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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 K0 (G; X ) ! K0 (X ) is surjective for any quasiprojective scheme X on which G acts if and only if Pic(G F E ) = 0 for all nite eld 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. 0
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The groups Kn0 (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 Gscheme X , i.e. a scheme X and a reductive algebraic group G acting on X over a eld F . The problem we study here is to compare the equivariant K -groups Kn0 (G; X ) and the ordinary ones Kn0 (X ) via the restriction homomorphism (1) res : Kn0 (G; X ) ! Kn0 (X ): It turns out that sometimes the group Kn0 (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 nite dimensional representations of the group H . Since in the latter category any object has nite length, the computation of the K -groups of this category is an easy task. For example, K0(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 HumboldtStiftung and the hospitality of the University of Bielefeld.
c 0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page
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kernel and cokernel of the homomorphism (1) sheds some light on the structure of Kn0 (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 nite eld 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 K0(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=G0, where G0 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 G0 is simply connected or the fundamental group 1(G) is torsion free. Let us brie y describe other results of the paper. 1. Split case. Let G be a split reductive group with 1(G) torsion free. We prove that for any G-scheme X there is a spectral sequence (Theorem 4.3) 2 = TorR(G) (Z; K 0 (G; X )) ) K 0 (X ): Ep;q p q p+q In particular, the restriction homomorphism (1) induces an isomorphism Z R(G) K00 (G; X ) ! K00 (X ): For example, K0(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) Kn0 (G; X ) ! Kn0 (X ): 2. Inner case. Assume that G is a reductive group of inner type with 1(G) torsion free. The idea is to use the variety Y of Borel subgroups in G as a base scheme instead of the eld F . Since for any point y 2 Y the group GF (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) 2 = TorK0(G;Y ) (K0 (Y ); K 0 (G; X )) ) K 0 (X ): Ep;q p q p+q 2
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 K0(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 K0(G=H ) is nitely generated (Theorem 7.7). In particular, the group K0(G) is nitely generated of Q -rank 1. It is also proved that the natural action of G on K00 (X ) (resp. Kn0 (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 Kn0 (G; X ) with Kn0 (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) Kn0 (G; X ) ! Kn0 (B ; X ) is an isomorphism (Proposition 4.1). In the second step we prove that the restriction homomorphism Kn0 (B ; X ) ! Kn0 (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 Kn0 (T ; X ) and Kn0 (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 de ned over the ground eld does not exist, we work over the variety of Borel subgroups. The main technical tool we use to come back to the ground eld is the local-to-global principle developed in 2.8. We use the word scheme for a separated scheme of nite type over a eld. If X is a scheme over F and L=F is a eld extension, then the scheme X F L we simply denote by XL. By Xsep we denote XFsep , where Fsep is a separable closure of F . For a scheme X by Kn0 (X ) (resp. Kn(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 ane group scheme of nite type over a eld. I thank M.Rost and I.Panin for helpful discussions. 1. Algebraic groups In this section we collect de nitions and preliminary results. 3
1.1. Fundamental group. Let G be an algebraic group de ned over an algebraically closed eld F . A loop in G is a group homomorphism (co-character) p : G m;F ! G. Two loops p and p0 are called conjugate if there is g 2 G(F ) such that p0(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 SL2;F ! G (we identify G m;F with the maximal torus of diagonal matrices in SL2;F by t 7! diag(t; t?1)). If p and p0 are two loops with commuting images, then the product pp0 given by (pp0 )(t) = p(t)p0(t) is also a loop. The fundamental group 1 (G) of the group G is the group de ned by generators and relations as follows. Generators are loops in G. For a loop p we denote by [p] the corresponding generator in 1(G). The relations are the following: 1. [p] = [p0] if p and p0 are conjugate; 2. [p] = 1 if p is contractible; 3. [pp0] = [p] [p0 ] if the images of p and p0 commute. Example 1.1. Let G be a commutative algebraic group. Since conjugate loops are equal and contractible loops are trivial (SL2;F coincides with its commutant), the group 1 (G) equals the group of co-characters G = Hom(G m;F ; G) of G. A group homomorphism G ! G0 clearly induces a homomorphism of fundamental groups 1 (G) ! 1 (G0). Let G be a reductive group, T G be a maximal torus. The inclusion T G induces a group homomorphism : T = 1(T ) ! 1 (G): Denote by T = Hom(T; G m;F ) the root system of G and by T the dual system of co-roots [8]. For any 2 denote by G the commutant of the centralizer of ker( ) in G. It is a semisimple group of rank 1 [8, Ch.IX]. Hence there is an epimorphism SL2;F ! G such that the composition G m;F ,! SL2;F ! G ,! G coincides with the co-root followed by the inclusion T ,! G, and thus the loop in G is contractible. Therefore, the map factors through the homomorphism ' : T=A() ! 1 (G); where A() is the subgroup in T generated by all co-roots 2 . Proposition 1.2. The homomorphism ' : T =A() ! 1 (G) is an isomorphism. 4
Proof. We de ne a homomorphism : 1(G) ! T =A() on generators as follows. Let p : G m;F ! G be a loop. Since im(p) is a torus in G, there exists g 2 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 ([p]) = Int(g) p + A(): In order to prove that is well de ned assume that Int(g0)(im(p)) T for some g0 in G(F ). Consider the compositions p1 = Int(g) p and p2 = Int(g0) p as the loops in T and denote im(pi) by Ti. Then p2 = Int(h) p1, where h = g0 g?1, and hence T2 = hT1 h?1 . Consider the centralizer H of the torus T1 in G. Since T1 = h?1T2 h h?1Th, clearly T and h?1Th are maximal tori in H and hence are conjugate in H by [8, Cor. A, p.135], i.e. there exists h0 2 H (F ) such that h?1Th = h0 Th0?1, or, equivalently, hh0 2 NG (T ) represents an element w in the Weyl group W . The group W acts naturally on T . Since p2 = Int(h) p1 = Int(hh0) p1 = w(p1), it suces to show that p ? w(p) 2 A() for all p 2 T. We may assume that w = w is the re ection with respect to a root 2 . The statement we need follows from the formula p ? w (p) = h ; pi 2 A; where w is the re ection with respect to . The next step is to check that the map , de ned on generators, factors through relations. It follows immediately from the de nition of that ([p]) = ([p0]) for conjugate loops p and p0. Let p be a contractible loop in G, i.e. p coincides with the composition
,! SL2;F !q G for some group homomorphism q. Replacing p by its conjugate we may assume that im(p) T . Since the group SL2;F coincides with its commutant, the image of p belongs also to the commutant G0 of G. Let : Ge0 ! G0 be the universal covering (with a simply connected group Ge0). Since SL2;F is a simply connected group, the homomorphism q lifts to Ge0, i.e. p equals a composition G m;F
0 p : G m;F ,! SL2;F !q~ Ge0 ! G ,! G: 5
Let T 0 = G0 \ T and Te0 = ?1(T 0). Then Te0 is a maximal torus in Ge0 and the image of the composition q~ e0 G m;F ,! SL2;F ! G belongs to Te0. Hence, the co-character p 2 T belongs to the image of Te0 ! T0 ! T which coincides with A() since Ge0 is a simply connected group. Assume nally that the images of loops p and p0 commute and hence generate a subtorus T 0 G. There exists g 2 G(F ) such that gT 0g?1 T . Modifying p and p0 by Int(g), we may assume that their images belong to T . Then the image of pp0 also belongs to T and the relation (pp0) = (p) + (p0) holds. Thus, we have proved that is a well de ned map being clearly the inverse to '. Now let G be an algebraic group over an arbitrary eld F . The fundamental group 1 (G) of G is, by de nition, the group 1(GFalg ), where Falg is an algebraic closure of F . Corollary 1.3. The fundamental group 1 (G) of a reductive group G is an abelian nitely generated group. It is nite i G is a semisimple group. It is trivial i G is a simply connected semisimple group. Remark 1.4. If G is an algebraic group over the eld of complex numbers, then 1 (G) coincides with the ordinary fundamental group of the corresponding Lie group [1, Th.5.47]. The algebraic de nition 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, be the lattice of abstract weights, T . Then the ( nite) group 1 (G) is dual to the abstract group =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 : Ge ! G viewed as a diagonalizable group scheme. More precisely, let M = =T , then 1(G) = Hom(M; Q =Z), C = ker() = Spec F [M ], in particular, C (F ) = Hom(M; F ). Proposition 1.6. Let G be a reductive group, G0 be the commutant of G. Then 1 (G0) = 1 (G)tors. Proof. The torus T 0 = T \ G0 is a maximal one in G0 , A() T0 T . Since T=T 0 is a torus, it follows that T =T0 = (T=T 0) is a torsion-free 6
group. Hence, 1 (G0) = T0 =A() = (T=A())tors = 1(G)tors: Corollary 1.7. The fundamental group 1 (G) is torsion free i G0 is a simply connected group. 1.2. Factorial groups. An algebraic group G over a eld F is called factorial if for any nite eld extension E=F the Picard group Pic(GE ) is trivial. Example 1.8. Let T be an algebraic torus over F , L=F be a nite Galois splitting eld, = Gal(L=F ). The torus T is factorial i T is co asque, i.e. H 1(0 ; T (L)) = 0 for any subgroup 0 (see [4]). A quasi-trivial torus (i.e. a torus of the form GL1;C for an etale F -algebra C ) is factorial.
