Connective K-theory and Adams operations

We investigate the relations between the Grothendieck group of coherent modules of an algebraic variety and its Chow group of algebraic cycles modulo rational equivalence. Those are in essence torsion phenomena, which we attempt to control by considering the action of the Adams operations on the Brown-Gersten-Quillen spectral sequence and related objects, such as connective K_0-theory. We provide elementary arguments whenever possible. As applications, we compute the connective K_0-theory of the following objects: (1) the variety of reduced norm one elements in a central division algebra of prime degree; (2) the classifying space of the split special orthogonal group of odd degree.


Introduction
The goal of the paper is to illustrate the usefulness of the connective K 0 -groups of an algebraic variety X and Adams operations for the study of relations between K-theory and the Chow groups of X.
For every integer i, denote M i (X) the abelian category of coherent O X -modules with dimension of support at most i. We have a filtration (M i (X)) of the category M(X) of all coherent O X -modules such that M i (X) = 0 if i < 0 and M i (X) = M(X) if i ≥ dim(X).
The K-groups of M(X) are denoted K ′ n (X). The exact couple (D r,s , E r,s ) of homological type with D 1 r,s = K r+s (M r (X)) and E 1 r,s = x∈X (r) where X (r) denotes the set of points in X of dimension r, yields the Brown-Gersten-Quillen (BGQ) spectral sequence with respect to the topological filtration K ′ n (X) (i) = Im(K n (M i (X)) → K n (M(X))) on K ′ n (X). The group K ′ 0 (X) coincides with the Grothendieck group of coherent O X -modules. The terms E 2 i,−i = CH i (X) of the second page are the Chow groups of classes of dimension i algebraic cycles on X. The natural surjective homomorphism ϕ i : CH i (X) → → K ′ 0 (X) (i/i−1) := K ′ 0 (X) (i) /K ′ 0 (X) (i−1) takes the class [Z] of an integral closed subvariety Z ⊂ X of dimension i to the class of O Z . The kernel of ϕ i is covered by the images of the differentials in the spectral sequence with target in CH i (X).
The groups CK i (X) := D 2 i+1,−i−1 = Im(K 0 (M i (X)) → K 0 (M i+1 (X))) are the connective K 0 -groups of X (see [1]). These groups are related to the Chow groups via exact sequences In the present paper we study differentials in the spectral sequence with target in the Chow groups via the connective K 0 -groups. In Sections 2 and 3 we introduce and study the notion of an endo-module associated with an algebraic variety that locates a part of the BGQ spectral sequence near the zero diagonal. In Section 4 we introduce an approach based on the Adams operations of homological type on the Grothendieck group. Compatibility of the Adams operations with the differentials in the spectral sequence was proved in [12,Corollary 5.5] with the help of heavy machinery of higher K-theory. We give an elementary proof of the compatibility with the differential coming to the zero diagonal of the spectral sequence. The Adams operations are applied in Section 5 to the study of the kernel of the homomorphism ϕ i , and of the relations between the Grothendieck group and its graded group with respect to the topological filtration.
In Section 6 we consider the endo-module arising form the equivariant analog of the BGQ spectral sequence. As an example we compute the connective K 0 -groups of the classifying space of the special orthogonal group O + n with n odd and as an application compute the differentials in the spectral sequence. We use the following notation in the paper. We fix a base field F . A variety is a separated scheme of finite type over F . The residue field of a variety at a point x is denoted by F (x), and the function field of an integral variety X by F (X). The tangent bundle of a smooth variety X is denoted by T X .

Endo-modules
Definition 2.1. Let R be a commutative ring and B • a Z-graded R-module. An endomorphism of B • of degree 1 is (an infinite) sequence of R-module homomorphisms . . .
We call the pair (B • , β • ) an endo-module over R. If β • is clear from the context, we simply write B • for (B • , β • ).
We have exact sequences and (an infinite) diagram of R-module homomorphisms: We define the derived endo-module (B (1) Then the derivatives of A i 's and C i 's are: r,s , E 1 r,s ) be an exact couple of R-modules (see [21, §5.9]) such that D 1 r,s = 0 if r + s < 0. The exact sequences The associated group A i coincides with the image of the first homomorphism in the exact sequence and i,−i on the first page of the spectral sequence associated with the exact couple factors into the composition The derived endomodule of B • arises the same way from the derived exact couple. It follows that the differential δ (s) in the sth derivative B (s) • correspond to the differentials in the (s + 1)th page of the spectral sequence.
