COMPARISON OF EQUIVARIANT AND ORDINARY K-THEORY OF ALGEBRAIC VARIETIES

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1 Series Logo Volume 00, Number 00, Xxxx 19xx COMPARISON OF EQUIVARIANT AND ORDINARY K-THEORY OF ALGEBRAIC VARIETIES A. S. MERKURJEV Abstract. It is proved that for a reductive group G the natural homomorphism K 0(G; X) K 0(X) is surjective for any quasiprojective scheme X on which G acts if and only if Pic(G F E) = 0 for all finite field extensions E/F. If G splits, a spectral sequence with the E 2 -term related to the equivariant K -theory of X, which converges to the ordinary K -groups of X, is constructed. The groups K n (X) are known at least for the following classes of varieties X: 1. Homogeneous projective varieties [14]. 2. Varieties of simply connected semisimple algebraic groups and principal homogeneous spaces over these groups [10], [15]. 3. Toric varieties and toric models [11]. In all these cases there is a reductive algebraic group acting nontrivially on a variety. The starting line of the present paper is a G- scheme X, i.e. a scheme X and a reductive algebraic group G acting on X over a field F. The problem we study here is to compare the equivariant K-groups K n (G; X) and the ordinary ones K n (X) via the restriction homomorphism (1) res : K n (G; X) K n (X). It turns out that sometimes the group K n(g; X) is easy to compute. For example, if X = G/H is a homogeneous variety with the natural action of G, where H G is a subgroup, then the category of G-modules on X is equivalent to the category of finite dimensional representations of the group H. Since in the latter category any object has finite length, the computation of the K-groups of this category is an easy task. For example, K 0 (G; G/H) = R(H) is the representation ring of H, which is a free abelian group generated by the classes of irreducible representations of H. Therefore, the knowledge of the The author gratefully acknowledges the support of the Alexander von Humboldt- Stiftung and the hospitality of the University of Bielefeld. 1 c 0000 American Mathematical Society /00 $ $.25 per page

2 kernel and cokernel of the homomorphism (1) sheds some light on the structure of K n (X). A simple argument (see the proof of Theorem 6.4) shows that the surjectivity of (1) for n = 0 and all X implies that the Picard group Pic(G F E) is trivial for any finite field extension E/F (we call such groups G factorial). It turns out that the converse statement holds for quasi-projective varieties (Theorem 6.4): If G is factorial and X is quasi-projective, then the homomorphism (1) is surjective for n = 0. In particular, for any subgroup H G, the group K 0 (G/H) is generated by the classes of vector bundles (G V )/H over G/H for all linear representations H GL(V ) (Corollary 6.5). We also prove that (1) is a split surjection for any n if X is a smooth projective variety (Theorem 6.7). In the case when the torus G/G, where G is the commutant of G, splits (e.g. G is a semisimple or split group), the condition for G to be factorial is equivalent to saying that G is simply connected or the fundamental group π 1 (G) is torsion free. Let us briefly describe other results of the paper. 1. Split case. Let G be a split reductive group with π 1 (G) torsion free. We prove that for any G-scheme X there is a spectral sequence (Theorem 4.3) Ep,q 2 = T orr(g) p (Z, K q (G; X)) K p+q (X). In particular, the restriction homomorphism (1) induces an isomorphism Z R(G) K 0(G; X) K 0(X). For example, K 0 (G/H) Z R(G) R(H) for any subgroup H G. If X is a smooth projective variety, then the spectral sequence degenerates and the homomorphism (1) induces an isomorphism (Theorem 4.2) Z R(G) K n (G; X) K n (X). 2. Inner case. Assume that G is a reductive group of inner type with π 1 (G) torsion free. The idea is to use the variety Y of Borel subgroups in G as a base scheme instead of the field F. Since for any point y Y the group G F (y) is split, one can use some results of the split case. In particular, for any G-scheme X over Y we construct a spectral sequence (Theorem 5.3) Ep,q 2 = T ork 0(G;Y ) p (K 0 (Y ), K q (G; X )) K p+q (X ). 2

3 This result enables us to describe the kernel of the restriction homomorphism (1) in terms of Tits algebras of the group G (Theorem 5.8). At the end of the section we compute the group K 0 (G) for any adjoint semisimple group G of inner type. As an application we prove that for any reductive group G and a connected subgroup H G the group K 0 (G/H) is finitely generated (Theorem 7.7). In particular, the group K 0 (G) is finitely generated of Q-rank 1. It is also proved that the natural action of G on K 0(X) (resp. K n (X)) for any quasi-projective (resp. smooth projective) G-scheme X is trivial (Proposition 7.10). We study the restriction homomorphism (1) in three steps. First of all, we compare K n(g; X) with K n(b; X), where B G is a Borel subgroup (assuming G split), using the technique developed by Panin in [14] which involves a motivic category C(G). We prove that the natural homomorphism R(B) R(G) K n(g; X) K n(b; X) is an isomorphism (Proposition 4.1). In the second step we prove that the restriction homomorphism K n (B; X) K n (T ; X), where T B is a maximal torus, is an isomorphism, using a version of the homotopy invariance theorem proved by R.Thomason [21]. In the last step we compare K n (T ; X) and K n (X) by using the spectral sequence associated to a family of closed subschemes, constructed by M.Levine in [10]. This sequence generalizes the localization exact sequence in the equivariant algebraic K-theory. In the general case, when a Borel subgroup defined over the ground field does not exist, we work over the variety of Borel subgroups. The main technical tool we use to come back to the ground field is the local-to-global principle developed in 2.8. We use the word scheme for a separated scheme of finite type over a field. If X is a scheme over F and L/F is a field extension, then the scheme X F L we simply denote by X L. By X sep we denote X Fsep, where F sep is a separable closure of F. For a scheme X by K n(x) (resp. K n (X)) we denote the K-groups of the category of coherent (resp. locally free coherent) sheaves on X [16]. An algebraic group is a smooth affine group scheme of finite type over a field. I thank M.Rost and I.Panin for helpful discussions. 1. Algebraic groups In this section we collect definitions and preliminary results. 3