Example 1.9. Let G be a semisimple algebraic group over F , : Ge ! G be the universal covering, C = ker(). By [17, Lemme 6.9] there is a natural isomorphism Pic(G) ' C . Hence G is factorial i G is simply connected. The following Proposition reduces the case of a reductive group to these two examples. Proposition 1.10. Let G be a reductive group over a eld F , G0 be the commutant of G. Then G is factorial i G0 and the torus G=G0 are factorial, i.e. G0 is simply connected and the torus G=G0 is co asque. Proof. For any eld extension E=F there is an exact sequence [17, Cor. 6.11] (2) (G0E ) ! Pic(G=G0)E ! Pic(GE ) ! Pic(G0E ):
If G0 and G=G0 are factorial groups, then it follows from the sequence that G is also factorial. Conversely, assume that G is factorial. Since G0 is semisimple, the rst term in (2) is trivial, hence Pic(G=G0)E =0 and G=G0 is factorial. The group Pic(Gsep ), being the direct limit of Pic(GE ), is trivial. By [17, Rem. 6.11.3], the right map in (2) is surjective if E = Fsep . Hence Pic(G0sep ) = 0 and G0 is simply connected and hence is also factorial. Remark 1.11. It follows from Corollary 1.7 and Example 1.9 that the fundamental group 1 (G) of a factorial reductive group G is torsion free. 7
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 = G0 G1 Gn = 1 such that Gi=Gi+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 G0 is the commutant of G, then G splits i G0 and G=G0 are split groups. A reductive group G over F is called quasi-split if it has a Borel subgroup de ned over F . Let G and H be algebraic groups over F . The group G is called a twisted form of H if Gsep and Hsep are isomorphic over Fsep . Let G be a reductive group over F , Gd be its split twisted form. Denote by Gd 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(Gdsep)), corresponding to G, belongs to the image of the map H 1(F; Gd(Fsep )) ! H 1(F; Aut(Gdsep )) induced by the conjugation homomorphism Int : Gd(Fsep ) ! Aut(Gdsep ): 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 1.12. Let G be a reductive group, G0 be the commutant of G. Then G is of inner type i G0 is of inner type and the torus G=G0 splits. Proof. Assume that G is of inner type. Then the cocycle ' : ? = Gal(Fsep =F ) ! Aut(Gdsep )
is of the form '( ) = Int(g ) for some g 2 Gd (Fsep ). Since the restriction of '( ) to the split twisted form (G0 )d = (Gd)0 of G0 is an inner automorphism and Gd(Fsep ) = (G0)d (Fsep), it follows that G0 is of inner type. The inner automorphism '( ) clearly induces the trivial automorphism on Gdsep=(G0)dsep. Hence, G=G0 is a split torus. Conversely, assume that G0 is of inner type and G=G0 is a split torus. Then the cocycle ' : ? ! Aut(Gdsep ), de ning G, being restricted to (G0)d, is of the form '( )jG0 = Int(g )jG0 for some g 2 Gd(Fsep ) = (G0 )d(Fsep ) and '( ) induces the trivial automorphism of Gd=(G0)d. Let Z d be the connected reduced center of Gd . Since the natural homomorphism Z d ! Gd=(G0)d is an isogeny, '( ) restricted to Z d is the 8
trivial automorphism. Hence, '( ) = Int(g ) since Gd is generated by (G0)d and Z d . Corollary 1.13. Let G be a reductive group of inner type. Then G is factorial i 1 (G) is torsion free. Proof. Follows from Proposition 1.10 and lemma 1.12. Example 1.14. The group GL1;A for a central simple F -algebra A is a factorial reductive group over F of inner type. 1.4. Category of representations. Let G be an algebraic group de ned over a eld F . Denote by Rep(G) the category of nite dimensional linear representations of G (called sometimes G-modules). It is an abelian category in which any object has nite length. Hence by [16, Th.4, Cor.1] a (3) Kn(Rep(G)) = Kn(EndG(V )op): V irred.
In particular, K0(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 2 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 eld extension L=F the exact functor Rep(G) ! Rep(GL); [V ] 7! [V F L] induces a ring homomorphism R(G) ! R(GL). If L=F is a nite extension, then we have an exact functor in opposite direction taking a GL-module over L to itself considered as a G-module over F . It induces the corestriction map R(GL) ! R(G). It is clear that the composition R(G) ! R(GL) ! R(G) coincides with the multiplication by the degree (L : F ). In particular, the homomorphism R(G) ! R(GL) 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 1.15. Let G be an algebraic group over a eld F , U be a unipotent normal subgroup in G. Then U acts trivially on any irreducible G-module V . 9
Proof. Since U is normal in G, the subspace of xed elements V U is a G-submodule in V . By the x point theorem [8, Th.17.5], over an algebraic closure of F , (V U )alg = ValgUalg 6= 0 if V 6= 0, hence V U 6= 0 and, therefore, V U = V since V is irreducible. Corollary 1.16. 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 1.17. Let D be a diagonalizable group over F . Since all irreducible representations of D are 1-dimensional [6, IV,x1, Prop.1.6], it follows from (3) that Kn(Rep(D)) ' Kn(F )[D]; where D = Hom(D; G m;F ). In particular, R(D) = Z[D] is a Laurent polynomial ring over Z. Example 1.18. 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), Kn(Rep(B )) ' Kn(Rep(B=U )) ' Kn(F )[(B=U )] ' Kn(F )[B ]; since (B=U ) = B and B=U is a split torus. In particular, R(B ) = Z[B ]. Example 1.19. 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 1.20. Let G be a split simply connected semisimple group over F . Denote by 1 ; : : : ; 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[1; : : : ; r ] (see [1, Th.6.41], [20, Lemma 3.1], [22]). Example 1.21. Let i be the class in R(GLn;F ) of the ith -exterior power representation of GLn;F . Then R(GLn;F ) = Z[1; : : : ; n?1; n 1]: Proposition 1.22. ([20, Th.1.3]) Let G be a split reductive group with 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 jW j. 10
Let G be a split reductive group de ned over a eld F , G0 be the commutant of G, Z be the connected reduced center of G. Denote by = G0 Z !?1 G. Choose a C the kernel of the natural isogeny : G def 0 0 Borel subgroup B G and set B = B \ G and B = (B ). Clearly, B 0 is a Borel subgroup in G0 and B = B 0 Z . Lemma 1.23. If 1(G) is torsion free, then the natural homomorphism ' : R(B ) R(G) R(G) ! R(B ) is an isomorphism. Proof. First of all we prove that ' is surjective. Since G=G0 = Z = B=B 0, the image of the restriction R(G) ! R(B ) contains the classes of all characters of B which are trivial on B 0. 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 0 = 1, it follows that (B ) is generated by (B=B 0) and (B=C ) = B , hence ' 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 jW (G)j = jW (G)j, so that ' 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 nite length. Set R(G; A) = K0(Rep(G; A)). Clearly, R(G; F ) = R(G) and for any eld extension L=F the canonical homomorphism R(G; A) ! R(GL; AL) is injective. Example 1.24. If G be a diagonalizable group over F , then the canonical homomorphism K0(A) Z[G] = K0 (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 2 G the weight subspace V in V is an A-submodule. 1.5. Tits algebras. Let G be a simply connected semisimple group over F and : G ! GL1;A, be an algebraic group homomorphism, where A is a central simple F -algebra. We call A a Tits algebra, if sep is an irreducible representation over Fsep . A Tits algebra A is uniquely determined, up to an isomorphism, by the representation 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 to C is given by the multiplication by some character 2 C = Hom(C; G m;F ). Moreover, 11
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 , but only on the restriction of to the center C . Thus, we get the Tits homomorphism : C ! Br(F ): A Tits algebra A is called a fundamental Tits algebra if sep is a fundamental representation. If G is a group of inner type, then the fundamental Tits algebra exists for any fundamental representation of Gsep. Example 1.25. The fundamental Tits algebras of SL1;A, where A is a central simple F -algebra of degree n, are the -powers iA of A, i = 1; 2; : : : ; n ? 1 (see [9]). 2. Equivariant K -theory The equivariant K -theory was developed by R.Thomason in [21]. 2.1. De nitions. Let G be a group scheme over a scheme Y . A scheme X over Y is called a G -scheme if an action morphism : G Y X ! X of the group scheme G on X over Y is given. A G -module M over X is a coherent OX -module M together with the isomorphism of OGX -modules (4) : (M ) ! p2(M ); (where p2 : G Y X ! X is a projection), satisfying the cocycle condition p23 () (idG ) () = (m idX )(); 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 Kn0 (G ; X ) def = Kn(M(G ; X )): The functor Kn0 (G ; ?) is contravariant with respect to at 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 Kn0 (G ; X ) = Kn0 (X ). Consider the full subcategory P (G ; X ) in M(G ; X ) consisting of locally free OX -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 Kn(G ; X ) def = Kn(P (G ; X )): The functor Kn(G ; ?) is contravariant with respect to G -morphisms of G -varieties. If G is a trivial group scheme, then Kn(G ; X ) = Kn(X ). 12
The inclusion of categories P (G ; X ) ,! M(G ; X ) induces a homomorphism Kn(G ; X ) ! Kn0 (G ; X ): Example 2.1. Let : G ! GL(V ) be a nite dimensional representation of an algebraic group G over a eld 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) ! K0 (G; Spec F ) = K00 (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) ! K0(G; X ), making the latter group (and also K00 (G; X )) a module over R(G). In particular, one can view characters of G as the elements of K0(G; X ) and K00 (G; X ). Let g : H ! G be a group scheme homomorphism over Y and X be a G -scheme over Y . The composition gid H Y X ?! G Y X ?! X makes X an H-scheme. Given a G -module M with the G -module structure de ned by the isomorphism in (4), one can introduce an Hmodule structure on M by (g id)(). Thus, we obtain exact functors M(G ; X ) ! M(H; X ); P (G ; X ) ! P (H; X )
inducing group homomorphisms Kn0 (G ; X ) ! Kn0 (H; X ); Kn(G ; X ) ! Kn(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 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
inducing an isomorphism Kn0 (G ; X ) ' Kn0 (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 Kn0 (G ; X ) ' Kn0 (GjZ ; X ): Example 2.3. Let G be a group scheme over Y , y 2 Y and X be a scheme over the residue eld F (y). Assume that X , considered as a scheme over Y via Spec F (y) ! Y , is a G -scheme. Then Kn0 (G ; X ) ' Kn0 (Gy ; X ); where Gy = G Y Spec F (y) is the ber of G over y. 2.2. Torsors. Let G be an algebraic group over F and p : X ! Y be a G-torsor. Then for any coherent OY -module M there is a natural G-module structure on p (M ) over X (see [11, Sec. 3]). Thus, there is an exact functor p0 : M(Y ) ! M(G; X ); M 7! p (M ): The following statement is well-known (see, for example [11, Prop. 33]). Proposition 2.4. The functor p0 is an equivalence of categories. In particular, the homomorphism p : Kn0 (Y ) ! Kn0 (G; X ), induced by p0 , is an isomorphism. By the faithfully at descent (see [13, Ch.1, Th. 2.23]), the assignment Z 7! X Y Z induces an equivalence of categories of ane schemes over Y and ane G-schemes over X . For an ane G-scheme V over X , the corresponding ane 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) 7! (p2(p); p1(p)x), where p1;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, x1, Prop. 5]). 14
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 eld 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 j i ?! Kn0 +1(G; U ) ?! Kn0 (G; Z ) ?! Kn0 (G; X ) ?! Kn0 (G; U ) ?! : : : ; where i : Z ! X and j : U ! X are imbeddings. 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 f : Kn0 (G; X ) ! Kn0 (G; E ) is an isomorphism. Corollary 2.8. If G acts linearly on the ane space A nF , then the projection p : X A nF ! X induces the inverse image isomorphism 0 p : Kn0 (G; X ) ! Kn(G; X A nF ) : Corollary 2.9. Let G act on A 1F by the multiplication by a character 2 G, i : X ' X f0g ,! X A 1F be the closed imbedding. Then the composition ?1
p) i Kn0 (G; X ) ?! Kn0 (G; X A 1F ) (?! Kn0 (G; X ) is the multiplication by 1 ? ?1 2 K0 (G; X ). Proof. Since pi = id, by Corollary 2.8, i is an isomorphism. Hence, by the projection formula, i (i(u)) = i (1) u for any u 2 Kn0 (G; X A 1F ). Thus, it suces to show that i (1) = 1 ? ?1 . But this follows from the exactness of the sequence of G-modules over X A 1F ?1
OX A 1F ! i (OX ) ! 0: 0 ! OX A 1F ?! Proposition 2.10. [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 fH g ,! X Y induces equivalences of categories P (G; X Y ) ! P (H ; X ); M(G; X Y ) ! M(H ; X ): 15
In particular, there are natural isomorphisms
0 Kn(G; X Y ) ! Kn(H ; X ); Kn0 (G; X Y ) ! Kn(H ; X ): Corollary 2.11. Let j : Y ! A nF be an open G-equivariant imbedding into the ane space A nF on which G acts linearly. Then the composition
(idX j ) 0 p 0 0 0 n K (G; X ) ?! K (G; X A ) ????! K (G; X Y ) ! K (H ; X ); n
n
F
n
n
where p : X A nF ! X is the projection, coincides with the restriction homomorphism. Proof. The composition of the rst 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 fH g ,! X Y , inducing the last isomorphism, with this projection is the identity.