For an endo-module of H. We would like to compute the subsequent factor modules There is a canonical surjective homomorphism and K ′ 0 (X) with a K 0 (X)-module structure. We will denote the latter by (a, b) → a · b, where a ∈ K 0 (X) and b ∈ K ′ 0 (X). For an integer i denote M i (X) the abelian category of coherent O X -modules of support dimension at most i.
Definition 3.1. We define an endo-module (B • (X), β • ) over Z associated with X as follows. Set where the colimit is taken over all closed subvarieties Z ⊂ X of dimension at most i with respect to the push-forward homomorphisms [16, §7]). The localization exact sequence [16, §7] looks then as follows: is the group of algebraic cycles of dimension i. The groups A i (X) associated with the endo-module B • (X) are given then by There are also homomorphisms f * : C i (Y, 1) → C i (X, 1), defined by letting the homomorphism F (y) × → F (x) × be trivial unless f (y) = x, in which case it is given by the norm of the finite degree field extension F (y)/F (x) (see [3, §1.4]). We have If f : Y → X is a flat morphism of relative dimension r, there are homomorphisms f * : B i (X) → B i+r (Y ). There are also homomorphisms f * : C i (X, 1) → C i+r (Y, 1), defined by letting the homomorphism F (x) × → F (y) × be trivial unless f (y) = x, in which case it is given by the inclusion F (x) ⊂ F (y) (see [3, §1.7]). We have
Definition 3.5. The derivatives B i (X) (1) of B i (X) are the connective K-groups CK i (X) and C i (X) (1) are the Chow groups CH i (X) of classes of cycles of dimension i (see [1]).
We have the exact sequences i−1 's are called the Bott homomorphisms. We can view the graded group CK • (X) as a module over the polynomial ring Z[β]. It follows from the definition that of an integral closed subvariety Z ⊂ X of dimension i to the class of O Z . The relations between the groups CK i (X), CH i (X) and K ′ 0 (X) (i) are given by a commutative diagram CK i (X) . The goal is to study the homomorphisms ϕ i . Recall that C i (X) (1) = CH i (X) and the groups C i (X) (s) are inductively defined via the exact sequences Proposition 3.7. The homomorphism ϕ i factors as the composition Remark 3.8. The groups B i (X) and B i (X) (1) = CK i (X) (but not B i (X) (s) with s > 1), viewed as generalized homology theories, satisfy the localization property (see [13,Definition 4.4.6]). The derivatives B i (X) (s) (but not B i (X)) satisfy homotopy property (see [13,Definition 5.1.3]) if s ≥ 1. Thus, the first derivative (the connective K-theory) is the only derivative that satisfies both localization and homotopy properties.

Generators for
Definition 3.9. Let L be a line bundle (locally free coherent O X -module of constant rank 1) over a variety X, and s ∈ H 0 (X, L) a section. We denote by Z(s) the closed subscheme of X whose ideal is the image of s ∨ : L ∨ → O X , and by D(s) its open complement. The section s is called regular if the morphism s : O X → L (or equivalently s ∨ : L ∨ → O X ) is injective. In this case, the immersion Z(s) → X is an effective Cartier divisor.
If s is a regular section of a line bundle L over X, the exact sequence of O X -modules We also view x/y (resp. y/x) as a regular function on D(y) (resp. D(x)). Mapping u to that function induces an isomorphism between A 1 = Spec(F [u]) and D(y) (resp. D(x)).

