EQUIVARIANT PRETHEORIES AND INVARIANTS OF TORSORS

STEFAN GILLE AND KIRILL ZAINOULLINE

Abstract. In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev’s (equivariant Ktheory) spectral sequence. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the J-invariant of E and in the case of Grothendieck’s K_{0} – indexes of the respective Tits algebras.

1. Introduction

In the present paper we introduce and study the notion of a (graded) equivariant pretheory. Roughly speaking, it is defined to be a contravariant functor from the category of G-varieties, where G is an algebraic group, to (graded) abelian groups which satisfies localization and homotopy invariance properties. All known examples of equivariant oriented cohomology theories (equivariant Chow groups, K-theory, algebraic cobordism, etc.) are pretheories in our sense.

We generalize the equivariant Chow groups of Edidin-Graham by introducing equivariant (co)homology theory with coefficients in a Rost cycle module. We also prove a version of Merkurjev’s equivariant K-theory spectral sequence for equivariant cycle homology. This provides many new examples of equivariant pretheories.

One of the key results of Karpenko-Merkurjev [16, Thm. 6.4] tells us that the characteristic subring of the Chow ring of a variety of Borel subgroups of a split linear algebraic group G is contained in the image of the restriction map, i.e. always consists of rational cycles. This fact plays a fundamental role in computations of canonical/essential dimensions, discrete motivic invariants of G and in the study of splitting properties of G-torsors.

In the present paper we generalize this result to an arbitrary equivariant pretheory (see Theorem 4.4). In particular, we obtain versions of [16, Thm. 6.4] for Grothendieck’s K_{0} and algebraic cobordism Ω of Levine-Morel.

As an application we define for any equivariant pretheory h and G-torsor E a commutative ring ĥ_{B}(E) (see Def. 4.5). If E is generic and h is either the Chow ring CH^{∗} or Grothendieck’s K_{0} or algebraic cobordism Ω, this ring coincides with the cohomology ring h(G) of G. In general, it is always a quotient of h(G) which in the case of the Chow ring is related to the motivic J-invariant of E and in the case of K_{0} – to the indexes of the Tits algebras of E. This provides a fascinating link between these two discrete invariants, totally unrelated at the first sight: one observes that the p-exceptional degrees of V. Kac for Chow groups [15] play the same role as the maximal Tits indexes for K_{0} [22].

2000 Mathematics Subject Classification. Primary 20G15; Secondary 19L47, 14F43.

Key words and phrases. Torsor, Equivariant cohomology, Rost cycle module.

The paper is organized as follows: In the first two sections we introduce the notion of an equivariant pretheory and provide several examples including equivariant cycle (co)homology. In Section 4 we generalize the result of Karpenko-Merkurjev to an arbitrary equivariant pretheory. In the last section we provide applications to equivariant oriented cohomology theories (Chow groups, Grothendieck’s K_{0} and algebraic cobordism of Levine-Morel). Appendix is devoted to the construction of a spectral sequence for cycle homology which generalizes the long exact localization sequence.

1.1 (Notations). Unless otherwise indicated, all schemes/varieties are defined over the base field k. By a scheme over a field k (k-scheme) we mean a reduced separated Noetherian scheme over k. By a variety over a field k (k-variety) we mean a quasiprojective scheme over k (note that it has to be of finite type over k). If l/k is a field extension and X is a k-scheme, we define X_{l} = X ×_{Spec k} Spec l to be the respective base change. By pt we denote Spec k.

By an algebraic group we mean an affine smooth group scheme over k. By a subgroup we always understand a closed algebraic subgroup. By an action of an algebraic group G on a scheme X we mean a morphism G×_{Spec k} X → X of schemes over k (all group actions are assumed to be on the left), subject to the usual axioms, see [23, Def. 0.3]. By a G-scheme we mean a scheme X endowed with an action of an algebraic group G.

We denote by G-Sm_{k} the category of smooth G-varieties over k with equivariant G-morphisms. A localization of a smooth variety over k is called essentially smooth. We denote by G-Ess_{k} the category of essentially smooth G-schemes over k with Gequivariant flat morphisms. We denote by 𝔄𝔟 the category of abelian groups.

2. Equivariant pretheories.

In the present section we introduce the notion of a (graded) equivariant pretheory and provide several examples.

Let G be an algebraic group over a field k. Consider a contravariant functor from the category of smooth G-varieties over k to the category of abelian groups

h_{G}: G-Sm_{k} ⟶ 𝔄𝔟, X ↦ h_{G}(X).

Given X, Y ∈ G-Ess_{k} and a G-equivariant map f: X → Y the induced functorial map h_{G}(Y) → h_{G}(X) is called a pull-back and is denoted by f_{G}^{∗}.

2.1. Definition. The functor h_{G}: G-Sm_{k} → 𝔄𝔟 is called a G-equivariant pretheory over k if it satisfies the following two axioms:

H. (homotopy invariance) For a G-equivariant map p: 𝔸_{k}^{n} → pt (where G acts trivially on pt) the induced pull-back

p_{G}^{∗}: h_{G}(pt) ⟶ h_{G}(𝔸_{k}^{n})

is an isomorphism.

L. (localization) For a smooth G-variety X and a G-equivariant open embedding ι: U ↪ X the induced pull-back

ι_{G}^{∗}: h_{G}(X) ⟶ h_{G}(U)

is surjective.

Let U be a G-scheme over k such that U is the localization of a smooth irreducible G-variety X with respect to G-equivariant open embeddings f_{ij}: U_{j} → U_{i}, U_{i} ⊂ X, i.e. U = \underleftarrow{lim}_{f_{ij}} U_{i}. Observe that U is essentially smooth over k.

Let h̄_{G}(U) denote the induced colimit \underrightarrow{lim}_{(f_{ij})_{G}^{∗}} h_{G}(U_{i}). Note that the canonical maps h_{G}(U_{i}) → h̄_{G}(U) are surjective by the localization property (L).

2.2. Definition. We call h_{G} an essential G-equivariant pretheory if h_{G} can be extended to the category G-Ess_{k} of essentially smooth G-schemes over k with Gequivariant flat morphisms, i.e.

h_{G}: G-Ess_{k} ⟶ 𝔄𝔟,

such that the following additional axiom holds:

C. Given U as above, the map induced by flat pull-backs h_{G}(U_{i}) → h_{G}(U)

h̄_{G}(U) → h_{G}(U)

is surjectve. Note that (C) holds if and only if the induced pull-back h_{G}(U_{i}) → h_{G}(U) is

surjective for some i.

2.3. Example (Equivariant K-theory). We recall definitions and basic properties of equivariant K-groups as defined by Thomason [27], see also the survey article [21] of Merkurjev.

Let G be an algebraic group over k and let X be a smooth G-variety. Then the category P(G, X) of locally free G-modules on X (in the sense of Mumford [23, I, §3]) is an exact category. Following Thomason [27] one defines the i-th Gequivariant K-group K_{i}(G, X) as Quillen’s i-th K-group of the exact category P(G, X).

Let h_{G}(X) = K_{0}(G, X). Then according to [21, Thm. 2,7 and Lem. 4.1] it satisfies localization and homotopy invariance, and by [9, 52.F] it satisfies (C). Hence, it provides an example of an essential G-equivariant pretheory.

2.4. Example (Equivariant cobordism). This theory has been recently defined by Heller and Malagón-López [14].

Assume that char(k) = 0. Consider the ring Ω_{∗}(X) of algebraic cobordism of a smooth k-variety X as defined by Levine and Morel [19]. Since Ω_{i}(X) does not vanish for i big enough (as Chow groups do) one can not copy word by word the definition of equivariant Chow groups given by Edidin and Graham [8], see also Section 3.

Instead Heller and Malagón-López consider [14] (what they call) good systems of representations. These are families of pairs (V_{i}, U_{i})_{i∈ℕ} of vector spaces with U_{i} ⊆ V_{i} endowed with an action of an algebraic group G such that

(i) G acts freely on U_{i} and U_{i} → U_{i}/G is a G-torsor,

(ii) V_{i+1} = V_{i} ⊕ W_{i} for some k-subspace W_{i}, such that U_{i} ⊕ W_{i} ⊆ U_{i+1},

(iii) sup dim V_{i} = ∞, and (iv) codim_{V_{i}}(V_{i} \U_{i}) < codim_{V_{i+1}}(V_{i+1} \U_{i+1}), where we consider V_{i} as an affine

space over k. Observe that assumption (i) ensures that the quotient X ×^{G} U := (X ×_{k} U)/G is a quasi-projective variety over k (see [8, Prop. 23]). Moreover, it is smooth over k by the descent, since X ×_{k} U → X ×^{G} U is faithfully flat.

Let G be connected. Then the n-th equivariant cobordism group of a smooth G-variety X is defined by

Ω_{n}^{G}(X) := \underleftarrow{lim}_{i} Ω_{n−dim G+dim U_{i}}(X ×^{G} U_{i}) .