For any representation : H ! GL(V ) one can associate the Gvector bundle P = (G V )=H over Y = G=H , where the H -action on G V is given by h (g; v) = (gh?1; (h)(v)). Corollary 2.12. The assignment 7! [P] induces an isomorphism
R(H ) ! K0 (G; Y ) ' K00 (G; Y ): 2.4. Exact sequence. Let G be an algebraic group over F . Assume we are given a nontrivial character : G ! G m;F . Denote the kernel of by H . Proposition 2.13. Let X be a G-scheme. Then there is an exact
sequence
1? 0 res K 0 (H ; X ) ! : : : : ! Kn0 +1 (H ; X ) ! Kn0 (G; X ) ?! Kn(G; X ) ?! n Proof. We consider X A 1F as a G-scheme with respect to the action 1
of G on A F given by the multiplication by . In the localization exact sequence
j i ! Kn0 (G; X ) ?! Kn0 (G; X A 1F ) ?! Kn0 (G; X G m;F ) ! : : : where i : X = X f0g ,! X A 1F and j : X G m;F ,! X A 1F are imbeddings, the second term is identi ed with Kn0 (G; X ) by Corollary 2.8 and the third with Kn0 (H ; X ) since G=H ' G m;F as G-schemes (see Proposition 2.10). With this identi cations i = 1 ? ?1 and j = res
by Corollaries 2.9 and 2.11.
16
2.5. Solvable groups acting on schemes. Lemma 2.14. 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 1F . Then for any G-scheme X the restriction map Kn0 (G; X ) ! Kn0 (H ; X ) is an isomorphism. Proof. Since the group Aut(A 1F ) consists of linear automorphisms, the action of G on A 1F , induced by an isomorphism G=H ' A 1F , is given by g y = (g)y + (g) where is a character of G and is a regular function on G. Consider the trivial linear vector bundle V = X A 1F over X as a Gscheme with the action natural on X and given by the multiplication by the character on A 1F . One can view V as an (additive) group scheme over X. Then the G-scheme E = X (G=H ) ' X A 1F can be considered as a V -torsor over X . Denote the action of V on E by . The formula (with (x; y) 2 V and (x; y0) 2 E ) g ((x; y) (x; y0)) = g (x; y + y0) = (gx; (g)(y + y0) + (g)) = (gx; (g)y) (gx; (g)y0 + (g)) = [g (x; y)] [g (x; y0)] 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 0 p : Kn0 (G; X ) ! Kn(G; X (G=H )): Composing this map with the isomorphism 0 Kn0 (G; X (G=H )) ! Kn(H ; X ) (Proposition 2.10) we get the restriction homomorphism by Corollary 2.11. Corollary 2.15. Let B be a split solvable group, T B be a maximal torus, X be a B -scheme. Then the restriction homomorphism Kn0 (B ; X ) ! Kn0 (T ; X ) is an isomorphism. Corollary 2.16. Let U be a split unipotent group, X be a U -scheme. Then the restriction homomorphism Kn0 (U ; X ) ! Kn0 (X ) is an isomorphism. 2.6. Unimodular form. Let G be a split reductive group de ned over a eld 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 ) = K00 (G; G=B ) ! K00 (G; Spec F ) = R(G) 17
is well de ned. Consider the R(G)-bilinear form on R(B ): hu; vi = hu; viG = p (u v): Proposition 2.17. Let G be a split reductive group with 1 (G) torsion free. Then the bilinear form h ; iG is unimodular. Proof. We use the notation of Lemma 1.23. Since G0 is a simply connected semisimple group (Corollary 1.7), the bilinear form h ; iG0 is unimodular ([7], [14, Th. 8.1.]). Since R(G) = R(G0 ) Z R(Z ) and R(B ) = R(B 0 ) Z R(Z ); it follows that h ; iG = h ; iG0 R(Z ) is an unimodular form. Lemma 1.23 implies that h ; iG = h ; iG R(G) R(G): Since h ; iG is unimodular, the determinant d of h ; iG 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). 2.7. The category C (G). Let G be an algebraic group over a eld 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 OX -module such that g(am) = ga gm for all a 2 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 OX - and G- modules. Set Kn0 (G; X; A) = Kn(M(G; X; A)) : The functor Kn0 (G; ?; A) is contravariant with respect to at Gmorphisms and is covariant with respect to projective G-morphisms of G-schemes. The group Kn0 (G; X; F ) coincides with Kn0 (G; X ). Consider also the full subcategory P (G; X; A) in M(G; X; A) consisting of locally free OX -modules. The K -groups of the category P (G; X; A) we denote by Kn(G; X; A) = Kn(P (G; X; A)) : The group Kn(G; X; F ) coincides with Kn(G; X ). Example 2.18. 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) ! K0(G; Spec F; A) = K00 (G; Spec F; A): 18
If H G is a subgroup, then as in Proposition 2.10 there exists a natural isomorphism K0(G; G=H; A) ! K0 (H ; Spec F; A) = R(H; A): Lemma 2.19. Let X be a quasi-projective smooth G-scheme. Then for any M 2 M(G; X; A) there is a surjection P !! M with P 2 P (G; X; A). Proof. By [21, Lemma 5.6] there is a surjection f 0 : P 0 !! M with P 0 2 P (G; X ). If we consider P = A F P 0, then f : P ! M , de ned by f (a p0) = af 0(p0 ), is the desired surjection. The Resolution Theorem [16, Th. 3] then gives Proposition 2.20. (compare [21, Cor. 5.8]) For any quasi-projective smooth G-scheme X over F the natural homomorphism Kn(G; X; A) ! Kn0 (G; X; A) is an isomorphism. In [14] I.Panin has de ned 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 de ned as follows: HomC(G) ((X; A); (Y; B )) = K0 (G; X Y; Aop F B ): If u : (X; A) ! (Y; B ) and v : (Y; B ) ! (Z; C ) are two morphisms in C (G), then their composition is de ned by the formula v u = v u def = p13 (p23 (v) B p12 (u)); where p12 , p13 and p23 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 K00 (G; X X; Aop F A) = K0 (G; X X; Aop F A) = EndC(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 2 Z there is the functor KnZ : C (G) ! Ab; taking a pair (X; A) to Kn0 (G; Z X; A) and a morphism v 2 HomC(G) ((X; A); (Y; B )) = K0 (G; X Y; Aop F B ) to KnZ (v) : Kn0 (G; Z X; A) ! Kn0 (G; Z Y; B ) 19
given by the formula
KnZ (v)(u) = v u (see [14, 6.1]). The functor KnSpec F we will simply denote by Kn.
