Riemann-Roch for Tensor Powers

by

Bernhard Köck

Abstract. Mapping a locally free module on a scheme to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behavior of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups.

Introduction

Let f: X →Y be a projective morphism between Noetherian schemes X and Y. Then, mapping a locally free O_{X}-module F to the Euler characteristic ∑_{i≥0}(−1)^{i}[R^{i}f_{∗}(F)] induces, under certain additional assumptions, a push-forward homomorphism

f_{∗} : K_{0}(X) → K_{0}(Y)

between the Grothendieck groups of X and Y (see [FL]). The fundamental task in Riemann-Roch theory is to compute this homomorphism f_{∗}.

For any locally free O_{X}-module F, the symmetric group Σ_{l} acts on the l-th tensor power F^{⊗l} by permuting the factors. Using the “binomial theorem”

(F ⊕ G)^{⊗l} ≅ ⊕_{i=0}^{l} Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(F^{⊗i} ⊗ G^{⊗(l−i)}),

the association F ↦ F^{⊗l} can be canonically extended to a map

τ^{l} : K_{0}(X) → K_{0}(Σ_{l}, X)

from K_{0}(X) to the Grothendieck group K_{0}(Σ_{l}, X) of all locally free Σ_{l}-modules on X (see section 1). We call τ^{l} the l-th tensor power operation. Similarly, we have an external tensor power operation

τ^{l} : K_{0}(X) → K_{0}(Σ_{l}, X^{l})

where X^{l} denotes the l-fold fibred product of X with itself over Y.

In this paper, we study the following Riemann-Roch problem: How does τ^{l} behave with respect to f_{∗}? We give the following answers to this problem.

Theorem A (Künneth formula). Let f be flat. Then the following diagram commutes:

K_{0}(X) ⟶^{τ^{l}} K_{0}(Σ_{l}, X^{l})
f_{∗} ↓  ↓ f_{∗}^{l}
K_{0}(Y) ⟶^{τ^{l}} K_{0}(Σ_{l}, Y).

Here, f_{∗}^{l} denotes the equivariant push-forward homomorphism associated with the projection f^{l} : X^{l} → Y.

In the following theorems, let H_{X,l} denote the Σ_{l}-module on X of rank l − 1 given by the short exact sequence

0 ⟶ H_{X,l} ⟶ O_{X}[{1, …, l}] ⟶^{sum} O_{X} ⟶ 0

in which O_{X}[{1, …, l}] denotes the direct sum of l copies of O_{X} together with the obvious Σ_{l}-action and sum denotes the summation map. Furthermore, we set λ_{−1}(F) := ∑_{i≥0}(−1)^{i}[Λ^{i}(F)] for any locally free Σ_{l}-module F on X.

Theorem B (Riemann-Roch formula for closed immersions). Let f be a regular closed immersion with conormal sheaf C. Then the following diagram commutes:

K_{0}(X) ⟶^{λ_{−1}(C⊗H_{X,l})·τ^{l}} K_{0}(Σ_{l}, X)
f_{∗} ↓  ↓ f_{∗}
K_{0}(Y)  ⟶^{τ^{l}}  K_{0}(Σ_{l}, Y).

In the following two theorems, let l be a prime and let τ^{l} denote also the composition

K_{0}(X) →^{τ^{l}} K_{0}(Σ_{l}, X) →^{Res_{C_{l}}^{Σ_{l}}} K_{0}(C_{l}, X) ⟶^{can} K_{0}(C_{l}, X)/(O_{X}[C_{l}])

where C_{l} denotes the cyclic subgroup of Σ_{l} generated by the cycle ⟨1, …, l⟩.

Theorem C (Riemann-Roch formula without denominators for principal G-bundles). Let f be a principal G-bundle for some finite group G with l |̸ ord(G). Then the following diagram commutes:

K_{0}(X) ⟶^{τ^{l}} K_{0}(C_{l}, X)/(O_{X}[C_{l}])
f_{∗} ↓  ↓ f̄_{∗}
K_{0}(Y) ⟶^{τ^{l}} K_{0}(C_{l}, Y)/(O_{Y} [C_{l}]).

Theorem D (Riemann-Roch formula for a general f). The following diagram commutes:

K_{0}(X) ⟶^{λ_{−1}(T_{f}^{∨}·H_{X,l})^{−1}·τ^{l}} K_{0}(C_{l}, X)[l^{−1}]/(O_{X}[C_{l}])
f_{∗} ↓  ↓ f̄_{∗}
K_{0}(Y)  ⟶^{τ^{l}}  K_{0}(C_{l}, Y)[l^{−1}]/(O_{Y} [C_{l}]).

Here, T_{f}^{∨} denotes the cotangential element associated with f.

The proof of Theorem A (see section 2) essentially consists of a lengthy reduction to the “base change isomorphism”

f_{∗}^{l}(F_{1} ⊠ … ⊠ F_{l}) ≅ f_{∗}(F_{1}) ⊗ … ⊗ f_{∗}(F_{l}).

In this reduction, we will not use the Künneth spectral sequences developed in [EGA] (as one might expect) but only some standard K-theoretical results of [Q]. For the proof of Theorem B (see Theorem 4.1), we may assume by using the deformation to the normal cone that f is a so-called elementary embedding. In this case, the proof is based on the multiplicativity of τ^{l} (see Proposition 1.7) and on the formula

τ^{l}(λ_{−1}(F)) = λ_{−1}(F) · λ_{−1}(F ⊗ H_{X,l})

which is proved in Corollary 1.11. (In [Ko7], we will give another proof of Theorem B which does not use the deformation to the normal cone.) For the proof of Theorem C and Theorem D (for smooth f), we generalize a new idea of Nori (see [Ra]): In section 3, we prove the formula

Δ_{∗} (λ_{−1}(Ω_{X/Y} ⊗ H_{X,l})^{−1}) = 1 in K_{0}(C_{l}, X^{l})[l^{−1}]/(O_{X^{l}}[C_{l}])

where Δ : X → X^{l} denotes the diagonal. (This formula can also be formulated for (non-projective) smooth morphisms f between affine schemes and can then even be strengthened to a Σ_{l}-version, see Remark 3.8). Together with the Künneth formula, this formula implies Theorem D for smooth f (see Theorem 4.2). (A G-equivariant version of) Theorem C will be proved in Theorem 4.9. This proof is based on the following (G-equivariant) version of the formula mentioned above:

Δ_{∗}(1) = 1 in K_{0}(C_{l} × G, X^{l})/Ind_{{1}}^{C_{l}}(K_{0}(G, X^{l}))

(see the proof of Theorem 4.9).

If l is a prime, we have the following fundamental relation between τ^{l} and the l-th Adams operation ψ^{l}: For all x ∈ K_{0}(X), we have:

τ^{l}(x) = ψ^{l}(x) in K_{0}(C_{l}, X)/(O_{X}[C_{l}])

(see Proposition 1.13). Furthermore, the element λ_{−1}(F ⊗ H_{X,l}) equals the l-th Bott element θ^{l}(F) in K_{0}(C_{l}, X)/(O_{X}[C_{l}]) (see Proposition 3.2). Hence, Theorem D already follows from the classical Adams-Riemann-Roch formula for f (see [FL]). Conversely, Theorem D implies the Adams-Riemann-Roch formula if X and Y are ℂschemes (see section 4). Thus, if in addition f is smooth, Nori’s idea yields a new proof of the Adams-Riemann-Roch-formula. If f is a principal G-bundle such that l |̸ ord(G), Theorem C implies a version without denominators of the (equivariant) Adams-Riemann-Roch formula ψ^{l} ∘ f_{∗} = f_{∗} ∘ ψ^{l} (see Corollary 4.10). One should be able to prove such formulas for arbitrary étale morphism similarly to the paper [FM] of Fulton and MacPherson.

Using Grayson’s construction in [Gr], one can define (external) tensor power operations also for higher K-groups (see section 1). Then, Theorem A and Theorem C hold for higher K-groups, too. If the conjecture on shuffle products in [Ko2] is true, then Theorem B and Theorem D hold for higher K-groups, too. Theorem D further holds if f is smooth or if the relation “ψ^{l} = τ^{l}” holds also for higher K-groups. Usually, all assertions concerning higher K-groups will be formulated together with the corresponding assertions for K_{0}. However, mostly, the proofs will be separated such that a reader interested only in K_{0} does not need to know very much about higher K-theory. Moreover, the proofs for higher K-groups are sometimes a little bit sketchy as we refer oftentimes to analogous situations and argumentations already existing in the literature.

In characteristic p, the p-th Adams operation is identical to the pull-back homomorphism associated with the absolute Frobenius morphism. This well-known relation can be considered as a substitute for the relation “ψ^{p} = τ^{p}” which is extremely weak in characteristic p; for instance, we have K_{0}(C_{p}, 𝔽_{p},)/(𝔽_{p}[C_{p}]) ≅ ℤ/pℤ, and Theorem D for Y = Spec(𝔽_{p}) and l = p even becomes trivial. Starting from this analogy, we develop and investigate the following question in section 5, which may be considered as the analogue of the formula in Theorem 3.1 mentioned above: Let f be smooth and let F : X → X_{Y} denote the relative Frobenius morphism. Does then the following formula hold:

F_{∗}(θ^{p}(Ω_{X/Y})^{−1}) = 1 in K_{0}(X_{Y})[p^{−1}]?

Acknowledgments. I would like to thank S. P. Dutta, R. Hübl, and G. Seibert for discussions about this question. Furthermore, I thank D. Grayson for his warm hospitality during my stay at the University of Illinois at Urbana-Champaign where the final part of this work was done. Finally, I thank the referee for reading the paper carefully and suggesting many corrections and improvements.

Notations. Let G be an (abstract) group and Y a scheme. Then, a G-scheme over Y is a Y -scheme X together with a homomorphism G → Aut_{Y} (X)^{op} of groups; i.e., G acts on X from the right by Y -automorphisms. A G-module on the G-scheme X is an O_{X}-module F together with homomorphisms g : g^{∗}(F) → F, g ∈ G, which satisfy the usual homomorphism properties; i.e., G acts on F from the left. The exact category of all locally free O_{X}-modules (respectively, of all locally free G-modules on X) of finite rank is denoted by P(X) (respectively, by P(G, X)). The corresponding higher K-groups (see [Q]) are denoted by K_{q}(X) and K_{q}(G, X), q ≥ 0. Forgetting the G-structure yields canonical homomorphisms K_{q}(G, X) → K_{q}(X), q ≥ 0. If G acts on X trivially, we have natural maps K_{q}(X) → K_{q}(G, X), q ≥ 0, which are right inverse to the forgetful maps. For any subgroup H of G and for any H-module F on the G-scheme X, let Ind_{H}^{G}(F) denote the G-module ⊕_{r∈R} r^{∗}(F) on X where R ⊆ G is a system of representatives for G/H and where G acts on Ind_{H}^{G}(F) as follows: For any pair of elements g ∈ G, r ∈ R, let s ∈ R and h ∈ H be the uniquely determined elements such that gr = sh. Then g acts on the direct summand r^{∗}(F) of Ind_{H}^{G}(F) by virtue of the composition

g^{∗}(r^{∗}(F)) = (gr)^{∗}(F) = (sh)^{∗}(F) = s^{∗}(h^{∗}(F)) ⟶^{s^{∗}(h)} s^{∗}(F).

It is easy to prove that Ind_{H}^{G}(F) does not depend on the chosen system R of representatives (up to a canonical isomorphism). For any l ≥ 0, Σ_{l} denotes the group of permutations of the set I_{l} := {1, …, l}. For any Y -scheme X, the l-fold fibred product X^{l} := X ×_{Y} … ×_{Y} X is considered as a Σ_{l}-scheme by

σ(x_{1}, …, x_{l}) := (x_{σ(1)}, …, x_{σ(l)})

(for x_{1}, …, x_{l} ∈ X and σ ∈ Σ_{l}). For any Σ_{l}-scheme X, O_{X}[I_{l}] denotes the O_{X}-module

⊕^{l} O_{X} together with the obvious Σ_{l}-action. The Σ_{l}-module H_{X,l} on any Σ_{l}-scheme X is defined by the short exact sequence

0 → H_{X,l} → O_{X}[I_{l}] →^{Σ} O_{X} → 0.

The cyclic subgroup of Σ_{l} generated by the cycle c := ⟨1, …, l⟩ ∈ Σ_{l} is denoted by C_{l}. H_{X,l} is isomorphic to coker(Δ : O_{X} → O_{X}[I_{l}]) as a C_{l}-module and, if l is invertible on X, even as a Σ_{l}-module.

§1 (External) Tensor Power Operations

Let f : X → Y be a morphism between Noetherian schemes. In this section, we will construct natural external tensor power operations

τ^{l} : K_{q}(X) → K_{q}(Σ_{l}, X^{l}), q ≥ 0, l ≥ 0,

such that, for q = 0, the class [F] of any locally free O_{X}-module F is mapped to the class [F^{⊠l}] of the external tensor power F^{⊠l} where Σ_{l} acts on F^{⊠l} by permuting the factors. In particular, for f = id, we obtain tensor power operations

τ^{l} : K_{q}(Y) → K_{q}(Σ_{l}, Y), q ≥ 0, l ≥ 0.

We will show in Proposition 1.7 that τ^{l} is multiplicative (on Grothendieck groups). Furthermore, we will prove that, for any locally free O_{Y} -module F the element τ^{l}(λ_{−1}([F])) is divisible by the element λ_{−1}([F]) in K_{0}(Σ_{l}, Y) (see Corollary 1.11). If l is a prime, we finally establish a fundamental relation between the tensor power operation

K_{q}(Y) ⟶^{τ^{l}} K_{q}(Σ_{l}, Y) ⟶^{Res_{C_{l}}^{Σ_{l}}} K_{q}(C_{l}, Y)

and the Adams operation ψ^{l} on K_{q}(Y) (see Proposition 1.13) and we will explicitly compute the operation τ^{l} : K_{1}(ℂ) → K_{1}(C_{l}, ℂ) (see Proposition 1.14).

We now begin to construct τ^{l}. For any l ≥ 0, let F_{l}(P(X)) denote the category of all chains F_{1} ↪ … ↪ F_{l} of admissible monomorphisms in P(X). Here, a monomorphism F ↪ G in P(X) is called admissible if the quotient G/F is locally free. We have functors

× : P(Σ_{k}, X^{k}) × P(Σ_{l}, X^{l}) → P(Σ_{k+l}, X^{k+l}), (F, G) ↦ Ind_{Σ_{k}×Σ_{l}}^{Σ_{k+l}}(F ⊠ G),

(for all k, l ≥ 0) and

F_{l}(P(X)) → P(Σ_{l}, X^{l}), (F_{1} ↪ … ↪ F_{l}) ↦ ∑_{σ∈Σ_{l}} F_{σ(1)} ⊠ … ⊠ F_{σ(l)},

(for all l ≥ 0). Here, F ⊠ G denotes the external tensor product p^{∗}(F) ⊗ q^{∗}(G) for p : X^{k+l} → X^{k} and q : X^{k+l} → X^{l} the projections to the first k and last l components, respectively. It is considered as a Σ_{k} × Σ_{l}-module on the Σ_{k+l}-scheme X^{k+l} in the obvious way. Similarly, the sum ∑_{σ∈Σ_{l}} F_{σ(1)}⊠…⊠F_{σ(l)} is considered as a Σ_{l}-submodule of the external tensor power F_{l}^{⊠l}. It is easy, but tedious, to prove that this sum of submodules is a locally free O_{X^{l}}-module and that these functors satisfy the axioms (E1) through (E5) in section 7 of Grayson’s paper [Gr] (see also section 2 in [Ko2], last example in [Gr], and Remark 1.6).