This is well defined, see [14, Cor. 3.4], and the functor

h_{G}: X ⟼ ⨁_{n∈ℤ} Ω_{n}^{G}(X)

satisfies the localization and homotopy invariance axioms by [loc.cit. Thm. 4.2 and Cor. 4.6]. Hence, it provides an example of a G-equivariant pretheory.

A further example is the equivariant Chow-theory of Edidin and Graham [8]. We consider this later (see Example 3.16) when we take a closer look at equivariant cycle (co)homology.

There is also a graded version of a G-equivariant pretheory

2.5. Definition. A pair of varieties (X, U) is called a G-pair if X ∈ G-Sm_{k} and U ⊆ X is a G-equivariant open subvariety. Consider the category of G-pairs over k with G-equivarant morphisms of pairs.

A contravariant functor

(X, U) ⟼ h_{G}^{∗}(X, U)

from the category of G-pairs to graded abelian groups is called a graded G-equivariant pretheory if it satisfies (H) homotopy invariance, and for any G-pair (X, U) there is a long exact localization sequence

\xymatrix{… \ar[r] & h_{G}^{i}(X) \ar[r]^{ι_{G}^{∗}} & h_{G}^{i}(U) \ar[r]^{∂} & h_{G}^{i+1}(X, U) \ar[r] & …,}

where ι: U ↪ X is the corresponding G-equivariant open embedding, and we have set h_{G}^{∗}(Y) := h_{G}^{∗}(Y, Y). It is called a graded essential G-equivariant pretheory if given an inverse limit (X , U) = \underleftarrow{lim}_{i}(X_{i}, U_{i}) of G-equivariant open embeddings of pairs, there is the induced surjection

\underrightarrow{lim}_{i} h_{G}^{∗}(X_{i}, U_{i}) ⟶ h_{G}^{∗}(X , U).

3. Equivariant cycle (co)homology

In this section we generalize the equivariant Chow groups of Edidin and Graham [8]. This theory has been considered for the cycle module Galois cohomology by Guillot [13]. We will use freely Rost’s [26] theory of cycle modules for which we refer also to the book [9] of Elman, Karpenko and Merkurjev, as well as to the article of Déglise [5] where several important properties of the generalized “intersection” product in cycle cohomology are proven (defined in [26, Sect. 14]).

Since Rost’s theory for algebraic spaces is not yet developed we have to restrict ourseives to quasi-projective schemes, i.e. to varieties. This assumption guarantees that certain quotients by groups actions which we consider here do exist.

3.1 (Equivariant cycle homology). To fix notations we recall briefly the definition of cycle homology. A cycle module over the field k is a (covariant) functor M_{∗} from the category of field extensions of k to the category of graded abelian groups subject to several axioms, see [26, Sects. 1,2]. The prototype of such a functor is Milnor K-theory K_{∗}^{M}, and by the very definition M_{∗}(E) = ⨁_{i∈ℤ} M_{i}(E) is a graded

K_{∗}^{M}(E)-module for all field extensions E ⊇ k.

Given a k-variety X (not necessarily smooth) and a cycle module M_{∗} over k Rost [26] has defined a complex, the so called cycle complex (generalizing a construction of Kato [17] for Milnor K-theory):

\xymatrix{… \ar[r] & ⨁_{x∈X_{(2)}} M_{n+2}(k(x)) \ar[r]^{d_{2}} & ⨁_{x∈X_{(1)}} M_{n+1}(k(x)) \ar[r]^{d_{1}} & ⨁_{x∈X_{(0)}} M_{n}(k(x)),}

where X_{(i)} ⊆ X denotes the set of points of dimension i in X. We denote this complex C_{•}(X, M_{n}) and consider it as a homological complex with the direct sum ⨁_{x∈X_{(i)}} M_{n+i}(k(x)) in degree i.

The i-th cycle homology group H_{i}(X, M_{n}) of M_{n} over X is then defined as H_{i}(C_{•}(X, M_{n})). Note that there is a natural isomorphism H_{i}(X, K_{−i}^{M}) ≃ CH_{i}(X) for all i ≥ 0, where we have set K_{−i}^{M} ≡ 0 for i < 0.

To introduce the equivariant cycle homology we adapt the definition of equivariant Chow groups due to Edidin and Graham [8], see also Guillot [13] and Totaro [28].

Let G be an algebraic group over k of dimension s and X a G-variety. To define the i-th cycle homology group with coefficients in the cycle module M_{∗} we chose a linear representation V of G, such that there is an open subscheme U ↪ V with codim_{V} (V \ U) ≥ c = dim X on which G acts freely. By shrinking U we can moreover assume that U ⟶ U/G is a principal bundle. The later assumption assures that X ×^{G} U := (X ×_{k} U)/G exists in the category of k-varieties, see [8, Prop. 23] (recall that we assume that X is quasi-projective, see 1.1). We call the pair (U, V) an (X, G)-admissible pair for the G-variety X. Note that for a finite number of G-varieties there always exist a pair (U, V) which is admissible for all of them.

3.2. Definition. Let G be an algebraic group over k. Then the i-th G-equivariant cycle homology group with values in the cycle module M_{∗} over k is defined as

H_{i}^{G}(X, M_{∗}) := H_{i+l−s}(X ×^{G} U, M_{∗−(l−s)}) ,

where s = dim G and (U, V) is a (X, G)-admissible pair with dim V = l and codim_{V} V \ U ≥ dim X.

It remains to check that this definition does not depend on the choice of the (X, G)-admissible pair (U, V). Since H_{j}(Y, M_{∗}) = 0 for any k-variety Y if j > dim Y this can be proven as for equivariant Chow groups using (the so called) Bogomolov’s double filtration argument, see [8] or [28]. We recall briefly the details:

Let U_{1} ⊂ V_{1} be another (X, G)-admissible pair with l_{1} = dim V_{1}. Then there exists an open subvariety W of V_{1} ⊕ V, which contains U_{1} ⊕ V and V_{1} ⊕ U as open subvarieties, and such that G acts on W with principal bundle quotient W/G. Then the quotient X ×^{G} W exists in the category of varieties.

We use the following fact from [8, Prop. 2 and Lem. 1], which is not hard to verify if U ⟶ U/G is a trivial G-torsor and follows in general by descent from the “trivial” case.

3.3. Lemma. Let G, U, V and X be as above and f: X → Y be a G-morphism. Denote f ×^{G} id_{U}: X ×^{G} U → Y ×^{G} U the induced morphism.

Consider the following properties P of f: vector bundle, flat, smooth, proper, regular immersion, open or closed immersion. Then if f as property P implies that also f ×^{G} id_{U} has property P.

3.4. We also need the following fact. Let

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

be a cartesian square of G-varieties and (U, V) a pair which is admissible for all varieties in the square. Then there is a natural and unique morphism

X′ ×^{G} U ⟶ X ×^{G} U ×_{Y ×^{G}U} Y′ ×^{G} U ,

which is an isomorphism if U ⟶ U/G is a trivial G-torsor. Hence by descent, see e.g. [10, Thm. 2.55 and Lem. 4.44], it is an isomorphism in general.

The morphism

id_{X} ×^{G}p_{U}: X ×^{G} (V_{1} ⊕ U) ⟶ X ×^{G} U ,

where p_{U}: V_{1} ⊕ U ⟶ U is the projection, and the inclusion

id_{X} ×^{G}ι_{U}: X ×^{G} (V_{1} ⊕ U) ⟶ X ×^{G} W ,

where ι_{U}: W ↪ V_{1} ⊕ U, induce homomorphisms

\xymatrix{H_{i+l_{1}+l−s}(X ×^{G} W, M_{n−l_{1}−l+s}) \ar@{-->}[dr]^{(id_{X} ×^{G}ι_{U})^{∗}} \\  & _{H_{i+l_{1}+l−s}(X ×^{G} (V_{1} ⊕ U), M_{n−l_{1}−l+s}) .} \\ H_{i+l−s}(X ×^{G} U, M_{n−l+s}) \ar@{-->}[ur]_{(id_{X} ×^{G}p_{U})^{∗}}}

Both maps are isomorphisms. The first since the dimension of the closed complement of X ×^{G} (V_{1} ⊕ U) in X ×^{G} W is smaller than i + l_{1} + l − s, and the second by homotopy invariance (recall that id_{X} ×^{G}p_{U} is a vector bundle by Lemma 3.3). Similarly we have an isomorphism

H_{i+l_{1}+l−s}(X ×^{G} W, M_{n−l_{1}−l+s}) ⟶^{≃} H_{i+l_{1}−s}(X ×^{G} U_{1}, M_{n−l_{1}+s}) ,

and hence both pairs define natural isomorphic groups.