It follows from (3.2) and (3.3) that the subgroup
Composing with the morphism A 1 ≃ D(y) ⊂ P 1 (using Notation 3.11), we obtain a morphism U → P 1 . We denote by S the closure in P 1 of the image of the latter morphism, endowed with the reduced scheme structure. Consider the graph of the morphism U → S as a closed subset of U × S, and let Y ′ be its closure in Z × S, endowed with the reduced scheme structure. By Chow's lemma [5, (5 If dim S = 0 (i.e. a is constant), then b = g * c for some c ∈ F (S) × , and the morphism g has relative dimension i + 1, so that, by (3.3) and (3.4) Otherwise S = P 1 , and g has relative dimension i. Using Notation 3.11, we have b = g * (x/y). By Lemma 3.12 and (3.4), we have in B i (Y ) The flatness of g implies that the sections s 1 := g * x and s 2 := g * y of g * O(1) are regular, and satisfy Z(

Homological Adams operations
4.1. K-theory with supports. Definition 4.1. Let X be a variety and Y ⊂ X a closed subscheme. We consider the category of chain complexes of locally free coherent O X -modules The full subcategory consisting of those complexes whose homology is supported on Y will be denoted by C Y (X). We define the group K Y 0 (X) as the free abelian group generated by the elements [E • ], where E • runs over the isomorphism classes of objects in C Y (X), modulo the following relations: . When P is a locally free coherent O X -module and i ∈ N, we will denote the complex Let X be a variety and Y ⊂ X a closed subscheme. There is a bilinear map ; (a, β) → a · β, such that for any locally free coherent O X -modules P and E • ∈ C Y (X) we have We will need the following basic compatibilities, which may be verified at the level of modules (before applying the functor K ′ 0 ). Lemma 4.3. Let X be a variety and Y, Z closed subschemes of X.
. Let X be a regular variety and Y ⊂ X a closed subscheme. Then the following map is an isomorphism: Let L be a line bundle over X, and s ∈ H 0 (X, L) a section. We will denote by K(s) the complex of locally free coherent O X -modules concentrated in degrees 1, 0.
The homology of K(s) is supported on Z(s), so that we have a class [K(s)] ∈ K Z(s) 0 (X). If the section s is regular, then  (4.2)). This element is visibly independent of s.

4.2.
Bott's class. From now on we fix a nonzero integer k.
Thus we are reduced to assuming that L ∨ is generated by its global sections. Pulling back along the associated morphism X → P n , we reduce to X = P n and L = O P n (−1). We prove by induction on n that (1 − [L]) n+1 = 0 ∈ K 0 (X). There is a regular section s of L ∨ such that Z(s) = P n−1 and L| Z(s) = O P n−1 (−1). Let i : Z(s) → X be the immersion. By (3.10) and the projection formula, we have in That element vanishes by induction. Since the natural homomorphism K 0 (X) → K ′ 0 (X) is an isomorphism [16, §7.1], the claim follows.
By Lemma 4.8 and the splitting principle, there is a unique way to assign to each vector bundle E over a variety X an element θ k (E) ∈ K 0 (X) so that: ]. We deduce, using Lemma 4.8 and the splitting principle, that θ k (E) is invertible in K 0 (X)[1/k] for any vector bundle E over a variety X. Thus for every variety X, the association E → θ k (E) extends uniquely to a map For any variety X, we have θ k (1) = τ k (0) = k, and therefore (4.10) θ k (n) = k n for any n ∈ Z ⊂ K 0 (X).

Adams operations.
The classical Adams operation ψ k : K 0 (−) → K 0 (−) is defined using the splitting principle by the following conditions: This construction may be refined to obtain an operation on the K-theory with supports: Definition 4.11. Let X be a regular variety and Y ⊂ X a closed subscheme. Then the group K Y 0 (X) defined in (4.1) coincides with the one considered in [18], as they are both canonically isomorphic to K ′ 0 (Y ). Thus the construction of [18] yields an Adams operation ψ k : K Y 0 (X) → K Y 0 (X). The following properties follow from the construction given in [18].
Lemma 4.12. Let X be a regular variety and Y ⊂ X a closed subscheme.
(a) If f : X ′ → X is a morphism and X ′ is regular, then ). Definition 4.13. Any quasi-projective variety X may be embedded as a closed subscheme of a smooth quasi-projective variety W . By Lemma 4.4, there is a unique homomorphism . It follows from the Adams Riemann-Roch theorem without denominators that this operation is independent of the choice of W , and that it commutes with proper push-forward homomorphisms (see [18,Théorème 7]).
We now explain how to remove the assumption of quasi-projectivity. An envelope is a proper morphism Y → X such that for each integral closed subscheme Z ⊂ X, there is an integral closed subscheme W ⊂ Y such that the induced morphism W → Z is birational. Any base change of an envelope is an envelope, and the composition of two envelopes is an envelope [3,Lemma 18.3 (2) (3)].
