω_{i}^{G}(X)_{j′} ↠ Ω_{i}^{G}(X)_{j}.

Proof. Choose a good pair (V_{j′}, U_{j′}) for j′. Then it is clearly a good pair for j too. Moreover, there is a natural surjection

\frac{Ω_{i+l−g} (X ×^{G} U_{j′})}{F_{d+l−g−j′}Ω_{i+l−g} (X ×^{G} U_{j′})} ↠ \frac{Ω_{i+l−g} (X ×^{G} U_{j′})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j′})}.

On the other hand, the left and the right terms are Ω_{i}^{G}(X)_{j′} and Ω_{i}^{G}(X)_{j} respectively by Lemma 4.2. □

Definition 4.4. Let X be a k-scheme of dimension d with a G-action. For any i ∈ ℤ, we define the equivariant algebraic cobordism of X to be

Ω_{i}^{G}(X) = \underleftarrow{lim}_{j} Ω_{i}^{G}(X)_{j}.

The reader should note from the above definition that unlike the ordinary cobordism, the equivariant algebraic cobordism Ω_{i}^{G}(X) can be non-zero for any i ∈ ℤ. We set

Ω_{∗}^{G}(X) = ⨁_{i∈ℤ} Ω_{i}^{G}(X).

If X is an equi-dimensional k-scheme with G-action, we let Ω_{G}^{i}(X) = Ω_{d−i}^{G}(X) and Ω_{G}^{∗}(X) = ⊕_{i∈ℤ} Ω_{G}^{i}(X). We shall denote the equivariant cobordism Ω_{G}^{∗}(k)

of the ground field by S(G). This is also called the algebraic cobordism of the classifying space of G and often written as Ω^{∗}(BG).

Remark 4.5. If G is the trivial group, we can take the good pair (V_{j}, V_{j}) for every j where V_{j} is any l-dimensional k-vector space. In that case, we get

Ω_{i+l}(X ×^{G} V_{j}) ≅ Ω_{i+l}(X×V_{j}) which is isomorphic to Ω_{i}(X) by the homotopy invariance of the non-equivariant cobordism. Moreover, F_{d+l−j}Ω_{i+l}(X × V_{j}) is isomorphic to F_{d−j}Ω_{i}(X) by Proposition 3.9 and this last term is zero for all large j. In particular, we see from (4.1) and the definition of the equivariant cobordism that there is a canonical isomorphism Ω_{∗}^{G}(X) ⟶^{≅} Ω_{∗}(X).

Remark 4.6. Let X be a G-scheme and let H be a closed normal subgroup of G with quotient W. If (V_{j}, U_{j}) is a good pair for the G-action for any j ≥ 0,

then W naturally acts on the mixed quotient X_{j} = X ×^{H} U_{j} and hence it acts on Ω_{∗}(X_{j}). Since W acts on X_{j} by automorphisms, it keeps the niveau filtration invariant. In particular, it acts on Ω_{∗}^{H}(X)_{j}. It is clear that if (V′_{j}, U′_{j}) is another good pair, then the isomorphisms in (4.2) and (4.3) are W -equivariant. In other words, the W -action on Ω_{∗}(X_{j}) does not depend on the choice of good pairs. Furthermore, for j′ ≥ j, we can choose a good pair for j′ and that makes the maps in the inverse system {Ω_{∗}(X_{j})}_{j≥0} W -equivariant. We conclude that W acts on the equivariant cobordism Ω_{∗}^{H}(X). One example of such a situation is where H is a maximal torus in a linear algebraic group and G is its normalizer. The quotient W is then the Weyl group. In that case, Ω_{∗}^{H}(X) becomes a ℤ[W]-module.

Remark 4.7. It is easy to check from the above definition of the niveau filtration that if X is a smooth and irreducible k-scheme of dimension d, then F_{j}Ω_{i}(X) = F^{d−j}Ω^{d−i}(X), where F^{•}Ω^{∗}(X) is the coniveau filtration used in [9]. Furthermore, one also checks in this case that if G acts on X, then

(4.4)  Ω_{G}^{i}(X) = \underleftarrow{lim}_{j} \frac{Ω^{i} (X ×^{G} U_{j})}{F^{j}Ω^{i} (X ×^{G} U_{j})},

where (V_{j}, U_{j}) is a good pair corresponding to any j ≥ 0. Thus the above definition 4.4 of the equivariant cobordism coincides with that of [9] for smooth schemes.

Remark 4.8. As is evident from the above definition (see Example 6.6), the equivariant cobordism Ω_{i}^{G}(X) can not in general be computed in terms of the algebraic cobordism of one single mixed space. This makes these groups more complicated to compute than the equivariant Chow groups, which can be computed in terms of a single mixed space. This also motivates one to ask if the equivariant cobordism can be defined in such a way that they can be calculated using one single mixed space in a given degree. It follows however from Lemmas 4.2 and 4.3 that for a given i and j, each component of the projective system {Ω_{i}^{G}(X)_{j}}_{j≥0} can be computed using a single mixed space.

4.1. Change of groups. If H ⊂ G is a closed subgroup of dimension h, then any l-dimensional good pair (V_{j}, U_{j}) for G-action is also a good pair for the

induced H-action. Moreover, for any X ∈ V_{G} of dimension d, X ×^{H} U_{j} → X ×^{G} U_{j} is an étale locally trivial G/H-fibration and hence a smooth map (cf. [3, Theorem 6.8]) of relative dimension g − h. This induces the inverse system of pull-back maps

Ω_{i}^{G}(X)_{j} = \frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j})} → h

→ \frac{Ω_{i+l−h} (X ×^{H} U_{j})}{F_{d+l−h−j}Ω_{i+l−h} (X ×^{H} U_{j})} = Ω_{i}^{H}(X)_{j}

and hence a natural restriction map

(4.5)  r_{H,X}^{G} : Ω_{∗}^{G}(X) → Ω_{∗}^{H}(X).

Taking H = {1} and using Remark 4.5, we get the forgetful map

(4.6)  r_{X}^{G} : Ω_{∗}^{G}(X) → Ω_{∗}(X)

from the equivariant to the non-equivariant cobordism. Since r_{H,X}^{G} is obtained as a pull-back under the smooth map, it commutes with any projective pushforward and smooth pull-back (cf. Theorem 5.2). We remark here that although the definition of r_{H,X}^{G} uses a good pair, it is easy to see as in Lemma 4.2 that it is independent of the choice of such good pairs.

4.2. Fundamental class of cobordism cycles. Let X ∈ V_{G} and let Y ⟶^{f} X be a morphism in V_{G} such that Y is smooth of dimension d and f is projective. For any j ≥ 0 and any l-dimensional good pair (V_{j}, U_{j}), [Y_{G} ⟶^{f_{G}} X_{G}] is an ordinary cobordism cycle of dimension d + l − g by Lemma 5.1 and hence defines an element α_{j} ∈ Ω_{d}^{G}(X)_{j}. Moreover, it is evident that the image of α_{j′} is α_{j} for j′ ≥ j. Hence we get a unique element α ∈ Ω_{d}^{G}(X), called the

G-equivariant fundamental class of the cobordism cycle [Y ⟶^{f} X]. We also see from this more generally that if [Y ⟶^{f} X, L_{1}, ⋯, L_{r}] is as above with each L_{i} a G-equivariant line bundle on Y, then this defines a unique class in Ω_{d−r}^{G}(X). It is interesting question to ask under what conditions on the group G, the equivariant cobordism group Ω_{∗}^{G}(X) is generated by the fundamental classes of G-equivariant cobordism cycles on X. It turns out that this question indeed has a positive answer if G is a split torus by [20, Theorem 4.11].

5. Some properties of equivariant cobordism

In this section, we establish some basic properties of equivariant algebraic cobordism that are analogous to the non-equivariant case. We begin with the following elementary result. This will be used in the sequel for the morphisms between mixed quotients.

Lemma 5.1. Let f : X → Y be a projective G-equivariant map in V_{G} with free G-actions such that Y/G is quasi-projective. Then X/G ∈ V_{k} and the induced map \overline{f} : X/G → Y/G of quotients is projective.

Proof. It follows from our assumption and [12, Proposition 7.1] that X/G exists and that \overline{f} : X′ = X/G → Y/G = Y′ is a morphism in V_{k}. Furthermore, the square

\xymatrix{X \ar[r] \ar[d]_{f} & X′ \ar[d]^{\overline{f}} \\ Y \ar[r] & Y′}

is Cartesian. Since both the horizontal maps are the principal G-bundles, they are smooth and surjective. Since proper maps have smooth descent (in fact fpqc descent), we see that \overline{f} : X′ → Y′ is proper. Since these schemes are quasiprojective, we leave it as an exercise to show that \overline{f} is also quasi-projective and hence must be projective. □

Theorem 5.2. The equivariant algebraic cobordism satisfies the following properties.

(i) Functoriality : The assignment X ↦ Ω_{∗}^{G}(X) is covariant for projective maps and contravariant for smooth maps in V_{G}. It is also contravariant for l.c.i. morphisms in V_{G}. Moreover, for a fiber diagram

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

in V_{G} with f projective and g smooth, one has g^{∗} ∘ f_{∗} = f′_{∗} ∘ g′^{∗} : Ω_{∗}^{G}(X) → Ω_{∗}^{G}(Y′). (ii) Homotopy : If f : E → X is a G-equivariant vector bundle, then f^{∗} :

Ω_{∗}^{G}(X) ⟶^{≅} Ω_{∗}^{G}(E).

(iii) Chern classes : For any G-equivariant vector bundle E ⟶^{f} X of rank r, there are equivariant Chern class operators c_{m}^{G}(E) : Ω_{∗}^{G}(X) → Ω_{∗−m}^{G}(X) for 0 ≤ m ≤ r with c_{0}^{G}(E) = 1. These Chern classes have same functoriality properties as in the non-equivariant case. Moreover, they satisfy the Whitney sum formula. (iv) F ree action : If G acts freely on X with quotient Y, then Ω_{∗}^{G}(X) ⟶^{≅} Ω_{∗}(Y).