4 1.1. Fundamental group. Let G be an algebraic group defined over an algebraically closed field F. A loop in G is a group homomorphism (co-character) p : G m,f G. Two loops p and p are called conjugate if there is g G(F ) such that p (t) = g p(t) g 1 for all t in G m,f. A loop p is called contractible if p extends to a group homomorphism SL 2,F G (we identify G m,f with the maximal torus of diagonal matrices in SL 2,F by t diag(t, t 1 )). If p and p are two loops with commuting images, then the product pp given by (pp )(t) = p(t)p (t) is also a loop. The fundamental group π 1 (G) of the group G is the group defined by generators and relations as follows. Generators are loops in G. For a loop p we denote by [p] the corresponding generator in π 1 (G). The relations are the following: 1. [p] = [p ] if p and p are conjugate; 2. [p] = 1 if p is contractible; 3. [pp ] = [p] [p ] if the images of p and p commute. Example 1.1. Let G be a commutative algebraic group. Since conjugate loops are equal and contractible loops are trivial (SL 2,F coincides with its commutant), the group π 1 (G) equals the group of co-characters G = Hom(G m,f, G) of G. A group homomorphism G G clearly induces a homomorphism of fundamental groups π 1 (G) π 1 (G ). Let G be a reductive group, T G be a maximal torus. The inclusion T G induces a group homomorphism θ : T = π 1 (T ) π 1 (G). Denote by Φ T = Hom(T, G m,f ) the root system of G and by Φ T the dual system of co-roots [8]. For any α Φ denote by G α the commutant of the centralizer of ker(α) in G. It is a semisimple group of rank 1 [8, Ch.IX]. Hence there is an epimorphism SL 2,F G α such that the composition G m,f SL 2,F G α G coincides with the co-root α followed by the inclusion T G, and thus the loop α in G is contractible. Therefore, the map θ factors through the homomorphism ϕ : T /A(Φ) π 1 (G), where A(Φ) is the subgroup in T generated by all co-roots α Φ. Proposition 1.2. The homomorphism ϕ : T /A(Φ) π 1 (G) is an isomorphism. 4

5 Proof. We define a homomorphism ψ : π 1 (G) T /A(Φ) on generators as follows. Let p : G m,f G be a loop. Since im(p) is a torus in G, there exists g G(F ), such that g im(p) g 1 T. Then the composition Int(g) p can be considered as an element in T. We set ψ([p]) = Int(g) p + A(Φ). In order to prove that ψ is well defined assume that Int(g )(im(p)) T for some g in G(F ). Consider the compositions p 1 = Int(g) p and p 2 = Int(g ) p as the loops in T and denote im(p i ) by T i. Then p 2 = Int(h) p 1, where h = g g 1, and hence T 2 = ht 1 h 1. Consider the centralizer H of the torus T 1 in G. Since T 1 = h 1 T 2 h h 1 T h, clearly T and h 1 T h are maximal tori in H and hence are conjugate in H by [8, Cor. A, p.135], i.e. there exists h H(F ) such that h 1 T h = h T h 1, or, equivalently, hh N G (T ) represents an element w in the Weyl group W. The group W acts naturally on T. Since p 2 = Int(h) p 1 = Int(hh ) p 1 = w(p 1 ), it suffices to show that p w(p) A(Φ) for all p T. We may assume that w = w α is the reflection with respect to a root α Φ. The statement we need follows from the formula p w α (p) = α, p α A, where w α is the reflection with respect to α. The next step is to check that the map ψ, defined on generators, factors through relations. It follows immediately from the definition of ψ that ψ([p]) = ψ([p ]) for conjugate loops p and p. Let p be a contractible loop in G, i.e. p coincides with the composition G m,f SL 2,F q G for some group homomorphism q. Replacing p by its conjugate we may assume that im(p) T. Since the group SL 2,F coincides with its commutant, the image of p belongs also to the commutant G of G. Let π : G G be the universal covering (with a simply connected group G ). Since SL 2,F is a simply connected group, the homomorphism q lifts to G, i.e. p equals a composition p : G m,f SL 2,F 5 q G π G G.

6 Let T = G T and T = π 1 (T ). Then T is a maximal torus in G and the image of the composition G m,f SL 2,F q G belongs to T. Hence, the co-character p T belongs to the image of T T T which coincides with A(Φ) since G is a simply connected group. Assume finally that the images of loops p and p commute and hence generate a subtorus T G. There exists g G(F ) such that gt g 1 T. Modifying p and p by Int(g), we may assume that their images belong to T. Then the image of pp also belongs to T and the relation ψ(pp ) = ψ(p) + ψ(p ) holds. Thus, we have proved that ψ is a well defined map being clearly the inverse to ϕ. Now let G be an algebraic group over an arbitrary field F. The fundamental group π 1 (G) of G is, by definition, the group π 1 (G Falg ), where F alg is an algebraic closure of F. Corollary 1.3. The fundamental group π 1 (G) of a reductive group G is an abelian finitely generated group. It is finite iff G is a semisimple group. It is trivial iff G is a simply connected semisimple group. Remark 1.4. If G is an algebraic group over the field of complex numbers, then π 1 (G) coincides with the ordinary fundamental group of the corresponding Lie group [1, Th.5.47]. The algebraic definition of the fundamental group, we present here, does not depend on the choice of a maximal torus. That makes evident the functorial properties of the fundamental group. Remark 1.5. Let G be a semisimple group, Λ be the lattice of abstract weights, Λ T. Then the (finite) group π 1 (G) is dual to the abstract group Λ/T (with respect to the pairing between T and T ) which, in its turn, being considered as a constant algebraic group, is dual to the kernel C of the universal covering π : G G viewed as a diagonalizable group scheme. More precisely, let M = Λ/T, then π 1 (G) = Hom(M, Q/Z), C = ker(π) = Spec F [M], in particular, C(F ) = Hom(M, F ). Proposition 1.6. Let G be a reductive group, G be the commutant of G. Then π 1 (G ) = π 1 (G) tors. Proof. The torus T = T G is a maximal one in G, A(Φ) T T. Since T/T is a torus, it follows that T /T = (T/T ) is a torsion-free 6

7 group. Hence, π 1 (G ) = T /A(Φ) = (T /A(Φ)) tors = π 1 (G) tors. Corollary 1.7. The fundamental group π 1 (G) is torsion free iff G is a simply connected group Factorial groups. An algebraic group G over a field F is called factorial if for any finite field extension E/F the Picard group Pic(G E ) is trivial. Example 1.8. Let T be an algebraic torus over F, L/F be a finite Galois splitting field, Π = Gal(L/F ). The torus T is factorial iff T is coflasque, i.e. H 1 (Π, T (L)) = 0 for any subgroup Π Π (see [4]). A quasi-trivial torus (i.e. a torus of the form GL 1,C for an étale F -algebra C) is factorial. Example 1.9. Let G be a semisimple algebraic group over F, π : G G be the universal covering, C = ker(π). By [17, Lemme 6.9] there is a natural isomorphism Pic(G) C. Hence G is factorial iff G is simply connected. The following Proposition reduces the case of a reductive group to these two examples. Proposition Let G be a reductive group over a field F, G be the commutant of G. Then G is factorial iff G and the torus G/G are factorial, i.e. G is simply connected and the torus G/G is coflasque. Proof. For any field extension E/F there is an exact sequence [17, Cor. 6.11] (2) (G E ) Pic(G/G ) E Pic(G E ) Pic(G E ). If G and G/G are factorial groups, then it follows from the sequence that G is also factorial. Conversely, assume that G is factorial. Since G is semisimple, the first term in (2) is trivial, hence Pic(G/G ) E =0 and G/G is factorial. The group Pic(G sep ), being the direct limit of Pic(G E ), is trivial. By [17, Rem ], the right map in (2) is surjective if E = F sep. Hence Pic(G sep) = 0 and G is simply connected and hence is also factorial. Remark It follows from Corollary 1.7 and Example 1.9 that the fundamental group π 1 (G) of a factorial reductive group G is torsion free. 7