The following statement is a slight generalization of [14, Th. 6.6]. Proposition 2.21. Let G be a split reductive group over a eld F with 1 (G) torsion free, B G be a Borel subgroup, Y = G=B , u1; u2; : : : ; um 2 R(B ) = K0(G; Y ) be a free R(G)-base of R(B ). Then the element u = (ui ) 2 K0(G; Y; F m) = R(B )m de nes an isomorphism Fm ! Y in the category C (G). Proof. Let v1 ; v2 ; : : : ; vm be the dual R(G)-base of R(B ) with respect to the unimodular bilinear form h ; iG (see Proposition 2.17). The element v = (vi) 2 K0(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 suces to show that the R(G)module K0 (G; Y Y ) is generated by m2 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 1(G) torsion free. Let G be a reductive quasi-split group de ned over F with 1 (G) torsion free, B G be a Borel subgroup, T B be a maximal torus de ned over F . The Galois group ? acts on the character group T (Fsep), on the normalizer NGsep (Tsep ) and hence on the Weyl group W . Denote by uw 2 R(Bsep ), w 2 W , the Steinberg base of the free R(Gsep)-module R(Bsep) consisting of the classes of characters (Proposition 1.22). The Galois action on R(Bsep ) agrees with one on W , i.e. (uw ) = u w for any 2 ? = Gal(Fsep=F ) and w 2 W . Therefore, the homomorphism Bsep ! (G m;Fsep )jW j, given over Fsep by the characters uw , is ?-invariant and hence descents to a representation u : B ! GL1;A; where A is the etale F -algebra corresponding to the nite ?-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 K0(B ; Spec F; A) ' K0 (G; Y; A); where Y = G=B and the algebra A is considered with the trivial Gaction (see Example 2.18). Hence it de nes a morphism A ! Y in C (G), which we also denote by u. Proposition 2.22. The morphism u : A ! Y in C (G) is an isomorphism. 20
Proof. The dual base (vw ), w 2 W , of R(Bsep ) with respect to the unimodular form h ; iGsep de nes in an analogous way a morphism v : Y ! A. By the proof of Proposition 2.21, usep and vsep are mutually inverse isomorphisms. Since for Z = Spec F ,
EndC(G) (A) = K0(G; Z; A A) = R(G; A A) ,!
R(Gsep ; Asep Asep ) = K0(Gsep ; Zsep; Asep Asep ) = EndC(Gsep) (Asep ); (see subsection 1.4), one has v u = id. By [14, 11.10(4)], EndC(G) (Y ) = K0(G; Y Y ) ,! K0(Gsep ; Ysep Ysep) = EndC(Gsep) (Ysep); hence, u v = id. We will need the following Lemma in the smooth projective case. Lemma 2.23. Let G be an algebraic group, H G be a closed subgroup with a group homomorphism : 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; x0 ) = ((g)x; gx0). Assume that the restriction homomorphism K0 (G; X X ) ! K0 (H ; X X ) is surjective. Then the restriction homomorphism Kn0 (G; X ) ! Kn0 (H ; X ) is a split surjection. Proof. Denote by X_ the scheme X together with the G-action given by the formula g x = (g)x. Since the restriction map _ X ) = K0 (G; X_ X ) ! resG=H : HomC(G) (X; K0 (H ; X X ) = HomC(H ) (X; X ) _ X ) such that resG=H (v) = idX . is surjective, there is v 2 HomC(G) (X; Consider the diagram Kn (v) Kn0 (H ; X ) ???! Kn0 (G; X_ ) ???! Kn0 (G; X ) ?
res? y
?
res? y
Kn0 (H ; X ) Kn0 (H ; X ); where the square is commutative since resG=H (v) = idX . The equality res = id implies that the composition in the top row splits the restriction homomorphism. 21
2.8. Local-to-global principle. Let Y be a scheme over a eld F . Assume that we are given: (i) For any n 2 Z and any (locally closed) subscheme Z Y an abelian group An(Z ), which is trivial if n < 0; j (ii) For all subschemes Z1 ,! Z2 in Y , where j is an open imbedding, for any n, the inverse image group homomorphism j : An(Z2) ! An(Z1) such that id = id and (j1 j2 ) = j2 j1 for any composable open imbeddings j1 and j2 ; (iii) For all subschemes Z1 ,!i Z2 in Y , where i is a closed imbedding, for any n, the direct image group homomorphism i : An(Z1) ! An(Z2) and the connecting homomorphism : An(Z2 ? Z1) ! An?1(Z1). The collection of groups A = fAn(Z )g, inverse, direct images and connecting homomorphisms is called a local group over Y if for any Z1 and Z2 as in (iii) the in nite sequence j i An(Z2) ! (5) ! An(Z1) ! An(Z2 ? Z1 ) ! An?1 (Z1) ! : : : is exact (here j : Z2 ? Z1 ! Z2 is the open imbedding). Let y 2 Y be a point. The groups An(U ) for all non-empty open subsets U fyg (the latter is considered as a reduced closed subscheme in Y ) form a direct system with respect to inverse images. We set An(y) = lim A (U ): ??! n U fyg
It is clear that if y is a closed point then An(y) = An(fyg). Let B be another local group over Y . A morphism f : A ! B of local groups over Y is a collection of homomorphisms fn(Z ) : An (Z ) ! Bn(Z ) for all n 2 Z and Z Y , commuting with the direct, inverse images and the connecting homomorphisms. For any n 2 Z and y 2 Y the morphism f induces a group homomorphism fn(y) : An(y) ! Bn(y). Remark 2.24. Taking the empty subscheme for Zi in the exact sequence (5) one gets An(;) = 0. Example 2.25. 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) An (Z ) = Kn0 (G ; XjZ ) = Kn0 (GjZ ; XjZ ): Inverse, direct images and the localization sequence in the equivariant algebraic K -theory make A a local group over Y . Since the K 0 -groups commute with direct limits, one has An(y) = Kn0 (G ; Xy ) = Kn0 (Gy ; Xy ) 22
for any y 2 Y , where Gy and Xy are bers over y (see Example 2.3). Lemma 2.26. Let Z be a subscheme in Y , i : Zred ! Z be the natural closed imbedding. Then the direct image i : An(Zred) ! An(Z ) is an isomorphism. Proof. One takes Z1 = Zred and Z2 = Z in the exact sequence (5) (see also Remark 2.24). Theorem 2.27. (Local-to-global principle) Let f : A ! B be a morphism of local groups over Y . 1. If fn(y) : An(y) ! Bn(y) is an isomorphism for all n 2 Z and y 2 Y , then fn(Z ) is an isomorphism for any n 2 Z and Z Y . 2. If f0 (y) : A0 (y) ! B0(y) is a surjection for any y 2 Y , then f0 (Z ) is surjective for any Z Y . Proof. We prove that fn(Z ) : An (Z ) ! Bn(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 2.24. Assume that we have proved the statement for all subschemes of dimension less than the dimension of Z . We would like to prove that fn(Z ) is an isomorphism (resp. surjective if n = 0). We prove this statement by induction on the number of irreducible components of Z . Assume rst that Z is irreducible. By Lemma 2.26, we may assume that Z is reduced. Surjectivity of fn(Z ). Let z 2 Z be the generic point and v 2 Bn (Z ). Since fn(z) is a surjection, it follows that there exists a non-empty open subset U Z such that the image of v in Bn(U ) belongs to the image of fn(U ). Set Z 0 = Z ? U (considered as a reduced scheme). Since dim(Z 0) < dim(Z ), the left vertical homomorphism in the diagram j i An+1(U ) ???! An(Z 0 ) ???! An(Z ) ???! An(U ) ???! An?1(Z 0) ?
fn+1 (U )? y
?
fn (Z 0 )? y
?
fn (Z )? y
?
fn (U )? y
? ? y
j i Bn+1 (U ) ???! Bn(Z 0) ???! Bn(Z ) ???! Bn(U ) ???! Bn?1 (Z 0) 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 2 im(fn(Z )). Injectivity of fn(Z ). Let u 2 ker(fn (Z )). Since the homomorphism fn(z) : An(z) ! Bn(z) is injective, one can nd a non-empty open subset U Z such that the image of u in An(U ) is trivial. Set Z 0 = Z ? U (considered as a reduced scheme). In the diagram above the map fn(Z 0) is an isomorphism by the induction hypothesis since dim Z 0 < 23
dim Z and fn+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 0 be an irreducible component in Z and U = Z ? Z 0 . 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 fn (Z ) is an isomorphism (resp. surjective if n = 0). The following statement is the rst application of the local-to-global principle. Corollary 2.28. Let U be a group scheme and X be a G -scheme over a scheme Y . Assume that for any point y 2 Y the ber Uy is a split unipotent group. Then the restriction homomorphism res : Kn0 (U ; X ) ! Kn0 (X ) is an isomorphism. Proof. Consider the following local groups on Y : An(Z ) = Kn0 (U ; XjZ ) = Kn0 (UjZ ; XjZ ) and Bn(Z ) = Kn0 (XjZ ) for a subscheme Z Y . For any y 2 Y the homomorphism fn(y), where f : A ! B is the restriction morphism of local groups, coincides with resy : Kn0 (Uy ; Xy ) ! Kn0 (Xy ) (see Examples 2.2 and 2.3). Since Uy is a split unipotent group, by Corollary 2.16, resy 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 , 2 T be a character such that T =(Z ) is a torsion-free group. Then T 0 = ker( ) is a subtorus in T . Denote by : T ! T 0 a splitting of the imbedding T 0 ,! T . Proposition 3.1. Let X be a smooth projective T -scheme. Then the restriction homomorphism Kn0 (T ; X ) ! Kn0 (T 0 ; X ) is a split surjection. Proof. Since T=T 0 ' G m;F , by Corollary 2.8, Proposition 2.10 and the localization (Proposition 2.6), we have the following surjection 0 K00 (T ; X ) ! K0 (T ; X A 1F ) !! K00 (T ; X G m;F ) ' K00 (T 0; X ) which is nothing but the restriction map by Corollary 2.11. Considering instead of X the product X X with the following T -action: t(x; x0 ) = ((t)x; tx0 ), we get a surjective restriction homomorphism res : K00 (T ; X X ) !! K00 (T 0; X X ): 24
Hence one can apply the Lemma 2.23. Proposition 2.13 now implies Corollary 3.2. The sequence 1? 0 res K 0 (T 0 ; X ) ! 0 0 ! Kn0 (T ; X ) ?! Kn(T ; X ) ?! n is split exact. Let A be a commutative ring, a1; a2 ; : : : ; ar 2 A and M be an Amodule. The elements ai form an M -regular sequence if the multiplication by ai in M=(a1 + + ai?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 1 ; 2 : : : ; r be a Z-base of the character group B . Then the elements 1 ? 1 ; 1 ? 2 : : : ; 1 ? r 2 R(B ) form a Kn0 (B ; X )-regular sequence over the ring R(B ) and the restriction map induces an isomorphism Z R(B ) Kn0 (B ; X ) ' Kn0 (B ; X )=J (B ) Kn0 (B ; X ) = r
X 0 K 0 (B ; X )= (1 ? )K 0 (B ; X ) ! K (X ): n
i=1
i
n
n
Proof. Let T be a maximal torus in B . Since B = T and the restriction homomorphism Kn0 (B ; X ) ! Kn0 (T ; X ) is an isomorphism by Corollary 2.15, it is sucient to consider the case B = T . Our statement follows then from Corollary 3.2 by induction on the dimension of T.