3.5 (Pull-backs and push-forwards). Let now f: X ⟶ Y be a morphism of Gvarieties, and (U, V) and (U_{1}, V_{1}) two (X, G)- and (Y, G)-admissible pairs as above. We set d = dim X − dim Y. Then we have by the functorial properties of pushforward and pull-back maps in cycle homology (using Lemma 3.3 to see that the maps in question are defined) two commutative diagrams:

(i) If f is flat of constant relative dimension or Y is a smooth k-variety then

[figure: H_{∗+l−s}(Y ×^{G} U, M_{n−l+s}); (f×^{G}id_{U})^{∗}; ≃; H_{∗+d+l−s}(X ×^{G} U, M_{n−l−d+s}); ≃; H_{∗+l_{1}−s}(Y ×^{G} U_{1}, M_{n−l+s}); (f×^{G}id_{U_{1}} )^{∗}; H_{∗+d+l_{1−s}}(X ×^{G} U_{1}, M_{n−l_{1}−d+s})]

is a commutative diagram whose column arrows are natural isomorphisms, and (ii) If f is proper there is another commutative diagram

[figure: H_{∗+l−s}(X ×^{G} U, M_{n−l+s}); (f×^{G}id_{U})_{∗}; ≃; H_{∗+l−s}(X ×^{G} U, M_{n−l+s}); ≃; H_{∗+l_{1}−s}(X ×^{G} U_{1}, M_{n−l_{1}+s}); (f×^{G}id_{U_{1}} )_{∗}; H_{∗+l_{1}−s}(X ×^{G} U_{1}, M_{n−l_{1}+s}) ,]

whose column arrows are again natural isomorphisms.

Let now f: X → Y be a morphism of G-varieties and (U, V) a (G, X)- and (G, Y)-admissible pair.

(i) If either f is flat of constant relative dimension or Y is smooth, we define the pull-back morphism

f_{G}^{∗}: H_{i}^{G}(Y, M_{n}) ⟶ H_{i+d}^{G}(X, M_{n+d})

as

(f ×^{G} id_{U})^{∗}: H_{i+l−s}(Y ×^{G} U, M_{n−l+s}) ⟶ H_{i+l−s+d}(X, M_{n+l−s−d}) ,

where d = dim X − dim Y.

(ii) If f is proper, we define the push-forward morphism

f_{G∗}: H_{i}^{G}(X M_{n}) ⟶ H_{i}^{G}(Y, M_{n})

as

(f ×^{G} id_{U})_{∗}: H_{i}(X, M_{n}) ⟶ H_{i}(Y, M_{n}) .

With these definitions G-equivariant cycle homology is a contravariant functor on the category of smooth and G-varieties and also covariant for proper morphisms. Moreover, if

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

is a cartesian square with f proper and g flat of constant relative dimension or all varieties in the diagram are smooth over k then we have

f′_{G∗} ∘ g_{G}^{∗} = g_{G}^{′∗} ∘ f_{G∗} .

3.6 (Axioms). There is the localization sequence

\xymatrix{… \ar[r]^{ι_{G∗}} & H_{i}^{G}(X, M_{n}) \ar[r]^{j_{G}^{∗}} & H_{i}^{G}(X \ Z, M_{n}) \ar[r]^{∂} & H_{i−1}^{G}(Z, M_{n}) \ar[r]^{ι_{G∗}} & …}

for a closed G-embedding ι: Z ↪ X with open G-equivariant complement j: X \ Z ↪ X which follows from the localization sequence in ordinary cycle homology. And if π: E ⟶ X is a G-vector bundle of rank r, i.e. a G-linear bundle, then

π ×^{G} id_{U}: E ×^{G} U ⟶ X ×^{G} U

is a vector bundle, see [8, Lem. 1], and therefore the pull-back

π^{∗}: H_{i}^{G}(X, M_{n}) ⟶ H_{i+r}^{G}(E, M_{n−r})

is an isomorphism by homotopy invariance of cycle homology. Finally, H_{i}^{G}(X, M_{n}) can be extended to G-Ess_{k} and satisfies (C) by [9, 52.F] or [26, p.320].

3.7. Example. If f: X → Y = X/G is a G-torsor then we have a natural isomorphism

H_{i−s}(Y, M_{n+s}) ≃ H_{i}^{G}(X, M_{n})

where s = dim X − dim Y = dim G, for all i ∈ ℕ and n ∈ ℤ. This can be seen as follows, cf. [13, Expl. 2.3.2]:

We choose a (X, G)-admissible pair (U, V). Let l = dim V. Then the closed complement of X ×^{G} U = (X ×_{k} U)/G in (X ×_{k} V)/G has dimension less than i + l − s and therefore we have by the localization sequence an isomorphism

H_{i+l−s}((X ×_{k} V)/G, M_{n−(l−s)}) ⟶^{j^{∗}} H_{i+l−s}(X ×^{G} U, M_{n−(l−s)}) = H_{i}^{G}(X, M_{n}) ,

where j: (X ×_{k} U)/G ↪ (X ×_{k} V)/G is the corresponding open immersion (note that the target of j exists in the category of varieties since by assumption X → X/G is a G-torsor). By [8, Lem. 1] we know that (X ×_{k} V)/G → X/G is a vector bundle of rank l and so by homotopy invariance we have

H_{i+l−s}((X ×_{k} V)/G, M_{n−(l−s)}) ≃ H_{i−s}(X/G, M_{n+s}) .

3.8 (Restriction map). If G_{1} ⊆ G is a closed subgroup and X a G-variety over k, we can choose a (X, G)- and (X, G_{1})-admissible pair (U, V). Then we have a morphism of k-varieties (X ×_{k} U)/G_{1} → (X ×_{k} U)/G which induces (via pull-back) a homomorphism of equivariant cycle homology groups

res_{G_{1}}^{G}: H_{i}^{G}(X, M_{n}) ⟶ H_{i}^{G_{1}}(X, M_{n})

for any cycle module M_{∗} called restriction homomorphism. In particular if G_{1} is the trivial group we have a (forgetful) morphism H_{i}^{G}(X, M_{n}) ⟶ H_{i}(X, M_{n}) from G-equivariant cycle homology to ordinary cycle homology.

3.9 (The first Chern class). Let π: L → X be a G-equivariant line bundle with zero section σ: X → L. Then σ is a G-equivariant closed embedding and so induces a morphism σ ×^{G} id_{U}: X ×^{G} U → L ×^{G} U which is the zero section of the line bundle (see Lemma 3.3)

π ×^{G} id_{U}: L ×^{G} U ⟶ X ×^{G} U .

The first Chern class (or also called Euler class) of L is then defined as the operator

c_{1}(L)^{G} := (π_{G}^{∗})^{−1} ∘ σ_{G∗}: H_{∗}^{G}(X, M_{∗}) ⟶ H_{∗−1}^{G}(X, M_{∗+1}) .

This map commutes with push-forwards and pull-backs. More precisely, assume we have a cartesian square of G-equivariant morphisms, where L and L′ are Gequivariant line bundles over X and X′, respectively:

\xymatrix{L′ \ar[r]^{f′} \ar[d]_{π′} & L \ar[d]^{π} \\ X′ \ar[r]^{f} & X .}

Denote by σ and σ′ the zero sections of L and L′, respectively. These are also G-equivariant morphisms and so we get a cartesian square

\xymatrix{L′_{G} \ar[r]^{f′_{G}} \ar[d]^{π′_{G}} & L_{G} \ar[d]^{π_{G}} \\ X′_{G} \ar[r]^{f_{G}} \ar[u]^{σ′_{G}} & X_{G} , \ar[u]^{σ_{G}}}

where we have set Y_{G} := Y ×^{G} U for any G-variety Y, g_{G} := g ×^{G} id_{U} for all Gequivariant morphisms g: Y′ ⟶ Y, and assumed that the pair (U, V) is admissible for all varieties in question. A straightforward computation using this diagram, see [9, Prop. 53.3] for the analogous result in ordinary cycle homology, shows

c_{1}^{G}(L′) ∘ f_{G}^{∗} = f_{G}^{∗} ∘ c_{1}^{G}(L) if f is flat, and
c_{1}^{G}(L) ∘ f_{G∗} = f_{G∗} ∘ c_{1}^{G}(L′) if f is proper.

3.10. Remark. (i) The Chern class homomorphism

c_{1}^{G}(L): H_{i}^{G}(X, K_{−i}^{M}) = CH_{i}^{G}(X) ⟶ CH_{i−1}^{G}(X) = H_{i−1}^{G}(X, K_{−i+1}^{M})

coincides with the first Chern class defined in Edidin and Graham [8, Sect. 2.4].

(ii) As for ordinary cycle homology in [9] one can use the Euler class of a vector bundle to prove the projective bundle theorem and then use this to define the higher Chern classes in G-equivariant cycle homology. We leave this to the interested reader.

3.11 (An equivariant spectral sequence). We provide now a version of Merkurjev’s [21] equivariant K-theory spectral sequence for equivariant cycle homology.

Let T be a k-split torus of rank m, and χ: T → 𝔾_{m} a character. The algebraic group T acts via χ on the affine line 𝔸_{k}^{1} defining a T -equivariant line bundle 𝔸_{k}^{1} → pt which we denote by L(χ) (trivial action of T on the base point pt). If p: X → pt is a T -scheme we denote the pull-back p^{∗}L(χ) by L_{X}(χ). This is also a T -equivariant vector bundle.