8 1.3. Reductive groups of inner type. A connected solvable group G over F is called split if there is a series of subgroups G = G 0 G 1 G n = 1 such that G i /G i+1 is isomorphic either to G a,f or to G m,f for i = 0, 1,..., n 1 (see [2, Def. 15.1]). A reductive group G over F is called split if it has a maximal torus which splits over F (see [2, Th.18.7]). If G is the commutant of G, then G splits iff G and G/G are split groups. A reductive group G over F is called quasi-split if it has a Borel subgroup defined over F. Let G and H be algebraic groups over F. The group G is called a twisted form of H if G sep and H sep are isomorphic over F sep. Let G be a reductive group over F, G d be its split twisted form. Denote by G d the corresponding split adjoint group G/Center(G). The group G is of inner type if the class of the 1-cocycles in H 1 (F, Aut(G d sep)), corresponding to G, belongs to the image of the map H 1 (F, G d (F sep )) H 1 (F, Aut(G d sep)) induced by the conjugation homomorphism Int : G d (F sep ) Aut(G d sep). Since inner automorphisms act trivially on the center Z of G, it follows that Z is a diagonalizable group scheme. A quasi-split reductive group of inner type is split. Lemma Let G be a reductive group, G be the commutant of G. Then G is of inner type iff G is of inner type and the torus G/G splits. Proof. Assume that G is of inner type. Then the cocycle ϕ : Γ = Gal(F sep /F ) Aut(G d sep) is of the form ϕ(γ) = Int(g γ ) for some g γ G d (F sep ). Since the restriction of ϕ(γ) to the split twisted form (G ) d = (G d ) of G is an inner automorphism and G d (F sep ) = (G ) d (F sep ), it follows that G is of inner type. The inner automorphism ϕ(γ) clearly induces the trivial automorphism on G d sep/(g ) d sep. Hence, G/G is a split torus. Conversely, assume that G is of inner type and G/G is a split torus. Then the cocycle ϕ : Γ Aut(G d sep ), defining G, being restricted to (G ) d, is of the form ϕ(γ) G = Int(g γ ) G for some g γ G d (F sep ) = (G ) d (F sep ) and ϕ(γ) induces the trivial automorphism of G d /(G ) d. Let Z d be the connected reduced center of G d. Since the natural homomorphism Z d G d /(G ) d is an isogeny, ϕ(γ) restricted to Z d is the 8

9 trivial automorphism. Hence, ϕ(γ) = Int(g γ ) since G d is generated by (G ) d and Z d. Corollary Let G be a reductive group of inner type. Then G is factorial iff π 1 (G) is torsion free. Proof. Follows from Proposition 1.10 and lemma Example The group GL 1,A for a central simple F -algebra A is a factorial reductive group over F of inner type Category of representations. Let G be an algebraic group defined over a field F. Denote by Rep(G) the category of finite dimensional linear representations of G (called sometimes G-modules). It is an abelian category in which any object has finite length. Hence by [16, Th.4, Cor.1] (3) K n (Rep(G)) = K n (End G (V ) op ). V irred. In particular, K 0 (Rep(G)) is a free abelian group freely generated by the classes of irreducible representations. We denote this group by R(G). It has a ring structure with respect to the tensor product. Any character χ G, being a 1-dimensional representation, can be considered as an element of R(G). There is the rank homomorphism of rings rk : R(G) Z, taking the class of a G-module V to dim V. We denote its kernel by J(G). For a field extension L/F the exact functor Rep(G) Rep(G L ); [V ] [V F L] induces a ring homomorphism R(G) R(G L ). If L/F is a finite extension, then we have an exact functor in opposite direction taking a G L -module over L to itself considered as a G-module over F. It induces the corestriction map R(G L ) R(G). It is clear that the composition R(G) R(G L ) R(G) coincides with the multiplication by the degree (L : F ). In particular, the homomorphism R(G) R(G L ) is injective. A homomorphism f : H G of algebraic groups over F induces a ring homomorphism f : R(G) R(H). Thus, we can consider R(H) as a R(G)-module. If f is the imbedding of a subgroup H to G, the homomorphism f is called the restriction map. Lemma Let G be an algebraic group over a field F, U be a unipotent normal subgroup in G. Then U acts trivially on any irreducible G-module V. 9

10 Proof. Since U is normal in G, the subspace of fixed elements V U is a G-submodule in V. By the fix point theorem [8, Th.17.5], over an algebraic closure of F, (V U ) alg = V U alg alg 0 if V 0, hence V U 0 and, therefore, V U = V since V is irreducible. Corollary There is a natural bijection between the sets of irreducible representations of G and G/U. In particular, the natural map R(G/U) R(G) is an isomorphism. Example Let D be a diagonalizable group over F. Since all irreducible representations of D are 1-dimensional [6, IV, 1, Prop.1.6], it follows from (3) that K n (Rep(D)) K n (F )[D ], where D = Hom(D, G m,f ). In particular, R(D) = Z[D ] is a Laurent polynomial ring over Z. Example Let B be a split solvable group over F. Denote by U the unipotent radical of B. By Corollary 1.16, example 1.17 and (3), K n (Rep(B)) K n (Rep(B/U)) K n (F )[(B/U) ] K n (F )[B ], since (B/U) = B and B/U is a split torus. In particular, R(B) = Z[B ]. Example Let G be a split reductive group over F, T G be a split maximal torus. The Weyl group W acts naturally on R(T ) and the restriction homomorphism induces an isomorphism (see [22]) R(G) R(T ) W = Z[T ] W. Example Let G be a split simply connected semisimple group over F. Denote by ρ 1,..., ρ r the classes of the fundamental representations in R(G) (with the highest weights being fundamental ones, [8, Ch.XI]). Then R(G) is the polynomial ring Z[ρ 1,..., ρ r ] (see [1, Th.6.41], [20, Lemma 3.1], [22]). Example Let ρ i be the class in R(GL n,f ) of the i th -exterior power representation of GL n,f. Then R(GL n,f ) = Z[ρ 1,..., ρ n 1, ρ ±1 n ]. Proposition ([20, Th.1.3]) Let G be a split reductive group with π 1 (G) torsion free, T be a maximal split torus in G. Then there is a set of characters of T, indexed by elements of the Weyl group W, forming a base of R(G)-module R(T ). Thus, R(T ) is a free R(G)-module of rank W. 10