Corollary 3.4.
(
Kn0 (X ); if p = 0; 0; if p > 0. Proof. Since R(B ) is a Laurent polynomial ring in variables i , the elements 1? i 2 R(B ) form a R(B )-regular sequence, the result follows from [18, IV-7]. 3.1. Spectral sequence. Let B be a group scheme over a scheme Y and : B ! T def = G rm;Y be a group scheme homomorphism over Y . Denote the kernel of by U . Assume that for any point y 2 Y the ber y of is a surjective homomorphism and Uy is a split unipotent group over F (y). Let be given by characters 1 ; 2; : : : ; r in B . We de ne an action of B on the ane space A rY by the rule b x = y where yi = i(b)xi . Denote by Vi (i = 1; 2; : : : ; r) the coordinate hyperplane in A rY de ned TorpR(B) (Z; Kn0 (B ; X )) =
25
by the equation xi = 0. Clearly, Vi is a closed B-subvariety in A rY and T = A rY ? [ri=1 Vi. 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 Zi = X Y Vi in X Y A rY induce then the following spectral sequence 1 = a K 0 (B; X Y VI ) ) K 0 (B; X Y T ); Ep;q q p+q jI j=p
where VI = \i2I Vi. Consider the composition res K 0 (X ); Kn0 (B; X Y T ) ! Kn0 (U ; X ) ?! n where the rst map is induced by the restriction along the map X ' X Y Y ,! X Y T . Since By =Uy ' TF (y) , by the local-to-global principle (Theorem 2.27) and Proposition 2.10 it follows that the rst homomorphism is an isomorphism. By Corollary 2.28, the last map, and hence the composition, is also an isomorphism. 1 , note that VI is an ane space over Y , hence In order to compute Ep;q the inverse image Kq0 (B; X ) ! Kq0 (B; X Y VI ) is an isomorphism by Corollary 2.8 and Theorem 2.27. Thus, 1 = a K 0 (B; X ) eI Ep;q q jI j=p
1 is given by the and by [10, p.419] the di erential map d : Ep1+1;q ! Ep;q formula
(6)
d(x eI ) =
p X k=0
(?1)k (1 ? ?k 1 )x eI ?fik g ;
where I = fi0 < i1 < < ipg. Consider Kozsul complex C built of the module K0(B; Y )r over K0(B; Y ) and the system of elements 1 ? ?i 1 2 K0(B; Y ). More precisely, a Cp = K0 (B; Y ) eI jI j=p
and the di erential d : Cp+1 ! Cp given by the rule formally equal to (6) with x 2 K0 (B; Y ). 26
Lemma 3.5. The natural homomorphism Kn0 (Y )[B ] ! Kn0 (B; Y )
is an isomorphism. Proof. Since the natural homomorphism of the character groups T ! B is an isomorphism, By ' B for any y 2 Y and hence, by Theorem 2.27, it suces to consider the case Y = Spec(F ), B is a solvable split group over F , is a surjective homomorphism with a unipotent kernel. But this case is covered by Example 1.18. Now assume that Y is smooth. Lemma 3.5 shows that K0(B; Y ) is a Laurent polynomial ring over K0(Y ). The Kozsul complex gives then a resolution C ! K0(Y ) ! 0 of K0(Y ) by free K0 (B; Y )-modules. It is clear that the complex E1;q coincides with C K0 (B;Y ) Kq0 (B; X ): 2 , being the homology group of E 1 , equals Hence, the term Ep;q ;q TorpK0(B;Y ) (K0(Y ); Kq0 (B; X )): Proposition 3.6. Let B be a group scheme over a smooth scheme Y and : B ! G rm;Y be a group scheme homomorphism over Y . Assume that for any point y 2 Y the ber y of is a surjective homomorphism with a split unipotent kernel over F (y). Then there exists a spectral sequence 2 = TorK0(B;Y ) (K0 (Y ); K 0 (B; X )) ) K 0 (X ): Ep;q p q p+q 3.2. Case of factorial tori. We start with a simple case. Proposition 3.7. Let T be a quasi-trivial torus over a eld F . Then for any T -scheme X over F the restriction map res : K00 (T ; X ) ! K00 (X ) is surjective. Proof. Choose a separable F -algebra C such that T ' GL1;C and consider the natural action of T on the ane space A (C ) of the algebra C . The variety of the group GL1;C is open in A (C ) and T -invariant. Hence the natural homomorphism K00 (T ; X A (C )) ! K00 (T ; X GL1;C ) is a surjection (Proposition 2.6). The rst group is isomorphic to K00 (T ; X ) by Corollary 2.8, the latter { to K00 (X ) by Proposition 2.10 and the homomorphism { to the restriction map by Corollary 2.11. Thus, the restriction homomorphism in question is surjective. It turns out that this Proposition can be generalized to the case of factorial tori at least for quasi-projective X . We begin with a preliminary result. 27
Lemma 3.8. Let T be a factorial torus and U be a principal homogeneous space under T over F . Then K0 (U ) = Z 1.
Proof. Consider a smooth projective T -model X and a projective toric T -variety Y having U as an open orbit (see [3], [23]). By [11, Th. 5.7] there exist an etale F -algebra A and Q 2 Pic(XA), such that the class [P ] 2 K0(X; A) de nes a split surjection A ! X in the category C = C (1). Let L=F be a nite Galois splitting eld of T . As in [11, Sec. 7], x a TA-equivariant isomorphism X AL : f : Y AL ! We set P 0 = f (X (Q)) 2 Pic(YAL ); where X : XA ! X is the natural morphism, and B = EndYA (P ); where P = (Y ) (P 0) is a locally free sheaf of rank n = (L : F ) over YA. We can consider P as an element in K0(Y; B ). Hence it de nes a morphism B op ! Y in C which is a split surjection by [11, Th. 7.6]. Thus, we have the surjective homomorphism (7) K0(B op) !! K0(Y ); [M ] 7! [M B P ]: The class [B ] belongs to the image of a homomorphism ([11, Lemma 7.3]) A : Pic(TA) ! Br(A): Since T is a factorial torus, B splits, B ' Mn(A). Therefore, it follows from the surjectivity of (7) that K0 (Y ) is generated by the direct images of classes of sheaves from Pic(YE ) for all nite separable eld extensions E=F . Therefore, the analogous statement holds for the open subset U Y . But by [17, Prop. 6.10] there is an injection Pic(UE ) ,! Pic(TE ) = 0, hence Pic(UE ) = 0 and therefore K0(U ) = Z 1. Proposition 3.9. Let T be a factorial torus, X be a quasi-projective T -scheme. Then the restriction homomorphism K00 (T ; X ) ! K00 (X ) is surjective. Proof. Assume rst that there is a morphism p : X ! Y , making X a T -torsor over some scheme Y . Then by Proposition 2.4, the restriction homomorphism in question is isomorphic to the inverse image p : K00 (Y ) ! K00 (X ): By the local-to-global principle (Theorem 2.27), it suces to consider the case when Y = Spec(E ) for some eld extension E=F and hence 28
X is a principal homogeneous space under TE over E . But this case is covered by Lemma 3.8. In the general case choose an imbedding T ,! P with a quasi-trivial torus P = GL1;C . The torus T acts on P X by t (p; x) = (pt?1; tx). Consider the commutative diagram K00 (T?; X ) ???! K00 (T ; A (?C ) X ) ???! K00 (T ; ?P X ) res? y
res? y
res? y
K00 (X ) ???! K00 (A (C ) X ) ???! K00 (P X ): By Lemma 2.5, there is a T -torsor p : P X ! Y , hence the rst part of the proof shows that the right vertical map is surjective. By localization, the right horizontal arrows are surjections. Finally, the composition in the bottom row is an isomorphism since it has a splitting K00 (P X ) ! K00 (X ) being the inverse image with respect to the closed imbedding X = f1g X ,! P X of nite Tor-dimension (see [16, x7, 2.5]). Thus, the left vertical restriction homomorphism is surjective.