Remark 1.1. Similarly to the last example in [Gr], one can define natural power operations F_{l}(P(Σ_{k}, X^{k})) → P(Σ_{kl}, X^{kl}) for all k ≥ 1. However, these operations can essentially be reduced to the case k = 1 (see Remark 1.12). So, we will consider only the case k = 1 in order to avoid complicated notations.

Grayson’s construction in section 7 of [Gr] yields maps

τ^{l} : K_{q}(X) → K_{q}(Σ_{l}, X^{l}), q ≥ 0, l ≥ 0,

which we will call external tensor power operations. If f = id, we in particular obtain maps

τ^{l} : K_{q}(Y) → K_{q}(Σ_{l}, Y), q ≥ 0, l ≥ 0,

which we will call tensor power operations. In the following lemma, we describe τ^{l} on Grothendieck groups.

Lemma 1.2. For any l ≥ 0 and any locally free O_{X}-modules E, F, we have in K_{0}(Σ_{l}, X^{l}):

τ^{l}([E] − [F]) = ∑_{{a≥0,u≥0,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=l}}(−1)^{u} [Ind_{Σ_{a}×Σ_{b_{1}}×...×Σ_{bu}}^{Σ_{l}}(E^{⊠a}⊠F^{⊠b_{1}}⊠ … ⊠F^{⊠b_{u}})] .

In particular, we have

τ^{l}([E]) = [E^{⊠l}] in K_{0}(Σ_{l}, X^{l}).

Proof. This follows from section 8 in [Gr].

For any i, j ≥ 0, the functor × obviously induces a bilinear product

× : K_{0}(Σ_{i}, X^{i}) × K_{0}(Σ_{j}, X^{j}) → K_{0}(Σ_{i+j}, X^{i+j})

which we call cross product. The Abelian group ∏_{l≥0} K_{0}(Σ_{l}, X^{l}) together with the cross product

\begin{aligned}× : & ∏_{i≥0} K_{0}(Σ_{i}, X^{i}) × ∏_{j≥0} K_{0}(Σ_{j}, X^{j}) & → & ∏_{l≥0} K_{0}(Σ_{l}, X^{l}) \\ & ((x_{i})_{i≥0}, (y_{j})_{j≥0}) & ↦ & (∑_{i+j=l} x_{i} × y_{j})_{l≥0} \end{aligned}

is then a commutative ring with 1 (see also Proposition 2.1 on p. 12 in [Ho]).

Corollary 1.3. The map

τ := (τ^{l})_{l≥0} : (K_{0}(X), +) → (∏_{l≥0} K_{0}(Σ_{l}, X^{l}), ×)

is a homomorphism.

Proof. For any locally free O_{X}-modules E, F, we have

τ([E] − [F]) = ([E^{⊠l}])_{l≥0} × ([F^{⊠l}])_{l≥0}^{−1} = τ([E]) × τ([F])^{−1}

by section 2 in [Gr] and by Lemma 1.2. Furthermore, for all l ≥ 0, we have a canonical isomorphism

(E ⊕ F)^{⊠l} ≅ ⊕_{i=0}^{l} Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(E^{⊠i} ⊠ F^{⊠(l−i)})

of Σ_{l}-modules on X^{l}; i.e., we have

τ([E ⊕ F]) = τ([E]) × τ([F]).

This immediately proves Corollary 1.3.

Proposition 1.4. For any l ≥ 0 and 1 ≤ i ≤ l − 1, the following diagram commutes:

K_{0}(X)  ⟶^{τ^{l}}  K_{0}(Σ_{l}, X^{l})
(τ^{i},τ^{l−i}) ↓  ↓ Res_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}
K_{0}(Σ_{i}, X^{i}) × K_{0}(Σ_{l−i}, X^{l−i}) ⟶^{⊠} K_{0}(Σ_{i} × Σ_{l−i}, X^{i} × X^{l−i}).

If Conjecture’ on p. 289 in [Ko2] is true, then the corresponding diagrams of higher K-groups commute, too.

Proof. First, we prove the assertion for Grothendieck groups. For this, we define a double cross product ×× on ∏_{i,j} K_{0}(Σ_{i} × Σ_{j}, X^{i+j}) by

[E]××[F] := [Ind_{Σ_{i_{1}}×Σ_{j_{1}}×Σ_{i_{2}}×Σ_{j_{2}}}^{Σ_{i_{1}+i_{2}}×Σ_{j_{1}+j_{2}}}(E ⊠ F)]

for E ∈ P(Σ_{i_{1}}×Σ_{j_{1}}, X^{i_{1}+j_{1}}) and F ∈ P(Σ_{i_{2}}×Σ_{j_{2}}, X^{i_{2}+j_{2}}). As in Proposition 2.1 on p. 12 in [Ho], one shows that the Abelian group ∏_{i,j} K_{0}(Σ_{i} × Σ_{j}, X^{i+j}) together with ×× is a commutative ring with 1. Using Mackey’s subgroup theorem, one can show as in Theorem 1.2 on p. 8 in [Ho] that the restriction map

Res : (∏_{l≥0} K_{0}(Σ_{l}, X^{l}), ×) → (∏_{i,j} K_{0}(Σ_{i} × Σ_{j}, X^{i+j}), ××)

is a homomorphism. It follows from this, from Corollary 1.3, and from the definition of ×× that all maps in the diagram

K_{0}(X)  →^{τ}  ∏_{l} K_{0}(Σ_{l}, X^{l})
(τ,τ) ↓  ↓ Res
∏_{i} K_{0}(Σ_{i}, X^{i}) × ∏_{j} K_{0}(Σ_{j}, X^{j}) →^{⊠} ∏_{i,j} K_{0}(Σ_{i} × Σ_{j}, X^{i+j})

are homomorphisms. Thus, it suffices to show that, for all locally free O_{X}-modules E, the following equality holds:

Res(τ([E])) = τ([E]) ⊠ τ([E]) in ∏_{i,j} K_{0}(Σ_{i} × Σ_{j}, X^{i+j}).

This equality follows from the obvious fact that, for all i, j, the Σ_{i} × Σ_{j}-modules E^{⊠i} ⊠ E^{⊠j} and Res_{Σ_{i}×Σ_{j}}^{Σ_{i+j}}(E^{⊠(i+j)}) on X^{i+j} are isomorphic. For the proof of the corresponding assertion for the higher K-groups, we use the notations introduced in [Ko2]. Furthermore, we set P_{l} := P(Σ_{l}, X^{l}) for all l ≥ 0. By construction (see [Gr] or [Ko2]), the map τ^{l} : K_{q}(X) → K_{q}(Σ_{l}, X^{l}) is induced by the continuous map

|GP_{1}| →^{|τ^{l}|} |H^{l}P_{l}| →^{Ξ^{l}} |G^{l}P_{l}|.

Here, the first map is the geometric realization of a certain simplicial map GP_{1} → H^{l}P_{l} which is induced by the above functors and which we denote by τ^{l}, too. The second map is defined in sections 5 and 7 in [Gr]. It is easy, but a little bit tedious to check that the diagram

GP_{1}  ⟶^{τ^{l}}  H^{l}P_{l}
⊠l ↓  ↓ Res_{Σ_{1}}^{Σ_{l}}
G^{l}P(X^{l}) ⟶^{Sym} H^{l}P(X^{l})

of simplicial maps commutes (up to homotopy). Here, the simplicial map^{⊠l} is induced by the external tensor product, and Sym := Sym^{1,...,1} is the shuffle operation introduced in (the last remark of) section 3 in [Ko2]. Since the external tensor product and the shuffle operation are associative in the obvious sense, the diagram

GP_{1}
(^{⊠i},^{⊠(l−i)}) ↓  ↘ ^{⊠l}
G^{i}P(X^{i}) × G^{l−i}P(X^{l−i}) →^{⊠}  G^{l}P(X^{l})
Sym×Sym ↓  ↘ Sym
H^{i}P(X^{i}) × H^{l−i}P(X^{l−i}) →^{⊠} H^{i,l−i}P(X^{l}) →^{Sym^{i,l−i}} H^{l}P(X^{l})

of simplicial maps commutes (up to homotopy). This implies that the upper quadrangle in the diagram

|GP_{1}|  →^{|τ^{l}|}  |H^{l}P_{l}|
(|τ^{i}|,|τ^{l−i}|) ↓  ↙ Res ↓ Ξ^{l}
|H^{i}P_{i}| × |H^{l−i}P_{l−i}| →^{⊠} |H^{i,l−i}P_{i,l−i}| →^{|Sym^{i,l−i}|} |H^{l}P_{i,l−i}|  |G^{l}P_{l}|
Ξ^{i}×Ξ^{l−i} ↓  Ξ^{i,l−i} ↘  ↓ Ξ^{l}  ↙ Res
|G^{i}P_{i}| × |G^{l−i}P_{l−i}|  →^{⊠}  |G^{l}P_{i,l−i}|

commutes up to homotopy (P_{i,l−i} := P(Σ_{i} × Σ_{l−i}, X^{l})). The lower left and the right quadrangle commute for trivial reasons. Conjecture’ on p. 289 in [Ko2] implies that the lower middle triangle commutes up to homotopy. Thus, the exterior pentagon commutes, too. This proves Proposition 1.4 for higher K-groups.

Remark 1.5. The conjecture on shuffle products mentioned above has been proved in the case i = 1 by Nenashev (see [Ne]). Thus, Proposition 1.4 holds in the case i = 1 also for higher K-groups. Furthermore, it follows from this (or already from the corollary on p. 293 in [Ko2]) that the diagram

|GP_{1}| →^{|τ^{l}|}  |H^{l}P_{l}|  →^{Ξ^{l}}  |G^{l}P_{l}|
⊠l ↘  ↙ Res_{Σ_{1}}^{Σ_{l}}
|G^{l}P(X^{l})|

of continuous maps commutes up to homotopy; i.e., Ξ^{l} ∘ |τ^{l}| is an equivariant lift of the usual external tensor power operation ^{⊠l}. In particular, the composition

K_{q}(X) ⟶^{τ^{l}} K_{q}(Σ_{l}, X^{l}) ⟶^{Res_{Σ_{1}}^{Σ_{l}}} K_{q}(X^{l})

is identical to the map x ↦ p_{1}^{∗}(x) · . . . · p_{l}^{∗}(x) which, for q ≥ 1, is the zero map since products on higher K-groups are trivial.

Remark 1.6. Without using the above functors, the simplicial maps

τ^{l} : GP(X) → H^{l}P(Σ_{l}, X^{l}), l ≥ 0,

(cf. proof of Proposition 1.4) and then the (external) tensor power operations on higher K-groups can be constructed as follows: Similarly to section 3 in [Ko4], one shows that the composition

GP(X) ⟶^{⊠l} G^{l}P(X^{l}) ⟶^{Sym^{1,...,1}} H^{l}P(X^{l})

can be canonically lifted to a simplicial map τ^{l} : GP(X) → H^{l}P(Σ_{l}, X^{l}). This construction of τ^{l} has the advantage that Grayson’s axioms for power operations need not to be checked for the above functors.

Proposition 1.7.

(a) For any l ≥ 0 and any x, y ∈ K_{0}(X), we have:

τ^{l}(x · y) = τ^{l}(x) · τ^{l}(y) in K_{0}(Σ_{l}, X^{l}).

(b) Let q ≥ 1. For any x ∈ K_{0}(X) of the form x = [E] (E a locally free O_{X}-module) and for any y ∈ K_{q}(X), we have:

τ^{l}(x · y) = τ^{l}(x) · τ^{l}(y) in K_{q}(Σ_{l}, X^{l}).

If Conjecture’ on p. 289 in [Ko2] is true, then this equality holds for all x ∈ K_{0}(X).

Proof.

(a) Using Corollary 1.3, Frobenius reciprocity, and Proposition 1.4 successively, we obtain for all x_{1}, x_{2}, y ∈ K_{0}(X) and l ≥ 0:

τ^{l}(x_{1} + x_{2}) · τ^{l}(y) =

= ∑_{i=0}^{l} Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(τ^{i}(x_{1}) ⊠ τ^{l−i}(x_{2})) ⊗ τ^{l}(y)

= ∑_{i=0}^{l} Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}} ((τ^{i}(x_{1}) ⊠ τ^{l−i}(x_{2})) ⊗ Res_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(τ^{l}(y)))

= ∑_{i=0}^{l} Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}} ((τ^{i}(x_{1}) ⊗ τ^{i}(y)) ⊠ (τ^{l−i}(x_{2}) ⊗ τ^{l−i}(y)))

= l-th component of (τ(x_{1}) · τ(y)) × (τ(x_{2}) · τ(y)).

This computation shows that the map

\begin{matrix}K_{0}(X) × K_{0}(X) & → & (∏_{l≥0} K_{0}(Σ_{l}, X^{l}), ×) \\ (x, y) & ↦ & τ(x) · τ(y) := (τ^{l}(x) · τ^{l}(y))_{l≥0} \end{matrix}

is bilinear. By Corollary 1.3, the map

K_{0}(X) × K_{0}(X) → (∏_{l≥0} K_{0}(Σ_{l}, X^{l}), ×), (x, y) ↦ τ(xy),

is bilinear, too. Hence, it suffices to show the equality

τ([E]) · τ([F]) = τ([E ⊗ F]) in ∏_{l≥0} K_{0}(Σ_{l}, X^{l})

for all locally free O_{X}-modules E and F. This equality immediately follows from the obvious fact that, for all l ≥ 0, the Σ_{l}-modules E^{⊠l} ⊗ F^{⊠l} and (E ⊗ F)^{⊠l} are isomorphic.

(b) For any locally free O_{X}-modules F_{1}, …, F_{u} and for any b_{1}, …, b_{u} ∈ ℕ with b_{1} + … + b_{u} = l, we have a canonical isomorphism

E^{⊠l}⊗Ind_{Σ_{b_{1}}×...×Σ_{bu}}^{Σ_{l}}(F_{1}^{⊠b_{1}}⊠ … ⊠F_{u}^{⊠b_{u}}) ≅ Ind_{Σ_{b_{1}}×...×Σ_{bu}}^{Σ_{l}}((E⊗F_{1})^{⊠b_{1}}⊠ … ⊠(E⊗F_{u})^{⊠b_{u}})

of Σ_{l}-modules on X^{l} by Frobenius reciprocity. Similarly to Proposition 7.2 on p. 306 in [Ko2], one deduces the first assertion in (b) from this. The second assertion then follows as in (a).