11 Let G be a split reductive group defined over a field F, G be the commutant of G, Z be the connected reduced center of G. Denote by C the kernel of the natural isogeny π : G def = G Z G. Choose a Borel subgroup B G and set B = B G and B = π 1 (B). Clearly, B is a Borel subgroup in G and B = B Z. Lemma If π 1 (G) is torsion free, then the natural homomorphism ϕ : R(B) R(G) R(G) R(B) is an isomorphism. Proof. First of all we prove that ϕ is surjective. Since G/G = Z = B/B, the image of the restriction R(G) R(B) contains the classes of all characters of B which are trivial on B. The image of the natural map R(B) R(B) contains the classes of all characters of B which are trivial on C. Since C B = 1, it follows that (B) is generated by (B/B ) and (B/C) = B, hence ϕ is surjective. But by Proposition 1.22, R(B) over R(G) and R(B) over R(G) are free modules of the same rank W (G) = W (G), so that ϕ is an isomorphism. Let G be an algebraic group over F and A be a separable F -algebra. Denote by Rep(G, A) the category of G-modules V having an additional structure of left A-module such that the action of G and A commute. Any object in Rep(G, A) has finite length. Set R(G, A) = K 0 (Rep(G, A)). Clearly, R(G, F ) = R(G) and for any field extension L/F the canonical homomorphism R(G, A) R(G L, A L ) is injective. Example If G be a diagonalizable group over F, then the canonical homomorphism K 0 (A) Z[G ] = K 0 (A) R(G) R(G, A) is surjective. Indeed, if V is a G-module being also a left A-module, then for any character χ G the weight subspace V χ in V is an A-submodule Tits algebras. Let G be a simply connected semisimple group over F and ρ : G GL 1,A, be an algebraic group homomorphism, where A is a central simple F -algebra. We call A a Tits algebra, if ρ sep is an irreducible representation over F sep. A Tits algebra A is uniquely determined, up to an isomorphism, by the representation ρ sep. (For this and further results in this subsection see [22].) Denote by C the center of G. It is a group scheme of multiplicative type. The restriction of the homomorphism ρ to C is given by the multiplication by some character χ C = Hom(C, G m,f ). Moreover, 11

12 any character in C can be obtained in such a way. The class of the corresponding Tits algebra A in Br(F ) does not depend on the choice of ρ, but only on the restriction of ρ to the center C. Thus, we get the Tits homomorphism β : C Br(F ). A Tits algebra A is called a fundamental Tits algebra if ρ sep is a fundamental representation. If G is a group of inner type, then the fundamental Tits algebra exists for any fundamental representation of G sep. Example The fundamental Tits algebras of SL 1,A, where A is a central simple F -algebra of degree n, are the λ-powers λ i A of A, i = 1, 2,..., n 1 (see [9]). 2. Equivariant K-theory The equivariant K-theory was developed by R.Thomason in [21] Definitions. Let G be a group scheme over a scheme Y. A scheme X over Y is called a G-scheme if an action morphism θ : G Y X X of the group scheme G on X over Y is given. A G-module M over X is a coherent O X -module M together with the isomorphism of O G X -modules (4) ρ : θ (M) p 2 (M), (where p 2 : G Y X X is a projection), satisfying the cocycle condition p 23 (ρ) (id G θ) (ρ) = (m id X ) (ρ), where m : G Y G G is the multiplication (see [21]). The abelian category of G-modules over a G-scheme X we denote by M(G; X ). We set also K n def (G; X ) = K n (M(G; X )). The functor K n (G;?) is contravariant with respect to flat G-morphisms and is covariant with respect to projective G-morphisms of G-schemes [21, 1.5]. If G is a trivial group scheme, then K n(g; X ) = K n(x ). Consider the full subcategory P(G; X ) in M(G; X ) consisting of locally free O X -modules. This category is naturally equivalent to the category of vector G-bundles over X with the G-action compatible with one on X. We set K n (G; X ) def = K n (P(G; X )). The functor K n (G;?) is contravariant with respect to G-morphisms of G-varieties. If G is a trivial group scheme, then K n (G; X ) = K n (X ). 12

13 The inclusion of categories P(G; X ) M(G; X ) induces a homomorphism K n (G; X ) K n (G; X ). Example 2.1. Let µ : G GL(V ) be a finite dimensional representation of an algebraic group G over a field F. One can view the G-module V as a vector G-bundle over Spec F. Clearly, we obtain an equivalence of categories Rep(G) and P(G; Spec F ) = M(G; Spec F ). Hence there is a natural isomorphism R(G) K 0 (G; Spec F ) = K 0(G; Spec F ). Therefore, for any G-scheme X over F the inverse image map with respect to the structure morphism X Spec F is a ring homomorphism R(G) K 0 (G; X), making the latter group (and also K 0 (G; X)) a module over R(G). In particular, one can view characters of G as the elements of K 0 (G; X) and K 0 (G; X). Let g : H G be a group scheme homomorphism over Y and X be a G-scheme over Y. The composition H Y X g id G Y X θ X makes X an H-scheme. Given a G-module M with the G-module structure defined by the isomorphism ρ in (4), one can introduce an H- module structure on M by (g id) (ρ). Thus, we obtain exact functors M(G; X ) M(H; X ), P(G; X ) P(H; X ) inducing group homomorphisms K n (G; X ) K n (H; X ), K n(g; X ) K n (H; X ). In the particular case when H is a subscheme in G and g is the imbedding, we call these maps the restriction homomorphisms. Let f : Z Y be a morphism of schemes, X be a scheme over Z and G be a group scheme over Y. Assume that G acts on X over Y (we consider X as a scheme over Y via f). Denote by θ the action morphism G Y X X. Since there is a natural isomorphism G Y X (G Y Z) Z X, the scheme X over Z can be considered as a G Y Z-scheme. A G- module over X (considered as a scheme over Y ) at the same time is a G Y Z-module over X (considered as a scheme over Z). Hence, we have an equivalence of categories M(G; X ) M(G Y Z; X ) 13