(v) Exterior P roduct : There is a natural product map

Ω_{i}^{G}(X) ⊗_{ℤ} Ω_{i′}^{G}(X′) → Ω_{i+i′}^{G}(X × X′).

In particular, Ω_{∗}^{G}(k) is a graded algebra and Ω_{∗}^{G}(X) is a graded Ω_{∗}^{G}(k)-module for every X ∈ V_{G}. (vi) P rojection formula : For a projective map f : X′ → X in V_{G}^{S}, one has for x ∈ Ω_{∗}^{G}(X) and x′ ∈ Ω_{∗}^{G}(X′), the formula : f_{∗} (x′ · f^{∗}(x)) = f_{∗}(x′) · x.

Proof. Assume that the dimensions of X and Y are m and n respectively and let d = m−n be the relative dimension of a projective G-equivariant morphism f : X → Y. For a fixed j ≥ 0, let (V_{j}, U_{j}) be an l-dimensional good pair for j. Since f is projective, Lemma 5.1 implies that \overline{f} : X_{G} → Y_{G} is projective and hence by Theorem 2.1 and Lemma 3.3, there is a push-forward map

\frac{Ω_{i+l−g}(X_{G})}{F_{m+l−g−j}Ω_{i+l−g}(X_{G})} → \frac{Ω_{i+l−g}(Y_{G})}{F_{m+l−g−j}Ω_{i+l−g}(Y_{G})} = \frac{Ω_{i+l−g}(Y_{G})}{F_{n+l−g−(j−d)}Ω_{i+l−g}(Y_{G})}.

In particular, we get a compatible system of maps

Ω_{i}^{G}(X)_{j+d} → Ω_{i}^{G}(Y)_{j}.

Taking the inverse limits, one gets the desired push-forward map Ω_{i}^{G}(X) ⟶^{f_{∗}} Ω_{i}^{G}(Y). If f is smooth of relative dimension d, then \overline{f} : X_{G} → Y_{G} is also smooth of same relative dimension. Hence, we get a compatible system of pull-back maps

Ω_{i}^{G}(Y)_{j} ⟶^{\overline{f}}^{∗} Ω_{i+d}^{G}(X)_{j}. Taking the inverse limit, we get the desired pull-back map of the equivariant algebraic cobordism groups. If f is a l.c.i. morphism of G-schemes, the same proof applies using the existence of similar map in the non-equivariant case. The required commutativity of the pull-back and pushforward maps follows exactly in the same way from the corresponding result for the non-equivariant cobordism groups. To prove the homotopy property, let E ⟶^{f} X be a G-equivariant vector bundle of rank r. For any j ≥ 0, let (V_{j}, U_{j}) be a good pair for j. Then the map of mixed quotients E_{G} → X_{G} is a vector bundle of rank r (cf. [10, Lemma 1]). Hence by Proposition 3.9, the pull-back map Ω_{i}^{G}(X)_{j} → Ω_{i+r}^{G}(E)_{j} is an isomorphism. If j′ ≥ j, then we can choose a common good pair for both j and j′. Hence, we have a pull-back map of the inverse systems {Ω_{i}^{G}(X)_{j}} → {Ω_{i+r}^{G}(E)_{j}} which is an isomorphism at each level. Hence f^{∗} : Ω_{i}^{G}(X) → Ω_{i+r}^{G}(E) is an isomorphism. To define the Chern classes of an equivariant vector bundle E of rank r, we choose an l-dimensional good pair (V_{j}, U_{j}) and consider the vector bundle E_{G} → X_{G} as above and let c_{m,j}^{G} : Ω_{i+l−g}(X_{G}) → Ω_{i+l−g−m}(X_{G}) be the

non-equivariant Chern class as in [31, 4.1.7]. For a closed subscheme Z↪^{ι}X_{G}, the projection formula for the non-equivariant cobordism

(5.1)  c_{m,j}^{G}(E_{G}) ∘ ι_{∗} = ι_{∗} ∘ (c_{m,j}^{G} (ι^{∗}(E_{G})))

implies that c_{m,j}^{G}(E_{G}) descends to maps c_{m,j}^{G} : Ω_{i}^{G}(X)_{j} → Ω_{i−m}^{G}(X)_{j}. One shows as in Lemma 4.2 that this is independent of the choice of the good pairs. Furthermore, choosing a common good pair for j′ ≥ j, we see that c_{m,j}^{G} actually defines a map of the inverse systems. Taking the inverse limit, we get the Chern classes c_{m}^{G}(E) : Ω_{i}^{G}(X) → Ω_{i−m}^{G}(X) for 0 ≤ m ≤ r with c_{0}^{G}(E) = 1. The functoriality and the Whitney sum formula for the equivariant Chern classes are easily proved along the above lines using the analogous properties of the non-equivariant Chern classes. The statement about the free action follows from [9, Lemma 7.2] and Remark 4.5. We now show the existence of the exterior product of the equivariant cobordism which requires some work. Let d and d′ be the dimensions of X and X′ respectively. We first define maps

(5.2)  Ω_{i}^{G}(X)_{j} ⊗ Ω_{i′}^{G}(X′)_{j} → Ω_{i+i′}^{G}(X × X′)_{j} for j ≥ 0.

Let (V_{j}, U_{j}) be an l-dimensional good pair for j and let α = [Y ⟶^{f} X_{G}] and α′ =

[Y′ ⟶^{f′} X′_{G}] be the cobordism cycles on X_{G} and X′_{G} respectively. Using the fact that X ×U_{j} → X_{G} and X′ ×U_{j} → X′_{G} are principal G-bundles, we get the unique cobordism cycles [Ỹ → X × U_{j}] and [Ỹ′ → X′ × U_{j}] whose G-quotients

are the above chosen cycles. We define α ⋆ α′ = [Ỹ ×^{G} Ỹ′ → (X × X′)_{G}]. Note that (V_{j} × V_{j}, U_{j} × U_{j}) is a good pair for j of dimension 2l and (X × X′)_{G} is the quotient of X × X′ × U_{j} × U_{j} for the free diagonal action of G and α ⋆ α′ is a well defined cobordism cycle by Lemma 5.1. Suppose now that W ⟶^{p} X_{G} × □^{1} is a projective morphism from a smooth scheme W such that the composite map π : W → X_{G} × □^{1} → □^{1} is a double point degeneration with W_{∞} = π^{−1}(∞) smooth. Letting G act trivially on □^{1},

this gives a unique G-equivariant double point degeneration W̃ \underrightarrow{p̃} X ×U_{j} ×□^{1}

of G-schemes. This implies in particular that W̃ ×Ỹ′ ⟶^{p̃×f̃′} X ×X′ ×U_{j} ×U_{j} × □^{1} is also a G-equivariant double point degeneration whose quotient for the

free G-action gives a double point degeneration W̃ ×^{G} Ỹ′ ⟶^{q} (X × X′)_{G} × □^{1}. Moreover, it is easy to see from this that C(p)⋆α′ = C(q) (cf. (2.4)). Reversing the roles of X and X′ and using (2.4) and Theorem 2.3, we get the maps

Ω_{i+l−g}(X_{G}) ⊗ Ω_{i′+l−g}(X′_{G}) → Ω_{i+i′−2l−g} ((X × X′)_{G}) .

It is also clear from the definition of α ⋆ α′ and the niveau filtration that

{F_{d+l−g−j}Ω_{i+l−g}(X_{G}) ⊗ Ω_{i′+l−g}(X′_{G})} +

+ {Ω_{i+l−g}(X_{G}) ⊗ F_{d′+l−g−j}Ω_{i′+l−g}(X′_{G})} → → F_{d+d′+2l−g−j}Ω_{i+i′−2l−g−j} ((X × X′)_{G}) .

This defines the maps as in (5.2). One can now show as in Lemma 4.2 that these maps are independent of the choice of the good pairs. We get the desired exterior product as the composite map

Ω_{i}^{G}(X) ⊗_{ℤ} Ω_{i′}^{G}(X′) = \underleftarrow{lim}_{j} Ω_{i}^{G}(X)_{j} ⊗_{ℤ} \underleftarrow{lim}_{j} Ω_{i′}^{G}(X′)_{j}
(5.3)  → \underleftarrow{lim}_{j} (Ω_{i}^{G}(X)_{j} ⊗_{ℤ} Ω_{i′}^{G}(X′)_{j})
(5.4)  → \underleftarrow{lim}_{j} Ω_{i+i′}^{G}(X × X′)_{j} = Ω_{i+i′}^{G}(X × X′).

Finally for X smooth, we get the product structure on Ω_{G}^{∗}(X) via the composite

Ω_{G}^{∗}(X) ⊗_{ℤ} Ω_{G}^{∗}(X) → Ω_{G}^{∗}(X × X) ⟶^{Δ_{X}^{∗}} Ω_{G}^{∗}(X). The projection formula can now be proven by using the non-equivariant version of such a formula (cf. [31, 5.1.4]) at each level of the projective system {Ω_{G}^{i}(X)_{j}} and then taking the inverse limit. □

We now turn our attention to the question of the localization sequence in equivariant cobordism. In the topological context, Buhstaber-Miscenko [6, 7] defined the topological K-theory of an infinite CW -complex as the projective limit of the K-theory of finite skeleta. They suggested that this theory might not have the Gysin exact sequence. In [26], Landweber showed that such a phenomenon for the K-theory is also reflected in the complex cobordism. This makes us believe that one should not expect the full localization sequence for the equivariant algebraic cobordism considered here. On the positive side however, we can prove the following weaker result.