Lemma 4.14. Let f : Y → X be an envelope. Denote by p 1 , p 2 : Y × X Y → Y the two projections. Then the following sequence is exact The sequence is clearly a complex. We proceed by noetherian induction on X. Since push-forward homomorphisms along nilimmersions are bijective [16, §7, Proposition 3.1], we may assume that X is reduced. Assuming that X = ∅, we may find a closed subscheme X ′ X whose open complement U is such that f | U : V := f −1 U → U admits a section s : U → V (letting X 1 , . . . , X n be the irreducible components of X, we find Y 1 ⊂ Y birationally dominating X 1 ; then Y 1 → X 1 restricts to an isomorphism over a nonempty open subscheme U 1 of X 1 , and we set U = U 1 ∩ (X − (X 2 ∪ · · · ∪ X n ))). Let Y ′ = f −1 (X ′ ), and consider the commutative diagram with exact rows [16, §7.3] Each homomorphism (f | U ) * is surjective, since it admits a section s * . The homomorphism (f | X ′ ) * is surjective by induction, and a diagram chase shows that f * is surjective.
Chasing the above diagram, we see that c may be modified to satisfy additionally (f | X ′ ) * (c) = 0. By induction c is the image of an element of This concludes the proof. Since any variety X admits an envelope Y → X where Y is quasi-projective (see [ Using (4.12.a) and the surjectivity of push-forward homomorphisms along envelopes, we see that the Adams operation ψ k commutes with the restriction to any open subscheme.
Let k ′ ∈ Z − {0} and let X be a quasi-projective variety. Note that for any a ∈ K 0 (X) this is immediate when a is the class of a line bundle, and follows in general from the splitting principle. Combining (4.16) with (4.12.b), (4.12.c) and (4.3.b), we deduce that . By Proposition 4.15, this formula remains valid when X is an arbitrary variety.
Proof. Let i : X → W be a closed immersion, where W is smooth and quasi-projective. Then i is a regular closed immersion, let N be its normal bundle. The Gysin homomorphism i * : K X 0 (X) → K X 0 (W ) is by definition the unique map compatible with the isomor- (4.2)). By (4.12.d) and the Adams Riemann-Roch theorem (see [18,Théorème 3], where N should be replaced by N ∨ ), we have in Proof. Let us apply Lemma 4.22 and use its notation. By Lemma 4.4, there is an element Proposition 4.25. Let X be an integral quasi-projective variety of dimension d. Let L be a line bundle over X, and s 1 , s 2 regular sections of L. Then we may find a closed subscheme Z X containing Z(s 1 ) and Z(s 2 ) as closed subschemes, and such that Proof. Since the sections s 1 , s 2 are regular, we may find a nonempty open subscheme U of X which does not meet Z(s 1 ) ∪ Z(s 2 ). Then L| U is trivial. Shrinking U, we may Lemma 4.20. Let Z ′ be the reduced closed complement of U in X. The intersection of the ideal sheaves of Z ′ , Z(s 1 ), Z(s 2 ) in O X defines a closed subscheme Z ⊂ X whose open complement is U, and we have closed immersions j n : Z(s n ) → Z for n ∈ {1, 2}. Since θ k (L ∨ | U ) = k by (4.10), we have  [1/k], and i : Z → X is the closed immersion. By Lemma 4.7, the image σ ∈ K X 0 (X) of [K(s n )] ∈ K Z(sn) 0 (X) does not depend on n ∈ {1, 2}. For such n, we have in The statement follows.
Combining Propositions 4.25 and 3.13, we obtain:  (1) and K ′ 0 (X) (i/i−1) is a factor module of CH i (X). It follows from Proposition 5.1 that for every nonzero integer k and every a ∈ A i (X) is killed by k m (k s − 1) for some m ≥ 0. We consider supernatural numbers k ∞ (k s − 1) (see [17, I.1.3]) and write over all k > 1. For a prime integer p and integer i > 0 the group (Z/p i Z) × is cyclic of order (p − 1)p i−1 unless p = 2 and i ≥ 3 in which case this group is of exponent 2 i−2 . It follows that N s = 2 if s is odd and if s is even, where the product is taken over the set of all prime integers p such that p − 1 divides s (here v p is the p-adic valuation). For example, N 2 = 24, N 4 = 240, N 6 = 2520,. . . We proved the following: Proposition 5.2. Let s be a positive integer and X a variety. Then every element in the kernel of the homomorphism C i (X) (s) → → C i (X) (s+1) is killed by N s .