14 inducing an isomorphism K n(g; X ) K n(g Y Z; X ). Example 2.2. Let G be a group scheme over Y, Z Y be an open subscheme and X be a G-scheme over Z. Then there is a natural isomorphism K n(g; X ) K n(g Z ; X ). Example 2.3. Let G be a group scheme over Y, y Y and X be a scheme over the residue field F (y). Assume that X, considered as a scheme over Y via Spec F (y) Y, is a G-scheme. Then K n(g; X) K n(g y ; X), where G y = G Y Spec F (y) is the fiber of G over y Torsors. Let G be an algebraic group over F and p : X Y be a G-torsor. Then for any coherent O Y -module M there is a natural G-module structure on p (M) over X (see [11, Sec. 3]). Thus, there is an exact functor p 0 : M(Y ) M(G; X), M p (M). The following statement is well-known (see, for example [11, Prop. 33]). Proposition 2.4. The functor p 0 is an equivalence of categories. In particular, the homomorphism p : K n(y ) K n(g; X), induced by p 0, is an isomorphism. By the faithfully flat descent (see [13, Ch.1, Th. 2.23]), the assignment Z X Y Z induces an equivalence of categories of affine schemes over Y and affine G-schemes over X. For an affine G-scheme V over X, the corresponding affine scheme W over Y, so that V = X Y W, we will denote by V/G. Assume that a torus T acts on a quasi-projective scheme X and P is another torus containing T. Then T acts on P X by t(p, x) = (pt 1, tx). Lemma 2.5. There exists a T -torsor p : P X Y. Proof. If P is a split torus, then P = T S for some subtorus S P and the morphism p : P X S X, (p, x) (p 2 (p), p 1 (p)x), where p 1,2 are projections of P onto T and S, is clearly a T -torsor. In the general case one can use the descent argument since X is quasi-projective (see [19, Ch.III, 1, Prop. 5]). 14

15 2.3. Basic results in the equivariant K-theory. We formulate three propositions in the equivariant algebraic K-theory developed by R.Thomason in [21]. In all of them G is an algebraic group over a field F and X is a G-scheme. Proposition 2.6. [21, Th. 2.7] (Localization) Let Z X be a closed G-subscheme and U = X Z. Then there is an exact sequence K n+1 δ (G; U) K n i (G; Z) where i : Z X and j : U X are imbeddings. K n (G; X) j K n (G; U)..., Proposition 2.7. [21, Th. 4.1] (Homotopy invariance) Let L be a locally free G-module over X, V = Spec(S L) X be the associated vector bundle space with the induced G-action. Assume that f : E X is a torsor under the vector bundle space V (considered as a group scheme over X) and G acts on E so that f and the action map V X E E are G-equivariant. Then is an isomorphism. f : K n (G; X) K n (G; E) Corollary 2.8. If G acts linearly on the affine space A n F, then the projection p : X A n F X induces the inverse image isomorphism p : K n(g; X) K n(g; X A n F ). Corollary 2.9. Let G act on A 1 F by the multiplication by a character χ G, i : X X {0} X A 1 F be the closed imbedding. Then the composition K n(g; X) i K n (G; X A 1 F ) (p ) 1 K n(g; X) is the multiplication by 1 χ 1 K 0 (G; X). Proof. Since p i = id, by Corollary 2.8, i is an isomorphism. Hence, by the projection formula, i (i (u)) = i (1) u for any u K n(g; X A 1 F ). Thus, it suffices to show that i (1) = 1 χ 1. But this follows from the exactness of the sequence of G-modules over X A 1 F 0 O X A 1 F χ 1 O X A 1 F i (O X ) 0. Proposition [21, Prop. 6.2] (Faddeev-Shapiro Lemma) Let H be a closed subgroup in G, Y = G/H. Then the restriction along the H-map X X {H} X Y induces equivalences of categories P(G; X Y ) P(H; X), 15 M(G; X Y ) M(H; X).

16 In particular, there are natural isomorphisms K n (G; X Y ) K n (H; X), K n(g; X Y ) K n(h; X). Corollary Let j : Y A n F be an open G-equivariant imbedding into the affine space A n F on which G acts linearly. Then the composition K n p (G; X) K n (G; X An F ) (id X j) K n (G; X Y ) K n (H; X), where p : X A n F homomorphism. X is the projection, coincides with the restriction Proof. The composition of the first two homomorphism is the inverse image with respect to the projection X Y X. It remains to notice that the composition of the closed imbedding X X {H} X Y, inducing the last isomorphism, with this projection is the identity. For any representation ρ : H GL(V ) one can associate the G- vector bundle P ρ = (G V )/H over Y = G/H, where the H-action on G V is given by h (g, v) = (gh 1, ρ(h)(v)). Corollary The assignment ρ [P ρ ] induces an isomorphism R(H) K 0 (G; Y ) K 0 (G; Y ) Exact sequence. Let G be an algebraic group over F. Assume we are given a nontrivial character χ : G G m,f. Denote the kernel of χ by H. Proposition Let X be a G-scheme. sequence Then there is an exact K n+1 (H; X) δ K n 1 χ (G; X) K n res (G; X) K n (H; X).... Proof. We consider X A 1 F as a G-scheme with respect to the action of G on A 1 F given by the multiplication by χ. In the localization exact sequence K n (G; X) i K n (G; X A 1 F ) j K n (G; X G m,f )... where i : X = X {0} X A 1 F and j : X G m,f X A 1 F are imbeddings, the second term is identified with K n (G; X) by Corollary 2.8 and the third with K n (H; X) since G/H G m,f as G-schemes (see Proposition 2.10). With this identifications i = 1 χ 1 and j = res by Corollaries 2.9 and