4. Split case
4.1. Comparison with a Borel subgroup. Let G be a split reductive group over a eld F , B G be a Borel subgroup de ned over F , X be a G-scheme. The group Kn0 (G; X ) (resp. Kn0 (B ; X )) is a module over the representation ring R(G) (resp. R(B )). The restriction homomorphism Kn0 (G; X ) ! Kn0 (B ; X ) is a homomorphism of modules with respect to the restriction R(G) ! R(B ) and hence induces a R(B )-module homomorphism : R(B ) R(G) Kn0 (G; X ) ! Kn0 (B ; X ): Proposition 4.1. Assume that 1 (G) is torsion free. Then the homomorphism is an isomorphism. Proof. Set Y = G=B . By Proposition 2.21, there is an isomorphism in the category C (G): (8) Fm ! Y; de ned by elements ui 2 K0 (G; Y ) = R(B ) forming a base of R(B ) over R(G). Applying the functor (see 2.7) KnX : C (G) ! Ab; to the isomorphism (8), we obtain an isomorphism 0 Kn0 (G; X )m ! Kn(G; X Y ): 29
Identifying Kn0 (G; X )m with R(B ) R(G) Kn0 (G; X ) using the same elements ui we get the canonical isomorphism 0 R(B ) R(G) Kn0 (G; X ) ! Kn(G; X Y ): Composing this isomorphism with the canonical isomorphism (Proposition 2.10) 0 Kn0 (G; X Y ) ! Kn(B ; X ); we clearly get . 4.2. Projective case. Theorem 4.2. Let G be a split reductive group with 1 (G) torsion free, X be a smooth projective G-scheme. Then the natural homomorphism Z R(G) Kn0 (G; X ) ! Kn0 (X ) is an isomorphism. In other words, the restriction homomorphism Kn0 (G; X ) ! Kn0 (X ) is a surjection with the kernel J (G) Kn0 (G; X ). Proof. Let B be a Borel subgroup in G. By Corollary 3.4 and Proposition 4.1, we have canonical isomorphisms Kn0 (X ) ' Z R(B) Kn0 (B ; X ) '
R(B) (R(B ) R(G) Kn0 (G; X )) ' Z R(G) Kn0 (G; X ): 4.3. General case. Let X is a G-scheme, where G is a split reductive Z
group de ned over F with 1 (G) torsion free, B is a Borel subgroup in G. By Proposition 4.1, we have the canonical isomorphism R(B ) R(G) Kn0 (G; X ) ' Kn0 (B ; X ): Hence, since R(B ) is free R(G)-module by Proposition 1.22, one has (9) TorpR(G) (Z; Kn0 (G; X )) ' TorpR(B) (Z; Kn0 (B ; X )): Applying Proposition 3.6 to the group B over Y = Spec F and to the natural surjection B ! T = B=U , where U is the unipotent radical of B , one gets Theorem 4.3. Let G be a split reductive group de ned over F with 1(G) torsion free and X be a G-scheme. Then there exists a spectral sequence 2 = TorR(G) (Z; K 0 (G; X )) ) K 0 (X ): Ep;q p q p+q 30
Corollary 4.4. 1. The natural homomorphism Z R(G) K00 (G; X ) ! K00 (X ) is an isomorphism, i.e. K00 (X ) ' K00 (G; X )=J (G) K00 (G; X ).
2. There is an exact sequence K20 (X ) ! TorR2 (G) (Z; K00 (G; X )) ! Z R(G) K10 (G; X ) ! K10 (X ) ! TorR1 (G) (Z; K00 (G; X )) ! 0: 2 = TorR(G) (Z; Kq(F )). This Remark 4.5. If X = G, one has Ep;q p formula together with the fact that the spectral sequence degenerates gives the computation of Kn(G) (see [10]). Remark 4.6. If X is a smooth projective G-scheme, then it follows from Corollary 3.4 and (9) that ( 0 2 = TorR(G) (Z; K 0 (G; X )) = Kn (X ); if p = 0; Ep;q p n 0; if p > 0, in other words, the spectral sequence in Theorem 4.3 degenerates, i.e. 2 = 0 if p > 0. Ep;q Corollary 4.7. Let H G be a closed subgroup of a split reductive group G with 1 (G) torsion free. Then K0(G=H ) ' Z R(G) R(H ) ' R(H )=J (G) R(H ): In particular, the group K0 (G=H ) is generated by the classes of P = (G V )=H for all representations : H ! GL(V ). Example 4.8. Let H GLn;F be an imbedding. Then for the \classifying variety" X = GLn;F =H of H one has K0(X ) ' Z R(GLn;F ) R(H ): Proposition 4.9. Let G be a split reductive group over F . Then the kernel K0 (G)(1) of the rank homomorphism K0 (G) ! Z is a nite group. In particular, K0 (G) is a nitely generated group of Q -rank 1. Proof. Let G0 be the commutant of G, Ge0 ! G0 be the universal covering, Z be the connected reduced center of G. Consider the group G = Ge0 Z and denote by C the ( nite) kernel of the natural surjection G ! G. Since G ' G=C and 1 (G) is torsion free (Corollary 1.7), by Corollary 4.7, K0(G) ' Z R(G) R(C ) = R(C )=J (G) R(C ); hence K0(G)(1) = J (C )=J (G) R(C ). 31
Since C is a nite diagonalizable group, R(C ) = Z[C ] is a nitely generated group (see Example 1.17), hence K0 (G) is also nitely generated. Take any character 2 C and choose a character of a maximal torus of G such that jC = . By the representation theory [22], there is an irreducible representation : G ! GL(V ) with the weight . All the weights of di er by elements of the root group. Since C is a central subgroup in G, the restriction of all the weights of to C equal , hence, the image of [V ] under the restriction map R(G) ! R(C ) equals n where n = dim(V ). In particular, resG=C (n ? [V ]) = n (1 ? ) 2 J (C ). We have shown that the generators 1 ? of J (C ) have nite order in the factor-group K0 (G)(1) = J (C )=J (G) R(C ). Hence, K0 (G)(1) is a torsion and hence a nite group. Remark 4.10. In Theorem 7.7 we will generalize this statement to arbitrary reductive groups. 5. Inner case
5.1. Comparison with the Borel subscheme. Let G be a reductive subgroup of inner type. Denote by Y the variety of Borel subgroups in G. Usually we consider the trivial action of G on Y . Consider the subscheme B in the product G Y consisting of pairs (g; B ) such that g 2 B . Clearly B is a group scheme over Y , called the Borel subscheme in G Y . Let Gd be a split twisted form of the group G, B Gd be a Borel subgroup. Then Y d = Gd =B is a twisted form of Y and the scheme Bd = f(x; yB ) 2 Gd Y d such that x 2 yBy?1g is a twisted form of B. Let 2 B be a character. Consider the following morphism over Y d: ~ : Bd ! G m;Y d = G m;F Y d; (x; yB ) 7! ( (y?1xy); yB ): This map is well de ned. For if yB = y0B , then y0 = yb for some b 2 B , y0?1xy0 = b?1 (y?1xy)b and (y0?1 xy0) = (b?1 ) (y?1xy) (b) = (y?1xy): Clearly ~ is a character of the group scheme Bd over Y d and the correspondence 7! ~ induces a homomorphism B ! (Bd ). The adjoint group Gd = Gd =Center(Gd) acts naturally on Bd and Y d making ~ equivariant morphism, since for any g in Gd one has ~(g (x; yB )) = ~(gxg?1; gyB ) = ( (y?1xy); gyB ) = g ( (y?1xy); yB ) = g (~ (x; yB )): 32
Since G is an inner form of Gd, the cocycle de ning G takes values in Gd(Fsep ), hence the character ~ is a twisted form of some character : B ! G m;Y over Y , and the assignment ! de nes a homomorphism B ! B . Assume that 1 (G) is torsion free. Choose a base 1; 2; : : : ; m of R(B ) over R(Gd) consisting of characters in B (Proposition 1.22), where m is the order of the Weyl group W . Consider the homomorphism : K0(G; Y )m =X K0(G Y ; Y )m ! K0(B; Y ) (vi) 7! resGY=B (vi) i; where we view the characters i as elements in K0(B; Y ). For any y 2 Y the group GF (y) of inner type has a Borel subgroup By and hence is split. The bers (i)y form a base of R(By ) ' R(BF (y) ) ' R(B ) over R(GF (y)) ' R(GdF (y) ) ' R(Gd). Hence by the local-to-global principle (Theorem 2.27) and Proposition 4.1, it follows that is an isomorphism. We have proved Proposition 5.1. Let G be a reductive group of inner type with 1 (G) torsion-free. Then the K0 (G; Y )-module K0 (B; Y ) is free of rank jW j with the base i. The following Proposition is a global version of 4.1. Proposition 5.2. In the conditions of Proposition 5.1, for any Gscheme X over Y the natural homomorphism K0 (B; Y ) K0 (G;Y ) Kn0 (G; X ) ! Kn0 (B; X ) is an isomorphism. Proof. The homomorphism in question is identi ed with X : Kn0 (G; X )m ! Kn0 (B; X ); (vi) 7! res(vi) (i); where : X ! Y is the structure morphism. For any y 2 Y the group GF (y) is split. Hence, the local-to-global principle (Theorem 2.27) and the proof of Proposition 4.1 show that is an isomorphism. 5.2. Spectral sequence. Let G be a reductive group of inner type de ned over F with 1(G) torsion free, Y be the variety of Borel subgroups in G. Since K0(B; Y ) is free K0(G; Y )-module by Proposition 5.1, it follows from Proposition 5.2 that for any G-scheme X over Y , TorpK0(G;Y )(K0 (Y ); Kq0 (G; X )) ' TorpK0(B;Y )(K0 (Y ); Kq0 (B; X )): Now we can apply Proposition 3.6 to the Borel subscheme B over Y and to the group scheme homomorphism : B ! G rm;Y given by 33