Remark 1.8. One can find constructions and statements which are similar to those made so far for Grothendieck groups already in Hoffman’s book [Ho] (in particular, see Proposition 3.7 on p. 22 in [Ho]). However, he uses the simplifying facts that short exact sequences of vector bundles split and that, for instance, K_{0}(Σ_{i}, X^{i})⊗K_{0}(Σ_{j}, X^{j}) is isomorphic to K_{0}(Σ_{i} ×Σ_{j}, X^{i+j}). Since such facts are not available in our situation, Hoffman’s proofs have been adapted to our situation and included into this paper for the reader’s convenience.

Proposition 1.9. For any l ≥ 0 and any locally free O_{X}-module F, we have:

τ^{l}(−[F]) = (−1)^{l}[F_{sgn}^{⊠l}] in K_{0}(Σ_{l}, X^{l});

here, F_{sgn}^{⊠l} := F^{⊠l} ⊗ O_{X^{l},sgn} denotes the tensor product of the Σ_{l}-module F^{⊠l} on X^{l} with the sign representation O_{X^{l},sgn} of Σ_{l}.

Proof. First, let l ≥ 0 be fixed. The sum N := ∑_{i=1}^{l}[i] of the canonical basis elements [i], i = 1, …, l, in the representation O_{X^{l}}[I_{l}] is obviously Σ_{l}-invariant. Thus, the homomorphisms

d_{i} : Λ^{i}(O_{X^{l}}[I_{l}]) → Λ^{i+1}(O_{X^{l}}[I_{l}]), x ↦ N ∧ x,

i = 0, …, l − 1, are Σ_{l}-homomorphisms and define a complex

0 → O_{X^{l}} →^{d_{0}} O_{X^{l}}[I_{l}] →^{d_{1}} … →^{d_{l−2}} Λ^{l−1}(O_{X^{l}}[I_{l}]) →^{d_{l−1}} Λ^{l}(O_{X^{l}}[I_{l}]) → 0

of Σ_{l}-modules on X^{l}. Let p_{1} : O_{X^{l}}[I_{l}] → O_{X^{l}} denote the projection onto the [1]- component. Then one easily checks that the maps

\begin{matrix}Λ^{i+1}(O_{X^{l}}[I_{l}]) & → & Λ^{i}(O_{X^{l}}[I_{l}]) \\ f_{1} ∧ … ∧ f_{i+1} & ↦ & ∑_{a=1}^{i+1}(−1)^{a−1}p_{1}(f_{a}) f_{1} ∧ … ∧ f̂_{a} ∧ … ∧ f_{i+1} \end{matrix}

i = 1, …, l, are well-defined and form a (non-equivariant) homotopy between the identity and the zero map on this complex. Hence, the complex is exact. For any i = 0, …, l, we have an isomorphism

Λ^{i}(O_{X^{l}}[I_{l}]) ˜→ Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(O_{X^{i},sgn} ⊠ O_{X^{l−i}})

of Σ_{l}-modules on X^{l} which is defined as follows: For any ρ(1) < … < ρ(i) in I_{l}, the basis element [ρ(1)] ∧ … ∧ [ρ(i)] of Λ^{i}(O_{X^{l}}[I_{l}]) is mapped to the basis element [ρ] of Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(O_{X^{i},sgn} ⊠ O_{X^{l−i}}) where ρ denotes the (i, l − i)-shuffle permutation corresponding to ρ(1) < … < ρ(i). Tensoring the above complex with F_{sgn}^{⊠l} and using Frobenius reciprocity, we obtain the following exact sequence of Σ_{l}-modules on X^{l}:

0 → F_{sgn}^{⊠l} → Ind_{Σ_{1}×Σ_{l−1}}^{Σ_{l}}(F ⊠ F_{sgn}^{⊠(l−1)}) → …
→ Ind_{Σ_{l−1}×Σ_{1}}^{Σ_{l}}(F^{⊠(l−1)} ⊠ F) → F^{⊠l} → 0.

This shows that the element ((−1)^{l}[F_{sgn}^{⊠l}])_{l≥0} ∈ ∏_{l≥0} K_{0}(Σ_{l}, X^{l}) is an inverse of the element τ([F]) = ([F^{⊠l}])_{l≥0} with respect to the cross product. Now, Proposition 1.9 follows from Corollary 1.3.

As seen in Corollary 1.3, the construction of the external tensor power operation τ^{l} is based on the equivariant version

(E ⊕ F)^{⊠l} ≅ ⊕_{i=0}^{l} Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(E^{⊠i} ⊠ F^{⊠(l−i)})

of the binomial theorem (a + b)^{l} = ∑_{i=0}^{l} \binom{l}{i}a^{i}b^{l−i}. The following corollary contains the equivariant version of the binomial theorem (a − b)^{l} = ∑_{i=0}^{l}(−1)^{l−i}\binom{l}{i}a^{i}b^{l−i}.

Corollary 1.10. For any l ≥ 0 and any locally free O_{X}-modules E, F, we have:

τ^{l}([E] − [F]) = ∑_{i=0}^{l}(−1)^{l−i} [Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(E^{⊠i} ⊠ F_{sgn}^{⊠(l−i)})] in K_{0}(Σ_{l}, X^{l}).

Proof. This follows from Corollary 1.3 and Proposition 1.9.

The next corollary will be an essential ingredient in the proof of the Riemann-Roch formula for closed immersions (see Theorem 4.1). Let λ_{−1}(x) denote the element

λ_{−1}(x) := ∑_{i≥0}(−1)^{i}λ^{i}(x) ∈ K

for any x of finite λ-degree in a λ-ring K.

Corollary 1.11. Let f = id. For any l ≥ 0 and any locally free O_{Y} -module F, we have:

τ^{l}(λ_{−1}([F])) = λ_{−1}([F ⊗ O_{Y} [I_{l}]]) = λ_{−1}([F ⊗ H_{Y,l}]) · λ_{−1}([F]) in K_{0}(Σ_{l}, Y).

Proof. The second equality is clear since λ_{−1} is multiplicative. We first prove the first equation in the case rank(F) = 1. Then we have by Corollary 1.10:

τ^{l}(λ_{−1}([F])) = τ^{l}([O_{Y}] − [F])

= ∑_{i=0}^{l}(−1)^{l−i} [Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(F_{sgn}^{⊗(l−i)})]

= ∑_{i=0}^{l}(−1)^{l−i} [F^{⊗(l−i)} ⊗ Λ^{l−i}(O_{Y} [I_{l}])]

= λ_{−1}([F ⊗ O_{Y} [I_{l}]]) in K_{0}(Σ_{l}, Y).

(In the third equality, we have used the isomorphisms

Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(O_{Y} ⊗ O_{Y,sgn}) ≅ Λ^{l−i}(O_{Y} [I_{l}]), i = 0, …, l,

established in the proof of Proposition 1.9.) Now let rank(F) be arbitrary. By the splitting principle (cf. section 2.5 in [Ko1]), there is a morphism p : Y′ → Y such that p^{∗} : K_{0}(Y) → K_{0}(Y′) and p^{∗} : K_{0}(Σ_{l}, Y) → K_{0}(Σ_{l}, Y′) are finite free ring extensions and such that p^{∗}([F]) decomposes into a sum of classes of invertible O_{Y′}-modules. Since λ_{−1} and τ^{l} commute with p^{∗} and since they are multiplicative (see Proposition 1.7(a)), Corollary 1.11 now follows from the case rank(F) = 1 proved above.

Remark 1.12. Let Γ be a group. We suppose that Γ acts on X by Y -automorphisms. For any locally free O_{X}-module F and for any l ≥ 0, the l-th external tensor power F^{⊠l} then carries a natural action of the wreath product Σ_{l}⟨Γ ⟩ := Γ^{l} ⋊ Σ_{l}. Hence, as above, one can construct operations

τ^{l} : K_{q}(Γ, X) → K_{q}(Σ_{l}⟨Γ ⟩, X^{l}), q ≥ 0, l ≥ 0.

One easily verifies that all statements proved so far hold in this more general situation, too. Now let Γ be the symmetric group Σ_{k} for some k ≥ 1 and let f : X^{k} → Y be the projection associated with a morphism X → Y between Noetherian schemes. Then, induction with respect to the canonical embedding Σ_{l}⟨Σ_{k}⟩ ↪ Σ_{kl} yields homomorphisms

Ind_{Σ_{l}⟨Σ_{k}⟩}^{Σ_{kl}} : K_{q}(Σ_{l}⟨Σ_{k}⟩, X^{kl}) → K_{q}(Σ_{kl}, X^{kl}), q ≥ 0, l ≥ 0.

One easily verifies that the homomorphism ∏_{l≥0} Ind_{Σ_{l}⟨Σ_{k}⟩}^{Σ_{kl}} commutes with cross products. Finally, the composition

K_{q}(Σ_{k}, X^{k}) ⟶^{τ^{l}} K_{q}(Σ_{l}⟨Σ_{k}⟩, (X^{k})^{l}) ⟶^{Ind_{Σ_{l}⟨Σ_{k}⟩}^{Σ_{kl}}} K_{q}(Σ_{kl}, X^{kl})

corresponds to the power operation mentioned in Remark 1.1.

Let f = id again. We now establish a fundamental relation between the Adams operation ψ^{l} on K_{q}(Y) and the composition

K_{q}(Y) ⟶^{τ^{l}} K_{q}(Σ_{l}, Y) ⟶^{Res_{C_{l}}^{Σ_{l}}} K_{q}(C_{l}, Y)

which will be denoted by τ^{l} again. One should be able to prove this relation also for the higher K-groups without the additional assumptions made in the following proposition.

Proposition 1.13. Let l be a prime. Then the following diagram commutes:

K_{0}(Y) ⟶^{τ^{l}}  K_{0}(C_{l}, Y)
ψ^{l} ↓  ↓ can
K_{0}(Y) ⟶^{can} K_{0}(C_{l}, Y)/[O_{Y} [I_{l}]] · K_{0}(C_{l}, Y).

The corresponding diagrams of higher K-groups commute if l is invertible on Y and if Y is affine or a smooth quasi-projective scheme over an affine regular base S.

Proof. First, we prove the assertion for Grothendieck groups. For any 1 ≤ i ≤ l − 1, the action of C_{l} on Σ_{l}/(Σ_{i} ×Σ_{l−i}) has no fixed points; thus, there is a C_{l}-stable system of representatives in Σ_{l} for Σ_{l}/(Σ_{i}×Σ_{l−i}). This implies that, for any (Σ_{i}×Σ_{l−i})-module F, the element [Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(F)] is contained in the ideal of K_{0}(C_{l}, Y) generated by

[Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(O_{Y})] and then in the ideal generated by [O_{Y} [I_{l}]]. It now follows from Corollaries 1.3 and 1.10 that the composition

K_{0}(Y) ⟶^{τ^{l}} K_{0}(C_{l}, Y) ⟶^{can} K_{0}(C_{l}, Y)/[O_{Y} [I_{l}]] · K_{0}(C_{l}, Y)

is a homomorphism. Since the same holds for the composition

K_{0}(Y) ⟶^{ψ^{l}} K_{0}(Y) ⟶^{can} K_{0}(C_{l}, Y)/[O_{Y} [I_{l}]] · K_{0}(C_{l}, Y),

it suffices to show the equality

τ^{l}([F]) = ψ^{l}([F]) in K_{0}(C_{l}, Y)/[O_{Y} [I_{l}]] · K_{0}(C_{l}, Y)

for any locally free O_{Y} -module F. We choose a morphism p : Y′ → Y for F as in the proof of Corollary 1.11. Then, the homomorphism

p^{∗} : K_{0}(C_{l}, Y)/[O_{Y} [I_{l}]] · K_{0}(C_{l}, Y) → K_{0}(C_{l}, Y′)/[O_{Y′}[I_{l}]] · K_{0}(C_{l}, Y′)

is still injective by (4.C)(ii) on p. 28 in [Mat]. As in the proof of Corollary 1.11, it thus suffices to show the above equality for invertible O_{Y} -modules. In this case, both sides are equal to [F^{⊗l}]. We now prove the assertion for higher K-groups. For this, let ε denote the idempotent element ε := \frac{1}{l} ∑_{i=0}^{l}[c^{i}] ∈ O_{Y} [C_{l}], G the group of automorphisms of C_{l} and C_{l} ⋊ G the semi-direct product of C_{l} with G (defined by (c^{i}, σ) · (c^{j}, τ) := (c^{i+σj}, στ) for i, j ∈ ℤ/lℤ and σ, τ ∈ (ℤ/lℤ)^{×} ≅ G). For any (C_{l} ⋊ G)-module F on Y with εF = 0, the module F^{G} of G-fixed elements will be considered as a C_{l}-module with trivial C_{l}-action. Then, obviously, the map

α : F → F^{G} ⊗ O_{Y} [C_{l}], f ↦ ∑_{i=0}^{l−1} (∑_{σ∈G} σ(c^{−i}(f))) ⊗ [c^{i}],

is a homomorphism of C_{l}-modules on Y and the sequence

0 → F →^{α} F^{G} ⊗ O_{Y} [C_{l}] →^{Σ} F^{G} → 0

is a split short sequence of C_{l}-modules. We now consider the functors

P(C_{l} ⋊ G, Y) →  P(Y)  →^{can} P(C_{l}, Y)
F_{0} :  F  ↦  εF
F_{1} :  F  ↦ ((1 − ε)F)^{G}.

Then, the sequence above shows that the direct sum F_{0} ⊕ F_{1} ⊗ H_{Y,l} is isomorphic to the forgetful functor

P(C_{l} ⋊ G, Y) → P(C_{l}, Y).

Hence, the functors F_{0} and F_{1} are exact and induce homomorphisms

F_{0} and F_{1} : K_{q}(C_{l} ⋊ G, Y) → K_{q}(Y)

with F_{0}(y) + [H_{Y,l}] · F_{1}(y) = y in K_{q}(C_{l}, Y) for all y ∈ K_{q}(C_{l} ⋊ G, Y). In particular, the diagram

K_{q}(C_{l} ⋊ G, Y) ⟶^{Res_{C_{l}}^{C_{l}⋊G}}  K_{q}(C_{l}, Y)
F_{0}−F_{1} ↓  ↓ can
K_{q}(Y)  ⟶^{can} K_{q}(C_{l}, Y)/[O_{Y} [I_{l}]] · K_{q}(C_{l}, Y)

commutes. We now consider C_{l} ⋊ G as a subgroup of Σ_{l} in the obvious way. Then it is easy to verify (see also Remark 1.6) that the compositions

K_{q}(Y) ⟶^{τ^{l}} K_{q}(Σ_{l}, Y) ⟶^{Res_{C_{l}⋊G}^{Σ_{l}}} K_{q}(C_{l} ⋊ G, Y) ⟶^{F_{0},F_{1}} K_{q}(Y)

are equal to the cyclic power operations [0]_{l} and [1]_{l} constructed in section 3 of [Ko4]. By Corollary 2.2(c) on p. 142 in [Ko3] (applied to the case Γ = {1}), the difference between these compositions equals ψ^{l} if Y is affine. By Jouanolou’s trick (e.g., see Satz 4.4 on p. 211 in [Ko1]) and Quillen’s resolution theorem (in the form of Satz 2.1 on p. 195 in [Ko1]), the same holds if Y is a smooth quasi-projective scheme over an affine regular base S. This proves Proposition 1.13 for higher K-groups.