17 2.5. Solvable groups acting on schemes. Lemma Let G be an algebraic group over F, H be a closed subgroup in G such that the factor-variety G/H is isomorphic to A 1 F. Then for any G-scheme X the restriction map K n(g; X) K n(h; X) is an isomorphism. Proof. Since the group Aut(A 1 F ) consists of linear automorphisms, the action of G on A 1 F, induced by an isomorphism G/H A1 F, is given by g y = χ(g)y + ρ(g) where χ is a character of G and ρ is a regular function on G. Consider the trivial linear vector bundle V = X A 1 F over X as a G- scheme with the action natural on X and given by the multiplication by the character χ on A 1 F. One can view V as an (additive) group scheme over X. Then the G-scheme E = X (G/H) X A 1 F can be considered as a V -torsor over X. Denote the action of V on E by. The formula (with (x, y) V and (x, y ) E) g ((x, y) (x, y )) = g (x, y + y ) = (gx, χ(g)(y + y ) + ρ(g)) = (gx, χ(g)y) (gx, χ(g)y + ρ(g)) = [g (x, y)] [g (x, y )] shows that the -action of V on E commutes with one of G on V and E. Hence, by Proposition 2.7, the projection p : X (G/H) X induces an isomorphism p : K n (G; X) K n (G; X (G/H)). Composing this map with the isomorphism K n (G; X (G/H)) K n (H; X) (Proposition 2.10) we get the restriction homomorphism by Corollary Corollary Let B be a split solvable group, T B be a maximal torus, X be a B-scheme. Then the restriction homomorphism K n (B; X) K n (T ; X) is an isomorphism. Corollary Let U be a split unipotent group, X be a U-scheme. Then the restriction homomorphism K n (U; X) K n (X) is an isomorphism Unimodular form. Let G be a split reductive group defined over a field F, B be a Borel subgroup in G. Denote by p : G/B Spec F the structure morphism. Since G/B is a projective variety, the direct image p : R(B) = K 0(G; G/B) K 0(G; Spec F ) = R(G) 17

18 is well defined. Consider the R(G)-bilinear form on R(B): u, v = u, v G = p (u v). Proposition Let G be a split reductive group with π 1 (G) torsion free. Then the bilinear form, G is unimodular. Proof. We use the notation of Lemma Since G is a simply connected semisimple group (Corollary 1.7), the bilinear form, G is unimodular ([7], [14, Th. 8.1.]). Since R(G) = R(G ) Z R(Z) and R(B) = R(B ) Z R(Z), it follows that, G =, G R(Z) is an unimodular form. Lemma 1.23 implies that, G =, G R(G) R(G). Since, G is unimodular, the determinant d of, G is invertible in R(G) (we identify R(G) with a subring of R(G)). Thus, d is the class of a character of G being trivial on C and therefore d is the class of a character of G and hence is invertible in R(G) The category C(G). Let G be an algebraic group over a field F, A be a separable F -algebra on which G acts by algebra automorphisms. A G-A-module over a G-scheme X is a G-module M over X being also a left A F O X -module such that g(am) = ga gm for all a A, m in M and g in G. We consider the abelian category M(G; X, A) of G-A-modules and morphisms of A F O X - and G- modules. Set K n (G; X, A) = K n(m(g; X, A)). The functor K n (G;?, A) is contravariant with respect to flat G- morphisms and is covariant with respect to projective G-morphisms of G-schemes. The group K n(g; X, F ) coincides with K n(g; X). Consider also the full subcategory P(G; X, A) in M(G; X, A) consisting of locally free O X -modules. The K-groups of the category P(G; X, A) we denote by K n (G; X, A) = K n (P(G; X, A)). The group K n (G; X, F ) coincides with K n (G; X). Example Assume that G acts trivially on A. As in Example 2.1 we have an equivalence of categories Rep(G, A) P(G; Spec F, A) = M(G; Spec F, A). Hence there is a natural isomorphism R(G, A) K 0 (G; Spec F, A) = K 0(G; Spec F, A). 18

19 If H G is a subgroup, then as in Proposition 2.10 there exists a natural isomorphism K 0 (G; G/H, A) K 0 (H; Spec F, A) = R(H, A). Lemma Let X be a quasi-projective smooth G-scheme. Then for any M M(G; X, A) there is a surjection P M with P P(G; X, A). Proof. By [21, Lemma 5.6] there is a surjection f : P M with P P(G; X). If we consider P = A F P, then f : P M, defined by f(a p ) = af (p ), is the desired surjection. The Resolution Theorem [16, Th. 3] then gives Proposition (compare [21, Cor. 5.8]) For any quasi-projective smooth G-scheme X over F the natural homomorphism K n (G; X, A) K n(g; X, A) is an isomorphism. In [14] I.Panin has defined a motivic category C(G) whose objects are the pairs (X, A), where X is a smooth projective G-scheme over F and A is a separable F -algebra on which G acts by F -algebra automorphisms. Morphisms in C(G) are defined as follows: Hom C(G) ((X, A), (Y, B)) = K 0 (G; X Y, A op F B). If u : (X, A) (Y, B) and v : (Y, B) (Z, C) are two morphisms in C(G), then their composition is defined by the formula v u = v u def = p 13 (p 23(v) B p 12(u)), where p 12, p 13 and p 23 are projections from X Y Z to X Y, X Z and Y Z respectively. The identity endomorphism of (X, A) in C(G) is the class [A F O ], where X X is the diagonal, in K 0 (G; X X, Aop F A) = K 0 (G; X X, A op F A) = End C(G) (X, A) (see Proposition 2.20). We will simply write X for (X, F ) and A for (Spec F, A) in C(G). A group homomorphism f : H G induces a functor f : C(G) C(H). For any scheme Z over F and n Z there is the functor K Z n : C(G) Ab, taking a pair (X, A) to K n(g; Z X, A) and a morphism to v Hom C(G) ((X, A), (Y, B)) = K 0 (G; X Y, A op F B) K Z n (v) : K n(g; Z X, A) K n(g; Z Y, B) 19

20 given by the formula K Z n (v)(u) = v u (see [14, 6.1]). The functor K Spec F n we will simply denote by K n. The following statement is a slight generalization of [14, Th. 6.6]. Proposition Let G be a split reductive group over a field F with π 1 (G) torsion free, B G be a Borel subgroup, Y = G/B, u 1, u 2,..., u m R(B) = K 0 (G; Y ) be a free R(G)-base of R(B). Then the element u = (u i ) K 0 (G; Y, F m ) = R(B) m defines an isomorphism F m Y in the category C(G). Proof. Let v 1, v 2,..., v m be the dual R(G)-base of R(B) with respect to the unimodular bilinear form, G (see Proposition 2.17). The element v = (v i ) K 0 (G; Y, F m ) can be considered as a morphism Y F m in C(G). The fact that u and v are dual bases is equivalent to v u = id. In order to prove that u v = id it suffices to show that the R(G)- module K 0 (G, Y Y ) is generated by m 2 elements (see [14, Cor. 7.3]). It is proved in [14, Prop. 8.4] for a simply connected group G, but the proof goes through for a reductive group G with π 1 (G) torsion free. Let G be a reductive quasi-split group defined over F with π 1 (G) torsion free, B G be a Borel subgroup, T B be a maximal torus defined over F. The Galois group Γ acts on the character group T (F sep ), on the normalizer N Gsep (T sep ) and hence on the Weyl group W. Denote by u w R(B sep ), w W, the Steinberg base of the free R(G sep )-module R(B sep ) consisting of the classes of characters (Proposition 1.22). The Galois action on R(B sep ) agrees with one on W, i.e. γ(u w ) = u γw for any γ Γ = Gal(F sep /F ) and w W. Therefore, the homomorphism B sep (G m,fsep ) W, given over F sep by the characters u w, is Γ-invariant and hence descents to a representation u : B GL 1,A, where A is the étale F -algebra corresponding to the finite Γ-set W. We can view the left A-module A together with the B-action, given by b a = u(b)a, as an element of the group K 0 (B; Spec F, A) K 0 (G; Y, A), where Y = G/B and the algebra A is considered with the trivial G- action (see Example 2.18). Hence it defines a morphism A Y in C(G), which we also denote by u. Proposition The morphism u : A Y in C(G) is an isomorphism. 20