Let X be a G-variety, where G is an algebraic group over k, and T ⊆ G a split torus of rank m. Let χ_{1}, …, χ_{m} be a basis of the character group T^{∗} = Hom(T, 𝔾_{m}), and let T act on the affine space 𝔸_{k}^{m} = Spec k[x_{1}, …, x_{m}] by

t · (a_{1}, …, a_{m}) ⟼ (χ_{1}(t) · a_{1}, …, χ_{m}(t) · a_{m}) ,  (1)

and on 𝔸_{X}^{m} = X ×_{k} 𝔸_{k}^{m} diagonally. Let Z_{i} ⊂ 𝔸_{k}^{m} be the hyperplan defined by x_{i} = 0 for i = 1, …, m. Then X ×_{k} Z_{i} are T -subvarieties of 𝔸_{X}^{m} and therefore, see Lemma 3.3, we have closed subschemes

(X ×_{k} Z_{1}) ×^{G} U , …, (X ×_{k} Z_{m}) ×^{G} U

of 𝔸_{X} ×^{G} U, where (U, V) is a (𝔸_{X}^{m}, T)-admissible pair. Since U → U/T is a T -torsor (by assumption) we have

⋂_{j∉I}(X ×_{k} Z_{j}) ×^{G} U = (X ×_{k} Z_{I}) ×^{G} U

for all I ∈ {1, …, m}, where Z_{I} = ⋂_{j∉I} Z_{j}. From Example A.7 we get then a

convergent spectral sequence

Ẽ_{1}^{p,q} = ⨁_{|I|=p} H_{−q−m}^{T}(X ×_{k} Z_{I}, M_{n}) ⟹ H_{−p−q}^{T}(X ×_{k} T, M_{n})  (2)

for all cycle modules M_{∗}.

By Example 3.7 we have H_{−p−q}^{T}(X ×_{k} T, M_{n}) ≃ H_{−p−q−m}(X, M_{n+m}) and since

Z_{I} = 𝔸_{k}^{|I|} = Spec k[x_{i}, i ∈ I] ↪ 𝔸_{k}^{m}

is a T -equivariant vector bundle over the base the pull-back

π_{I T}^{∗} : H_{−q−m−|I|}^{T}(X, M_{n+|I|}) ⟶ H_{−q−m}^{T}(X ×_{k} Z_{I}, M_{n})

is an isomorphism, where π_{I}: X ×_{k} Z_{I} → X is the projection. Replacing q by q +m the spectral sequence takes therefore the following form

E_{1}^{p,q} = ⨁_{|I|=p} H_{−q−p}^{T}(X, M_{n+p}) ⟹ H_{−p−q}(X, M_{n+m}) .  (3)

We compute the differential

d_{1}^{p,q}: ⨁_{|I|=p} H_{−q−p}^{T}(X, M_{n+p}) ⟶ ⨁_{|J|=p+1} H_{−q−p−1}^{T}(X, M_{n+p+1}) .

If J ⊉ I the the IJ-component of d_{1}^{p,q} is zero. If J ⊃ I let J = {i_{1}, …, i_{p+1}} and I = J \{i_{r}} for some 1 ≤ r ≤ p+1. Then by Example A.7 the IJ-component of the differential d̃_{1}^{p,q} of the spectral sequence (2) is equal (−1)^{r−1}-times the push-forward along the closed embedding (see Lemma 3.3)

ι_{IJ} ×^{T} id_{U}: (X ×_{k} Z_{I}) ×^{T} U ↪ (X ×_{k} Z_{J}) ×^{T} U ,

where ι_{IJ} is the closed immersion X ×_{k} Z_{I} ↪ X ×_{k} Z_{J}. Using the above identification we have then a commutative diagram

\xymatrix{E_{1}^{p,q} = & H_{−q−p}^{T}(X, M_{n+p}) \ar[r]^{π_{I T}^{∗}}_{≃} \ar[d]_{d_{1}^{p,q}} & H_{−q}^{T}(X ×_{k} Z_{I}, M_{n}) \ar[d]^{(−1)^{r−1}ι_{IJ,T ∗}} \\ E_{1}^{p+1,q} = & H_{−q−p−1}^{T}(X, M_{n+p+1}) \ar[r]^{π_{J T}^{∗}}_{≃} & H_{−q}^{T}(X ×_{k} Z_{J}, M_{n}) ,}

and therefore

\begin{matrix}(−1)^{r−1}d_{1}^{p,q}(x) = & (π_{J T}^{∗})^{−1}[ι_{IJ,T ∗}(π_{I T}^{∗} (x))] \\ = & (π_{J T}^{∗})^{−1}[ι_{IJ,T ∗}(𝕀_{X×Z_{I}}^{T} ∩ ((π_{J} ×^{T} id_{U}) ∘ (ι_{IJ} ×^{T} id_{U})^{∗}(x)))] \\ = & [(π_{J T}^{∗})^{−1}(ι_{IJ,T ∗}(𝕀_{X×Z_{I}}^{T}))] ∩ x \\ = & c_{1}(L_{X}(χ_{r})) ∩ x \end{matrix}

for all x ∈ H_{−q−p}^{T}(X, M_{n+p}). The last equation since ι_{IJ} is the zero section of the pull-back of L_{X}(χ) along the projection X ×_{k} Z_{I} → X.

3.12 (Equivariant cycle cohomology). From now on all varieties are assumed to be smooth. Given a smooth equidimensional variety X we define its i-th cycle cohomology group as

H^{i}(X, M_{n}) := H_{dim X−i}(X, M_{n−dim X}) .

A pairing

K_{i}^{M}(E) × M_{j}(E) ⟶ M_{i+j}(E)

induces the so called intersection product of cycle cohomology groups

H^{i}(X, K_{m}^{M}) × H^{j}(X, M_{n}) ⟶ H^{i+j}(X, M_{m+n}) , (α, x) ⟼ α ∩ x ,

see [26, Sect. 14]. This generalizes the usual intersection product of Chow groups as defined for instance in the book [11] of Fulton.

3.13. Definition. It follows by descent (since U ⟶ U/G is a principal bundle) that X ×^{G} U is an equidimensional smooth variety, too. Hence we can define

H_{G}^{i}(X, M_{n}) := H_{dim X−i}^{G}(X, M_{n−dim X}) = H^{i}(X ×^{G} U, M_{n})

and call it the i-th G-equivariant cycle cohomology group of X with values in M_{∗}.

The pairing of H^{∗}(X ×^{G} U, K_{∗}^{M}) with H^{∗}(X ×^{G} U, M_{∗}) induces a pairing

H_{G}^{i}(X, K_{m}^{M}) × H_{G}^{j}(X, M_{n}) ⟶ H_{G}^{i+j}(X, M_{m+n}) , (α, x) ⟼ α ∩ x .

Since the pull-back of ordinary cycle cohomology groups respects the product, we can use Bogomolov’s double filtration argument to check that this definition does not depend (up to natural isomorphism) on the choice of an (X, G)-admissible pair.

It follows from the properties of the ordinary “intersection” product in cycle cohomology that H_{G}^{∗}(X, K_{∗}^{M}) becomes a skew-commutative ring with this product, i.e.

α ∩ β = (−1)^{(i+m)·(j+n)} · β ∩ α

for all α ∈ H_{G}^{i}(X, K_{m}^{M}) and β ∈ H_{G}^{j}(X, K_{n}^{M}), cf. [9, Prop. 56.4]. We denote the neutral element of this multiplication by 𝕀_{X}^{G}. This is the rational equivalence class of X ×^{G} U in H_{G}^{0}(X, K_{0}^{M}) = H^{0}(X ×^{G} U, K_{0}^{M}).

3.14 (Projection formulas). From the properties of the ordinary intersection product shown in [5, Prop. 5.9] we obtain the following:

Let f: X ⟶ Y be a morphism of equidimensional smooth G-varieties. Then (i) f_{G}^{∗}(β ∪ y) = f_{G}^{∗}(β) ∩ f_{G}^{∗}(y), and (ii) if f is proper the equations (projection formulas)

f_{G∗}(α ∩ f_{G}^{∗}(y)) = f_{G∗}(α) ∩ y and f_{G∗}(f_{G}^{∗}(β) ∩ x) = β ∩ f_{G∗}(x)

for all α ∈ H_{G}^{i}(X, K_{m}^{M}), β ∈ H_{G}^{i}(Y, K_{m}^{M}), x ∈ H_{G}^{j}(X, M_{n}), and y ∈ H_{G}^{j}(Y, M_{n}).

We denote by c_{1}(L) also the element c_{1}(L)(𝕀_{X}^{G}) ∈ CH_{G}^{1}(X). With this notation we have by the projection formula

c_{1}(L)(x) = c_{1}(L) ∩ x

for all x ∈ H_{G}^{i}(X, M_{n}) and all i ∈ ℕ and n ∈ ℤ if the variety X is smooth k.

This section can be summarized by the following (see 3.13 and 3.6)

3.15. Theorem. The functor X ↦ H_{G}^{∗}(X, M_{∗}) provides an example of a graded essential G-equivariant pretheory.

3.16. Example (Equivariant Chow groups). We have

H_{i}^{G}(X, K_{−i}^{M}) = CH_{i}^{G}(X) ,

where CH_{∗}^{G} denotes the G-equivariant Chow-theory of Edidin and Graham [8].