Now let k be an algebraically closed field of characteristic 0 and l be a prime. In the following proposition, we will explicitly describe the l-th tensor power operation

τ^{l} : K_{1}(k) ⟶^{τ^{l}} K_{1}(Σ_{l}, k) ⟶^{Res_{C_{l}}^{Σ_{l}}} K_{1}(C_{l}, k).

For this, we fix a non-trivial character χ : C_{l} → k^{×}. For any i = 0, …, l − 1, let V_{i} denote the one-dimensional representation of C_{l} given by the character χ^{i}. Then it is well-known that the homomorphism

\begin{matrix}(k^{×})^{ℤ/lℤ} & → & K_{1}(k[C_{l}]) = K_{1}(C_{l}, k) \\ (β_{0}, …, β_{l−1}) & ↦ & (V_{0}, β_{0}) ⊕ … ⊕ (V_{l−1}, β_{l−1}) \end{matrix}

is bijective. Here, as usual, the pairs (V, β), V a finite dimensional k[C_{l}]-module and β ∈ Aut_{k[C_{l}]}(V), are considered as generators of the group K_{1}(k[C_{l}]).

Proposition 1.14. Let l be a prime. Then the following diagram commutes:

K_{1}(k) →^{τ^{l}}  K_{1}(C_{l}, k)
∥  ∥
k^{×} →  (k^{×})^{ℤ/lℤ}
β ↦ (β^{l−1}, β^{−1}, …, β^{−1}).

Proof. Let K_{0}(ℤ, k[C_{l}]) denote the Grothendieck group of all pairs (V, β) as above. An obvious generalization of the construction of this section (see also Remark 1.12) yields a tensor power operation

τ^{l} : K_{0}(ℤ, k) → K_{0}(ℤ, k[C_{l}])

such that, for any vector space V over k and any β ∈ Aut_{k}(V), the class [(V, β)] is mapped to the class [(V^{⊗l}, β^{⊗l})] where V^{⊗l} is considered as a C_{l}-module as usual. By restricting, we obtain a tensor power operation

τ^{l} : K̃_{0}(ℤ, k) → K̃_{0}(ℤ, k[C_{l}])

between the reduced Grothendieck groups K̃_{0}(ℤ, k) := ker(K_{0}(ℤ, k) →^{can} K_{0}(k)) and K̃_{0}(ℤ, k[C_{l}]) := ker(K_{0}(ℤ, k[C_{l}]) →^{can} K_{0}(k[C_{l}])). As in Theorem 3.3 on p. 145 in [Ko3] one shows that the following diagram commutes:

K̃_{0}(ℤ, k) ⟶^{can} K_{1}(k)
τ^{l} ↓  ↓ τ^{l}
K̃_{0}(ℤ, k[C_{l}]) ⟶^{can} K_{1}(k[C_{l}]).

By an obvious generalization of Corollary 1.10, we have the following equality in K̃_{0}(ℤ, k[C_{l}]) for all β ∈ k^{×}:

τ^{l} ([(k, β)] − [(k, 1)]) =

= ∑_{i=0}^{l}(−1)^{l−i} [Res_{C_{l}}^{Σ_{l}} (Ind_{Σ_{i}×Σ_{l−i}}^{Σ_{l}}(k ⊗ k_{sgn}, β^{i}))]

= [(V_{0}, β^{l})] + ∑_{i=1}^{l−1}(−1)^{l−i} \frac{1}{l}\binom{l}{i}[(k[C_{l}], β^{i})] + (−1)^{l}[(V_{0,sgn}, 1)].

(The last equality follows from the fact that, for all i = 1, …, l−1, there is a C_{l}-stable system of representatives in Σ_{l} for Σ_{l}/(Σ_{i} × Σ_{l−i}).) Because of

∑_{i=1}^{l−1}(−1)^{l−i} \frac{1}{l}\binom{l}{i}i = ∑_{i=1}^{l−1}(−1)^{l−i} \frac{(l − 1)!}{(i − 1)!(l − i)!} = ∑_{i=0}^{l−2}(−1)^{l−i−1}\binom{l − 1}{i} = −1

and the defining relation “(V, α ∘ β) = (V, α) + (V, β)” of K_{1}-groups, we have

τ^{l}((k, β)) = τ^{l}((k, β) − (k, 1)) = (V_{0}, β^{l}) − (k[C_{l}], β) in K_{1}(C_{l}, k).

This proves Proposition 1.14.

§2 A Künneth Formula

Let l ∈ ℕ, and let Y be a Noetherian scheme on which each coherent module is a quotient of a locally free O_{Y} -module. Let f : X → Y be a flat projective local complete intersection morphism (cf. p. 86 in [FL]). Being the composition of the successive projections X^{i} → X^{i−1}, i = 1, …, l, the morphism

f^{l} : X^{l} → Y

is then of the same kind (see Proposition 3.12 on p. 87 in [FL]). By section 2.6 in [Ko1], we have push-forward homomorphisms

f_{∗} : K_{q}(X) → K_{q}(Y) and f_{∗}^{l} : K_{q}(Σ_{l}, X^{l}) → K_{q}(Σ_{l}, Y)

(for all q ≥ 0) whose definition will be recalled in the subsequent proofs. The aim of this section is to prove the following theorem:

Theorem 2.1 (Künneth formula). For any q ≥ 0, the following diagram commutes:

K_{q}(X) ⟶^{τ^{l}} K_{q}(Σ_{l}, X^{l})
f_{∗} ↓  ↓ f_{∗}^{l}
K_{q}(Y) ⟶^{τ^{l}} K_{q}(Σ_{l}, Y).

Proof. There is a factorization X ↪^{i} ℙ_{Y} (E) →^{p} Y of f into a regular closed immersion i and the projection of the projective space bundle associated with a locally free O_{Y} -module E. Since we have f_{∗} = p_{∗}∘i_{∗} and f_{∗}^{l} = p_{∗}^{l}∘i_{∗}^{l}, Theorem 2.1 immediately follows from the subsequent Propositions 2.2 and 2.3. As in the proof of Proposition 1.4, we will use the notations of [Ko2] in their proof.

Proposition 2.2. Let i : X ↪ X̃ be a regular closed immersion of flat quasiprojective schemes over Y. Then the following diagram commutes for all q ≥ 0:

K_{q}(X) ⟶^{τ^{l}} K_{q}(Σ_{l}, X^{l})
i_{∗} ↓  ↓ i_{∗}^{l}
K_{q}(X̃) ⟶^{τ^{l}} K_{q}(Σ_{l}, X̃^{l}).

Proof. Being the composition of the regular closed immersions

X^{l} = X × … × X × X ↪ X × … × X × X̃, …,
X × X̃ × … × X̃ ↪ X̃ × X̃ × … × X̃ = X̃^{l},

the morphism i^{l} : X^{l} → X̃^{l} is a regular closed immersion, too. Let P_{∞}(X̃) (respectively, P_{∞,fl}(X̃)) denote the exact category of all coherent OX̃-modules which possess a finite resolution by locally free OX̃-modules (respectively, which are in addition flat over Y). Similarly, let P_{∞}(Σ_{l}, X̃^{l}) and P_{∞,fl}(Σ_{l}, X̃^{l})) denote the exact categories of all coherent Σ_{l}-modules on X̃^{l} which, after forgetting the Σ_{l}-structure, are contained in P_{∞}(X̃^{l}) and P_{∞,fl}(X̃^{l}), respectively. By Lemma 2.2 in [Ko1] and Lemma 3.4(b) in [Ko5], all modules in P_{∞}(Σ_{l}, X̃^{l}) then possess a finite resolution by locally free Σ_{l}-modules on X̃^{l}. The inclusions

P(X̃) ↪ P_{∞,fl}(X̃) ↪ P_{∞}(X̃) and P(Σ_{l}, X̃^{l}) ↪ P_{∞,fl}(Σ_{l}, X̃^{l}) ↪ P_{∞}(Σ_{l}, X̃^{l})

induce homotopy equivalences between the geometric realizations of the corresponding (l-fold iterated) G-constructions by Quillen’s resolution theorem (see Corollary 1 on p. 109 in [Q]). Furthermore, since X is flat over Y, we have well-defined exact functors

i_{∗} : P(X) → P_{∞,fl}(X̃) and i_{∗}^{l} : P(Σ_{l}, X^{l}) → P_{∞,fl}(Σ_{l}, X̃^{l})

(see section 2.6 in [Ko1]). By definition, the maps

i_{∗} : K_{q}(X) → K_{q}(X̃) and i_{∗}^{l} : K_{q}(Σ_{l}, X^{l}) → K_{q}(Σ_{l}, X̃^{l})

are induced by the compositions

|GP(X)| →^{i_{∗}} |GP_{∞}(X̃)| ≃ |GP(X̃)|

and

|G^{l}P(Σ_{l}, X^{l}) →^{i_{∗}^{l}} |G^{l}P_{∞}(Σ_{l}, X̃^{l})| ≃ |G^{l}P(Σ_{l}, X̃^{l})|.

For any i, j ∈ ℕ and any E_{1} ∈ P_{∞,fl}(X̃^{i}), E_{2} ∈ P_{∞,fl}(X̃^{j}), the external tensor product E_{1} ⊠ E_{2} is obviously flat over Y, again. Furthermore, we have: If F_{i}^{·} → E_{i}, i = 1, 2, are finite locally free resolutions, then the total complex associated with the double complex F_{1}^{·} ⊠ F_{2}^{·} is a finite locally free resolution of E_{1} ⊠ E_{2} since E_{1} and E_{2} are flat over Y and since X̃ is flat over Y. Finally, the functor

P_{∞,fl}(X̃^{i}) × P_{∞,fl}(X̃^{j}) → P_{∞,fl}(X̃^{i+j}), (E_{1}, E_{2}) ↦ E_{1} ⊠ E_{2},

is obviously bi-exact. This implies that we have functors

× : P_{∞,fl}(Σ_{i}, X̃^{i}) × P_{∞,fl}(Σ_{j}, X̃^{j}) → P_{∞,fl}(Σ_{i+j}, X̃^{i+j})

and

F_{k}(P_{∞,fl}(X̃)) → P_{∞,fl}(Σ_{k}, X̃^{k})

as in section 1 which satisfy axioms (E1) through (E5) in section 7 of [Gr]. Now, Grayson’s construction in section 7 of [Gr] yields a simplicial map τ^{l} : GP_{∞,fl}(X̃) → H^{l}P_{∞,fl}(Σ_{l}, X̃^{l}) and then an l-th external tensor power operation

|GP_{∞,fl}(X̃)| →^{|τ^{l}|} |H^{l}P_{∞,fl}(Σ_{l}, X̃^{l})| →^{Ξ^{l}} |G^{l}P_{∞,fl}(Σ_{l}, X̃^{l})|

for P_{∞,fl}(X̃) which is compatible with the corresponding operation for P(X̃) defined in section 1. Finally, for any E_{1}, …, E_{l} ∈ P(X), the canonical homomorphism

i_{∗}(E_{1}) ⊠ … ⊠ i_{∗}(E_{l}) → i_{∗}^{l}(E_{1} ⊠ … ⊠ E_{l})

is bijective as one can easily check stalk-wise. Altogether, we have the following diagram of continuous maps in which the left two squares commute (up to homotopy) by the considerations above, the right two squares commute for trivial reasons, and the left and right lower vertical arrow are homotopy equivalences:

|GP(X)| →^{τ^{l}} |H^{l}P(Σ_{l}, X^{l})| →^{Ξ^{l}} |G^{l}P(Σ_{l}, X^{l})|
i_{∗} ↓  i_{∗}^{l} ↓  ↓ i_{∗}^{l}
|GP_{∞,fl}(X̃)| →^{τ^{l}} |H^{l}P_{∞,fl}(Σ_{l}, X̃^{l})| →^{Ξ^{l}} |G^{l}P_{∞,fl}(Σ_{l}, X̃^{l})|
↑  ↑  ↑
|GP(X̃)| →^{τ^{l}} |H^{l}P(Σ_{l}, X̃^{l})| →^{Ξ^{l}} |G^{l}P(Σ_{l}, X̃^{l})|.

This proves Proposition 2.2.

Proposition 2.3. Let p : ℙ := ℙ_{Y} (E) → Y be the projective space bundle associated with a locally free O_{Y} -module E. Then the following diagram commutes for all q ≥ 0:

K_{q}(ℙ) ⟶^{τ^{l}} K_{q}(Σ_{l}, ℙ^{l})
p_{∗} ↓  ↓ p_{∗}^{l}
K_{q}(Y) ⟶^{τ^{l}} K_{q}(Σ_{l}, Y).