Write Z (p) for the localization of Z by the prime ideal pZ. Note that if p is a prime divisor of N s , then p − 1 divides s. It follows from Proposition 5.
is an isomorphism if p − 1 does not divides s. We have proved: It follows from Proposition 3.7 that the kernel of ϕ i is killed by the product N 1 N 2 · · · N d−i−1 . Every prime divisor p of the product is such that p − 1 divides an integer s ≤ d − i − 1, hence p ≤ d − i. We have proved: Theorem 5.4. Let X be a variety of dimension d. Then for every i = 0, 1, . . . , d, the map ϕ i is an isomorphism when localized by (d − i)!.
Remark 5.5. If X is a smooth variety of dimension d, an application of Chern classes and Riemann-Roch theorem imply that (d − i − 1)! · Ker(ϕ i ) = 0 for every i > 0 (see [3,Example 15.3.6]). Proposition 5.6. Let X be a variety. Then the kernel of the Bott homomorphism CK i (X) → CK i+1 (X) is killed by N 1 N 2 · · · N i+1 for every i ≥ 0. In particular, the Bott homomorphism is injective when localized by (i + 2)!.
Proof. We need to prove that A i (X) (1) is killed by N 1 N 2 · · · N i+1 . By induction on i we show that A i (X) (s) is killed by N s N s+1 · · · N s+i for every s ≥ 1. The statement is clear if and hence is killed by N s by Proposition 5.2. By induction, A i−1 (X) (s+1) is killed by N s+1 · · · N s+i . The result follows.
Corollary 5.7. Let X be a variety of dimension d. Then the associated endo-module CK • (X) degenerates when localized by d!.

Direct sum decompositions.
Theorem 5.8. For every variety X and integer i ≥ 0, the homomorphism admits a section, compatibly with proper push-forward homomorphisms.
Proof. For every integer k > 1, let r k = k · i j=1 (k j − 1) ∈ Z[1/(i + 1)!]. If p > i + 1 is a prime integer and k > 1 is such that the congruence class k + pZ is a generator of (Z/pZ) × , then since p − 1 > i, the integer r k is not divisible by p. It follows that the elements r k for k > 1 generate the unit ideal in Z[1/(i + 1)!].
Let M = K ′ 0 (X) (i) [1/(i + 1)!]. For each integer k > 1, consider the endomorphism Let N = K ′ 0 (X) (i−1) [1/(i + 1)!]. It follows from Proposition 5.1 that each σ k vanishes on N[1/r k ] and coincides with the identity modulo N[1/r k ]. Thus for any k, k ′ > 1, we have σ k = σ k • σ k ′ and σ k ′ = σ k ′ • σ k on M[1/r k r k ′ ]. Since σ k commutes with σ k ′ by (4.17), we deduce that σ k and σ k ′ coincide on M[1/r k r k ′ ]. By Zariski descent, there is a unique endomorphism of M whose localization is σ k for each k > 1. That endomorphism vanishes on N and coincides with the identity modulo N, hence induces the required section. The functoriality follows from that of the operations ψ k . Theorem 5.8 provides a functorial decomposition Taking appropriate colimits, we get the following: . These isomorphisms are compatible with proper push-forward homomorphisms.
certainly admits a section, since its target is freely generated by the classes [O Z ] where Z runs over the d-dimensional irreducible components of X. Therefore, in fact . However, these isomorphisms are not compatible with proper push-forward homomorphisms in general. For instance, let X be the Severi-Brauer variety of a central division algebra of prime degree p over F . Then d = p − 1 and χ(X, F ).
Then d X | n X . It follows from Proposition 5.1 that n X | N 1 · · · N d · d X . Thus if p is a prime number, we have (here v p is the p-adic valuation) This bound coincides with that of [7, Theorem 5.1 (ii)] if d < p(p − 1), but is not sharp anymore if d ≥ p(p − 1), at least when char F = p (see [7, Theorem 5.1 (i)]).