Since H_{j}(Y, K_{−i}^{M}) = 0 for j ≤ i − 1 we know by the localization sequence that for a G-equivariant open embedding ι: W ↪ X the pull-back

ι_{G}^{∗}: CH_{i}^{G}(X) ⟶ CH_{i}^{G}(W)

is surjective. Identifying CH_{G}^{i}(X) with CH_{dim X−i}^{G}(X) we obtain that the functor

h_{G}: X ⟼ CH_{G}^{∗}(X) = ⨁_{i∈ℤ} CH_{G}^{i}(X)

provides an example of an essential G-equivariant pretheory.

4. Torsors and equivariant maps

In the present section we generalize the result of Karpenko and Merkurjev [16, Thm. 6.4] to an arbitrary equivariant pretheory. Our arguments follow closely the exposition of [16, §6].

Let S = GL(V) be the group of automorphisms of a finite dimensional k-vector space V. Let H be an algebraic subgroup of S. Consider S as a (left) H-variety.

Let h_{H} be a H-equivariant pretheory over k. Following the proof of [16, Prop. 6.2] we embed S into the affine space End_{k}(V) as a S-equivariant (and, hence, H-equivariant) open subset.

Let ϕ: S → pt denote the structure map. The induced pull-back ϕ_{H}^{∗} factors as the composite of pull-backs

h_{H}(pt) ⟶^{≃} h_{H}(End(V)) ↠ h_{H}(S),

where the first map is an isomorphism by homotopy invariance and the second map is surjective by the localization property. This proves that

4.1. Lemma. The induced pull-back ϕ_{H}^{∗} is surjective.

Let μ_{s}: S → S denote the right multiplication by s ∈ S(k). Since ϕ ∘ μ_{s} = φ as morphisms over k and μ_{s} is H-equivariant, we have (μ_{s})_{H}^{∗} ∘ ϕ_{H}^{∗} = ϕ_{H}^{∗}. Since ϕ_{H}^{∗} is surjective by Lemma 4.1, this proves that

4.2. Lemma. The induced pull-back (μ_{s})_{H}^{∗}: h_{H}(S) → h_{H}(S) is the identity.

Let G be an algebraic subgroup of S such that H ⊆ G ⊆ S so that G is considered as a (left) H-variety. Let E be a (left) G-variety over k and let η_{E}: Spec K → E denote its generic point, where K = k(E).

Consider the G-equivariant (and, hence, H-equivariant) map

ψ_{E}: G_{K} = G ×_{Spec k} Spec K ⟶^{(id,η_{E})} G ×_{Spec k} E ⟶ E

which takes the identity of G to the generic point of E. Suppose that there is a G-equivariant map ρ: E → S over k. Then there is a commutative diagram of H-equivariant maps

\xymatrix{G_{K} \ar[r]^{ψ_{E}} \ar[d]_{i} & E \ar[r]^{ρ} & S \\ S_{K} \ar[rr]^{μ_{ρ(η_{E})}} &  & S_{K} \ar[u]_{p}}

where the map i is the embedding, p is the projection S_{K} = S ×_{Spec k} Spec K → S to the first factor and the bottom horizontal map is the multiplication by ρ(η_{E}).

By the diagram the pull-back of the composite (ψ_{E})_{H}^{∗} ∘ρ_{H}^{∗} = (ρ∘ψ_{E})_{H}^{∗} coincides with the pull back (p ∘ μ_{ρ(η_{E})} ∘ i)_{H}^{∗}. Here the map (ψ_{E})_{H}^{∗}: h_{H}(E) → h̄_{H}(G_{K}) is the canonical map to the colimit. By Lemma 4.2 the latter coincides with the pull-back i_{H}^{∗} ∘ p_{H}^{∗}, hence, proving the following

4.3. Lemma. Let E be a G-variety together with a G-equivariant map ρ: E → S. Then we have

(ψ_{E})_{H}^{∗} ∘ ρ_{H}^{∗} = i_{H}^{∗} ∘ p_{H}^{∗}: h_{H}(S) → h̄_{H}(G_{K}).

We are now in position to prove the main result of this section

4.4. Theorem. Let H ⊂ G be algebraic groups and let h_{H}(−) be a H-equivariant pretheory. Then for any G-torsor E with K = k(E) we have

Im(φ_{H}^{∗}) ⊆ Im((ψ_{E})_{H}^{∗}) in h̄_{H}(G_{K}),

where φ: G_{K} → pt is the structure map.

Proof. By Lemma 4.1 we have

Im(φ_{H}^{∗}) = Im(i_{H}^{∗} ∘ p_{H}^{∗} ∘ ϕ_{H}^{∗}) = Im(i_{H}^{∗} ∘ p_{H}^{∗}).

Theorem then follows from Lemma 4.3 and the fact that there exists a finite dimensional k-vector space V and a G-equivariant map E ⟶ S = GL(V), see [16, Prop. 6.4]. □

4.5. Definition. Let H ⊂ G be algebraic groups over k and h_{H}(−) be an equivariant pretheory with values in the category of commutative rings. To each G-torsor E we associate a commutative ring

E ↦ h̄_{H}(G_{K}) ⊗_{h_{H}(E)} h̄_{H}(H_{K}),

where h̄_{H}(G_{K}) is the h_{H}(E)-module via (ψ_{E})_{H}^{∗} and h̄_{H}(H_{K}) is the h_{H}(E)-module

via the composite h_{H}(E) →^{(ψ_{E})_{H}^{∗}} h̄_{H}(G_{K}) → h̄_{H}(H_{K}) with the last map induced by the embedding H ⊂ G.

This ring will be denoted by ĥ_{H}(E) and will play the central role in the last section of this paper. In particular, it will be shown that for known examples of equivariant pretheories it is always a quotient of the cohomology ring h(G) of G.

4.6. Corollary. Let H ⊂ G be algebraic groups over k and let h_{H}(−) be an essential H-equivariant pretheory. Then there exists a field extension l/k and a G-torsor E over l with L = l(E) such that

Im(φ_{H_{l}}^{∗}) = Im((ψ_{E})_{H_{l}}^{∗}) in h̄_{H_{l}}(G_{L})

Proof. We fix an embedding G ⟶ S = GL(V) for some finite dimensional kvector space V. The quotient S ⟶ G\S (for the right action of G on S) is a (left) G-torsor. Let l be its function field and consider the cartesian square

\xymatrix{E \ar[r]^{ρ} \ar[d] & S \ar[d] \\ Spec l \ar[r] & G\S .}

Since S ⟶ G\S is a G-torsor the map E ⟶ Spec l is a G-torsor, too. The l-scheme E is a localization of S and, therefore, by (C) and localization property (L) the pull-back ρ_{H}^{∗}: h_{H}(S) → h̄_{H}(E) → h_{H}(E) is surjective. This implies that the pull-back ρ_{H_{l}}^{∗}: h_{H_{l}}(S_{l}) → h_{H_{l}}(E) is surjective. It remains to apply the proof of Theorem 4.4 over l and to observe that Im(φ_{H_{l}}^{∗}) = Im((ψ_{E})_{H_{l}}^{∗} ∘ρ_{H_{l}}^{∗}) = Im((ψ_{E})_{H_{l}}^{∗}).

□

4.7. Definition. A G-torsor E over k which satisfies the equality of Corollary 4.6 will be called a generic torsor with respect to the pretheory h_{H}(−). Note that in the proof we provided an example of a G-torsor which is generic over some field extension for all equivariant pretheories.

4.8. Example. Observe that for a generic G-torsor E over k we have

ĥ_{H}(E) = h̄_{H}(G_{K}) ⊗_{h_{H}(pt)} h̄_{H}(H_{K}),

where h̄_{H}(G_{K}) is an h_{H}(pt)-module via φ_{H}^{∗}, and for the trivial G-torsor G we have

ĥ_{H}(E) = h̄_{H}(G_{K}) ⊗_{h_{H}(G)} h̄_{H}(H_{K}).

5. Equivariant oriented cohomology

In the present section we apply Theorem 4.4 to the case of a B-equivariant oriented cohomology, where B is a Borel subgroup of a split semisimple linear algebraic group.

Let G be a split semisimple linear algebraic group of rank n over a field k and let T be a split maximal torus of G. Similarly to 3.11 consider the action (1) of T on the affine space 𝔸_{k}^{n} with weights χ_{1}, …, χ_{n} together with an action of T on G by left multiplication. Then T embeds into 𝔸_{k}^{n} = Spec k[x_{1}, …, x_{n}] as the complement of the coordinates hyperplanes Z_{i}, i = 1, …, n. Let V = 𝔸_{k}^{n} ×^{T} G be the associated vector bundle over G/T (see [3, p.22]). By definition V = L_{G/T} (χ_{1})⊕…⊕L_{G/T} (χ_{n}) and G = T ×^{T} G embeds into V as the complement of the union of zero-sections

V_{j} = ⨁_{j≠i} L_{G/T} (χ_{i}) = Z_{j} ×^{T} G , i = 1, …, n .

Note that e_{j}: V_{j} ↪ V is a smooth subvariety for every j.