Proof. Let R(ℙ) denote the exact category consisting of all locally free O_{ℙ}-modules F with R^{j}p_{∗}F(i) = 0 for all j ≥ 1 and i ≥ 0. By Lemma 1.13 on p. 141 in [Q], we have a well-defined exact functor p_{∗} : R(ℙ) → P(Y). The inclusion R(ℙ) ↪ P(ℙ) induces a homotopy equivalence |GR(ℙ)| → |GP(ℙ)| by Lemma 2.2 on p. 142 in [Q]. By definition, the map p_{∗} : K_{q}(ℙ) → K_{q}(Y) is induced by the composition

|GP(ℙ)| ≃ |GR(ℙ)| →^{p_{∗}} |GP(Y)|. The map p_{∗}^{l} : K_{q}(Σ_{l}, ℙ^{l}) → K_{q}(Σ_{l}, Y) is defined

as the composition r_{∗} ∘ i_{∗} where i : ℙ^{l} → ℝ is a (regular) closed Σ_{l}-embedding of ℙ^{l} into a projective space bundle ℝ associated with a locally free Σ_{l}-module on Y and r : ℝ → Y denotes the canonical projection (see also below). Now, let R(Σ_{l}, ℙ^{l}) denote the exact category consisting of all locally free Σ_{l}-modules F on ℙ^{l} for which R^{j}p_{∗}^{l}F(i) vanishes for all j ≥ 1 and i ≥ 0 and for which p_{∗}^{l}F(i) is a locally free O_{Y} -module for all i ≥ 0 (for this, we put F(i) := F ⊗ O(i)^{⊠l}). In particular, the functor p_{∗}^{l} : R(Σ_{l}, ℙ^{l}) → P(Σ_{l}, Y) is well-defined and exact. For any F_{1}, …, F_{l} ∈ R(ℙ), the higher direct image R^{j}p_{∗}^{l}(F_{1} ⊠ … ⊠ F_{l})(i) vanishes for all j ≥ 1 and i ≥ 0, and the module p_{∗}^{l}(F_{1} ⊠ … ⊠ F_{l})(i) ≅ p_{∗}F_{1}(i) ⊗ … ⊗ p_{∗}F_{l}(i) is locally free for all i ≥ 0 (by Proposition 9.3 on p. 255 in [H]). Similarly to the proof of Proposition 2.2, we can therefore define a simplicial map τ^{l} : GR(ℙ) → H^{l}R(Σ_{l}, ℙ^{l}) and an l-th external tensor power operation

|GR(ℙ)| →^{|τ^{l}|} |H^{l}R(Σ_{l}, ℙ^{l})| →^{Ξ^{l}} |G^{l}R(Σ_{l}, ℙ^{l})|

such that the following diagram commutes (up to homotopy):

|GP(ℙ)| →^{|τ^{l}|} |H^{l}P(Σ_{l}, ℙ^{l})| →^{Ξ^{l}} |G^{l}P(Σ_{l}, ℙ^{l})|
↑  ↑  ↑
|GR(ℙ)| →^{|τ^{l}|} |H^{l}R(Σ_{l}, ℙ^{l})| →^{Ξ^{l}} |G^{l}R(Σ_{l}, ℙ^{l})|
p_{∗} ↓  p_{∗}^{l} ↓  ↓ p_{∗}^{l}
|GP(Y)| →^{|τ^{l}|} |H^{l}P(Σ_{l}, Y)| →^{Ξ^{l}} |G^{l}P(Σ_{l}, Y)|.

Now, let r : ℝ := ℙ_{Y} (E^{⊗l}) → Y denote the projective space bundle associated with E^{⊗l}; it will be considered as a Σ_{l}-scheme over Y by virtue of the usual action of Σ_{l} on E^{⊗l}. The canonical Σ_{l}-epimorphism (p^{l})^{∗}(E^{⊗l})⟶O(1)^{⊠l} defines a regular closed Σ_{l}- immersion i : ℙ^{l} → ℝ (see section 1.3 in [Ko1] and Proposition 3.11 on p. 85 in [FL]). Let RP_{∞,fl}(Σ_{l}, ℝ) denote the exact category consisting of all coherent Σ_{l}-modules F on ℝ which possess a finite resolution by locally free O_{ℝ}-modules, which are flat over Y, and for which R^{j}r_{∗}F(i) vanishes for all j ≥ 1 and i ≥ 0. As in the proof of Proposition 2.2, we then have a well-defined exact functor i_{∗} : R(Σ_{l}, ℙ^{l}) → RP_{∞,fl}(Σ_{l}, ℝ). We now prove that the inclusion RP_{∞,fl}(Σ_{l}, ℝ) ↪ P_{∞,fl}(Σ_{l}, ℝ) induces a homotopy equivalence

|G^{l}RP_{∞,fl}(Σ_{l}, ℝ)| → |G^{l}P_{∞,fl}(Σ_{l}, ℝ)|

(here, P_{∞,fl}(Σ_{l}, ℝ) is defined as in the proof of Proposition 2.2). For this, let R_{n} denote the exact category consisting of all F ∈ P_{∞,fl}(Σ_{l}, ℝ) with F(n) ∈ RP_{∞,fl}(Σ_{l}, ℝ) (for any n ≥ 0). Thus we have R_{0} = RP_{∞,fl}(Σ_{l}, ℝ) and R_{0} ⊆ R_{1} ⊆ R_{2} ⊆ . . .. By Lemma 1.12 on p. 141 in [Q], we have ∪_{n≥0}R_{n} = P_{∞,fl}(Σ_{l}, ℝ). (Note that Lemma 1.12 in [Q] holds even for coherent Y -flat modules F. Quillen’s proof for locally free F need not be changed.) By property (9) on p. 104 in [Q], it therefore suffices to show that the inclusion R_{n} ↪ R_{n+1} induces a homotopy equivalence |G^{l}R_{n}| → |G^{l}R_{n+1}| for all n ≥ 0. The exact functors

u_{p} : R_{n+1} → R_{n}, F ↦ r^{∗}Λ^{p}((E^{⊗l})^{∨}) ⊗ F(p), p ≥ 1,

induce homomorphisms u_{p} : K_{q}(R_{n+1}) → K_{q}(R_{n}), p ≥ 1, (for all q ≥ 0). The functorial exact sequence

0 → F → r^{∗}(E^{⊗l})^{∨} ⊗ F(1) → … → F(rank(E^{⊗l})) → 0

(see sequence 2.2 on p. 107 in [FL]) together with Corollary 3 of Theorem 2 on p. 107 in [Q] imply that ∑_{p≥1}(−1)^{p−1}u_{p} is inverse to the canonical map K_{q}(R_{n}) → K_{q}(R_{n+1}). Hence, the map |G^{l}R_{n}| → |G^{l}R_{n+1}| is a homotopy equivalence. It finally follows from Lemma 1.13 on p. 141 in [Q] (which again holds more generally for coherent Y -flat modules) that we have well-defined exact functors r_{∗} : RP_{∞,fl}(Σ_{l}, ℝ) → P(Σ_{l}, Y) and r_{∗} : R(Σ_{l}, ℝ) → P(Σ_{l}, Y). Altogether, we now have the following (up to natural equivalence) commutative diagram of exact categories and exact functors in which all six inclusions in the upper right corner become homotopy equivalences after passing to the geometric realization of the (l-fold iterated) G-construction:

P(Σ_{l}, ℙ^{l}) →^{i_{∗}}  P_{∞}(Σ_{l}, ℝ)
↑  ↖
↑  P_{∞,fl}(Σ_{l}, ℝ)  ← P(Σ_{l}, ℝ)
↑  ↑
R(Σ_{l}, ℙ^{l}) →^{i_{∗}} RP_{∞,fl}(Σ_{l}, ℝ) ← R(Σ_{l}, ℝ)
p_{∗}^{l} ↘  r_{∗} ↓  ↙ r_{∗}
P(Σ_{l}, Y).

This diagram together with the previous diagram prove Proposition 2.3 since the upper left arrow in the previous diagram is a homotopy equivalence.

Remark 2.4. The following generalizations of Theorem 2.1 can be proved with the same methods:

(a) Let f : X → Y be a projective morphism. For any q ≥ 0, let K_{q}^{Y −fl}(X) denote the q-th K-group associated with the exact category consisting of all coherent Y - flat O_{X}-modules. Then one can naturally define a push-forward homomorphism f_{∗} : K_{q}^{Y −fl}(X) → K_{q}(Y) and an l-th external tensor power operation τ^{l} : K_{q}^{Y −fl}(X) → K_{q}^{Y −fl}(Σ_{l}, X^{l}) such that the following diagram commutes:

K_{q}^{Y −fl}(X) ⟶^{τ^{l}} K_{q}^{Y −fl}(Σ_{l}, X^{l})
f_{∗} ↓  ↓ f_{∗}^{l}
K_{q}(Y) ⟶^{τ^{l}} K_{q}(Σ_{l}, Y).

(b) Let Y be a Noetherian scheme and X_{1}, X_{2} flat Noetherian schemes over Y on which each coherent module is a quotient of a locally free module. Furthermore, let f : X_{2} → X_{1} be a projective local complete intersection morphism. Then the following diagram commutes for all q ≥ 0:

K_{q}(X_{2}) ⟶^{τ^{l}} K_{q}(Σ_{l}, X_{2}^{l})
f_{∗} ↓  ↓ f_{∗}^{l}
K_{q}(X_{1}) ⟶^{τ^{l}} K_{q}(Σ_{l}, X_{1}^{l}).

(Here, the fibred products X_{1}^{l} and X_{2}^{l} are formed over Y.)

§3 On a Certain Multiplicative Class in K_{0}(Σ_{l}, X)

Let l ∈ ℕ, and let f : X → Y be a smooth quasi-projective morphism between Noetherian schemes of relative dimension d. We assume furthermore that there exists an ample invertible O_{X}-module. Let Ω_{X/Y} denote the locally free sheaf of relative differentials, Δ : X → X^{l} the diagonal, and I the ideal of K_{0}(Σ_{l}, Y) generated by the elements [Λ^{i}(O_{Y} [I_{l}])], i = 1, …, l −1. The aim of this section is to prove the following theorem which, besides the Künneth formula Theorem 2.1, will be the main tool in the proof of the Riemann-Roch formula Theorem 4.2 for smooth morphisms.

Theorem 3.1. The element λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}]) ∈ K_{0}(Σ_{l}, X) is invertible in

K_{0}(Σ_{l}, X)[l^{−1}]/IK_{0}(Σ_{l}, X)[l^{−1}]

and in particular in K_{0}(C_{l}, X)[l^{−1}]/IK_{0}(C_{l}, X)[l^{−1}], and we have:

Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1}) = 1 in K_{0}(C_{l}, X^{l})[l^{−1}]/IK_{0}(C_{l}, X^{l})[l^{−1}].

In subsections 3.1 through 3.5, we prove some auxiliary results. The proof of Theorem 3.1 will be contained in subsection 3.6.

3.1 The Bott element θ^{l}(F) and the class λ_{−1}([F ⊗ H_{X,l}])

Let l ∈ ℕ and X a Σ_{l}-scheme. For any locally free Σ_{l}-module F on X, let θ^{l}(F) ∈ K_{0}(Σ_{l}, X) denote the l-th Bott element associated with F (e.g., see section 4 in [Ko5]). It is characterized by the following properties: If rank(F) = 1, then

θ^{l}(F) = [O_{X}] + [F] + … + [F^{⊗(l−1)}].

If F′ is another locally free Σ_{l}-module on X, then θ^{l}(F ⊕F′) = θ^{l}(F)·θ^{l}(F′). Finally, we have θ^{l}(p^{∗}F) = p^{∗}θ^{l}(F) in K_{0}(Σ_{l}, X′) for any morphism p : X′ → X between Σ_{l}-schemes.

If l is prime and C_{l} ⊆ Σ_{l} acts on X trivially, we have in K_{0}(C_{l}, X)/([O_{X}[C_{l}]]):

θ^{l}(F) · λ_{−1}([F]) = ψ^{l}(λ_{−1}([F]))
= τ^{l}(λ_{−1}([F]))
= λ_{−1}([F ⊗ H_{X,l}]) · λ_{−1}([F]).

Here, we have used the splitting principle (see section 2.5 in [Ko1]), Proposition 1.13 and Corollary 1.11, successively. The following proposition strengthens this computation:

Proposition 3.2. For any locally free Σ_{l}-module F on X, we have:

θ^{l}(F) = λ_{−1}([F ⊗ H_{X,l}]) in K_{0}(Σ_{l}, X)/ ([Λ^{i}(O_{X}[I_{l}])], i = 1, …, l − 1) .

Proof. For any i = 0, … l − 1, we have

[Λ^{i}(H_{X,l})] = λ^{i}([O_{X}[I_{l}]] − 1)

= ∑_{j=0}^{i}(−1)^{j} [Λ^{i−j}(O_{X}[I_{l}])]

= (−1)^{i} in K_{0}(Σ_{l}, X)/ ([Λ^{i}(O_{X}[I_{l}])], i = 1, …, l − 1)

since λ^{j}(−1) = (−1)^{j} for j ≥ 0. Hence, we have for all invertible Σ_{l}-modules F on X:

θ^{l}(F) = ∑_{i=0}^{l−1}[F^{⊗i}]

= ∑_{i=0}^{l−1}(−1)^{i}[F^{⊗i} ⊗ Λ^{i}(H_{X,l})]

= ∑_{i=0}^{l−1}(−1)^{i}[Λ^{i}(F ⊗ H_{X,l})]

= λ_{−1}([F ⊗ H_{X,l}]) in K_{0}(Σ_{l}, X)/ ([Λ^{i}(O_{X}[I_{l}])], i = 1, …, l − 1) .

As in Proposition 1.13, Proposition 3.2 now follows from this using the splitting principle.

3.2 On the homology of Koszul complexes

Let G be a group, and let i : X ↪ X̃ be a regular closed G-immersion of Noetherian G-schemes. Let I := ker(OX̃ → i_{∗}(O_{X})) denote the corresponding G-stable ideal in OX̃. Furthermore, let F be a locally free G-module on X̃ of rank n and ε : F⟶I an epimorphism of G-modules. For any k = 0, …, n, let H_{k}(Λ^{·}(F), d_{·}) denote the k-th homology module of the Koszul complex

0 → Λ^{n}(F) →^{d_{n}} … →^{d_{3}} Λ^{2}(F) →^{d_{2}} F →^{d_{1}=ε} OX̃ → 0

associated with the homomorphism ε : F ⟶ I ⊆ OX̃. Furthermore, let the locally free G-module E on X be defined by the short exact sequence

0 → E → i^{∗}(F) →^{i^{∗}(ε)} i^{∗}(I) = I/I^{2} → 0.

Lemma 3.3. For any k = 0, …, n, we have an isomorphism

H_{k}(Λ^{·}(F), d_{·}) ˜→ i_{∗}(Λ^{k}(E))

of G-modules on X̃.

Proof. We will prove stalk-wise that the adjunction homomorphism

ker(d_{k}) ⊆ Λ^{k}(F) → i_{∗}i^{∗}(Λ^{k}(F))

induces an isomorphism

H_{k}(Λ^{·}(F), d_{·}) ˜→ i_{∗}(Λ^{k}(E)).

This isomorphism is then compatible with the G-actions since this already holds for the adjunction homomorphism. So let x ∈ X̃. If x is not contained in i(X), then I_{x} = O_{X̃,x} and the Koszul complex (Λ^{·}(F)_{x}, d_{·}) is exact by Proposition 2.1(c) on p. 71 in [FL]. Thus we have H_{k}(Λ^{·}(F), d_{·})_{x} = 0 = i_{∗}(Λ^{k}(E))_{x}, as was to be shown. Now let x ∈ i(X). Then the ideal I := I_{x} of the Noetherian local ring A := O_{X̃,x} is generated by a regular sequence, say of length d. Let H be a free A-module of rank d and δ : H ⟶ I a surjective A-homomorphism. Since F := F_{x} and H are free, there are homomorphisms α : H → F and β : F → H such that the following diagrams commute:

\begin{aligned}H & & →^{α} & & F \\ & δ ↘ & & ↙ ε \\ & & I \end{aligned}  and  \begin{aligned}F & & →^{β} & & H \\ & ε ↘ & & ↙ δ \\ & & I. \end{aligned}

Since δ̄ : H/IH → I/I^{2} is bijective, the composition β̄ ∘ ᾱ : H/IH → H/IH is bijective, too. Using the Nakayama-Lemma, we obtain from this that β ∘ α : H → H is bijective. We thus have the decomposition F = H ⊕ K, where H is identified with α(H) and where K := ker(β). Since ε vanishes on K, the Koszul complex (Λ^{·}(F), d_{·}) is the tensor product of the Koszul complex (Λ^{·}(H), e_{·}) associated with the homomorphism δ : H → I ⊆ A and the complex (Λ^{·}(K), 0) with the trivial differential 0. Thus, we have by Proposition 2.1(a) on p. 71 in [FL]:

H_{k}(Λ^{·}(F), d_{·})_{x} ≅ H_{k}(Λ^{·}(F), d_{·})
≅ A/I ⊗ Λ^{k}(K) ≅ Λ^{k}(K/IK) ≅ Λ^{k}(E_{x}) ≅ i_{∗}(Λ^{k}(E))_{x},

as was to be shown.