21 Proof. The dual base (v w ), w W, of R(B sep ) with respect to the unimodular form, Gsep defines in an analogous way a morphism v : Y A. By the proof of Proposition 2.21, u sep and v sep are mutually inverse isomorphisms. Since for Z = Spec F, End C(G) (A) = K 0 (G; Z, A A) = R(G, A A) R(G sep, A sep A sep ) = K 0 (G sep ; Z sep, A sep A sep ) = End C(Gsep)(A sep ), (see subsection 1.4), one has v u = id. By [14, 11.10(4)], hence, u v = id. End C(G) (Y ) = K 0 (G; Y Y ) K 0 (G sep ; Y sep Y sep ) = End C(Gsep)(Y sep ), We will need the following Lemma in the smooth projective case. Lemma Let G be an algebraic group, H G be a closed subgroup with a group homomorphism π : G H being the identity on H, X be a smooth projective G-scheme. Consider the action of G G on X X given by g(x, x ) = (π(g)x, gx ). Assume that the restriction homomorphism K 0 (G; X X) K 0 (H; X X) is surjective. Then the restriction homomorphism K n (G; X) K n (H; X) is a split surjection. Proof. Denote by Ẋ the scheme X together with the G-action given by the formula g x = π(g)x. Since the restriction map res G/H : Hom C(G) (Ẋ, X) = K 0(G; Ẋ X) K 0 (H; X X) = Hom C(H) (X, X) is surjective, there is v Hom C(G) (Ẋ, X) such that res G/H(v) = id X. Consider the diagram K n π (H; X) K n (G; Ẋ) res K n (H; X) K K n(v) K n (G; X) res n (H; X), where the square is commutative since res G/H (v) = id X. The equality res π = id implies that the composition in the top row splits the restriction homomorphism. 21

22 2.8. Local-to-global principle. Let Y be a scheme over a field F. Assume that we are given: (i) For any n Z and any (locally closed) subscheme Z Y an abelian group A n (Z), which is trivial if n < 0; j (ii) For all subschemes Z 1 Z 2 in Y, where j is an open imbedding, for any n, the inverse image group homomorphism j : A n (Z 2 ) A n (Z 1 ) such that id = id and (j 1 j 2 ) = j2 j1 for any composable open imbeddings j 1 and j 2 ; i (iii) For all subschemes Z 1 Z 2 in Y, where i is a closed imbedding, for any n, the direct image group homomorphism i : A n (Z 1 ) A n (Z 2 ) and the connecting homomorphism δ : A n (Z 2 Z 1 ) A n 1 (Z 1 ). The collection of groups A = {A n (Z)}, inverse, direct images and connecting homomorphisms is called a local group over Y if for any Z 1 and Z 2 as in (iii) the infinite sequence (5) A n (Z 1 ) i A n (Z 2 ) j A n (Z 2 Z 1 ) δ A n 1 (Z 1 )... is exact (here j : Z 2 Z 1 Z 2 is the open imbedding). Let y Y be a point. The groups A n (U) for all non-empty open subsets U {y} (the latter is considered as a reduced closed subscheme in Y ) form a direct system with respect to inverse images. We set A n (y) = lim A n (U). U {y} It is clear that if y is a closed point then A n (y) = A n ({y}). Let B be another local group over Y. A morphism f : A B of local groups over Y is a collection of homomorphisms f n (Z) : A n (Z) B n (Z) for all n Z and Z Y, commuting with the direct, inverse images and the connecting homomorphisms. For any n Z and y Y the morphism f induces a group homomorphism f n (y) : A n (y) B n (y). Remark Taking the empty subscheme for Z i in the exact sequence (5) one gets A n ( ) = 0. Example Let X be a scheme over Y. Assume that a group scheme G over Y acts on X over Y. For any subscheme Z Y set (see Example 2.2) A n (Z) = K n (G; X Z) = K n (G Z; X Z ). Inverse, direct images and the localization sequence in the equivariant algebraic K-theory make A a local group over Y. Since the K -groups commute with direct limits, one has A n (y) = K n(g; X y ) = K n(g y ; X y ) 22

23 for any y Y, where G y and X y are fibers over y (see Example 2.3). Lemma Let Z be a subscheme in Y, i : Z red Z be the natural closed imbedding. Then the direct image is an isomorphism. i : A n (Z red ) A n (Z) Proof. One takes Z 1 = Z red and Z 2 = Z in the exact sequence (5) (see also Remark 2.24). Theorem (Local-to-global principle) Let f : A B be a morphism of local groups over Y. 1. If f n (y) : A n (y) B n (y) is an isomorphism for all n Z and y Y, then f n (Z) is an isomorphism for any n Z and Z Y. 2. If f 0 (y) : A 0 (y) B 0 (y) is a surjection for any y Y, then f 0 (Z) is surjective for any Z Y. Proof. We prove that f n (Z) : A n (Z) B n (Z) is an isomorphism (resp. surjective if n = 0) by induction on the dimension of a subscheme Z Y. The induction base, the case Z =, is clear by Remark Assume that we have proved the statement for all subschemes of dimension less than the dimension of Z. We would like to prove that f n (Z) is an isomorphism (resp. surjective if n = 0). We prove this statement by induction on the number of irreducible components of Z. Assume first that Z is irreducible. By Lemma 2.26, we may assume that Z is reduced. Surjectivity of f n (Z). Let z Z be the generic point and v B n (Z). Since f n (z) is a surjection, it follows that there exists a non-empty open subset U Z such that the image of v in B n (U) belongs to the image of f n (U). Set Z = Z U (considered as a reduced scheme). Since dim(z ) < dim(z), the left vertical homomorphism in the diagram δ i A n+1 (U) A n (Z ) An (Z) A n (U) A n 1 (Z ) f n(z) f n(u) f n+1 (U) f n(z ) B n+1 (U) δ B n (Z ) i Bn (Z) j j B n (U) δ δ B n 1 (Z ) is surjective and the right one is an isomorphism (in the case n = 0, since A 1 = 0 = B 1 ). Hence, by diagram chase, v im(f n (Z)). Injectivity of f n (Z). Let u ker(f n (Z)). Since the homomorphism f n (z) : A n (z) B n (z) is injective, one can find a non-empty open subset U Z such that the image of u in A n (U) is trivial. Set Z = Z U (considered as a reduced scheme). In the diagram above the map f n (Z ) is an isomorphism by the induction hypothesis since dim Z < 23