Proposition 5.3. Let X be a G-scheme of dimension d and let f : U ↪ X be a G-invariant open subscheme. Then the restriction map f^{∗} : Ω_{∗}^{G}(X) → Ω_{∗}^{G}(U) is surjective.

Proof. Let Z be the complement of U in X with the reduced induced closed subscheme structure and let g : Z ↪ X be the inclusion map. We fix integers i ∈ ℤ and j ≥ 0 and choose a good pair (V_{j}, U_{j}) of dimension l for j. Then we see that Z is a G-invariant closed subscheme of X and Z_{G} ⊆ X_{G} is a closed subscheme with the complement U_{G}. Hence by applying Theorem 3.5 at the appropriate levels of the niveau filtration and taking the quotients, we get an exact sequence

\frac{Ω_{i+l−g}(Z_{G})}{F_{d+l−g−j}Ω_{i+l−g}(Z_{G})} → \frac{Ω_{i+l−g}(X_{G})}{F_{d+l−g−j}Ω_{i+l−g}(X_{G})} → \frac{Ω_{i+l−g}(U_{G})}{F_{d+l−g−j}Ω_{i+l−g}(U_{G})} → 0.

If d′ = dim(Z), then F_{d′+l−g−j}Ω_{i+l−g}(Z_{G}) ⊆ F_{d+l−g−j}Ω_{i+l−g}(Z_{G}) and hence by Lemma 4.2, we get an exact sequence of inverse systems

(5.5)  Ω_{i}^{G}(Z)_{j} ⟶^{ϕ_{j}^{i}} Ω_{i}^{G}(X)_{j} → Ω_{i}^{G}(U)_{j} → 0.

Setting M_{j}^{i} and N_{j}^{i} to be the kernel and the image of the map ϕ_{j}^{i} respectively, (5.5) can be split into the short exact sequences of inverse systems

(5.6)  0 → M_{j}^{i} → Ω_{i}^{G}(Z)_{j} → N_{j}^{i} → 0;
(5.7)  0 → N_{j}^{i} → Ω_{i}^{G}(X)_{j} → Ω_{i}^{G}(U)_{j} → 0.

It follows from Lemma 4.3 that {Ω_{i}^{G}(Z)_{j}}_{j≥0} is an inverse system of surjective maps and hence so is {N_{j}^{i}}_{j≥0}. In particular, it satisfies the Mittag-Leffler (ML) condition. As a consequence, we get an exact sequence of inverse limits

\underleftarrow{lim}_{j} N_{j}^{i} → Ω_{i}^{G}(X) → Ω_{i}^{G}(U) → 0

and this proves the proposition. □

Proposition 5.4 (Morita Isomorphism). Let H ⊂ G be a closed subgroup and let X ∈ V_{H}. Then there is a canonical isomorphism

(5.8)  Ω_{∗}^{G} (G ×^{H} X) ⟶^{≅} Ω_{∗}^{H}(X).

Proof. Define an action of H × G on G × X by

(h, g) · (g′, x) = (gg′h^{−1}, hx) ,

and an action of H × G on X by (h, g) · x = hx. Then the projection map G × X ⟶^{p} X is (H × G)-equivariant which is a G-torsor. Hence by [9, Lemma 7.2],

the natural map Ω_{∗}^{H}(X) ⟶^{p^{∗}} Ω_{∗}^{H×G}(G × X) is an isomorphism. On the other

hand, the projection map G × X → G ×^{H} X is (H × G)-equivariant which is

an H-torsor. Hence we get an isomorphism Ω_{∗}^{G} (G ×^{H} X) ⟶^{≅} Ω_{∗}^{H×G} (G × X).

The proposition follows by combining these two isomorphisms. □

6. Computations

Let X be a k-scheme of dimension d with a G-action. We have seen above that unlike the situation of Chow groups, the cobordism group Ω_{i+l−g} (X ×^{G} U_{j})

is not independent of the choice of the l-dimensional good pair (V_{j}, U_{j}) even if j is large enough. This anomaly is rectified by considering the quotients of the cobordism groups of the good pairs by the niveau filtration. Our main result in this section is to show that if we suitably choose a sequence of good pairs {(V_{j}, U_{j})}_{j≥0}, then the above equivariant cobordism group can be computed without taking quotients by the niveau filtration. This reduction is often very helpful in computing the equivariant cobordism groups.

Theorem 6.1. Let {(V_{j}, U_{j})}_{j≥0} be a sequence of l_{j}-dimensional good pairs such that

(i) V_{j+1} = V_{j} ⊕ W_{j} as representations of G with dim(W_{j}) > 0 and (ii) U_{j} ⊕ W_{j} ⊊ U_{j+1} as G-invariant open subsets. (iii) codim_{V_{j+1}} (V_{j+1} \ U_{j+1}) > codim_{V_{j}} (V_{j} \ U_{j}). Then for any scheme X as above and any i ∈ ℤ, we have

\underleftarrow{lim}_{j} Ω_{i+l_{j}−g} (X ×^{G} U_{j}) ⟶^{≅} Ω_{i}^{G}(X).

Moreover, such a sequence {(V_{j}, U_{j})}_{j≥0} of good pairs always exists.

Proof. Let {(V_{j}, U_{j})}_{j≥0} be a sequence of good pairs as in the theorem. We have natural maps

(6.1) Ω_{i+l_{j+1}−g} (X ×^{G} U_{j+1}) ↠

↠ Ω_{i+l_{j+1}−g} (X ×^{G} (U_{j} ⊕ W_{j})) ⟵^{≅} Ω_{i+l_{j}−g} (X ×^{G} U_{j}) ,

where the first map is the restriction to an open subset and the second is the pull-back via a vector bundle. Taking the quotients by the niveau filtrations, we get natural maps (cf. proof of Lemma 4.2) (6.2)

[figure: \frac{Ω_{i+l_{j+1}−g}(X×^{G}U_{j+1})}{F_{d+l_{j+1}−g−j−1}Ω_{i+l_{j+1}−g}(X×^{G}U_{j+1})}; \frac{Ω_{i+l_{j+1}−g}(X×^{G}(U_{j}⊕W_{j}))}{F_{d+l_{j+1}−g−j−1}Ω_{i+l_{j+1}−g}(X×^{G}(U_{j}⊕W_{j}))}; \frac{Ω_{i+l_{j} −g}(X×^{G}U_{j})}{F_{d+l_{j} −g−j}Ω_{i+l_{j} −g}(X×^{G}U_{j})}; \frac{Ω_{i+l_{j} −g}(X×^{G}U_{j})}{F_{d+l_{j} −g−j−1}Ω_{i+l_{j} −g}(X×^{G}U_{j})}; ≅]

where the right vertical arrow is an isomorphism by Proposition 3.9. Setting

X_{j} = X ×^{G} U_{j}, we get natural maps

(6.3)  [figure: Ω_{i+l_{j+1}−g} (X_{j+1}); ν_{j}^{j+1}; Ω_{i+l_{j}−g} (X_{j}); \frac{Ω_{i+l_{j+1}−g}(X_{j+1})}{F_{d+l_{j+1}−g−j−1}Ω_{i+l_{j+1}−g}(X_{j+1})}; \frac{Ω_{i+l_{j} −g}(X_{j})}{F_{d+l_{j} −g−j}Ω_{i+l_{j} −g}(X_{j})}.]

Documenta Mathematica 17 (2012) 95–134

Since (V_{j}, U_{j}) is a good pair for each j, we see that \frac{Ω_{i+l_{j} −g}(X_{j})}{F_{d+l_{j} −g−j}Ω_{i+l_{j} −g}(X_{j})} ≅

Ω_{i}^{G}(X)_{j}. Hence, we only have to show that the map

(6.4)  \underleftarrow{lim}_{j} Ω_{i+l_{j}−g} (X_{j}) → \underleftarrow{lim}_{j} \frac{Ω_{i+l_{j}−g} (X_{j})}{F_{d+l_{j}−g−j}Ω_{i+l_{j}−g} (X_{j})}

is an isomorphism in order to prove the theorem. To prove (6.4), we only need to show that for any given j ≥ 0, the map

Ω_{i+l_{j′} −g} (X_{j′}) ⟶^{ν_{j}^{j′}} Ω_{i+l_{j}−g} (X_{j}) factors through

(6.5)  \frac{Ω_{i+l_{j′} −g} (X_{j′})}{F_{d+l_{j′} −g−j′}Ω_{i+l_{j′} −g} (X_{j′})} → Ω_{i+l_{j}−g} (X_{j}) for all j′ ≫ j.

However, it follows from (6.2) that ν_{j}^{j′} induces the map

\frac{Ω_{i+l_{j′} −g} (X_{j′})}{F_{d+l_{j′} −g−j′}Ω_{i+l_{j′} −g} (X_{j′})} → \frac{Ω_{i+l_{j}−g} (X_{j})}{F_{d+l_{j}−g−j′}Ω_{i+l_{j}−g} (X_{j})}.

On the other hand F_{d+l_{j}−g−j′}Ω_{i+l_{j}−g} (X_{j}) vanishes for j′ ≫ j. This proves (6.5) and hence (6.4). Finally, it follows easily from the proof of Lemma 4.2 (see also [39, Remark 1.4]) that a sequence of good pairs as in Theorem 6.1 always exists. □

As a simple corollary of Theorem 6.1, we get the following localization sequence for the equivariant cobordism in a special case.

Corollary 6.2. Let X be a G-scheme and let Z ⊆ X be a G-invariant closed subscheme with the complement U. Assume that there is a G-equivariant projective morphism p : X → Y whose restriction to Z is an isomorphism. Then there is a short exact sequence

(6.6)  0 → Ω_{∗}^{G}(Z) → Ω_{∗}^{G}(X) → Ω_{∗}^{G}(U) → 0.