the characters i 2 B such that i 2 B form a Z-base. For any y 2 Y the bers ( i)y form a Z-base of By, hence y is a surjective homomorphism and Uy = ker y is a split unipotent group since GF (y) is split. Theorem 5.3. Let G be a reductive group of inner type de ned over F with 1 (G) torsion free and Y be the variety of Borel subgroups in G. Then for any G-scheme X over Y there is a spectral sequence 2 = TorK0 (G;Y ) (K0 (Y ); K 0 (G; X )) ) K 0 (X ): Ep;q p q p+q Corollary 5.4. The natural homomorphism K0(Y ) K0(G;Y ) K00 (G; X ) ! K00 (X ) is an isomorphism. Remark 5.5. The groups Kn(G) for a simply connected group G of inner type were computed by I.Panin in [15]. This result can be also obtained by using the spectral sequence in Theorem 5.3. 5.3. Semisimple case. Let G be a simply connected semisimple group of inner type, C be the center of G ( nite diagonalizable group scheme over F ), Gd be a split twisted form of G. We identify C with the center of Gd . Denote by Gd the adjoint group Gd =C . Choose a Borel subgroup B Gd de ned over F . Let : Gd ! GL(V ) be an irreducible representation. Denote its highest weight (with respect to B ) by 2 B . Consider the line bundle L = (Gd A 1F )=B over Y d = Gd=B , where the B -action on A 1F is given by , and the vector bundle M d = L F V over Y d where L is the dual line bundle. There are two actions of Gd on M d . The rst action is induced by one on V given by and trivial on L. In this case Gd acts over Y d. The second action of Gd on M d is given by one of Gd on V with d and by the natural action on L . Since C acts trivially, this action induces an Gd-action on M d . Note that these two actions commute. Since G is of inner type, the variety Y d descents to the variety Y of Borel subgroups in G, the vector bundle M d descents to a vector bundle M over Y and the rst action descents to an action of G on M over Y . The vector bundle M together with the G-action depends only on , we denote it by M (). The class of M () in K0 (G; Y ) we denote by [M ()]. Proposition 5.6. Let 1; 2 ; : : : ; r be the fundamental representations of Gd . Then the natural homomorphism K0(Y )[X1 ; X2; : : : ; Xr ] ! K0 (G; Y ) 34
taking indeterminate Xi to [M (i )], is an isomorphism, in particular, K0(G; Y ) is a polynomial ring over K0(Y ). Proof. By the local-to-global principle (Theorem 2.27), it is sucient to show that for any y 2 Y the ring K0(G; Spec F (y)) is the polynomial ring over K0(F (y)) with the indeterminates being the classes of the fundamental representations. But since G is split over F (y), it follows from Example 1.20. Let : G ! GL1;A be a group homomorphism, where A is a central simple F -algebra. We denote by A the algebra A together with the G-action given by the formula ga = (g) a (g)?1; where a 2 A and g is in G. We simply write A for the algebra A together with the trivial G-action. Let X be a G-scheme. The exact functor (being an equivalence of categories) M(G; X; A) ! M(G; X; A); taking M 2 M(G; X; A) to the same A OX -module M with the G-action given by g m = (g) gm; induces an isomorphism 0 Kn0 (G; X; A) ! Kn(G; X; A): The image of an element u 2 Kn(G; X; A) in Kn(G; X; A) we denote by u. Let M ! Q be an injective resolution of M 2 M(G; X; A). Then clearly M ! Q is an injective resolution in M(G; X; A), hence Rif( M ) ' (Ri f(M )). We have proved Lemma 5.7. Let f : X ! X 0 be a projective morphism of G-schemes. Then f (u) = (f (u)) 2 K00 (G; X; A) for any u 2 K00 (G; X; A). Denote by hui the image of an element u 2 Kn0 (G; X; A) under the homomorphism for : Kn0 (G; X; A) ! Kn0 (G; X ) induced by the forgetful functor M(G; X; A) ! M(G; X ): Theorem 5.8. Let G be a simply connected semisimple group of inner type, X be a G-scheme. Then the kernel of the natural surjection K00 (G; X ) ! K00 (X ) is generated by the elements hui?hui for all fundamental Tits representations : G ! GL1;A and all u 2 K00 (G; X; A). 35
Proof. Let Y be the variety of Borel subgroups in G. By Corollary 5.4, we have canonical isomorphisms K00 (X Y ) ' K0 (Y ) K0(G;Y ) K00 (G; X Y ) ' K00 (G; X Y )=I (G) K00 (G; X Y ); res K (Y )). where I (G) = ker(K0 (G; Y ) ?! 0 Consider the natural projection p : X Y ! X . Since p ([OX Y ]) = [OX ] (see [5]), by the projection formula, p p = id, hence res K 0 (X )) = p (I (G) K 0 (G; X Y )): ker(K00 (G; X ) ?! 0 0 By Proposition 5.6, K0 (G; Y ) is the polynomial ring over K0(Y ) in variables [M ()] for all fundamental Tits representations . Hence the ideal I (G) is generated by [M ()] ? [M ()], where bar indicates the vector bundle with the trivial G-action. Therefore, the kernel in question is generated by the di erences hp(M () N )i ? hp(M ()
N )i for all N 2 M(G; X Y ). It remains to notice that by Lemma 5.7, (p (M () N )) = p ( (M () N )) = p (M () N ): Corollary 5.9. Assume that for any any fundamental Tits representation : G ! GL1;A the multiplication homomorphism K0 (A) K00 (G; X ) ! K00 (G; X; A) is surjective. Then the natural map K00 (G; X )=J (G) K00 (G; X ) = Z R(G) K00 (G; X ) ! K00 (X ) is an isomorphism. Proof. It suces to show that for any u 2 K00 (G; X; A) the element hui ? hui belongs to J (G) K00 (G; X ). We may assume that u = [V F M ] for some M 2 M(G; X ) and an A-module V considered with the trivial G-action. Then u = [V M ], where the G-action on V is given by g v = (g)v and hui?hui = v [M ], where v = h[V ]i?h[V ]i clearly belongs to J (G). As an application we compute the Grothendieck group of an adjoint group of inner type. Theorem 5.10. Let G be an adjoint semisimple group of inner type over F , : Ge ! G be the universal covering, C = ker(). Then the natural homomorphism R(C )=J (Ge) R(C ) = Z R(Ge) R(C ) ! K0(G) is an isomorphism. 36
Proof. For any central simple F -algebra A there is the commutative diagram with horizontal isomorphisms (Corollary 2.12, Example 2.18) K0(A) R(C ) ???! K0 (A) K0 (Ge; G) ? ? y
? ? y
R(C; A) ???! K0(Ge; G; A): Since G is of inner type, C is a diagonalizable nite group scheme, hence the left vertical homomorphism is surjective (see Example 1.24). Therefore, we can apply Corollary 5.9 to the Ge-variety G. Consider the isomorphism in the Theorem in detail. First of all, note that R(C ) ' Z[C ]. For any character 2 C consider the line bundle P = (Ge A 1F )=C on G, where the C -action is given by c (~g; x) = (~gc?1; (c)x) and denote by n the greatest common divisor of dimensions of all representations of Gesep over Fsep with the restriction on C given by the multiplication by . These numbers depend only on the Dynkin diagram of G and were computed in [12]. Corollary 5.11. The assignment 7! [P ] induces an isomorphism Z[C ]=A ! K0 (G); where A is the subgroup in Z[C ] generated by n ?1 ind ( ?1) ( ? ) for all ; 2 C (here : C ! Br(F ) is the Tits map). In particular, the group K0 (G) is generated by the classes of linear vector bundles P , 2 C . Proof. By the representation theory of semisimple groups [22], one can identify R(Ge) with the subring in R(Gesep) generated by all u = ind ( ) [V ], where V is an irreducible Gesep-module such that C acts on V by the multiplication by 2 C . Hence the ideal J (Ge) Z[C ] in Z[C ] is generated by resG=C e (u) ? rank(u) = dim(V ) ind ( ) ( ? 1) and therefore coincides with the group A. Example 5.12. Let (V; q) be a quadratic form over F of dimension 2n + 1, G = SO(V; q) be the special orthogonal group. Then Ge = Spin(V; q) is the universal covering group, C = 2 , C = f0; g and n = 2n (see [12]). The fundamental Tits algebra corresponding to the character is the even Cli ord algebra C0 (V; q). Hence K0(SO(V; q)) = Z 1 (Z=2n ind C0(V; q)Z) ([P ] ? 1): 37
6. General case Assume rst that G is a quasi-split reductive group over F with 1(G) torsion free. Choose a Borel subgroup B G, a maximal torus T B , de ned over F , and set Y = G=B . By Proposition 2.22, there is an etale F -algebra A and an element u 2 K0 (G; Y; A) inducing an isomorphism u : A ! Y in C (G). Applying the functor K0X : C (G) ! Ab; where X is a G-scheme, to the isomorphism u, we get an isomorphism : K00 (G; X; A) ! K00 (G; X Y ); x 7! p(x) A q(u); where p : X Y ! X and q : X Y ! Y are projections. Lemma 6.1. The following diagram is commutative K00 (G; X; A) ???! K00 (G; X Y ) ???! K00 (B ; X ) ? ?for y
K00 (G; X )
res ?!
? ?res y
K00 (X )
( is de ned by Proposition 2.10). Proof. Let L 2 M(G; X; A); then ([L]) = [p(L) A q(u)]: The composition res is given by the restriction along the morphism r : X = X fB g ,! X Y . Since p r = id and q r passes through Spec F , it follows that (res )[L] = [L A A] = [L] 2 K00 (X ); since the restriction of u in K0(A) equals [A]. On the other hand, res(for([L])) = [L] 2 K00 (X ). Corollary 6.2. The image of the restriction K00 (B ; X ) ! K00 (X ) is contained in the image of res : K00 (G; X ) ! K00 (X ).