Let now h(−) be an oriented cohomology theory in the sense of [19], i.e. a contravariant functor from the category of smooth varieties over k to the category of graded commutative rings satisfying certain axioms. In particular, if X is a k-variety with an open subvariety ι: U ↪ X there is an exacts sequence

\xymatrix{h(Z) \ar[r]^{j_{∗}} & h(X) \ar[r]^{ι^{∗}} & h(U) \ar[r] & 0}

where j: Z = X \ U ↪ X is the closed complement of U, and there is also a first Chern class which we denote by c_{1}^{h}.

Having such a theory h(−) we get from this localization sequence (by induction) an exact sequence

⨁_{j=1}^{n} h(V_{j}) ⟶^{⊕_{j}(e_{j})_{∗}} h(V) → h(G) → 0

By the properties of the first Chern class we have

(e_{j})_{∗}(1_{V_{j}}) = c_{1}^{h}(L_{V} (χ_{j}))

which implies that the image of ⊕_{j}(e_{j})_{∗} is an ideal generated by the first Chern classes c_{1}^{h}(L_{V} (χ_{j})), j = 1, …, n.

Let B be a Borel subgroup of G containing T and let G/B be the variety of Borel subgroups. By [1, Thm.10.6 of Ch.III] the composite of projections V → G/T → G/B is a chain of affine bundles. Therefore, by the homotopy invariance there is an isomorphism h(G/B) →^{≃} h(V) compatible with the Chern classes and we obtain the following

5.1. Proposition. There is an isomorphism of rings

h(G) ≃ h(G/B)/(c_{1}^{h}(L_{G/B}(χ_{1})), …, c_{1}^{h}(L_{G/B}(χ_{n}))) ,

where χ_{1}, …, χ_{n} is a basis of the character group T^{∗}.

Let h_{B}(−) be an B-equivariant pretheory to the category of commutative rings such that

(i) h_{B}(E) = h(E/B) for every G-torsor E, where h(−) is an oriented cohomology in the sense of [19].

(ii) h̄_{B}(B_{K}) = h(pt) and h̄_{B}(G_{K}) ≃ h(G/B).

Then the ring ĥ_{B}(E) = h̄_{B}(G_{K}) ⊗_{h_{B}(E)} h̄_{B}(B_{K}) introduced in Definition 4.5 can be identified with a quotient of h̄_{B}(G_{K}) ≃ h(G/B) modulo the ideal generated by non-constant elements from the image of the restriction (ψ_{E})_{B}^{∗}: h(E/B) → h(G/B).

Consider now the map φ_{B}^{∗}: h_{B}(pt) ⟶ h̄_{B}(G_{K}) ≃ h(G/B). By Theorem 4.4 Im(φ_{B}^{∗}) ⊆ Im((ψ_{E})_{B}^{∗}), hence, ĥ_{B}(E) can be identified with a quotient of the factor ring h(G/B)/I, where I denotes the ideal generated by elements from the image of φ_{B}^{∗} which are in the kernel of the augmentation.

Then by Proposition 5.1 and Corollary 4.6 we obtain the following

5.2. Corollary. Assume that the image of φ_{B}^{∗} is generated by the Chern classes c_{1}^{h}(L_{G/B}(χ_{i})) of line bundles associated to the characters χ_{i} ∈ T^{∗} (i = 1 … n). Then ĥ_{B}(E) is a quotient of h(G/B)/I ≃ h(G). Moreover, if h_{B}(−) is essential and E is generic, then

ĥ_{B}(E) ≃ h(G).

5.3. Example (Chow groups and the J-invariant). Consider the equivariant Chow groups h_{B}(−) = CH^{B}(−). Let E be a G-torsor. The ring h_{B}(pt) can be identified with the symmetric algebra S(T^{∗}) and the map

φ_{B}^{∗}: S(T^{∗}) = h_{B}(pt) ⟶ h̄_{B}(G_{K}) = CH(G/B)

coincides with the characteristic map for Chow groups studied in [6]. So its image is generated by the first Chern classes c_{1}(L_{G/B}(χ_{i})) of the respective line bundles.

The map (ψ_{E})_{B}^{∗} coincides with the restriction map

res: CH(E/B) ⟶ CH(G/B),

where E/B is the twisted form of G/B by means of E and the map

S(T^{∗}) = h_{B}(pt) ⟶ h_{B}(B_{K}) = CH(pt) = ℤ

is the augmentation map. If E is generic, then we have

ĥ_{B}(E) ≃ CH(G/B) ⊗_{S(T^{∗})} ℤ ≃ CH(G).

where the last isomorphism follows by Corollary 5.2. For an arbitrary G-torsor E the ring

ĥ_{B}(E) = CH(G/B) ⊗_{Im(res)} ℤ

is a quotient ring of CH(G/B) modulo the ideal generated by non-constant elements from the image of the restriction CH(E/B) → CH(G/B).

Observe that the characteristic map φ_{B}^{∗} is not surjective in general. However, its image is a subgroup of finite index in CH(G/B) measured by the torsion index τ of G. This implies that for a G-torsor E we have

ĥ_{B}(E) ⊗_{ℤ} ℚ ≃ ℚ.

If p | τ, then there is an isomorphism

ĥ_{B}(E) ⊗_{ℤ} ℤ/p ≃ \frac{ℤ/p [x_{1}, …, x_{r}]}{(x_{1}^{p^{j_{1}}} , …, x_{r}^{p^{jr}})},

where (j_{1}, …, j_{r}) is the J-invariant of G twisted by E as defined in [25]. Observe that j_{i} ≤ k_{i}, i = 1 … r, where k_{i} are defined via the p-exceptional degrees introduced by Kac [15], and for a generic torsor E we have equalities j_{i} = k_{i} for each i.

5.4. Example (Grothendieck’s K_{0} and indexes of the Tits algebras). Consider the equivariant K_{0}-groups h_{B}(−) = K_{0}(B, −). Let E be a G-torsor. The ring h_{B}(pt) can be identified with the integral group ring ℤ[T^{∗}] and with the representation ring Rep T of T, i.e.

h_{B}(pt) = ℤ[T^{∗}] = Rep T.

The map

φ_{B}^{∗}: ℤ[T^{∗}] = h_{B}(pt) ⟶ h̄_{B}(G_{K}) ≃ K_{0}(G/B)

coincides with the characteristic map 𝔠 for K_{0} studied in [6] and again its image is generated by the first Chern classes.

As before the map (ψ_{E})_{B}^{∗} coincides with the restriction map

res: K_{0}(E/B) ⟶ K_{0}(G/B),

and applying 4.4 we obtain the following K_{0}-analogue of the Karpenko-Merkurjev result:

5.5. Corollary. Let E be a G-torsor over k and let E/B be a twisted form of G/B by E. Then

(i) 𝔠(ℤ[T^{∗}]) ⊆ res(K_{0}(E/B)); (ii) there exists a G-torsor E over some field extension of k such that

𝔠(ℤ[T^{∗}]) = res(K_{0}(E/B)).

According to a result of Panin [24] the image of the restriction map is given by the sublattice

{i_{w,E} · g_{w}}_{w∈W} ,

where W is the Weyl group of G, {g_{w}}_{w∈W} is the Steinberg basis of K_{0}(G/B) and {i_{w,E}} are indexes of the respective Tits algebras.

Corollary 5.5 implies that there exists a maximal set of indexes {m_{w}}_{w∈W} such that

• i_{w,E} ≤ m_{w} for every w ∈ W and every torsor E;

• there exists E such that i_{w,E} = m_{w} for every w ∈ W;

• the image of the characteristic map φ_{B}^{∗}(ℤ[T^{∗}]) coincides with the sublattice

{m_{w} · g_{w}}_{w∈W}, hence, providing a way to compute m_{w}. The indexes m_{w} are called the maximal Tits indexes. They have been extensively

studied by Merkurjev [20] and Merkurjev, Panin and Wadsworth [22]. They are closely related to the dimensions of irreducible representations of G. Comparing with the case of Chow groups one observes that the maximal Tits indexes in K_{0} play the same role as the p-exceptional degrees in Chow groups.

Since the map ℤ[T^{∗}] = h_{B}(pt) ⟶ h̄_{B}(B_{K}) = K_{0}(pt) = ℤ is the augmentation map, for a generic torsor E we have

ĥ_{B}(E) = K_{0}(G/B) ⊗_{ℤ[T^{∗}]} ℤ ≃ K_{0}(G),

where the last isomorphism follows by Corollary 5.2. Hence, for an arbitrary Gtorsor E

ĥ_{B}(E) = K_{0}(G/B) ⊗_{Im(res)} ℤ

is the quotient ring of K_{0}(G/B) modulo the ideal generated by elements from the image of the restriction K_{0}(E/B) → K_{0}(G/B) which are in the kernel of the augmentation.