Proof. Let {(V_{j}, U_{j})}_{j≥0} be a sequence of good pairs as in Theorem 6.1. The localization sequence for the ordinary algebraic cobordism yields for any i ∈ ℤ, an exact sequence of inverse systems

(6.7) Ω_{i+l_{j}−g} (Z ×^{G} U_{j}) → Ω_{i+l_{j}−g} (X ×^{G} U_{j}) → Ω_{i+l_{j}−g} (U ×^{G} U_{j}) → 0.

It follows from Lemma 5.1 that there are projective morphisms Z ×^{G} U_{j} ⟶^{f_{j}}

X ×^{G} U_{j} ⟶^{p_{j}} Y ×^{G} U_{j} such that p_{j} ∘ f_{j} is an isomorphism. This implies that the map p_{j}_{∗} ∘ f_{j}_{∗} is an isomorphism. In other words, (6.8) is in fact a short exact sequence of inverse systems (6.8)

0 → Ω_{i+l_{j}−g} (Z ×^{G} U_{j}) → Ω_{i+l_{j}−g} (X ×^{G} U_{j}) → Ω_{i+l_{j}−g} (U ×^{G} U_{j}) → 0.

We have moreover seen in (6.1) that {Ω_{i+l_{j}−g}(Z ×^{G} U_{j})}_{j≥0} is an inverse system of surjective maps. It follows that (6.8) remains short exact after taking limit, which proves (6.6). □

Another consequence of Theorem 6.1 is that for a linear algebraic group G acting on a scheme X of dimension d, the forgetful map r_{X}^{G} : Ω_{∗}^{G}(X) → Ω_{∗}(X) (cf. (4.6)) can be easily shown to be analogous to the one used in [10, Subsection 2.2] for the Chow groups. This interpretation of the forgetful map has some interesting applications in the computation of the non-equivariant cobordism using the equivariant techniques (cf. [20], [21]). So let {(V_{j}, U_{j})}_{j≥0} be a sequence of good pairs as in Theorem 6.1. We choose a k-rational point x ∈ U_{0} and let x_{j} be its image in U_{j}/G under the natural

map U_{0} → U_{0}/G → U_{j}/G. Setting X_{j} = X ×^{G} U_{j}, this yields a commutative diagram

(6.9)  \xymatrix{X × U_{j} \ar[r]^{p_{j}} \ar[d]_{π_{j}} & X_{j} \ar[r]^{ϕ_{j}} \ar[d]_{ψ_{j}} & X_{j+1} \ar[d]^{ψ_{j+1}} \\ U_{j} \ar[r] & U_{j}/G \ar[r] & U_{j+1}/G}

such that the left square is Cartesian and

(6.10)  X ≅ π_{j}^{−1}(x) ⟶^{≅} ψ_{j}^{−1}(x_{j}) ⟶^{≅} ψ_{j+1}^{−1}(x_{j+1}).

Let ν_{j} : ψ_{j}^{−1}(x_{j}) ↪ X_{j} be the closed embedding. Notice that since U_{j}/G is smooth and ψ_{j} is flat, it follows that ν_{j} is a regular closed embedding (hence an l.c.i. morphism). Using the identification in (6.10), we get maps ν_{j}^{∗} : Ω_{∗}(X_{j}) → Ω_{∗}(X) such that ν_{j}^{∗} ∘ ϕ_{j}^{∗} = ν_{j+1}^{∗}. Taking the limit over j ≥ 0, this yields for any i ∈ ℤ, a restriction map

(6.11)  r_{X}^{G} : Ω_{i}^{G}(X) = \underleftarrow{lim}_{j≥0} Ω_{i}^{G}(X)_{j} → Ω_{i}(X).

Corollary 6.3. The maps r_{X}^{G}, r_{X}^{G} : Ω_{i}^{G}(X) → Ω_{i}(X) coincide.

Proof. Using the construction of the map r_{X}^{G} in (4.6) and the diagram (6.9), it suffices to show that for any i ∈ ℤ, the natural maps

\frac{Ω_{i}(X)}{F_{d−j}Ω_{i}(X)} ← \frac{Ω_{i} (X × V_{j})}{F_{d+l_{j}−j}Ω_{i} (X × V_{j})} → \frac{Ω_{i} (X × U_{j})}{F_{d+l_{j}−j}Ω_{i} (X × U_{j})}

are isomorphisms for all j ≫ 0. Here, the first map is the restriction induced by the section corresponding to the rational point x_{j} ∈ U_{j}/G, and the second map is the restriction to an open subset. The existence of the first map follows from Proposition 3.9. The assertion that these two maps are isomorphisms follows immediately from Corollary 3.6 and Proposition 3.9. □

Remark 6.4. It follows from Corollary 6.3 that the map r_{X}^{G} does not depend on the choice of the k-rational point x ∈ U_{0}.

6.1. Graded vs. completed cobordism rings. Another consequence of Theorem 6.1 is that it allows us to explain the relation between the graded (or noncomplete) and the nongraded (or complete) versions of the equivariant cobordism rings of smooth G-schemes. If {(V_{j}, U_{j})}_{j≥0} is a sequence of l_{j}- dimensional good pairs as in Theorem 6.1, then it can be easily checked from the proof of this theorem that the expression

(6.12)  { \atop Ω_{G}^{∗}(X)} = \underleftarrow{lim}_{j} Ω^{∗} (X ×^{G} U_{j})

is well-defined and there is a natural map ι_{X} : Ω_{G}^{∗}(X) → { \atop Ω_{G}^{∗}(X)}.

Furthermore, the surjectivity of the map Ω^{∗} (X ×^{G} U_{j+1}) → Ω^{∗} (X ×^{G} U_{j})

(cf. (6.1)) implies that ι_{X} identifies { \atop Ω_{G}^{∗}(X)} as the completion of Ω_{G}^{∗}(X) with respect to the linear topology given by the decreasing filtration

F^{j}Ω_{G}^{∗}(X) = Ker (Ω_{G}^{∗}(X) ↠ Ω^{∗} (X ×^{G} U_{j})) .

6.2. Formal group law in equivariant cobordism. Let G be a linear algebraic group over k acting on a scheme X of dimension d. We have seen before that the equivariant line bundles on X give rise to the equivariant Chern class operators on Ω_{∗}^{G}(X). Below, we write down an expression for the equivariant Chern class of the tensor product of two such line bundles. Let {(V_{j}, U_{j})}_{j≥0} be a sequence of l_{j}-dimensional good pairs as in Theorem 6.1.

Letting X_{j} = X ×^{G} U_{j}, we see that for every j ≥ 0, Ω_{∗} (X_{j}) = ⨁_{i∈ℤ} Ω_{i+l_{j}−g}(X_{j})

is an 𝕃-module and for j′ ≥ j, there is a natural surjection Ω_{∗} (X_{j′}) ↠ Ω_{∗} (X_{j}) of 𝕃-modules. Given G-equivariant line bundles L, M on X, we get line bundles L_{j}, M_{j} on

X_{j}, where L_{j} = L ×^{G} U_{j} for j ≥ 0. The formal group law of the non-equivariant cobordism yields

c_{1} ((L ⊗ M)_{j}) =

= c_{1} (L_{j} ⊗ M_{j}) = c_{1}(L_{j}) + c_{1}(M_{j}) + ∑_{i,i′≥1} a_{i,i′} (c_{1}(L_{j}))^{i} ∘ (c_{1}(M_{j}))^{i′} .

Note that if (x_{j}) ∈ Ω_{i}^{G}(X), then the evaluation of the operator c_{1}^{G}(L)(x_{j}) at any level j ≥ 0 is a finite sum above. Taking the limit over j ≥ 0 and noting that the sum (and the product) in the equivariant cobordism groups are obtained by taking the limit of the sums (and the products) at each level of the inverse system, we get the same formal group law for the equivariant Chern classes:

(6.13) c_{1}^{G}(L ⊗ M) = c_{1}^{G}(L) + c_{1}^{G}(M) + ∑_{i,i′≥1} a_{i,i′} (c_{1}^{G}(L))^{i} ∘ (c_{1}^{G}(M))^{i′} .

Note that the coefficients a_{i,i′} are homogeneous elements of 𝕃 and can be considered as elements of S(G) under the natural inclusion of graded rings 𝕃 ↪ S(G) . One should also observe that unlike the case of ordinary cobordism, the evaluation of the above sum on any given equivariant cobordism cycle may no longer be finite. In other words, the equivariant Chern classes are not in general locally nilpotent.

6.3. Cobordism ring of classifying spaces. Let R be a Noetherian ring and let A = ⊕_{j∈ℤ}A_{j} be a ℤ-graded R-algebra with R ⊆ A_{0}. Recall that the

graded power series ring S^{(n)} = ⊕_{i∈ℤ}S_{i} is a graded ring such that S_{i} is the set

of formal power series of the form f(t) = ∑_{m(t)∈C}a_{m(t)}m(t) such that a_{m(t)} is

a homogeneous element of A of degree |a_{m(t)}|and |a_{m(t)}| + |m(t)| = i. Here, C is the set of all monomials in t = (t_{1}, ⋯, t_{n}) and |m(t)| = i_{1} + ⋯ + i_{n} if m(t) = t_{1}^{i_{1}} ⋯ t_{n}^{i_{n}}. We call |m(t)| to be the degree of the monomial m(t). We shall write the above graded power series ring as A[[t]]_{gr} to distinguish it from the usual formal power series ring A[[t]]. Notice that if A is only non-negatively graded, then S^{(n)} is nothing but the standard polynomial ring A[t_{1}, ⋯, t_{n}] over A. It is also easy to see that S^{(n)} is indeed a graded ring which is a subring of the formal power series ring A[[t_{1}, ⋯, t_{n}]]. The following result summarizes some basic properties of these rings. The proof is straightforward and is left as an exercise.