3.3 Σ_{l}-invariant sections of L^{⊠l}

Let f : X → Y be a morphism between Noetherian schemes, L an invertible O_{X}- module, and l ∈ ℕ.

Lemma 3.4. If L is generated by global sections, then the l-th external tensor power L^{⊠l} on X^{l} is generated by Σ_{l}-invariant global sections.

Proof. Let p_{1}, …, p_{l} : X^{l} → X denote the projections, and let x ∈ X^{l}. We inductively define sections s_{1}, …, s_{N} ∈ Γ (X, L) and subsets M_{1}, …, M_{N} of I_{l} = {1, …, l} as follows: Let j ∈ ℕ_{0} and let s_{1}, …, s_{j} and M_{1}, …, M_{j} be already defined. If M_{1} ∪ … ∪ M_{j} = I_{l}, then we set N := j. Otherwise, there exists an s_{j+1} ∈ Γ (X, L) with

M_{j+1} := {i ∈ I_{l}\(M_{1} ∪ … ∪ M_{j}) : s_{j+1}(p_{i}(x)) ≠ 0} ≠ ∅

by assumption. Now we set

s := ∑_{σ∈Σ_{l}/Σ(M_{1},...,M_{N})} ⊗_{i∈I_{l}} s_{r_{σ^{−1}(i)}} ∈ Γ (X^{l}, L^{⊠l})

where r_{i} := j for i ∈ M_{j} and where Σ(M_{1}, …, M_{N}) denotes the subgroup of those permutations σ ∈ Σ_{l} which satisfy σ(M_{j}) ⊆ M_{j} for all j = 1, …, N. Note that the sum does not depend on the chosen system of representatives in Σ_{l} for Σ_{l}/Σ(M_{1}, …, M_{N}). Thus, s is a Σ_{l}-invariant section of L^{⊠l}. Furthermore, we have

s(x) = ⊗_{i∈I_{l}} s_{r_{i}}(p_{i}(x)) ≠ 0

by construction. This proves Lemma 3.4.

3.4 The Σ_{l}-structure of the conormal sheaf associated with the diagonal Δ : X → X^{l}

Let f : X → Y be a separated smooth morphism between Noetherian schemes. Let I := ker(O_{X^{l}} → Δ_{∗}(O_{X})) denote the Σ_{l}-stable ideal in O_{X^{l}} associated with the diagonal Δ : X → X^{l}.

Lemma 3.5. We have an isomorphism

I/I^{2} ≅ Ω_{X/Y} ⊗ H_{X,l}

of Σ_{l}-modules on X; here, the sheaf Ω_{X/Y} of relative differentials is considered as a Σ_{l}-module with trivial Σ_{l}-action.

Proof. The O_{X}-module H_{X,l} is free with basis [i] − [i + 1], i = 1, …, l − 1, (see Notations). For any i = 1, …, l − 1, the projection p_{i,i+1} : X^{l} → X × X defines an O_{X}-homomorphism p_{i,i+1}^{∗} : Ω_{X/Y} → I/I^{2}. We define an O_{X}-homomorphism

α : Ω_{X/Y} ⊗ H_{X,l} → I/I^{2}

by α(ω⊗([i]−[i+1])) := p_{i,i+1}^{∗}(ω) for ω ∈ Ω_{X/Y} and i ∈ {1, …, l−1}. The short exact

sequences of conormal sheaves associated with the composition X ⟶^{Δ_{2}} X × X ⟶^{1×Δ_{2}} X × X × X ⟶^{1×1×Δ_{2}} … ⟶ X^{l} (see Proposition 3.4 on p. 79 in [FL]) show that α is an O_{X}-isomorphism. The following computation shows that α is compatible with the Σ_{l}-structures: Let i ∈ {1, …, l − 1} and σ ∈ Σ_{l}. We suppose that σ(i + 1) > σ(i). (If σ(i + 1) < σ(i), one has a similar computation.) Let p_{1}, …, p_{l} : X^{l} → X denote the projections. For any local section x in O_{X}, we then have:

ασ (dx ⊗ ([i] − [i + 1])) = α (dx ⊗ ([σ(i)] − [σ(i + 1)]))

= α ( ∑_{j=σ(i)}^{σ(i+1)−1} dx ⊗ ([j] − [j + 1]))

= ∑_{j=σ(i)}^{σ(i+1)−1}(p_{j}^{∗}(x) − p_{j+1}^{∗}(x)) + I^{2}

= (p_{σ(i)}^{∗}(x) − p_{σ(i+1)}^{∗}(x)) + I^{2}

= σ(p_{i}^{∗}(x) − p_{i+1}^{∗}(x)) + I^{2} = σα (dx ⊗ ([i] − [i + 1])) .

Since Ω_{X/Y} is locally generated by the total differentials dx, x ∈ O_{X}, this computation shows the required equality σα = ασ.

3.5 On the equivariant structure of the ideal I in O_{X^{l}} associated with the diagonal Δ : X → X^{l}

Let f : X → Y be a quasi-projective morphism between Noetherian schemes (in the sense of [H], p. 103) and L an invertible O_{X}-module which is very ample relative to Y (in the sense of [H], p. 120). Let I := ker(O_{X^{l}} → Δ_{∗}(O_{X})) denote the Σ_{l}-stable ideal in O_{X^{l}} associated with the diagonal Δ : X → X^{l}.

Proposition 3.6. There is an epimorphism

α : ⊕^{n} H_{X^{l},l}⟶IL^{⊠l}

of C_{l}-modules on X^{l} for some n > 0.

Proof. Let x_{0}, …, x_{r} be the global sections of L associated with an embedding i : X ↪ ℙ_{Y}^{r} with i^{∗}(O(1)) ≅ L. For any j = (j_{1}, …, j_{l}) ∈ {0, …, r}^{l}, let the homomorphism

α_{j} : O_{X^{l}}[I_{l}] → L^{⊠l}

of C_{l}-modules on X^{l} be defined by [i] ↦ c^{i}(x_{j_{1}} ⊗ … ⊗ x_{j_{l}}) for i = 1, …, l (where c denotes the generator ⟨1, …, l⟩ of C_{l}). By restricting, the map α_{j} obviously induces a homomorphism

α_{j} : H_{X^{l},l} → IL^{⊠l}

of C_{l}-modules on X^{l}. We set n := (r + 1)^{l} = #{0, …, r}^{l} and

α := (α_{j})_{j} : ⊕^{n} H_{X^{l},l} → IL^{⊠l}.

We are now going to show that α is surjective. For this, we may assume that Y = Spec(A) is affine and that i : X → ℙ_{Y}^{r} is a closed immersion. By Corollary 5.16(a) on p. 119 in [H], we may furthermore assume that X = Proj(S) where S = ⊕_{d≥0}S_{d} is a graded algebra with S_{0} = A which is generated by the elements x_{0}, …, x_{r} ∈ S_{1}. By Exercise 5.11 on p. 125 in [H], we then have X^{l} ≅ Proj(S^{l}) where

S^{l} := ⊕_{d≥0} S_{d} ⊗ … ⊗ S_{d},

and I is the sheaf associated with the homogeneous ideal I := ker(S^{l} → S). For any j ∈ {0, …, r}^{l}, the map α_{j} : H_{X^{l},l} → IL^{⊠l} corresponds to the homomorphism

\begin{aligned}α_{j} : & ⊕_{i=1}^{l−1} S^{l}([i] − [i + 1]) & → & I[1] \\ & [i] − [i + 1] & ↦ & c^{i}(x_{j_{1}} ⊗ … ⊗ x_{j_{l}}) − c^{i+1}(x_{j_{1}} ⊗ … ⊗ x_{j_{l}}). \end{aligned}

Since the open affine subsets D(x_{k_{1}} ⊗ … ⊗ x_{k_{l}}), k = (k_{1}, …, k_{l}) ∈ {0, …, r}^{l}, form a covering of Proj(S^{l}), it suffices to show that, for any k ∈ {0, …, r}^{l} and for any homogeneous f ∈ I, the element (x_{k_{1}} ⊗ … ⊗ x_{k_{l}})^{N} · f is contained in the image of α = (α_{j})_{j} for sufficiently large N (see also Exercise 5.10 on p. 125 in [H]). Now let (k_{1}, …, k_{r}) ∈ {0, …, r}^{l} be fixed. For any homogeneous f_{1}, …, f_{l} ∈ S of degree d, we have:

(x_{k_{1}} ⊗ … ⊗ x_{k_{l}})^{d} · (f_{1} ⊗ … ⊗ f_{l})
= (x_{k_{1}}^{d} ⊗ … ⊗ x_{k_{l−1}}^{d} ⊗ f_{l}) · (f_{1} ⊗ … ⊗ f_{l−1} ⊗ x_{k_{l}}^{d} − x_{k_{l}}^{d} ⊗ f_{1} ⊗ … ⊗ f_{l−1})
+ (x_{k_{1}}x_{k_{l}})^{d} ⊗ (x_{k_{2}}^{d}f_{1}) ⊗ … ⊗ (x_{k_{l−1}}^{d}f_{l−2}) ⊗ f_{l−1}f_{l}.

We can make a similar computation for the error term (x_{k_{1}}x_{k_{l}})^{d} ⊗ (x_{k_{2}}^{d}f_{1}) ⊗ … ⊗ (x_{k_{l−1}}^{d}f_{l−2}) ⊗ f_{l−1}f_{l} in place of f_{1} ⊗ … ⊗ f_{l}. After (l − 1) such computations, one sees that, for some sufficiently large N, there are elements h_{1}, …, h_{l} ∈ S which do not depend on f_{1}, …, f_{l} such that the element

(x_{k_{1}} ⊗ … ⊗ x_{k_{l}})^{N}(f_{1} ⊗ … ⊗ f_{l}) − h_{1} ⊗ … ⊗ h_{l−1} ⊗ h_{l}f_{1} · . . . · f_{l}

is a linear combination of elements of the form h − c(h), h ∈ S^{l}. Hence, for any homogeneous f which is contained in the ideal I, the element (x_{k_{1}} ⊗ … ⊗ x_{k_{l}})^{N} · f is a linear combination of elements of the form (1 − c)(u_{1} ⊗ … ⊗ u_{l}), where u_{1}, …, u_{l} are monomials in the x_{0}, …, x_{r} of the same degree. Using the identity

(1 − c)(uv) = u((1 − c)v) + ((1 − c)u)v

(for homogeneous u, v ∈ S^{l}), we finally obtain that (x_{k_{1}} ⊗ … ⊗ x_{k_{l}})^{N} · f is contained in the image of α, as was to be shown.

Remark 3.7. If X = Spec(B) and Y = Spec(A) are affine, then there exists even a Σ_{l}-epimorphism

α : ⊕^{n} H_{X^{l},l} → I

on X^{l} for some n > 0.

Proof. Let x_{1}, …, x_{n} be generators of the A-algebra B. For any j = 1, …, n, let the B^{⊗l}-homomorphism

α_{j} : B^{⊗l}[I_{l}] → B^{⊗l}

be defined by [i] ↦ 1 ⊗ … ⊗ 1 ⊗ x_{j} ⊗ 1 ⊗ … ⊗ 1 where x_{j} is put at the i-th place (for any i = 1, …, l). Obviously, the map α_{j} is compatible with the Σ_{l}-structures. As in Proposition 3.6, one easily shows that α := (α_{j})_{j=1,...,n} induces an epimorphism

⊕^{n} H_{X^{l},l} → I.

3.6 Proof of Theorem 3.1

The element λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}]) is invertible in K_{0}(Σ_{l}, X)[l^{−1}]/IK_{0}(Σ_{l}, X)[l^{−1}] by Lemma 4.3 in [Ko5] and by Proposition 3.2 since the elements λ^{i}([Ω_{X/Y}]) − \binom{d}{i}, i ≥ 0, are nilpotent by Proposition 1.5 on p. 52 and Lemma 3.1 on p. 119 in [FL]. Now let L be a very ample invertible O_{X}-module relative to f. Let M denote the invertible Σ_{l}-module (L^{⊠l})^{∨} on X^{l}. By Lemma 3.4, there is an epimorphism

β : ⊕^{n} M ⟶ O_{X^{l}}

of Σ_{l}-modules on X^{l} for some n > 0. The Koszul complex

0 → Λ^{n}(⊕^{n} M) → … → ⊕^{n} M → O_{X^{l}} → 0

associated with this epimorphism is exact by Proposition 2.1(c) on p. 71 in [FL]. Thus we have (1 − [M])^{n} = λ_{−1}([⊕^{n} M]) = 0, i.e., 1 − [M] is nilpotent in K_{0}(Σ_{l}, X^{l}). By Proposition 3.2, we have

λ_{−1}([M ⊗ H_{X^{l},l}]) = θ^{l}(M) = l + ([M] − 1) + … + ([M]^{l−1} − 1)

in K_{0}(Σ_{l}, X^{l})/IK_{0}(Σ_{l}, X^{l}); thus, the element λ_{−1}([M ⊗ H_{X^{l},l}]) is invertible in

K_{0}(Σ_{l}, X^{l})[l^{−1}]/IK_{0}(Σ_{l}, X^{l})[l^{−1}].

We set I := ker(O_{X^{l}} → Δ_{∗}(O_{X})). By Proposition 3.6, there is an epimorphism

⊕^{n}(M ⊗ H_{X^{l},l})⟶I

of C_{l}-modules on X^{l} for some n > 0. Let the locally free C_{l}-module E on X be defined by the short exact sequence

0 → E → Δ^{∗} (⊕^{n}(M ⊗ H_{X^{l},l})) → Δ^{∗}(I) = I/I^{2} → 0.

Then we have in K_{0}(C_{l}, X) by Lemma 3.5:

λ_{−1}([E]) · λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}]) = λ_{−1}([E]) · λ_{−1}([I/I^{2}])

= λ_{−1} ([Δ^{∗} (⊕^{n}(M ⊗ H_{X^{l},l}))]) = Δ^{∗} (λ_{−1}([M ⊗ H_{X^{l},l}]))^{n} .