5.6. Example (Equivariant algebraic cobordism). Consider the equivariant algebraic cobordism h_{B}(−) = Ω^{B}(−). Let E be a G-torsor. The completion h_{B}(pt)^{∧} of h_{B}(pt) at the augmentation ideal, (the kernel of h_{B}(pt) → h_{B}(B)) can be identified with the formal group ring 𝕃[[T^{∗}]]_{U} introduced in [2, Def. 2.4 and 2.7], where 𝕃 is the Lazard ring and U denotes the universal formal group law.

The map

φ_{B}^{∗}: 𝕃[[T^{∗}]]_{U} = h_{B}(pt)^{∧} ⟶ h̄_{B}(G_{K}) = Ω(G/B)

coincides with the characteristic map of [2, Def. 10.2] and its image is generated by the first Chern classes.

The map (ψ_{E})_{B}^{∗} coincides with the restriction map

res: Ω(E/B) ⟶ Ω(G/B),

where E/B is the twisted form of G/B by means of E and the map 𝕃[[T^{∗}]]_{U} = h_{B}(pt)^{∧} ⟶ h_{B}(B_{K}) = Ω(pt) = 𝕃 is the augmentation map. By Corollary 5.2 for an arbitrary G-torsor E we have

ĥ_{B}(E) = Ω(G/B) ⊗_{Im(res)} 𝕃.

is a quotient of the ring Ω(G/B) modulo the image of the restriction Ω(E/B) → Ω(G/B) from the kernel of the augmentation.

Appendix A. A spectral sequence

Let X be a k-scheme, Z_{1}, …, Z_{m} closed subschemes of X (with reduced structure), and M_{∗} a cycle module over k. We construct in this section a spectral sequence

which converges to the cycle homology of M_{∗} over the open complement of ⋃_{i=1}^{m} Z_{i}

in X generalizing the long exact localization sequence (the case m = 1). A reader familiar with Levine [18, Sect. 1] will notice an analogy with Levine’s construction of a similar spectral sequence for Quillen K-theory which should have an explanation in the theory of model categories.

Let m be a positive integer and \underline{m} the set of subsets of \{1, 2, …, m\} ⊂ ℕ. We set further \underline{0} = ∅. We consider \underline{m} as a category with Mor\underline{m}(I, J) = \{∅\} (one element set) if I ⊆ J and ∅ otherwise.

A.1. Definition. A m-cube of complexes is a functor

K_{∗}: \underline{m} ⟶ K^{b}(𝔄𝔟) I ⟼ K_{I}

from \underline{m} into the category of bounded (homological) complexes of abelian groups. If I ⊆ J we denote r_{IJ}^{K} the induced morphism K_{I} ⟶ K_{J}, and set r_{IJ}^{K} = 0 if I ⊈ J. We observe that r_{JL}^{K} · r_{IJ}^{K} = r_{IL}^{K} if I ⊆ J ⊆ K since I ↦ K_{I} is a functor.

A morphism of m-cubes is a natural transformation.

For brevity of notation we define if l = |J| = |I| + 1 a morphism ϵ_{IJ}^{K} as follows: If I ⊄ J we set ϵ_{IJ}^{K} = 0 and if J = {i_{1} < i_{2} < … < i_{l}} and I = J \ {i_{d}} for some 1 ≤ d ≤ l we set ϵ_{IJ}^{K} = (−1)^{d−1} · r_{IJ}^{K}. The signs are chosen, such that the matrix product

(ϵ_{JL})_{|J|=p+1,|L|=p+2} · (ϵ_{IJ})_{|I|=p,|J|=p+1}  (4)

is zero.

With a m-cube of complexes K_{∗} we can associate two (m−1)-cubes of complexes (as long as m ≥ 2). First we have the (m − 1)-cubes of complexes K̃_{∗} which is the restriction of K_{∗} to \underline{m − 1}, and second we have K′_{∗}. The latter is defined as the composition of \underline{m − 1} ⟶ \underline{m}, J ⟼ J ∪ \{m\}, with K_{∗}, i.e.

K′_{∗}: \underline{m − 1} ⟶ D^{b}(𝔄𝔟) J ⟼ K_{J∪\{m\}} .

A.2. Example. Let X be a k-scheme with closed subschemes Z_{1}, …, Z_{m}, and M_{∗} a cycle module. We set Z_{I} := ⋂_{j∉I} Z_{j} for all I in \underline{m}. We have than an m-cube of

complexes

K_{∗}: C_{•}(X, Z_{1}, …, Z_{m}, M_{n}, ∗): I ⟼ C_{•}(Z_{I}, M_{n}) ,

where for I ⊆ J the morphism r_{IJ}^{C} is the push-forward along the inclusion of closed subschemes ⋂_{j∉I} Z_{j} ↪ ⋂_{j∉J} Z_{j}. Then we have

K′_{∗} = C_{•}(X, Z_{1}, …, Z_{m−1}, M_{n}, ∗)

and

K̃_{∗} = C_{•}(Z_{m}, Z_{1} ∩ Z_{m} …, Z_{m−1} ∩ Z_{m}, M_{n}, ∗) .

Note also that K_{∅} = C_{•}( ⋂_{i=1}^{m} Z_{i}, M_{n}) and K_{{1,...,m}} = C_{•}(X, M_{n}).

A.3 (The functors 𝔠𝔣_{≥i}). Recall first the cone of a morphism of complexes f: K_{•} ⟶ L_{•}. This is the complex cone f which is given in degree i + 1 and i by:

\xymatrix{ &  & ^{( \begin{matrix}d_{i}^{K} & f_{i} \\ 0 & −d_{i+1}^{L} \end{matrix} )} \ar[dr] \\ … \ar[r] & K_{i} ⊕ L_{i+1} &  & K_{i−1} ⊕ L_{i} \ar[r] & ….}

We have then a (so called) exact triangle K_{•} ⟶^{f} L_{•} ⟶ cone f ⟶ K_{•}[1] (with obvious morphisms on the right).

We define inductively a functor 𝔠𝔣_{≥i} form the category of m-cubes of complexes to K^{b}(𝔄𝔟) for all i ∈ ℤ. Let K_{∗} be such a m-cube. Then we set 𝔠𝔣_{≥i} K_{∗} = 0 if i ≥ m + 1 and

𝔠𝔣_{≥m} K_{∗} := K_{{1,...,m}} .

We have then the morphism of complexes

Θ_{m}^{K} := ∑_{i=1}^{m} ϵ_{I{1,...,m}}: ⨁_{|I|=m−1} K_{I} ⟶ K_{{1,...,m}} = 𝔠𝔣_{≥m} K_{∗}

and define 𝔠𝔣_{≥m−1} K_{∗} to be the cone of this morphism.

The composition

⨁_{|I|=m−2} K_{I} \underrightarrow{(ϵ_{IJ})_{I,J}} ⨁_{|J|=m−1} K_{J} ⟶^{Θ_{m}^{K}} K_{\{1,...,m\}}

is the zero morphism, see (4), and therefore induces a morphism of complexes

Θ_{m−1}^{K}: ⨁_{|I|=m−2} K_{I}[1] ⟶ 𝔠𝔣_{≥m−1} K_{∗} = cone Θ_{m}^{K} .

We set then 𝔠𝔣_{≥m−2} := cone Θ_{m−1}^{K}. Let now p ≤ m − 3. Then by (descending) induction we have a morphism of complexes

Θ_{p+2}^{K}: ⨁_{|L|=p+1} K_{L}[m − p − 2] ⟶ 𝔠𝔣_{≥p+2} K_{∗}

such that we have Θ_{p+2}^{K} · (ϵ_{JL}[m − p − 2])_{|J|=p,|L|=p+1} = 0. Therefore there exists

Θ_{p+1}^{K}: ⨁_{|J|=p} K_{J}[m − p − 1] ⟶ 𝔠𝔣_{≥p+1} K_{∗} = cone Θ_{p+2}^{K} ,

such that the following diagram commutes:

\xymatrix{ & ⨁_{|J|=p} K_{J}[m − p − 2] \ar[d]^{(ϵ_{JL}[m−p−2])_{J,L}} \ar@{-->}[dl]_{Θ_{p+1}^{K}[−1]} \\ 𝔠𝔣_{≥p+1}[−1] \ar[r] & ⨁_{|L|=p+1} K_{L}[m − p − 2] \ar[r]^{Θ_{p+2}^{K}} & 𝔠𝔣_{≥p+2} K_{∗} .}

More precisely, the morphism of complexes Θ_{p+1}^{K}[p + 1 − m] is in degree t given by

\xymatrix{ & ^{( \begin{matrix}((ϵ_{JL})_{JL})_{t} \\ 0 \end{matrix} )} \ar[dr] \\ ⨁_{|J|=p} K_{J t} &  & ⨁_{|L|=p+1} K_{L t} ⊕ 𝔠𝔣_{≥p+2}[p + 1 − m]_{t} = 𝔠𝔣_{≥p+1}[p + 1 − m]_{t} .}

Therefore we have by (4) that Θ_{p+1}^{K} · (ϵ_{IJ}[m − p − 1])_{I,J} = 0 which finishes the induction step and the definition of 𝔠𝔣_{≥p} K_{∗} for p ≥ 0.

If p ≤ 0 we set 𝔠𝔣_{≥p} K_{∗} := 𝔠𝔣_{≥0} K_{∗}.