Lemma 6.5. (i) There are inclusions of rings A[t_{1}, ⋯, t_{n}] ⊂ S^{(n)} ⊂ A[[t_{1}, ⋯, t_{n}]], where the first is an inclusion of graded rings. (ii) These inclusions are analytic isomorphisms with respect to the t-adic topology. In particular, the induced maps of the associated graded rings

A[t_{1}, ⋯, t_{n}] → Gr_{(t)}S^{n} → Gr_{(t)}A[[t_{1}, ⋯, t_{n}]]

are isomorphisms.

(iii) S^{(n−1)}[[t_{n}]]_{gr} ⟶^{≅} S^{(n)}.

(iv) \frac{S^{(n)}}{(t_{i_{1}} ,··· ,t_{ir})} ⟶^{≅} S^{(n−r)} for any n ≥ r ≥ 1, where S^{(0)} = A.

(v) The sequence {t_{1}, ⋯, t_{n}} is a regular sequence in S^{(n)}. (vi) If A = R[x_{1}, x_{2}, ⋯] is a polynomial ring with |x_{i}| < 0 and lim_{i→∞} |x_{i}| = −∞,

then S^{(n)} ⟶^{≅} \underleftarrow{lim}_{i} R[x_{1}, ⋯, x_{i}][[t]]_{gr}.

Examples 6.6. In the following examples, we compute Ω^{∗}(BG) = Ω_{G}^{∗}(k) for some classical groups G over k. These computations follow directly from the definition of equivariant cobordism and suitable choices of good pairs. We first consider the case when G = 𝔾_{m} is the multiplicative group. For any j ≥ 1, we choose the good pair (V_{j}, U_{j}), where V_{j} is the j-dimensional representation of 𝔾_{m} with all weights −1 and U_{j} is the complement of the origin. We see then that U_{j}/𝔾_{m} ≅ ℙ_{k}^{j−1}. Let ζ be the class of c_{1}(O(−1))(1) ∈ Ω^{1}(ℙ_{k}^{j−1}). The projective bundle formula for the ordinary algebraic cobordism implies that (Ω_{G}^{i})_{j} = ⨁_{0≤p≤j−1}𝕃^{i−p}ζ^{p}. Taking the inverse limit over j ≥ 1, we

find from this that for i ∈ ℤ,

Ω_{𝔾_{m}}^{i}(k) = ∏_{p≥0}𝕃^{i−p}ζ^{p}.

It particular, if x = n ; ∑_{j=1} x_{i_{j}} is a sum of homogeneous elements of Ω^{∗}(B𝔾_{m}),

then we get a natural map

(6.14)  Ω^{∗}(B𝔾_{m}) → 𝕃[[t]]_{gr}

x = (x_{i_{1}} = ∏ a_{p}^{i_{1}}ζ^{p}, ⋯, x_{i_{n}} = ∏ a_{p}^{i_{n}}ζ^{p}) ↦ ∑_{p≥0} ( ∑_{1≤j≤n}a_{p}^{i_{j}} ) t^{p},

which is an isomorphism of graded 𝕃-algebras. Observe that { \atop Ω^{∗}(B𝔾_{m})} (cf. (6.12)) is the formal power series ring 𝕃[[t]]. For a general split torus T of rank n, we choose a basis {χ_{1}, ⋯, χ_{n}} of the character group T̂. This is equivalent to a decomposition T = T_{1} × ⋯ × T_{n} with each T_{i} isomorphic to 𝔾_{m} and χ_{i} is a generator of T̂_{i}. Let L_{χ} be the one-dimensional representation of T, where T acts via χ. For any j ≥ 1, we

take the good pair (V_{j}, U_{j}) such that V_{j} = n ; ∏_{i=1} L_{χ_{i}}^{⊕j}, U_{j} = n ; ∏_{i=1} (L_{χ_{i}}^{⊕j} \ {0}) and

T acts on V_{j} by (t_{1}, ⋯, t_{n})(x_{1}, ⋯, x_{n}) = (χ_{1}(t_{1})(x_{1}), ⋯, χ_{n}(t_{n})(x_{n})). It is then easy to see that U_{j}/T ≅ X_{1} × ⋯ × X_{n} with each X_{i} isomorphic to ℙ_{k}^{j−1}.

Moreover, the T -line bundle L_{χ_{i}} gives the line bundle L_{χ_{i}} ×^{T_{i}} (L_{χ_{i}}^{⊕j} \ {0}) → X_{i} which is O(±1). Letting ζ_{i} be the first Chern class of this line bundle, the projective bundle formula for the non-equivariant cobordism shows that

Ω_{T}^{i} (k) = ∏_{p_{1},··· ,p_{n}≥0}𝕃^{i−( n ; ∑_{i=1}p_{i})}ζ_{1}^{p_{1}} ⋯ ζ_{n}^{p_{n}},

which is isomorphic to the set of formal power series in {ζ_{1}, ⋯, ζ_{n}} of degree i with coefficients in 𝕃. It particular, one concludes as in the rank one case above that

Proposition 6.7. Let {χ_{1}, ⋯, χ_{n}} be a chosen basis of the character group of a split torus T of rank n. The assignment t_{i} ↦ c_{1}^{T} (L_{χ_{i}}) yields a graded 𝕃-algebra isomorphism

𝕃[[t_{1}, ⋯, t_{n}]]_{gr} → Ω^{∗}(BT).

For G = GL_{n}, we can take a good pair for j to be (V_{j}, U_{j}), where V_{j} is the vector space of n × p matrices with p > n with GL_{n} acting by left multiplication, and U_{j} is the open subset of matrices of maximal rank. Then the mixed quotient is the Grassmannian Gr(n, p). We can now calculate the cobordism ring of Gr(n, p) using the projective bundle formula (by standard stratification technique) and then we can use the similar calculations as above to get a natural isomorphism

(6.15)  Ω^{∗}(BGL_{n}) → 𝕃[[γ_{1}, ⋯, γ_{n}]]_{gr}

of graded 𝕃-algebras, where γ_{i}’s are the elementary symmetric polynomials in t_{1}, ⋯, t_{n} that occur in (6.14). Another way to obtain the isomorphism (6.15) is to observe that the Weyl group of GL_{n} is the permutation group S_{n} and t_{GL_{n}} = 1, where t_{G} denotes the torsion index of a connected reductive group G. It follows from [15, Theorem 3.7] that we can assume the base field to be the field of complex numbers. Subsequently, it follows from [15, Proposition 4.8] that the natural map Ω^{∗}(BGL_{n}) → (Ω^{∗}(BT))^{S_{n}} = 𝕃[[γ_{1}, ⋯, γ_{n}]]_{gr} is an isomorphism. Using the same argument, on obtains an isomorphism Ω^{∗}(BSL_{n}) ⟶^{≅} 𝕃[[γ_{2}, ⋯, γ_{n}]]_{gr}.

Remark 6.8. The cobordism rings of BGL_{n} and BSL_{n} have also been written down by Deshpande in [9, Section 4]. His expressions depend on the assumption that these groups are isomorphic to the complex cobordism. As the reader will find in Subsection 7.3, there may not in general exist a map from the algebraic to the complex equivariant cobordism although such a map does exist for a classifying space BG (cf. Corollary 7.7). Moreover, it is not clear when such a map is an isomorphism. We refer the reader to [15, Theorem 3.7] for a result in this direction.

7. Comparison with other equivariant cohomology theories

In this paper, we fix the following notation for the tensor product while dealing with inverse systems of modules over a commutative ring. Let A be a commutative ring with unit and let {L_{n}} and {M_{n}} be two inverse systems of A-modules with inverse limits L and M respectively. Following [38], one defines the topological tensor product of L and M by

(7.1)  Lˆ⊗_{A}M := \underleftarrow{lim}_{n} (L_{n}⊗_{A}M_{n}).

In particular, if D is an integral domain with quotient field F and if {A_{n}} is an inverse system of D-modules with inverse limit A, one has Aˆ⊗_{D}F = \underleftarrow{lim}_{n} (A_{n}⊗_{D}F). The examples ℤ̂_{(p)} = \underleftarrow{lim}_{n} ℤ/p^{n} and ℤ[[x]] ⊗_{ℤ} ℚ →

\underleftarrow{lim}_{n} \frac{ℤ[x]}{(x^{n})}⊗_{ℤ}ℚ = ℚ[[x]] show that the map A ⊗_{D} F → Aˆ⊗_{D}F is in general nei-

ther injective nor surjective. We shall denote Aˆ⊗_{D}F in the sequel by A_{F} to simplify the notations. If R is a ℤ-graded ring and if M and N are two R-graded modules, then recall that M ⊗_{R} N is also a graded R-module given by the quotient of M ⊗_{R_{0}} N by the graded submodule generated by the homogeneous elements of the type ax ⊗ y − x ⊗ ay where a, x and y are the homogeneous elements of R, M and N respectively. If all the graded pieces M_{i} and N_{i} are the limits of inverse systems {M_{i}^{λ}} and {N_{i}^{λ}} of R_{0}-modules, we define the graded topological tensor product as M ˆ⊗_{R}N = ⨁_{i∈ℤ} (M ˆ⊗_{R}N)_{i}, where

(7.2)  (M ˆ⊗_{R}N)_{i} = \underleftarrow{lim}_{λ} ( ⨁_{j+j′=i} \frac{M_{j}^{λ} ⊗_{R_{0}} N_{j′}^{λ}}{(ax ⊗ y − x ⊗ ay)}) .

Notice that this reduces to the ordinary tensor product of graded R-modules if the underlying inverse systems are trivial.

7.1. Comparison with equivariant Chow groups. Let X be a k-scheme of dimension d with a G-action. It was shown by Levine and Morel [31] that there is a natural map Ω_{∗}(X) → CH_{∗}(X) of graded abelian groups which is a ring homomorphism if X is smooth. Moreover, this map induces a graded isomorphism

(7.3)  Ω_{∗}(X) ⊗_{𝕃} ℤ ⟶^{≅} CH_{∗}(X).