Hence, we have in K_{0}(C_{l}, X^{l})[l^{−1}]/IK_{0}(C_{l}, X^{l})[l^{−1}]:

Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1})

= Δ_{∗} (\frac{λ_{−1}([E])}{Δ^{∗} (λ_{−1}([M ⊗ H_{X^{l},l}]))^{n}} ) = \frac{Δ_{∗}(λ_{−1}([E]))}{λ_{−1} ([⊕^{n}(M ⊗ H_{X^{l},l})])} = 1.

Here, the second equality follows from the projection formula; the last equality follows from Lemma 3.3 since Δ is regular and all homology modules, kernels, and images in the Koszul complex associated with ⊕^{n}(M ⊗ H_{X^{l},l}) ⟶ I ⊆ O_{X^{l}} are contained in the category P_{∞}(C_{l}, X^{l}) (defined as in section 2).

Remark 3.8. If X and Y are affine, then the equality

Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1}) = 1

holds even in K_{0}(Σ_{l}, X^{l})[l^{−1}]/IK_{0}(Σ_{l}, X^{l})[l^{−1}].

Proof. This can be proved as Theorem 3.1 by using Remark 3.7 in place of Proposition 3.6.

Remark 3.9. If l is a prime, then the ideal IK_{0}(C_{l}, Y) in K_{0}(C_{l}, Y) is generated by the element [O_{Y} [I_{l}]].

Proof. This follows from the fact that the exterior powers Λ^{i}(O_{Y} [I_{l}]), i = 1, …, l−1, are free O_{Y} [C_{l}]-modules (see Proposition 1.1 in [Ko6]).

§4 Riemann-Roch Formulas for Tensor Power Operations

Let l ∈ ℕ, and let f : X → Y be a projective morphism as in section 2. Whilst we considered the l-th external tensor power operation for X in section 2, we now consider the l-th tensor power operation τ^{l} : K_{q}(X) → K_{q}(Σ_{l}, X). The aim of this section is to prove the Riemann-Roch formula

τ^{l}(f_{∗}(x)) = f_{∗} (λ_{−1}(T_{f}^{∨} · H_{X,l})^{−1} · τ^{l}(x))

for x ∈ K_{q}(X) (T_{f}^{∨} ∈ K_{0}(X) denotes the cotangential element). In general, this formula is valid only in K_{q}(C_{l}, Y)[l^{−1}] modulo a certain subgroup (see Theorem 4.2). It is then essentially equivalent to the usual Adams-Riemann-Roch formula for f (by Proposition 1.13, Proposition 3.2 and Lemma 4.3). If f is smooth, we prove Theorem 4.2 by generalizing an idea of Nori (see [Ra]), and we in particular obtain a new and simple proof of the usual Adams-Riemann-Roch formula for f in the case X and Y are ℂ-schemes. Furthermore, we prove the following strengthened versions: If f is a regular closed immersion, the formula above holds even in K_{q}(Σ_{l}, Y) (see Theorem 4.1); here, the multiplier λ_{−1}(T_{f}^{∨} · H_{X,l})^{−1} equals λ_{−1}([C ⊗ H_{X,l}]) where C denotes the conormal sheaf. If l is a prime and f a principal G-bundle for some finite group G with l |̸ ord(G), then the formula above holds even in K_{q}(C_{l}, Y)/[O_{Y} [C_{l}]] · K_{q}(C_{l}, Y), i.e., without inverting l (see Theorem 4.9); here, the multiplier λ_{−1}(T_{f}^{∨} · H_{X,l})^{−1} is 1. In case of ℂ-schemes, we deduce from (a G-equivariant version of) this result a version without denominators of the (G-equivariant) Adams-Riemann-Roch formula ψ^{l}f_{∗} = f_{∗}ψ^{l} (see Corollary 4.10).

Theorem 4.1. Let X̃ be a Noetherian scheme on which each coherent OX̃-module is a quotient of a locally free OX̃-module. Furthermore, let i : X ↪ X̃ be a regular closed immersion with the conormal sheaf C. Then the following diagram commutes:

K_{0}(X) ⟶^{λ_{−1}([C⊗H_{X,l}])·τ^{l}} K_{0}(Σ_{l}, X)
i_{∗} ↓  ↓ i_{∗}
K_{0}(X̃)  ⟶^{τ^{l}}  K_{0}(Σ_{l}, X̃).

If Conjecture’ on p. 289 in [Ko2] is true, then the corresponding diagrams of higher K-groups commute, too.

Proof. First, let X̃ = ℙ_{X}(E ⊕ O_{X}) where E is a locally free O_{X}-module, and let i : X ↪ X̃ be the zero section embedding. Let D denote the universal hyperplane sheaf on X̃. For any y ∈ K_{0}(X̃), we then have in K_{0}(Σ_{l}, X̃):

τ^{l}(i_{∗}(i^{∗}(y)) = τ^{l}(i_{∗}(1) · y) (projection formula)
= τ^{l}(i_{∗}(1)) · τ^{l}(y) (Proposition 1.7(a))
= τ^{l}(λ_{−1}([D])) · τ^{l}(y) (Lemma 6.2 on p. 142 in [FL])
= λ_{−1}([D]) · λ_{−1}([D ⊗ H_{X̃,l}]) · τ^{l}(y) (Corollary 1.11)
= i_{∗}(1) · λ_{−1}([D ⊗ H_{X̃,l}]) · τ^{l}(y) (Lemma 6.2 on p. 142 in [FL])
= i_{∗} (λ_{−1}([i^{∗}(D ⊗ H_{X̃,l})]) · τ^{l}(i^{∗}(y))) (projection formula)
= i_{∗} (λ_{−1}([C ⊗ H_{X,l}]) · τ^{l}(i^{∗}(y))) (Proposition 3.2(b) on p. 78 in [FL]).

Since i^{∗} : K_{0}(ℙ_{X}(E ⊕ O_{X})) → K_{0}(X) is surjective, this computation proves Theorem 4.1 for K_{0} in the case of a zero section embedding. The general case follows from this by using the deformation to the normal cone as, for example, in Theorem 6.3 on p. 142 in [FL]. For higher K-groups, Theorem 4.1 can be proved similarly by using Proposition 1.7(b) in place of Proposition 1.7(a).

Remark. Applying the forgetful map K_{0}(Σ_{l}, X̃) → K_{0}(X̃) to the formula proved in Theorem 4.1 yields the following weak version: For all x ∈ K_{0}(X), we have

i_{∗}(x)^{l} = i_{∗}(x^{l} · λ_{−1}([C])^{l−1}) in K_{0}(X̃).

This weak version also follows from the self intersection formula (see Corollaire 2.8 on p. 436 in [SGA 6]).

Now, let Y be a Noetherian scheme on which an ample invertible O_{Y} -module exists, and let f : X → Y be a projective local complete intersection morphism. Let T_{f}^{∨} ∈

K_{0}(X) denote the cotangential element: For any decomposition X →^{i} X̃ →^{p} Y of f into a regular closed immersion i and a smooth morphism p, we have

T_{f}^{∨} = [i^{∗}(Ω_{X̃/Y})] − [C_{X/X̃}] in K_{0}(X)

by Proposition 7.1 on p. 145 in [FL]. Let I be the ideal of K_{0}(C_{l}, Y) generated by the elements [Λ^{i}(O_{Y} [I_{l}])], i = 1, …, l−1. By Theorem 3.1, the element λ_{−1}([Ω_{X̃/Y} ⊗H_{X̃,l}]) is invertible in K_{0}(C_{l}, X̃)[l^{−1}]/IK_{0}(C_{l}, X̃)[l^{−1}]. We set

λ_{−1}(T_{f}^{∨} · [H_{X,l}])^{−1} := i^{∗} (λ_{−1}([Ω_{X̃/Y} ⊗ H_{X̃,l}])^{−1}) · λ_{−1}([C_{X/X̃} ⊗ H_{X,l}])

in K_{0}(C_{l}, X)[l^{−1}]/IK_{0}(C_{l}, X)[l^{−1}]. This definition does obviously not depend on the chosen decomposition of f as above. Moreover, it is of course possible to define the expression λ_{−1}(x · [H_{X,l}]) for all x ∈ K_{0}(X), but we will not need this generalization.

Theorem 4.2. The following diagram commutes:

K_{0}(X) ⟶^{λ_{−1}(T_{f}^{∨}·[H_{X,l}])^{−1}·τ^{l}} K_{0}(C_{l}, X)[l^{−1}]/IK_{0}(C_{l}, X)[l^{−1}]
f_{∗} ↓  ↓ f̄_{∗}
K_{0}(Y)  ⟶^{τ^{l}}  K_{0}(C_{l}, Y)[l^{−1}]/IK_{0}(C_{l}, Y)[l^{−1}].

The corresponding diagrams of higher K-groups commute if one of the following conditions holds:

(a) f is smooth.

(b) Conjecture’ on p. 289 in [Ko2] is true.

(c) The number l is a prime, and the relation between τ^{l} and ψ^{l} mentioned in Proposition 1.13 holds for the higher K-groups of X and Y.

Proof. If l is a prime, the assertion for K_{0} and the assertion for the higher K-groups in case of the condition (c) follow from the Adams-Riemann-Roch theorem (see Theorem 7.6 on p. 149 in [FL] and Theorem 4.5, Remark 4.6(b) in [Ko5]) by using Propositions 1.13 and 3.2. Now, let f be smooth. In this case, we prove Theorem 4.2 for K_{0} (once more) and for the higher K-groups by generalizing Nori’s idea (see [Ra]) to our situation. For this, let Δ : X → X^{l} denote the diagonal and f^{l} : X^{l} → Y the projection. By τ^{l} we denote both the tensor power operation K_{q}(X) → K_{q}(C_{l}, X) and the external tensor power operation K_{q}(X) → K_{q}(C_{l}, X^{l}). For any x ∈ K_{q}(X), we then have in K_{q}(C_{l}, X^{l})[l^{−1}]/IK_{q}(C_{l}, X^{l})[l^{−1}]:

Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1} · τ^{l}(x))
= Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1} · Δ^{∗}(τ^{l}(x))) (since Δ^{∗}(τ^{l}(x)) = τ^{l}(x))
= Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1}) · τ^{l}(x) (projection formula)
= τ^{l}(x) (Theorem 3.1).

For any x ∈ K_{q}(X), we thus have in K_{q}(C_{l}, Y)[l^{−1}]/IK_{q}(C_{l}, Y)[l^{−1}]:

f_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1} · τ^{l}(x))
= f_{∗}^{l}Δ_{∗} (λ_{−1}([Ω_{X/Y} ⊗ H_{X,l}])^{−1} · τ^{l}(x)) (since f = f^{l} ∘ Δ)
= f_{∗}^{l}(τ^{l}(x)) (see above)
= τ^{l}(f_{∗}(x)) (Theorem 2.1).

Together with Theorem 4.1 this implies Theorem 4.2 also for a general f (for K_{0} and for higher K-groups in the case (b)) since a general f is the composition of regular closed embedding with a smooth morphism.

If the canonical map K_{q}(Y) → K_{q}(C_{l}, Y)/IK_{q}(C_{l}, Y) is injective (and if the diagram in Proposition 1.13 commutes for the higher K-groups of X and Y), then, conversely, the Adams-Riemann-Roch theorem obviously follows from Theorem 4.2. If, in addition, f is smooth, Nori’s idea in particular yields a quick proof of the Adams-Riemann-Roch theorem which, in contrast to [FL], does not use the deformation to the normal cone and the explicit computations for so-called elementary embeddings and projections. The following considerations deal with the question under which conditions the map K_{q}(Y) → K_{q}(C_{l}, Y)/IK_{q}(C_{l}, Y) is injective.

Lemma 4.3. Let l be a prime, q ≥ 0, and Y a scheme on which l is invertible. Then we have: If the map K_{q}(Y) → K_{q}(C_{l}, Y), y ↦ [H_{Y,l}]·y, is injective, then the canonical map K_{q}(Y) → K_{q}(C_{l}, Y)/IK_{q}(C_{l}, Y) is injective, too.

Proof. By Remark 3.9, the ideal I is generated by the element [O[C_{l}]]. Furthermore, we have [O_{Y} [C_{l}]] · K_{q}(C_{l}, Y) = [O_{Y} [C_{l}]] · K_{q}(Y) by Frobenius reciprocity. Finally, we have K_{q}(Y)∩([H_{Y,l}·K_{q}(Y)) = 0 since the functor which maps a locally free C_{l}-module F to the module F^{C_{l}} of C_{l}-fixed elements induces a homomorphism K_{q}(C_{l}, Y) → K_{q}(Y) which is the identity on K_{q}(Y) and which vanishes on [H_{Y,l}] · K_{q}(Y). This immediately implies Lemma 4.3.

Corollary 4.4. The canonical map K_{q}(Y) → K_{q}(C_{l}, Y)/IK_{q}(C_{l}, Y) is injective if K_{q}(Y) has no (l − 1)-torsion or if there exists a primitive l-th root of unity on Y.

Proof. If K_{q}(Y) has no (l − 1)-torsion, the composition

\begin{matrix}K_{q}(Y) & → & K_{q}(C_{l}, Y) & →^{can} & K_{q}(Y) \\ y & ↦ & [H_{Y,l}] · y \end{matrix}

is injective; hence, also the first map is injective. If there exists a primitive l-th root

of unity on Y, we have K_{q}(C_{l}, Y) ≅ ⊕^{l} K_{q}(Y), and each of the non-trivial projections K_{q}(C_{l}, Y) → K_{q}(Y) is left-inverse to the map K_{q}(Y) → K_{q}(C_{l}, Y), y ↦ [H_{Y,l}] · y. Now, Lemma 4.3 proves Corollary 4.4.

Example 4.5. Let l be a prime.

(a) Let Y = Spec(𝔽_{l}). Then we have K_{0}(C_{l}, Y)/([O[C_{l}]]) ≅ ℤ/lℤ. In particular, the canonical map K_{0}(Y) → K_{0}(C_{l}, Y)/IK_{0}(C_{l}, Y) is not injective.

(b) Let Y = Spec(ℂ). Then K_{0}(C_{l}, Y)/([O_{Y} [C_{l}]]) is isomorphic to ℤ[ζ_{l}] where ζ_{l} denotes a primitive l-th root of unity.

Now, let f : X → Y be a morphism between Noetherian schemes and G a finite group which acts on X by Y -automorphisms. We recall the following definition of [SGA 1]:

Definition 4.6. The Y -scheme X is called a principal G-bundle if f is faithfully flat and the morphism

G × X → X ×_{Y} X, (γ, x) ↦ (x, γ(x)),

is an isomorphism.

Here, we have used the following notation: For any finite set M and for any scheme Z, let M × Z denote the disjoint union of #M copies of Z. If M is G-set, we will consider M × Z as a G-scheme in the obvious way.