A.4 (An exact triangle). By construction we have then for all 0 ≤ p ≤ m a commutative diagram

[figure: ⨁ ; \begin{matrix}|I| = p − 2 \\ m ∉ I \end{matrix} K_{I∪\{m\}}[m − p]; Θ_{p−1}^{K′}; ⨁_{|J|=p−1}K_{J} [m − p]; Θ_{p}^{K}; ⨁ ; \begin{matrix}|L| = p − 1 \\ m ∉ I \end{matrix} K_{L}[m − p]; ΘK̃ ; p; 𝔠𝔣_{≥p−1} K′_{∗}; 𝔠𝔣_{≥p} K_{∗}; (𝔠𝔣_{≥p} K̃_{∗})[1] ,]

whose lower row is an exact triangle and whose upper row is a short split exact sequence of complexes (with obvious morphisms) for all p ≥ −1.

Shifting the diagram of the lemma for p = m−1 to the left we get a commutative diagram in the bounded derived category of complexes of abelian groups:

[figure: ⨁ ; \begin{matrix}|I| = m − 2 \\ m ∉ I \end{matrix} K_{I}; 0; ΘK̃ ; m−1; ⨁_{|J|=m−2}K_{J} [1]; Θ_{m−1}^{K}; ⨁ ; \begin{matrix}|L| = m − 3 \\ m ∉ I \end{matrix} K_{L∪\{m\}}[1]; Θ_{m−2}^{K′}; K_{\{1,...,m−1\}}; Θ; 𝔠𝔣_{≥m−2} K′_{∗}; 𝔠𝔣_{≥m−1} K_{∗} ,]

where the arrow

Θ: K_{{1,...,m−1}} = 𝔠𝔣_{≥m−1} K̃_{∗} ⟶

𝔠𝔣_{≥m−2} K′_{∗} = cone ( ⨁ ; \begin{matrix}|J| = m − 2 \\ m ∉ J \end{matrix} K_{J} ⟶^{Θ_{m−1}^{K′}} K_{\{1,...,m\}})

is induced by ϵ_{{1,...m−1}{1,...m}}: K_{{1,...m−1}} ⟶ K_{{1,...m}}.

A.5. Definition. The cofiber of the m-cube of complexes K_{∗} is the complex

𝔠𝔣 K_{∗} := 𝔠𝔣_{≥0} K_{∗} .

The assignment K_{∗} ↦ 𝔠𝔣 K_{∗} is a covariant functor from the category of m-cubes of complexes to K^{b}(𝔄𝔟).

A.6 (The spectral sequence). By the very definition of the complexes 𝔠𝔣_{≥p} K_{∗} we have exact triangles

𝔠𝔣_{≥p+1} K_{∗} ⟶ 𝔠𝔣_{≥p} K_{∗} ⟶ ⨁_{|I|=p} K_{I}[m − p]

for any m-cube of complexes K_{∗} and all p ≥ 0. The associated long exact homology sequences constitute an exact couple and so we get a convergent spectral sequence of cohomological type

E_{1}^{p,q}(K_{∗}) := H_{−p−q} ( ⨁_{|I|=p} K_{I}[m − p]) ⟹ H_{−p−q}(𝔠𝔣 K_{∗}) .

(Note that the complexes K_{I} and so also 𝔠𝔣_{≥p} K_{∗} are all bounded.) By construction the IJ-component H_{−p−q}(K_{I}[m − p]) ⟶ H_{−p−q}(K_{J}[m − p]) of the differential d_{1}^{p,q}: E_{1}^{p,q}(K_{∗}) ⟶ E_{1}^{p+1,q}(K_{∗}) is equal H_{−p−q}(ϵ_{IJ}[m − p]).

A.7. Example. Let X, Z_{1}, …, Z_{m}, K_{∗}, and M_{∗} be as in example A.2. We set

W_{l} := ⋃_{j=1}^{l} Z_{j} for 1 ≤ l ≤ m and C_{•}(Y) := C_{•}(Y, M_{n}) for any finite type k-scheme Y.

The pull-back along the open immersion X \ W_{m} ↪ X induces a morphism of complexes 𝔠𝔣_{≥m} K_{∗} = C_{•}(X) ⟶ C_{•}(X \ W_{m}). The composition of the morphism with Θ_{m}^{K} is zero given a morphism of complexes 𝔠𝔣_{≥m−1} K_{∗} ⟶ C_{•}(X \ W_{m}). Composing this morphism with Θ_{m−1}^{K} is again zero and hence induce a morphism from 𝔠𝔣_{≥m−2} K_{∗} to C_{•}(X \ W_{m}). Proceeding further we finally get a morphism of complexes

γ: 𝔠𝔣 K_{∗} ⟶ C_{•}(X \ W_{m}) .

Similarly we have morphisms of complexes γ̃ : 𝔠𝔣 K̃_{∗} ⟶ C_{•}(Z_{m} \ W_{m−1}) and γ′: 𝔠𝔣 K′_{∗} ⟶ C_{•}(X \ W_{m−1}). We claim that this is a quasi-isomorphism. This is obvious for m = 1. Let m ≥ 2. Then by A.4 we have a commutative diagram whose rows are exact triangles

\xymatrix{𝔠𝔣 K̃_{∗} \ar[r] \ar[d]^{γ̃} & 𝔠𝔣 K′_{∗} \ar[r] \ar[d]^{γ′} & 𝔠𝔣 K_{∗} \ar[d]_{γ} \\ C_{•}(Z_{m} \ W_{m−1}) \ar[r]^{ι_{∗}} & C_{•}(X \ W_{m−1}) \ar[r]^{j^{∗}} & C_{•}(X \ W_{m}) ,}

where ι: Z_{m}\W_{m−1} ↪ X \W_{m−1} and j: X \W_{m−1} ↪ X \W_{m} are to the respective subschemes corresponding open respectively closed immersions. The claim follows from this diagram by induction.

Hence we have a convergent spectral sequence

E_{1}^{p,q}(X, Z_{1}, …, Z_{m}, M_{n}) := ⨁_{|I|=p} H_{−q−m}(Z_{I}, M_{n}) ⟹ H_{−p−q}(U, M_{n}) ,

where U = X \ ⋃_{i=1}^{m} Z_{i}. The IJ-component of the differential is equal 0 if I ⊄ J

and equal (−1)^{d−1} times the push-forward along the closed immersion Z_{I} ↪ Z_{J} if J = {i_{1} < … < i_{l}} and I = J \ {i_{d}}.

A.8. Remark. This spectral sequence applies also to Voevodsky’s [29] motivic cohomology of a homotopy invariant Nisnevich sheaf with transfers F. By Déglise’s Thèse [4] one can associate to such a sheaf a cycle module F̂_{∗}, such that there is a natural isomorphism H_{Nis}^{i}(X, F) ≃ H^{i}(X, F̂_{0}) for all i ∈ ℕ. (Vice versa, if M_{∗} is a cycle module then X ↦ H^{i}(X, M_{n}) is a homotopy invariant Nisnevich sheaf with transfers.)

A.9. Example. Let X be a topological space with closed subspaces Z_{1}, …, Z_{m}. As in the book [7] of Dold we denote by SX and S(X, A) the singular and relative singular complex of X and the pair (X, A) with A ⊆ X a closed subset, respectively. Let H_{∗}(X) and H_{∗}(X, A) be the homology groups of these complexes, i.e. the (relative) singular homology of the space X and the pair (X, A), respectively.

We set (as above) Z_{I} := ⋂_{j∉I} Z_{j} for subsets I ⊆ {1, …, m}, and denote by ι_{IJ}

the embedding Z_{I} ↪ Z_{J} if I ⊆ J.

The map I ↦ SZ_{I} is then a m-cube of complexes and we get by the same reasoning as in Example A.7 a convergent spectral sequence of cohomological type

E_{1}^{p,q}(X, Z_{1}, …, Z_{m}) := ⨁_{|I|=p} H_{−q−m}(Z_{I}) ⟹ H_{−p−q}(X, ⋃_{i=1}^{m} Z_{i}) ,

where the IJ-component of the differential d_{1}^{p,q}: E_{1}^{p,q} ⟶ E_{1}^{p+1,q} is zero if I ⊄ J and equal (−1)^{r−1} · H_{−q−m}(ι_{IJ}) if J = {i_{1}, …, i_{p}, i_{p+1}} and I = J \ {i_{r}} for some 1 ≤ r ≤ p + 1.

Acknowledgments. We would like to thank Roland Lötscher for several helpful comments on quotients of group scheme actions. The first author has been supported by the Deutsche Forschungsgemeinschaft, GI 706/1-2 and GI 706/2-1. The second author has been supported by the NSERC Discovery 385795-2010, Accelerator Supplement 396100-2010 and the Early Researcher Award grants.

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Stefan Gille, Mathematisches Institut, Universität München, Theresienstrasse 39, 80333 München, Germany

   E-mail address: gille@mathematik.uni-muenchen.de

Kirill Zainoulline, Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa ON K1N 6N5, Canada

   E-mail address: kirill@uottawa.ca