Recall from [39] and [10] that the equivariant Chow groups of X are defined

as CH_{i}^{G}(X) = CH_{i+l−g} (X ×^{G} U), where (V, U) is an l-dimensional good pair

corresponding to d−i+1. It is known that CH_{i}^{G}(X) is well-defined and can be non-zero for any −∞ < i ≤ d. We set CH_{∗}^{G}(X) = ⨁_{i} CH_{i}^{G}(X). If X is equi-

dimensional, we let CH_{G}^{i}(X) = CH_{d−i}^{G}(X) and set CH_{G}^{∗}(X) = ⨁_{i≥0} CH_{G}^{i}(X).

Notice that in this case, CH_{G}^{i}(X) is same as CH^{i} (X ×^{G} U), where (V, U) is

an l-dimensional good pair corresponding to i + 1. If we fix i ∈ ℤ and choose an l-dimensional good pair (V_{j}, U_{j}) corresponding to j ≥ max(0, d − i + 1), the universality of the algebraic cobordism gives a

unique map Ω_{i+l−g} (X ×^{G} U_{j}) → CH_{i+l−g} (X ×^{G} U_{j}). By Lemma 3.7, this

map factors through

(7.4)  \frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{i+l−g−1}Ω_{i+l−g} (X ×^{G} U_{j})} → CH_{i+l−g} (X ×^{G} U_{j}) .

Since j ≥ d − i + 1 by the choice, we have d + l − g − j ≤ i + l − g − 1 and hence we get the map (7.5)

Ω_{i}^{G}(X)_{j} = \frac{Ω_{i+l−g} (X ×^{G} U_{j})}{F_{d+l−g−j}Ω_{i+l−g} (X ×^{G} U_{j})} → CH_{i+l−g} (X ×^{G} U_{j}) = CH_{i}^{G}(X).

Documenta Mathematica 17 (2012) 95–134

It is easily shown using the proof of Lemma 4.2 that this map is independent of the choice of the good pair (V_{j}, U_{j}). Taking the inverse limit over j ≥ 0, we get a natural map Ω_{i}^{G}(X) → CH_{i}^{G}(X) and hence a map of graded abelian groups

(7.6)  Φ_{X} : Ω_{∗}^{G}(X) → CH_{∗}^{G}(X)

which is in fact a map of graded 𝕃-modules. Notice that the right side of (7.5) does not depend on j as long as j ≫ 0. If X is equi-dimensional, we write the above map cohomologically as Ω_{G}^{∗}(X) → CH_{G}^{∗}(X).

Example 7.1. Let T be a split torus of rank n over k. It follows from Proposition 6.7 that Ω^{∗}(BT) is isomorphic to the graded power series ring 𝕃[[t_{1}, ⋯, t_{n}]]_{gr}. One knows that CH^{∗}(BT) is isomorphic to the polynomial ring ℤ[t_{1}, ⋯, t_{n}] (cf. [10, 3.2]). And the map Φ_{k} : Ω^{∗}(BT) → CH^{∗}(BT) in this case is the obvious map 𝕃[[t_{1}, ⋯, t_{n}]]_{gr} → ℤ[t_{1}, ⋯, t_{n}] obtained by killing the ideal 𝕃^{<0}.

Proposition 7.2. The map Φ_{X} induces an isomorphism of graded 𝕃-modules

Φ_{X} : Ω_{∗}^{G}(X)ˆ⊗_{𝕃}ℤ ⟶^{≅} CH_{∗}^{G}(X).

Proof. Let {(V_{j}, U_{j})}_{j≥0} be a sequence of l_{j}-dimensional good pairs as in Theorem 6.1. It follows from (7.3) that for any i ∈ ℤ, there is a short exact sequence (7.7)

0 → (𝕃^{<0}Ω_{∗}(X_{j}) ∩ Ω_{i+l_{j}−g}(X_{j})) → Ω_{i+l_{j}−g}(X_{j}) → CH_{i+l_{j}−g}(X_{j}) → 0.

By comparing this exact sequence for j′ ≥ j ≥ 0, using the surjection Ω_{i+l_{j+1}−g}(X_{j+1}) → Ω_{i+l_{j}−g}(X_{j}) as in (6.1) and using the localization sequences for the cobordism and Chow groups, we find that the map

(𝕃^{<0}Ω_{∗}(X_{j+1}) ∩ Ω_{i+l_{j+1}−g}(X_{j+1})) → (𝕃^{<0}Ω_{∗}(X_{j}) ∩ Ω_{i+l_{j}−g}(X_{j}))

is surjective for each j ≥ 0. Taking the limit in (7.7) and using Theorem 6.1, we get a short exact sequence

0 → \underleftarrow{lim}_{j≥0} (𝕃^{<0}Ω_{∗}(X_{j}) ∩ Ω_{i+l_{j}−g}(X_{j})) → Ω_{i}^{G}(X) → CH_{i}^{G}(X) → 0.

Observe here that the inverse system {CH_{i+l−g}(X_{j})}_{j≥0} is eventually constant with CH_{i}^{G}(X) as its limit. We now take the direct sum over i ∈ ℤ to get the desired isomorphism Ω_{∗}^{G}(X)ˆ⊗_{𝕃}ℤ ⟶^{≅} CH_{∗}^{G}(X). □

Let C(G) = CH_{G}^{∗}(k) denote the equivariant Chow ring of the field k. The following is the equivariant analogue of (7.3).

Corollary 7.3. For a k-scheme X with a G-action, the natural map

Ω_{∗}^{G}(X)⊗_{S(G)}C(G) → CH_{∗}^{G}(X)

is an isomorphism of C(G)-modules. This is a ring isomorphism if X is smooth.

Proof. It is clear that the above map is a ring homomorphism if X is smooth. So we only need to prove the first assertion. But this follows directly from the isomorphisms Ω_{∗}^{G}(X)⊗_{S(G)}C(G) ≅ Ω_{∗}^{G}(X)⊗_{S(G)} (S(G)ˆ⊗_{𝕃}ℤ) ≅ Ω_{∗}^{G}(X)ˆ⊗_{𝕃}ℤ using Proposition 7.2. □

7.2. Comparison with equivariant K-theory. It was shown by Levine and Morel in [30, Corollary 11.11] that the universal property of the algebraic cobordism implies that there is a canonical isomorphism of oriented cohomology theories

(7.8)  Ω^{∗}(X) ⊗_{𝕃} ℤ[β, β^{−1}] ⟶^{≅} K_{0}(X)[β, β^{−1}]

in the category of smooth k-schemes. This was later generalized to a complete algebraic analogue of the Conner-Floyd isomorphism

MGL^{∗} ⊗_{𝕃} ℤ[β, β^{−1}] ⟶^{≅} K_{∗}(X)[β, β^{−1}]

between the motivic cobordism and algebraic K-theory by Panin, Pimenov and Röndigs [33]. Since the equivariant cobordism is a Borel style cohomology theory, one can not expect an equivariant version of the isomorphism (7.8) even with the rational coefficients. However, we show here that the equivariant Conner-Floyd isomorphism holds after we base change the above by the completion of the representation ring of G with respect to the ideal of virtual representations of rank zero. In fact, it can be shown easily that such a base change is the minimal requirement. In Theorem 7.4, all cohomology groups are considered with rational coefficients (cf. Section 8). For a linear algebraic group G, let R(G) denote the representation ring of G. Let I denote the ideal of of virtual representations of rank zero in R(G) and let { \atop R(G)} denote the associated completion of R(G). Let { \atop C(G)} denote the completion of C(G) with respect to the augmentation ideal of algebraic cycles of positive codimensions. For a scheme X with G-action, let K_{0}^{G}(X) denote the Grothendieck group of G-equivariant vector bundles on X. By [11, Theorem 4.1], there is a natural ring isomorphism { \atop R(G)} ⟶^{≅} { \atop C(G)} given by the equivariant Chern character. We identify these two rings via this isomorphism. In particular, the maps S(G) ↠ C(G) → { \atop C(G)} yield a ring homomorphism

S(G) → { \atop R(G)}.

Theorem 7.4. Let X be a smooth scheme with a G-action. Then, with rational coefficients, there is a natural isomorphism of rings

Ψ_{X} : Ω_{G}^{∗}(X)⊗_{S(G)}{ \atop R(G)} ⟶^{≅} K_{0}^{G}(X) ⊗_{R(G)} { \atop R(G)}.

Proof. By [17, Theorem 1.2], there is a Chern character isomorphism K_{0}^{G}(X) ⊗_{R(G)} { \atop R(G)} ⟶^{≅} CH^{∗}(X) ⊗_{C(G)} { \atop C(G)} of cohomology rings. Thus, we

only need to show that the map Ω_{G}^{∗}(X)⊗_{S(G)}{ \atop C(G)} → CH^{∗}(X) ⊗_{C(G)} { \atop C(G)} is an isomorphism. However, we have

\begin{aligned}Ω_{G}^{∗}(X)⊗_{S(G)}{ \atop C(G)} & ≅ & (Ω_{G}^{∗}(X)⊗_{S(G)}C(G)) ⊗_{C(G)} { \atop C(G)} \\ & ≅ & CH_{G}^{∗}(X) ⊗_{C(G)} { \atop C(G)}, \end{aligned}

where the last isomorphism follows from Corollary 7.3. This finishes the proof. □

7.3. Comparison with complex cobordism. Let G be a complex Lie group acting on a finite CW -complex X. We define the equivariant complex cobordism ring of X as

(7.9)  MU_{G}^{∗}(X) := MU^{∗} (X ×^{G} EG)

where EG → BG is universal principal G-bundle over the classifying space BG of G. If E′G → B′G is another such bundle, then the projection (X ×

EG × E′G)/G → X ×^{G} EG is a fibration with contractible fiber. In particular, MU_{G}^{∗}(X) is well-defined. Moreover, if G acts freely on X with quotient X/G,

then the map X ×^{G} EG → X/G is a fibration with contractible fiber EG and hence we get MU_{G}^{∗}(X) ≅ MU^{∗}(X/G). For a linear algebraic group G over ℂ acting on a ℂ-scheme X, let H_{G}^{∗}(X, A) denote the (equivariant) cohomology of the complex analytic space X(ℂ) with coefficients in the ring A.