24 dim Z and f n+1 (U) is surjective as was proved above for all irreducible schemes of dimension at most dim Z. Hence, by diagram chase, u = 0. Let now Z be an arbitrary subscheme in Y, Z be an irreducible component in Z and U = Z Z. The number of irreducible components of U is less than ones of Z. Applying the induction hypothesis and the 5-lemma to the commutative diagram above one sees that f n (Z) is an isomorphism (resp. surjective if n = 0). The following statement is the first application of the local-to-global principle. Corollary Let U be a group scheme and X be a G-scheme over a scheme Y. Assume that for any point y Y the fiber U y is a split unipotent group. Then the restriction homomorphism is an isomorphism. res : K n (U; X ) K n (X ) Proof. Consider the following local groups on Y : A n (Z) = K n(u; X Z ) = K n(u Z ; X Z ) and B n (Z) = K n(x Z ) for a subscheme Z Y. For any y Y the homomorphism f n (y), where f : A B is the restriction morphism of local groups, coincides with res y : K n (U y; X y ) K n (X y) (see Examples 2.2 and 2.3). Since U y is a split unipotent group, by Corollary 2.16, res y is an isomorphism. Hence by the local-to-global principle, res is also an isomorphism. 3. Solvable case Let T be a split torus over F, χ T be a character such that T /(Z χ) is a torsion-free group. Then T = ker(χ) is a subtorus in T. Denote by π : T T a splitting of the imbedding T T. Proposition 3.1. Let X be a smooth projective T -scheme. Then the restriction homomorphism K n (T ; X) K n (T ; X) is a split surjection. Proof. Since T/T G m,f, by Corollary 2.8, Proposition 2.10 and the localization (Proposition 2.6), we have the following surjection K 0 (T ; X) K 0 (T ; X A1 F ) K 0 (T ; X G m,f ) K 0 (T ; X) which is nothing but the restriction map by Corollary Considering instead of X the product X X with the following T -action: t(x, x ) = (π(t)x, tx ), we get a surjective restriction homomorphism res : K 0(T ; X X) K 0(T ; X X). 24

25 Hence one can apply the Lemma Proposition 2.13 now implies Corollary 3.2. The sequence is split exact. 0 K n 1 χ (T ; X) K n res (T ; X) K n (T ; X) 0 Let A be a commutative ring, a 1, a 2,..., a r A and M be an A- module. The elements a i form an M-regular sequence if the multiplication by a i in M/(a a i 1 )M is injective for all i = 1, 2,..., n. Proposition 3.3. Let B be a split solvable group, X be a B-scheme and χ 1, χ 2..., χ r be a Z-base of the character group B. Then the elements 1 χ 1, 1 χ 2..., 1 χ r R(B) form a K n(b; X)-regular sequence over the ring R(B) and the restriction map induces an isomorphism Z R(B) K n(b; X) K n(b; X)/J(B) K n(b; X) = r K n (B; X)/ (1 χ i )K n (B; X) K n (X). i=1 Proof. Let T be a maximal torus in B. Since B = T and the restriction homomorphism K n (B; X) K n (T ; X) is an isomorphism by Corollary 2.15, it is sufficient to consider the case B = T. Our statement follows then from Corollary 3.2 by induction on the dimension of T. Corollary 3.4. T or R(B) p (Z, K n(b; X)) = { K n (X), if p = 0; 0, if p > 0. Proof. Since R(B) is a Laurent polynomial ring in variables χ i, the elements 1 χ i R(B) form a R(B)-regular sequence, the result follows from [18, IV-7] Spectral sequence. Let B be a group scheme over a scheme Y and χ : B T def = G r m,y be a group scheme homomorphism over Y. Denote the kernel of χ by U. Assume that for any point y Y the fiber χ y of χ is a surjective homomorphism and U y is a split unipotent group over F (y). Let χ be given by characters χ 1, χ 2,..., χ r in B. We define an action of B on the affine space A r Y by the rule b x = y where y i = χ i (b)x i. Denote by V i (i = 1, 2,..., r) the coordinate hyperplane in A r Y defined 25

26 by the equation x i = 0. Clearly, V i is a closed B-subvariety in A r Y and T = A r Y r i=1 V i. In [10] M.Levine has constructed a spectral sequence associated to a family of closed subschemes in a given scheme. This sequence generalizes the localization exact sequence. We adapt this sequence to the equivariant algebraic K-theory and also change indices making this sequence of homological type. Let X be a B-scheme over Y. The family of closed subsets Z i = X Y V i in X Y A r Y induce then the following spectral sequence Ep,q 1 = K q (B; X Y V I ) K p+q (B; X Y T ), I =p where V I = i I V i. Consider the composition K n (B; X Y T ) K n res (U; X ) K n (X ), where the first map is induced by the restriction along the map X X Y Y X Y T. Since B y /U y T F (y), by the local-to-global principle (Theorem 2.27) and Proposition 2.10 it follows that the first homomorphism is an isomorphism. By Corollary 2.28, the last map, and hence the composition, is also an isomorphism. In order to compute Ep,q 1, note that V I is an affine space over Y, hence the inverse image K q(b; X ) K q(b; X Y V I ) is an isomorphism by Corollary 2.8 and Theorem Thus, Ep,q 1 = K q (B; X ) e I I =p and by [10, p.419] the differential map d : Ep+1,q 1 Ep,q 1 is given by the formula p (6) d(x e I ) = ( 1) k (1 χ 1 k )x e I {i k }, k=0 where I = {i 0 < i 1 < < i p }. Consider Kozsul complex C built of the module K 0 (B; Y ) r over K 0 (B; Y ) and the system of elements 1 χ 1 i K 0 (B; Y ). More precisely, C p = K 0 (B; Y ) e I I =p and the differential d : C p+1 C p given by the rule formally equal to (6) with x K 0 (B; Y ). 26

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