It follows that the differentials are isomorphisms for all i > 0. As a consequence we get the following calculation: It implies that for every j = 0, 1, . . . , p − 2 we have a sequence of isomorphisms In particular, the natural homomorphism CK p+1 (G) → CH p+1 (G) is an isomorphism and hence the element τ in CK p+1 (G) is unique. Our calculation yields: . Thus by the splitting principle, we may assume that r = 1. Since CK i (X) is generated by the images of the push-forward homomorphisms CK i (Z) → CK i (X), where Z ⊂ X is closed subscheme of dimension at most i, we may assume that dim X ≤ i. Then the homomorphism β : CK i−1 (X) → CK i (X) = K ′ 0 (X) is injective, hence it will suffice to prove that id = [E] · (id − c 1 (E)) and c 1 (E ∨ ) = −[E] · c 1 (E) as endomorphisms of K ′ 0 (X). This follows at once from the formula c 1 (L) = 1 − [L ∨ ] ∈ K 0 (X), valid for any line bundle L over X (in particular for L = E and L = E ∨ ).
For every i ≥ 1, let Q i be the subgroup of CK i (BO + n ) generated by β j c i+j over all j ≥ 0. Write Q i even for the subgroup of Q i generated by β j c i+j with i + j even. Obviously, βQ i ⊂ Q i−1 and βQ i even ⊂ Q i−1 even . Proposition 6.2. For every odd i = 1, 3, . . . , n, (1) Q i−1 = Q i−1 even , (2) There is an elementc i ∈ c i + Q i even such that 2c i = 0 and βc i = 0. Proof. We proceed by descending induction on i. Let i = n. It follows from Lemma 6.1 that c n = −c n and c n−1 = c n−1 − nβc n , i.e., nβc n = 0. Settingc n = c n we deduce that 2c n = 0 and βc n = 0 since n is odd.
The group Q n−1 even is generated by c n−1 and Q n−1 is generated by c n−1 and βc n = 0, hence Q n−1 = Q n−1 even . (i + 2) ⇒ i: It follows from Lemma 6.1 and the induction hypothesis that 2c i ∈ βQ i+1 = βQ i+1 even = Q i even , thus 2c i = even j>i a j β j−i c j with a j ∈ Z. Mapping to R(G) we see that 2c K i = a j c K j in R(G). On the other hand, c K i ∈ R(G) = Z[c K 2 , c K 4 , . . . , c K n−1 ], hence all a j are even, therefore, 2c i ∈ 2Q i even . We deduce that there isc i ∈ c i + Q i even such that 2c i = 0. Lemma 6.1 for c i−1 yields iβc i ∈ β 2 Q i+1 = β 2 Q i+1 even ⊂ Q i−1 even and therefore, iβc i ∈ Q i−1 even . As Q i−1 even maps injectively to R(G) and βc i maps to zero (since βc i is 2-torsion and R(G) is torsion-free), we have iβc i = 0. But i is odd, hence βc i = 0.
It follows fromc i ∈ c i + Q i even and βc i = 0 that βc i ∈ βQ i even ⊂ Q i−1 even . Finally, Q i−1 = Zc i−1 + Zβc i + β 2 Q i+1 = Zc i−1 + Zβc i + β 2 Q i+1 even ⊂ Q i−1 even . Note that since the Bott map CK 1 (BG) → R(G) is injective and R(G) is torsion free, the elementc 1 is trivial. It follows from Proposition 6.2 that the ring CK(BG) is generated by c 2 ,c 3 , c 4 , . . . ,c n and β. Writec i = c i for all even i.
Under the natural homomorphism CK(BG) → CH(BG) the classc i goes to c CH i . It is immediate that the natural diagonal homomorphism CK(BG) → R(G) × CH(BG) is injective. This implies two things: first the relations 2c i = 0 and βc i = 0 are the defining relation between thec i 's. In other words, CK(BG) = Z[c 2 ,c 3 , . . . ,c n , β]/(2c odd , βc odd ).
Second, the derived endo-module of B • (BG) degenerates, i.e., all nonzero differentials appear in the first derivative only and therefore, the second derivative degenerates.