Proposition 7.5. Assume that X ∈ V_{G}^{S} is such that H_{G}^{∗}(X, ℤ) is torsion-free. Then there is a natural homomorphism of graded rings

ρ_{X}^{G} : Ω_{G}^{∗}(X) → MU_{G}^{2∗}(X).

Proof. If {(V_{j}, U_{j})} is a sequence of good pairs as in Theorem 6.1, then the universality of the Levine-Morel cobordism gives a natural 𝕃-algebra map of inverse systems

Ω^{i} (X ×^{G} U_{j}) → MU^{2i} (X ×^{G} U_{j})

which after taking limits yields the map

(7.10)  Ω_{G}^{i}(X) = \underleftarrow{lim}_{j≥0} Ω^{i} (X ×^{G} U_{j}) → \underleftarrow{lim}_{j≥0} MU^{2i} (X ×^{G} U_{j}) .

On the other hand, it follows from [15, Lemma 3.2] (see also [39, Theorem 2.1]) that there is a Milnor exact sequence (7.11)

\xymatrix{0 \ar[r] & \underleftarrow{lim}_{j≥0}^{1} MU^{2i−1}(X ×^{G} U_{j}) \ar[r] & MU^{2i}(X ×^{G} EG) \ar[r] & \underleftarrow{lim}_{j≥0} MU^{2i}(X ×^{G} U_{j}) \ar[r] & 0.}

Moreover, it follows from our assumption and [26, Corollary 1] that the first term of this exact sequence vanishes. This yields the natural map ρ_{X}^{G} : Ω_{G}^{i}(X) → MU_{G}^{2i}(X). □

It follows from the proof of [26, Corollary 1] that the first term in (7.11) always vanishes if we work over the rationals. We can thus imitate the proof of Proposition 7.5 to see that there is a natural map Ω_{G}^{∗}(X)_{ℚ} → MU_{G}^{2∗}(X)_{ℚ}. Combining this with Proposition 7.2 (with rational coefficients), one concludes the following.

Corollary 7.6. For any X ∈ V_{G}^{S}, there is a natural map of graded 𝕃_{ℚ}-algebras

ρ_{X}^{G} : Ω_{G}^{∗}(X)_{ℚ} → MU_{G}^{2∗}(X)_{ℚ}.

In particular, there is a natural ring homomorphism

\overline{ρ}_{X}^{G} : CH_{G}^{∗}(X)_{ℚ} → MU_{G}^{2∗}(X)_{ℚ} ˆ⊗_{𝕃_{ℚ}}ℚ

which factors the cycle class map CH_{G}^{∗}(X) → H_{G}^{2∗}(X, ℚ).

Corollary 7.7. There is a natural morphism Ω^{∗}(BG) → MU^{2∗}(BG) of graded 𝕃-algebras. In particular, there is a natural ring homomorphism CH^{∗}(BG) → MU^{2∗}(BG)ˆ⊗_{𝕃}ℤ which factors the cycle class map CH^{∗}(BG) → H^{2∗}(BG, ℤ).

Proof. The first assertion follows immediately from (7.10), (7.11) and [26, Theorem 1] using the fact that BG is homotopy equivalent to the classifying space of its maximal compact subgroup. The second assertion follows from the first and Proposition 7.2 using the identification 𝕃 ⟶^{≅} MU^{∗}. □

Remark 7.8. The map CH^{∗}(BG) → MU^{2∗}(BG)ˆ⊗_{𝕃}ℤ has also been constructed by Totaro [39] by a different method.

We shall study the above realization maps in more detail in the next section.

8. Reduction of arbitrary groups to tori

The main result of this section is to show that with the rational coefficients, the equivariant cobordism of schemes with an action of a connected linear algebraic group can be written in terms of the Weyl group invariants of the equivariant cobordism for the action of the maximal torus. This reduces the problems about the equivariant cobordism to the case where the underlying group is a torus. We draw some consequences of this for the cycle class map from the rational Chow groups to the complex cobordism groups of classifying spaces. We first prove some reduction results about the equivariant cobordism which reflect the relations between the G-equivariant cobordism and the equivariant cobordism for actions of subgroups of G. The results of this section are used in [19] and [21] to compute the non-equivariant cobordism ring of flag varieties and flag bundles.

Proposition 8.1. Let G be a connected reductive group over k. Let B be a Borel subgroup of G containing a maximal torus T over k. Then for any X ∈ V_{G}, the restriction map

(8.1)  Ω_{∗}^{B} (X) ⟶^{r_{T,X}^{B}} Ω_{∗}^{T} (X)

is an isomorphism.

Proof. By Proposition 5.4, we only need to show that

(8.2)  Ω_{∗}^{B} (B ×^{T} X) ≅ Ω_{∗}^{B} (X) .

By [8, XXII, 5.9.5], there exists a characteristic filtration B^{u} = U_{0} ⊇ U_{1} ⊇ ⋯ ⊇ U_{n} = {1} of the unipotent radical B^{u} of B such that U_{i−1}/U_{i} is a vector group, each U_{i} is normal in B and T U_{i} = T ⋉ U_{i}. Moreover, this filtration also implies that for each i, the natural map B/T U_{i} → B/T U_{i−1} is a torsor under the vector bundle U_{i−1}/U_{i} × B/T U_{i−1} on B/T U_{i−1}. Hence, the homotopy invariance (cf. Theorem 5.2) gives an isomorphism

Ω_{∗}^{B} (B/T U_{i−1} × X) ⟶^{≅} Ω_{∗}^{B} (B/T U_{i} × X) .

Composing these isomorphisms successively for i = 1, ⋯, n, we get

Ω_{∗}^{B} (X) ⟶^{≅} Ω_{∗}^{B} (B/T × X) .

The canonical isomorphism of B-varieties B ×^{T} X ≅ B/T × X and Proposition 5.4 together now prove (8.2) and hence (8.1). □

Proposition 8.2. Let H be a possibly non-reductive group over k. Let H = L⋉H^{u} be the Levi decomposition of H (which exists since k is of characteristic zero). Then the restriction map

(8.3)  Ω_{∗}^{H} (X) ⟶^{r_{L,X}^{H}} Ω_{∗}^{L} (X)

is an isomorphism.

Proof. Since the ground field is of characteristic zero, the unipotent radical H^{u} of H is split over k. Now the proof is exactly same as the proof of Proposition 8.1, where we just have to replace B and T by H and L respectively. □

Notation: All results in the rest of this section will be proven with the rational coefficients. In order to simplify our notations, an abelian group A from now on will actually mean the ℚ-vector space A ⊗_{ℤ} ℚ, and an inverse limit of abelian groups will mean the limit of the associated ℚ-vector spaces. In particular, all cohomology groups will be considered with the rational coefficients and Ω_{i}^{G}(X) will mean

Ω_{i}^{G}(X) := \underleftarrow{lim}_{j} (Ω_{i}^{G}(X)_{j} ⊗_{ℤ} ℚ) .

Notice that this is same as Ω_{i}^{G}(X)ˆ⊗_{ℤ}ℚ in our earlier notation.

8.1. The motivic cobordism theory. Before we prove our main results of this section, we recall the theory of motivic algebraic cobordism MGL_{∗,∗} introduced by Voevodsky in [41]. This is a bi-graded ring cohomology theory in the category of smooth schemes over k. Levine has recently shown in [28] that MGL_{∗,∗} extends uniquely to a bi-graded oriented Borel-Moore homology theory MGL′_{∗,∗} on the category of all schemes over k. This homology theory has exterior products, homotopy invariance, localization exact sequence and Mayer-Vietoris among other properties (cf. [loc. cit., Section 3]). Moreover, the universality of Levine-Morel cobordism theory implies that there is a unique map

ϑ : Ω_{∗} → MGL′_{2∗,∗}

of oriented Borel-Moore homology theories. Our motivation for studying the motivic cobordism theory in this text comes from the following result of Levine.

Theorem 8.3 ([29]). For any X ∈ V_{k}, the map ϑ_{X} is an isomorphism.

We recall from [28] that for a smooth k-scheme X, there is a Hopkins-Morel spectral sequence

(8.4)  E_{2}^{p,q}(n) = CH^{n−q}(X, 2n − p − q) ⊗ 𝕃^{q} ⇒ MGL^{p+q,n}(X)

which is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence in complex cobordism. If X is possibly singular, we embed it as a closed subscheme of a smooth scheme M. Then, the functoriality of the above spectral sequence with respect to an open immersion yields a spectral sequence

E_{2}^{p,q}(n) = CH_{X}^{n−q}(M, 2n − p − q) ⊗ 𝕃^{q} ⇒ MGL_{X}^{p+q,n}(M)

of cohomology with support. Since the higher Chow groups and the motivic cobordism groups of M with support in X are canonically isomorphic to the higher Chow groups and the Borel-Moore motivic cobordism groups of X (cf. [1], [29, Section 3]), the above spectral sequence is identified with

(8.5)  E_{p,q}^{2}(n) = CH_{n}(X, p) ⊗ 𝕃^{q} ⇒ MGL′_{2n+2q−p,n+q}(X).