For any i 1 denote by Bi the subgroup in B generated by T and the root subgroups U for all roots with l( ) i where l is the length function (see [8, 29.2]). Clearly, Bi is a normal subgroup in B de ned over F and B1 = B , Br+1 = T , where r = rank(G). For any i 1 the factor-group Bi=Bi+1 is isomorphic to the additive group of the etale algebra corresponding to the nite Galois-set of roots on length i and hence is isomorphic to a direct product of several copies of G a;F . 38
Hence, there is a ltration between T and B by subgroups with the consequent factors isomorphic to G a;F . The Lemma 2.14 then implies Corollary 6.3. The homomorphism res : K00 (B ; X ) ! K00 (T ; X ) is an isomorphism. Now we prove the main result of this section Theorem 6.4. Let G be a reductive group de ned over a eld F . Then the following condition are equivalent: 1. G is factorial. 2. For any quasi-projective G-scheme X the restriction homomorphism K00 (G; X ) ! K00 (X ) is surjective. Proof. 1 ) 2. Assume rst that G is a quasi-split reductive group and choose a maximal torus T and a Borel subgroup B as above. Denote by G0 the commutant of G and by T 0 the torus T \G0 . It is a maximal torus in G0 with the module of characters isomorphic to the lattice of weights. Since the latter is a permutation Galois module (with fundamental weights forming a permutation base), the torus T 0 is quasi-trivial. Since G is factorial, the torus G=G0 is also factorial (see Proposition 1.10). Since G=G0 ' T=T 0, the exactness of the sequence of tori (10) 1 ! T 0 ! T ! T=T 0 ! 1 and [17, Cor. 6.11] imply that T is also a factorial torus. Hence by Proposition 3.9, Corollaries 6.2 and 6.3, the restriction homomorphism (1) is surjective. Consider now the general case. Let Y be the variety of Borel subgroups in G. Since for any point y 2 Y the group GF (y) is quasi-split, it follows by the local-to-global principle (Theorem 2.27) from the quasisplit case considered above that the restriction homomorphism (11) K00 (G; X Y ) ! K00 (X Y ) is surjective. Let p : X Y ! X be a projection. Since inverse and direct images commute with the restriction homomorphism and the composition p p is the identity (see the proof of Theorem 5.8), it follows that restriction homomorphism (1) is a direct factor of (11) and hence is surjective. 2 ) 1. Taking X = GE for some nite eld extension E=F , we have a surjective homomorphism Z = K0 (E ) = K00 (G; GE ) ! K00 (GE ); 39
i.e. K00 (GE ) = Z 1. Hence, the rst term of the topological ltration K00 (GE )(1) of K00 (GE ) (see [16, x7.5]), being the kernel of the rank homomorphism K00 (GE ) ! Z, is trivial, in particular, Pic(GE ) ' K00 (GE )(1=2) = 0; i.e. G is a factorial group. Corollary 6.5. Let H be a subgroup in G. Then K0 (G=H ) is generated by the classes of vector bundles (G V )=H over G=H for all linear representations H ! GL(V ). Remark 6.6. Using Proposition 3.7 instead of 3.9 one can drop the assumption for X to be quasi-projective in the case when the factor group G=G0 modulo the commutant is a quasi-trivial torus. For example, a semisimple group satis es this property. At the end of the section consider smooth projective case. Theorem 6.7. In the conditions of Theorem 6.4 let X be a smooth projective G-scheme over F . Then the restriction homomorphism Kn0 (G; X ) ! Kn0 (X ) is split surjective. Proof. Consider the smooth scheme X X with the action of G given by g(x; x0) = (x; gx0). By Theorem 6.4, the restriction homomorphism K00 (G; X X ) ! K00 (X X ) is surjective. Hence by Lemma 2.23, applied to the trivial subgroup of G, the restriction homomorphism Kn0 (G; X ) ! Kn0 (X ) is a split surjection.
7. Applications 7.1. Finiteness results. Assume rst that the base eld F is algebraically closed. Let G be a connected algebraic group over F and T be a maximal torus in G. Lemma 7.1. The restriction homomorphism R(G) ! R(T ) is nite and injective. Proof. By Corollary 1.16, we may assume that G is a reductive group. Then the restriction map induces an isomorphism R(G) ! R(T )W ; where W is the Weyl group of G (see example 1.19). Hence R(T ) is integral over R(G) and therefore the restriction homomorphism is nite, since R(T ) = Z[T ] is a nitely generated ring (Laurent polynomial ring over Z). 40
Lemma 7.2. Let S be a subtorus in G. Then the restriction homomorphism R(G) ! R(S ) is nite.
Proof. Let T be a maximal torus containing S . Then in the composition of the restriction maps R(G) ! R(T ) ! R(S ) the rst homomorphism is nite by Lemma 7.1 and the latter is surjective since R(S ) is generated by 1-dimensional representations and the restriction map T ! S is surjective. Assume now that F is an arbitrary eld. Lemma 7.3. Let G be a connected subgroup in GL(V ) for some vector space V over F of nite dimension. Then the restriction homomorphism R(GL(V )) ! R(G) is nite. Proof. Assume rst that F is algebraically closed. Denote by T a maximal torus in G. By Lemma 7.2 applied to GL(V ), the group R(T ) is a nite R(GL(V ))-module. Since R(GL(V )) is a noetherian ring (see Example 1.21), it follows from Lemma 7.1, that R(G) is also a nite R(GL(V ))-module. Now let F be an arbitrary eld. Since R(G) is a submodule in R(Galg ) over the noetherian ring R(GL(V )) = R(GL(Valg )), it follows that R(G) is a nite R(GL(V ))-module. Proposition 7.4. Let H be an algebraic group over an arbitrary eld F , G be a connected subgroup in H . Then the restriction homomorphism R(H ) ! R(G) is nite. Proof. Choose any imbedding H ,! GL(V ). By Lemma 7.3, the composition of the restriction maps R(GL(V )) ! R(H ) ! R(G) is a nite homomorphism. Hence, the latter homomorphism is also nite. Corollary 7.5. The group Z R(G) R(H ) = R(H )=J (G) R(H ) is nitely generated. Lemma 7.6. Let G be a reductive group de ned over a eld F . Then there exists an exact sequence 1 ! P ! Ge ! G ! 1 with a factorial reductive group Ge and a quasi-trivial torus P . 41
Proof. Case 1. Assume that G is a torus. Choose a asque resolution 1 ! S ! P1 ! G ! 1 of the torus G and a co asque resolution 1!S!P !V !1 of the torus S (see [4]). Consider the torus Ge de ned by the following commutative diagram
S ???! P1 ???! G
? ? y
P ???! ? ? y
? ? y Ge ? ? y
???! G
V: V Since P1 is a quasi-trivial torus and V is factorial (see Example 1.8), the torus G~ is also factorial by [17, Cor. 6.11]. Case 2. Assume that the commutant G0 of G is simply connected. Since G=G0 is a torus, by the rst case, there is an exact sequence 1 ! P ! T ! G=G0 ! 1 with a quasi-trivial torus P and a factorial torus T . The exact sequence we need is de ned by the following commutative diagram G0 G0 P ???!
? ? y Ge ? ? y
? ? y
???! G? ? y
P ???! T ???! G=G0; since the commutant Ge0 of G~ is isomorphic to the simply connected e G e0 is isomorphic to the factorial torus T (see Propogroup G0 and G= sition 1.10). Now consider the general case. Let Z be the connected reduced center of G and : Ge0 ! G0 be the universal covering of the commutant G0 of G. Denote by C the ( nite) kernel of the homomorphism Ge0 Z ! G, (x; z) 7! (x) z. Choose an imbedding C ,! P1 of C to a quasitrivial torus P1 and consider the group G1 de ned by the commutative 42
diagram
C? ???! Ge0 ? Z ???! G ? y
? y
P1 ???! G1 ???! G: The commutant of G1 is isomorphic to Ge0 and hence is simply connected. By the second case there exists an exact sequence 1 ! P2 ! Ge ! G1 ! 1 with a factorial reductive group Ge and a quasi-trivial torus P2. Since P1;2 are quasi-trivial, the left column in the commutative diagram P?2 P?2 ? y
P ???! ? ? y
? y Ge ? ? y
???! G
P1 ???! G1 ???! G splits and hence P is also a quasi-trivial torus. Theorem 7.7. Let G be a reductive group over F , H be a connected subgroup in G, X = G=H . Then the group K0 (X ) is nitely generated. Proof. Let 1 ! P ! Ge ! G ! 1 be an exact sequence from Lemma e H e . The sequence 7.6, He = ?1(H ), so that X ' G= j
He 1 ! P ! He ?! H!1 is exact, hence He is a connected algebraic group. By Theorem 6.4, the second map in the composition (see also Corollary 2.12) res K (X ) R(He ) ! K0(Ge; X ) ?! 0 is surjective. On the other hand, this composition factors as R(He ) ! Z R(Ge) R(He ) ! K0(X ) and the group in the middle is nitely generated by Corollary 7.5, hence K0(X ) is a nitely generated group. Corollary 7.8. Let G be a reductive group over F . Then the kernel K0(G)(1) of the rank homomorphism K0(G) ! Z is a nite group. In particular, K0 (G) is a nitely generated group of Q -rank 1. 43
Proof. Since Gsep is a split group, by Proposition 4.9, K0 (Gsep)(1) is a nite group. The usual restriction-corestriction argument shows that the kernel of the natural homomorphism K0(G)(1) ! K0 (Gsep)(1) is a torsion and hence a nite group by Theorem 7.7. Remark 7.9. The group K0 (G) for an algebraic torus G was computed in terms of generators and relations in [11]. 7.2. Group action on K 0-groups. Let G be an algebraic group and X be a G-scheme over F . For any g 2 G(F ) denote the automorphism x 7! g x of X by g . The group G(F ) acts naturally on Kn0 (X ) by the inverse images g . Proposition 7.10. Let G be a reductive group. Then 1. For any quasi-projective G-scheme X the group G(F ) acts trivially on K00 (X ). 2. For any smooth projective G-scheme X the group G(F ) acts trivially on Kn0 (X ) for any n 0. Proof. By Lemma 7.6 there exists an exact sequence 1 ! P ! Ge ! G!1 with a factorial reductive group Ge and a quasi-trivial torus P . It follows from the exact sequence
(F ) Ge(F ) ?! G(F ) ! H 1(F; P (Fsep)) and the triviality of H 1(F; P (Fsep)) (the Hilbert 90 Theorem) that the homomorphism (F ) is surjective. Hence, we can replace G by Ge and assume that G is factorial. By the de nition of a G-module M (see 2.1), the isomorphism : (M ) ! p2(M ); where : G X ! X is an action morphism, induces an isomorphism of two compositions res M(X ) ???! M(G; X ) ???! M(G X ); p2
hence by [16, x2] the compositions res K 0 (X ) ???! Kn0 (G; X ) ???! Kn0 (G X ) n p2
coincide. For any g 2 G(F ) denote by "g the morphism X ! G X , x 7! (g; x). Then clearly p2 "g = idX and "g = g . The inverse image "g 44
is de ned since "g is of nite Tor-dimension (see [16, x7, 2.5]). Thus, we have "g p2 = id and "g = g on Kn0 (X ), hence res = "g p2 res = "g res = g res : Kn0 (G; X ) ! Kn0 (X ): If X is a quasi-projective, then by Theorem 6.4, the restriction homomorphism res is surjective for n = 0, hence g = id. In the case of smooth projective X the restriction is surjective for any n (see Theorem 6.7), hence again g = id. Corollary 7.11. The classes of rational points from the same G(F )orbit in X (F ) coincide in K00 (X ). In particular, the classes of rational points of G in K0(G) are pairwise equal.
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