Now, suppose that a finite group G acts on X. By embedding X equivariantly in a smooth G-scheme M, the formula

MGL′_{p,q}(X) := Hom_{SH(k)} (Σ_{T}^{∞}M/(M − X), Σ^{p′,q′}MGL)

(where p′ = 2dim(M) − p, q′ = dim(M) − q) shows that G acts naturally on MGL′_{p,q}(X). It also acts on the higher Chow groups CH_{p}(X, q) likewise, where we just have to replace MGL in (8.5) by the Eilenberg-MacLane spectrum Hℤ. Recall furthermore that MGL and Hℤ are ring spectra and the spectral sequence (8.4) (and hence (8.5)) is obtained by first showing that there is a natural quotient morphism of ring spectra MGL → Hℤ. In particular, for any G-scheme X, (8.5) is a spectral sequence of ℤ[G]-modules. The reader will notice that the rational coefficients have not been used so far and Theorem 8.3 as well as the spectral sequence (8.5) hold integrally.

We shall now use rational coefficients everywhere and draw some consequences of Theorem 8.3 and (8.5). Since the functor of taking “G-invariants” is exact on the category of ℚ[G]-modules, the spectral sequence (8.5) over the rationals, yields the spectral sequence of G-invariants

(8.6)  E′_{p,q}^{2}(n) = (CH_{n}(X, p))^{G} ⊗ 𝕃^{q} ⇒ (MGL′_{2n+2q−p,n+q}(X))^{G} .

Recall that a connected and reductive group G over k is said to be split, if it contains a split maximal torus T over k such that G is given by a root datum relative to T. One knows that every connected and reductive group containing a split maximal torus is split (cf. [8, Chapter XXII, Proposition 2.1]). In such a case, the normalizer N of T in G and all its connected components are defined over k and the quotient N/T is the Weyl group W of the corresponding root datum. As an application of the spectral sequences (8.5) and (8.6), we get the following.

Lemma 8.4. Let G be a connected reductive group with split maximal torus T and the associated Weyl group W. Let G act freely on a scheme X. Then, with rational coefficients, the pull-back map Ω_{∗}(X/G) → (Ω_{∗}(X/T))^{W} (up to a shift) is an isomorphism.

Proof. It follows from [17, Corollary 3.9] (see also [22, Corollary 8.9]) that in the case under consideration, the natural map CH_{∗}(X/G, p) → (CH_{∗}(X/T, p))^{W} is an isomorphism for all p ≥ 0. We can thus apply the spectral sequences (8.5) and (8.6) to conclude that the map MGL′_{∗,∗}(X/G) → (MGL′_{∗,∗}(X/T))^{W} is an isomorphism. The lemma now follows from this isomorphism and Theorem 8.3.

□

Remark 8.5. The proof of Lemma 8.4 is based on the existence of the Atiyah-Hirzebruch spectral sequence in the motivic cobordism. There is no published proof of the existence of this spectral sequence, though it has been presented by the authors during various seminars. We can give another proof of the above lemma without using the spectral sequence as follows. It was proven by Levine and Morel in [31, Theorems 4.1.28, 4.5.1] that there is a morphism of oriented Borel-Moore homology theories Ω_{∗} → CH_{∗}[t]^{(t)} which is an isomorphism with rational coefficients. Here, CH_{∗}[t] is the polynomial module CH_{∗}[t_{1}, t_{2}, ⋯] in infinitely many variables with deg(t_{i}) = i. Recall also that CH_{∗}[t]^{(t)} is same as CH_{∗}[t] as CH_{∗}(k)-module. Applying the above isomorphism to X/T and X/G and using the isomorphism CH_{∗}(X/G) ⟶^{≅} (CH_{∗}(X/T))^{W} (cf. [10, Proposition 9]), we immediately get that the map Ω_{∗}(X/G) → (Ω_{∗}(X/T))^{W} is an isomorphism.

Recall from Remark 4.6 that if a connected reductive group G acts on a scheme X, then the Weyl group W acts on Ω_{∗}^{T} (X) where T is a maximal torus of G.

Theorem 8.6. ^{1} Let G be a connected linear algebraic group and let L be a Levi subgroup of G with a split maximal torus T. Let W denote the Weyl group of L with respect to T. Then for any X ∈ V_{G}, the natural map

(8.7)  Ω_{∗}^{G}(X) → (Ω_{∗}^{T} (X))^{W}

is an isomorphism with rational coefficients.

Proof. By Proposition 8.2, we can assume that G = L and hence G is a connected reductive group with split maximal torus T. We choose a sequence of l_{j}-dimensional good pairs {(V_{j}, U_{j})} as in Theorem 6.1 for the G-action. Then, this is also a sequence of good pairs for the action of

T. Setting X_{H}^{j} = {X ×^{H} U_{j}} for any closed subgroup H ⊆ G, we see that {X_{T}^{j} } is a sequence of W -schemes, each term of which has a free W -action. It follows from Lemma 8.4 (or Remark 8.5) that the smooth pull-back map

(8.8)  Ω_{i+l_{j}−g} (X_{G}^{j}) → (Ω_{i+l_{j}−n} (X_{T}^{j} ))^{W}

is an isomorphism, where dim(G) = g and dim(T) = n. Since the action of W on the inverse system {Ω_{i+l_{j}−n} (X_{T}^{j} )}_{j} induces the

similar action on the inverse limit and since the inverse limit commutes with taking the W -invariants, we get

(8.9)  \underleftarrow{lim}_{j} Ω_{i+l_{j}−g} (X_{G}^{j}) ⟶^{≅} (\underleftarrow{lim}_{j} Ω_{i+l_{j}−n} (X_{T}^{j} ))^{W} .

Since the left and the right terms are same as Ω_{i}^{G}(X) and (Ω_{i}^{T} (X))^{W} respectively by choice of our good pairs and Theorem 6.1, we conclude that Ω_{i}^{G}(X) ⟶^{≅} (Ω_{i}^{T} (X))^{W}. This completes the proof of the theorem. □

Corollary 8.7. Let X ∈ V_{G} be as in Theorem 8.6. Then the restriction map

(8.10)  Ω_{∗}^{G} (X)_{ℚ} ⟶^{r_{T,X}^{G}} Ω_{∗}^{T} (X)_{ℚ}

is a split monomorphism which is natural for the morphisms in V_{G}. In particular, if H is any closed subgroup of G, then there is a split injective map

(8.11)  Ω_{∗}^{H} (X)_{ℚ} ⟶^{r_{T,X}^{G}} Ω_{∗}^{T} (G ×^{H} X)_{ℚ}.

Proof. The first statement follows directly from Theorem 8.6, where the splitting is given by the trace map. The second statement follows from the first and Proposition 5.4. □

1It has been shown recently in [15, Proposition 4.8] that the map S(G) → S(T)^{W} is an isomorphism over ℤ[t_{G}^{−1}], where t_{G} is the torsion index of G.

Before we apply Theorem 8.6 to study the rational cobordism rings of classifying spaces, we need the following topological analogue, which is much simpler to prove. Recall from (7.9) that if G is a complex Lie group and X is a finite CW - complex with a G-action, then its equivariant complex cobordism is defined as

(8.12)  MU_{G}^{∗}(X) := MU^{∗} (X ×^{G} EG) .

Theorem 8.8. Let G be a complex Lie group with a maximal torus T and Weyl group W. Then for any X as above, the natural map

(8.13)  MU_{G}^{∗}(X) → (MU_{T}^{∗}(X))^{W}

is an isomorphism with rational coefficients.

Proof. As in the proof of Theorem 8.6, we can reduce to the case when G is reductive. It follows from the above definition of the equivariant complex cobordism and the similar definition of the equivariant singular cohomology of X, plus the Atiyah-Hirzebruch spectral sequence in topology that there is a spectral sequence

(8.14)  E_{2}^{p,q} = H_{G}^{p}(X, ℚ) ⊗_{ℚ} MU^{q} ⇒ MU_{G}^{p+q}(X).

Since the Atiyah-Hirzebruch spectral sequence degenerates rationally, we see that the above spectral sequence degenerates too. Since one knows that H_{G}^{∗}(X) ≅ (H_{T}^{∗} (X))^{W} (cf. [4, Proposition 1]), the corresponding result for the cobordism follows. □

Theorem 8.9. For a connected linear algebraic group G over ℂ, the degree doubling map ρ^{G} : Ω^{∗}(BG) → MU^{∗}(BG) (cf. Corollary 7.7) of 𝕃-algebras, is an isomorphism with rational coefficients. In particular, the natural map of ℚ-algebras

CH^{∗}(BG)_{ℚ} ⟶^{\overline{ρ}^{G}} MU^{∗}(BG)ˆ⊗_{𝕃}ℚ

is an isomorphism.

Proof. To prove the first isomorphism, we can use Theorems 8.6 and 8.8 to reduce to the case of a torus. But this case is already known even with the integer coefficients (cf. (6.14) and [39]). The second isomorphism follows from the first and Proposition 7.2. □

Remark 8.10. The map \overline{ρ}^{G} : CH^{∗}(BG) → MU^{∗}(BG)ˆ⊗_{𝕃}ℤ was found by Totaro in [39] even before Levine and Morel discovered their algebraic cobordism. It was conjectured that the map \overline{ρ}^{G} should be an isomorphism with the integer coefficients for a connected complex algebraic group G. Totaro modified this conjecture to an expectation that \overline{ρ}^{G} should be an isomorphism after localization at a prime p such that MU^{∗}(BG)_{(p)} is concentrated in even degree. The above theorem proves the isomorphism in general with the rational coefficients. We also remark that the map MU^{∗}(BG)ˆ⊗_{𝕃}ℚ → H^{∗}(BG, ℚ) is an isomorphism (cf. [38]). The above result then shows that the cycle class map for the classifying space is an isomorphism with the rational coefficients. One wonders if the techniques of this paper could be applied to the algebraic version of the Brown-Peterson cobordism theory to prove the Totaro’s modified conjecture. We do not know the answer yet.

Acknowledgments. The author would like to thank the referee for his/her comments and suggestions, which were immensely helpful in improving the contents and the presentation of this paper.

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Amalendu Krishna School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Colaba Mumbai India amal@math.tifr.res.in