Remark 4.7. By Proposition 2.6 on p. 115 in [SGA 1], X is a principal G-bundle if and only if f is finite, Y = X/G, and the inertia groups of all points in X are trivial. In this case, f is étale (see Corollaire 2.3 on p. 113 in [SGA 1]), and the exact functor f_{∗} obviously induces homomorphisms

f_{∗} : K_{q}(X) → K_{q}(Y) and f_{∗} : K_{q}(G, X) → K_{q}(G, Y)

for all q ≥ 0.

Example 4.8. An extension R ⊆ S of Noetherian commutative rings is a Galois extension with Galois group G (in the sense of Definition 1.5 on p. 2 in [Gre]) if and only if f : Spec(S) → Spec(R) is a principal G-bundle. For instance, the extension of the rings of integers in a Galois extension of number fields is a Galois extension if and only if it is unramified (see Theorem 4.1 on p. 18 in [Gre]).

In the following Theorem 4.9, we will consider both the tensor power operation τ^{l} : K_{q}(X) → K_{q}(C_{l}, X) and the equivariant version τ^{l} : K_{q}(G, X) → K_{q}(C_{l} × G, X) which is defined in the obvious way (see also Remark 1.12). The proof of Theorem 4.9 is similar to the proof of Theorem 4.2; however, we will not use the results of sections 2 and 3, but more or less only the very definition of τ^{l}.

Theorem 4.9. Let l be a prime with l |̸ ord(G). Let f : X → Y be a principal G-bundle or the canonical projection associated with X = (G/G′) × Y where G′ is a subgroup of G. Then the diagram

K_{q}(H, X) ⟶^{τ^{l}} K_{q}(C_{l} × H, X)/[O_{X}[C_{l}]] · K_{q}(C_{l} × H, X)
f_{∗} ↓  ↓ f̄_{∗}
K_{q}(H, Y) ⟶^{τ^{l}} K_{q}(C_{l} × H, Y)/[O_{Y} [C_{l}]] · K_{q}(C_{l} × H, Y)

commutes for all subgroups H of G and all q ≥ 0.

Proof. The first (big) part of the proof is to describe the (C_{l} × G)-structure of the fibred product X^{l} where C_{l} acts by cyclic permutations as usual (see Notations) and where G acts diagonally. First, let f : X → Y be a principal G-bundle. We identify G^{l−1} with {1} × G^{l−1} ⊆ G^{l} and define an action of C_{l} on G^{l−1} by

c^{i}(g_{1}, …, g_{l}) := g_{i+1}^{−1} · (g_{i+1}, …, g_{l}, g_{1}, …, g_{i})
= (1, g_{i+1}^{−1}g_{i+2}, …, g_{i+1}^{−1}g_{l}, g_{i+1}^{−1}, g_{i+1}^{−1}g_{2}, …, g_{i+1}^{−1}g_{i})

(for i ∈ {0, …, l − 1} and (g_{1}, …, g_{l}) ∈ G^{l−1}). We furthermore define an action of C_{l} on G^{l−1} × X by

c^{i}(g, x) := (c^{i}(g), g_{i+1}(x))

(for g = (g_{1}, …, g_{l}) ∈ G^{l−1} and x ∈ X). Finally, we define a (right) action of G on G^{l−1} × X by

γ(g, x) := (γ^{−1}gγ, γ(x))

(for g ∈ G^{l−1}, γ ∈ G, and x ∈ X). One easily checks that G^{l−1} × X together with these actions is a (C_{l} × G)-scheme and that the morphism

G^{l−1} × X → X^{l}, ((g_{1}, …, g_{l}), x) ↦ (g_{1}(x), …, g_{l}(x)),

is an isomorphism of (C_{l} × G)-schemes. The action of G on G^{l−1} by conjugation obviously induces an action on the set (G^{l−1}\{(1, …, 1)})/C_{l} of C_{l}-orbits. We will now show that there is a G-stable system of representatives in G^{l−1}\{(1, …, 1)} for this set of orbits. For this, it suffices to show that, for any g ∈ G^{l−1}\{(1, …, 1)}, the map

C_{l} × {γ^{−1}gγ : γ ∈ G} → G^{l−1}, (c^{i}, h) ↦ c^{i}(h),

is injective. So let i, j ∈ {0, …, l − 1} and η, θ ∈ G with c^{i}(η^{−1}gη) = c^{j}(θ^{−1}gθ). We may assume that j = 0 and η = 1. Then we have c^{i}(g) = θ^{−1}gθ, hence c^{ir}(g) = θ^{−r}gθ for all r ≥ 0; i.e., the orbit of g under the subgroup of G generated by θ is the set N := {c^{0}(g), c^{i}(g), c^{2i}(g), …}. Thus, the number #N of elements in N divides the order of θ. Since #N ∈ {1, l} and since l |̸ ord(G), we have #N = 1, i.e., c^{i}(g) = g. This implies i = 0 because, otherwise, g would be fixed under the action of C_{l} and would hence be of the form (γ^{0}, γ^{1}, …, γ^{l−1}) for some γ ∈ G with γ^{l} = 1 which means that g would be equal to (1, …, 1) since l |̸ ord(G). Now, let M ⊆ G^{l−1}\{(1, …, 1)} be a G-stable system of representatives for

(G^{l−1}\{(1, …, 1})/C_{l}.

We consider C_{l} × M × X as a (C_{l} × G)-scheme by virtue of

(c^{i}, γ)(c^{j}, g, x) := (c^{i+j}, γ^{−1}gγ, γ(x))

(for i, j ∈ {0, …, l − 1}, γ ∈ G. g ∈ M, and x ∈ X). Then the morphism

C_{l} × M × X → (G^{l−1}\{(1, …, 1)}) × X, (c^{i}, g, x) ↦ (c^{i}(g), g_{i}(x)),

is obviously an isomorphism of (C_{l} × G)-schemes. Altogether, we now have the following isomorphisms of (C_{l} × G)-schemes:

X^{l} ≅ Δ(X) ∐(G^{l−1}\{(1, …, 1)}) × X ≅ Δ(X) ∐ C_{l} × M × X

where Δ : X → X^{l} denotes the diagonal. Now, let X = (G/G′) × Y where G′ is a subgroup of G. Let Δ : G′\G → (G′\G)^{l} denote the diagonal. Similarly as above, we show that, for any g ∈ (G′\G)^{l} \ Δ(G′\G), the map

C_{l} × gG → (G′\G)^{l}, (c^{i}, h) ↦ c^{i}(h),

is injective. So let i, j ∈ {0, …, l − 1} and η, θ ∈ G with c^{i}(gη) = c^{j}(gθ). We may assume that j = 0 and η = 1. Then we have c^{i}(g) = gθ, hence c^{ir}(g) = gθ^{r} for all r ≥ 0; i.e., the orbit of g under the subgroup of G generated by θ is the set N := {c^{0}(g), c^{i}(g), c^{2i}(g), …}. Thus, the number #N of elements in N divides the order of θ. Since #N ∈ {1, l} and since l |̸ ord(G), we have #N = 1, i.e., c^{i}(g) = g. This implies i = 0 because, otherwise, we would have g ∈ Δ(G′\G). As above, this implies that there exists a G-stable subset M ⊆ (G′\G)^{l} \ Δ(G′\G) such that the map

C_{l} × M → (G′\G)^{l} \ Δ(G′\G), (c^{i}, h) ↦ c^{i}(h),

is bijective. Hence, we have isomorphisms

X^{l} ≅ (G′\G)^{l} × Y ≅ Δ(X) ∐ C_{l} × M × Y

of (C_{l} × G)-schemes. Now, let f be a principal G-bundle or X = (G′\G) × Y. If follows for instance from the established decompositions of X^{l} that, for any locally free O_{X}-module E, the direct image Δ_{∗}(E) is a locally free O_{X^{l}}-module. The functor Δ_{∗} thus induces a homomorphism

Δ_{∗} : K_{0}(C_{l} × H, X) → K_{0}(C_{l} × H, X^{l})

for all subgroups H of G. Furthermore, it follows from the established decompositions of X^{l} that the homomorphism Ind_{{1}}^{C_{l}} : K_{0}(H, X^{l}\Δ(X)) → K_{0}(C_{l} × H, X^{l}\Δ(X)) is surjective. This implies that the homomorphism

Δ_{∗} : K_{0}(C_{l} × H, X)/([O_{X}[C_{l}]]) → K_{0}(C_{l} × H, X^{l})/Ind_{{1}}^{C_{l}}(K_{0}(H, X^{l}))

is an isomorphism with

Δ_{∗}(1) = 1.

Now, let f^{l} : X^{l} → Y denote the projection. By the projection formula and the base change isomorphism (see Proposition 9.3 on p. 255 in [H]), we have a natural isomorphism f_{∗}(E_{1}) ⊗ … ⊗ f_{∗}(E_{l}) ≅ f_{∗}^{l}(E_{1} ⊠ … ⊠ E_{l}) for all locally free O_{X}-modules E_{1}, …, E_{l}. Hence, for all x ∈ K_{q}(H, X), we obtain the equality

τ^{l}(f_{∗}(x)) = f_{∗}^{l}(τ^{l}(x)) in K_{q}(C_{l} × H, Y).

Furthermore, we obviously have:

f_{∗}^{l} (Ind_{{1}}^{C_{l}}(K_{0}(H, X^{l})) · K_{q}(C_{l} × H, X^{l})) ⊆ [O_{Y} [C_{l}]] · K_{q}(C_{l} × H, Y).

As in the proof of Theorem 4.2, we now have:

τ^{l}(f_{∗}(x)) = f_{∗}^{l}(τ^{l}(x)) = f_{∗}^{l}(τ^{l}(x) · Δ_{∗}(1)) = f_{∗}^{l}(Δ^{∗}(τ^{l}(x))) = f_{∗}(τ^{l}(x))

in K_{q}(C_{l} × H, Y)/[O_{Y} [C_{l}]] · K_{q}(C_{l} × H, Y). This proves Theorem 4.9.

Corollary 4.10. Let Y be a ℤ[l^{−1}][ζ_{l}]-scheme. Then we have for all x ∈ K_{0}(H, X):

ψ^{l}(f_{∗}(x)) = f_{∗}(ψ^{l}(x)) in K_{0}(H, Y).

The corresponding assertion for higher K-groups is true if the relation between τ^{l} and ψ^{l} mentioned in Proposition 1.13 holds for the (higher equivariant) K-groups of X and Y.

Proof. This follows from Theorem 4.9 and obvious generalizations of Proposition 1.13 and Corollary 4.4.

Remark 4.11. Let G′ be a subgroup of G and Y a Noetherian scheme. In section 6 in [Ko5], we have raised the following question: Under which conditions does the induction formula

ψ^{l}(Ind_{G′}^{G}(x)) = Ind_{G′}^{G}(ψ^{l}(x))

hold in K_{q}(G, Y) for x ∈ K_{q}(G′, Y)? In Theorem 6.2 in [Ko5], we have proved that, for all l, this formula holds in a certain completion of the group K_{q}(G, Y)[l^{−1}]. From Corollary 4.10 it now follows as in Theorem 6.2 in [Ko5] that this formula holds already in K_{q}(G, Y) if l is a prime with l |̸ ord(G) and Y is a ℤ[l^{−1}][ζ_{l}]-scheme (and if, in case q ≥ 1, the relation between ψ^{l} and τ^{l} mentioned in Proposition 1.13 holds).

§5 Speculations in Characteristic p

For any scheme Z of characteristic p, the p-th Adams operation ψ^{p} on K_{0}(Z) equals the pull-back homomorphism F_{Z}^{∗} associated with the absolute Frobenius morphism F_{Z}. This well-known fact (e.g., see Proposition 2.15 on p. 64 in [Ko4]) should be considered as a substitute for the relation “ψ^{p} = τ^{p}” established in Proposition 1.13 which, in characteristic p, still holds but is extremely weak (cf. Example 4.5(a)). In this section, we will formulate and investigate an assertion (see Question 5.2) which may be considered as a substitute for Theorem 3.1. In doing so, we will find many analogies between the previous sections and the considerations in this section.

So let p be a prime and f : X → Y a morphism between Noetherian 𝔽_{p}-schemes X and Y. As a substitute for the external tensor power operation τ^{l} : K_{0}(X) → K_{0}(Σ_{l}, X^{l}), we consider the pull-back homomorphism F_{Y}^{∗} : K_{0}(X) → K_{0}(X_{Y}) where X_{Y} and F_{Y} : X_{Y} → X are defined by the following cartesian commutative diagram:

X_{Y}  ⟶^{F_{Y}} X
f_{Y} ↓  ↓ f
Y  ⟶^{F_{Y}} Y.

The following proposition may be viewed as an analogue of Theorem 2.1:

Proposition 5.1. Let Y be a Noetherian scheme on which each coherent O_{Y} -module is a quotient of a locally free O_{Y} -module. Furthermore, let f : X → Y be a flat projective local complete intersection morphism. Then the following diagram commutes:

K_{0}(X) ⟶^{F_{Y}^{∗}} K_{0}(X_{Y})
f_{∗} ↓  ↓ (f_{Y} )_{∗}
K_{0}(Y) ⟶^{F_{Y}^{∗}} K_{0}(Y).

Proof. This follows from the excess intersection formula (see Theorem 1.3 on p. 155 in [FL]). Note that the excess conormal sheaf in our situation is trivial since f is flat.

As a substitute for the diagonal Δ : X → X^{l}, we consider the relative Frobenius morphism F : X → X_{Y} which is defined by the following commutative diagram:

[figure: X; f; F_{X}; F; X_{Y}; f_{Y}; F_{Y}; X; f; Y; F_{Y}; Y]

Now, let f be smooth. Then, a theorem of Kunz (e.g., see Theorems 15.7 and 15.5 in [Ku]) says that F is (finite and) flat. This fact may be considered as an analogue of the fact that the closed immersion Δ is regular if f is smooth. As a substitute for the pushforward homomorphism Δ_{∗}, we in particular have the push-forward homomorphism

F_{∗} : K_{0}(X) → K_{0}(X_{Y})

(induced by the exact functor F_{∗}). We now assume furthermore that there exists an ample invertible O_{X}-module. Then the Bott element θ^{p}(Ω) associated with the module Ω := Ω_{X/Y} of relative differentials is invertible in K_{0}(X)[p^{−1}] (e.g., see beginning of subsection 3.6). In view of Proposition 3.2, the following question may be considered as an analogue of Theorem 3.1:

Question 5.2. Do we have F_{∗}(θ^{p}(Ω)^{−1}) = 1 in K_{0}(X_{Y})[p^{−1}]?

The following proposition may be considered as an analogue of the equality

λ_{−1} ([⊕^{n}(M ⊗ H_{X^{l},l})]) · (Δ_{∗}(θ^{p}(Ω)^{−1}) − 1) = 0

established in the proof of Theorem 3.1.

Proposition 5.3. The element λ_{−1}([F_{Y}^{∗}(Ω)]) = λ_{−1}([Ω_{X_{Y} /Y}]) ∈ K_{0}(X_{Y}) annihilates the difference F_{∗}(θ^{p}(Ω)^{−1}) − 1.

Proof. Using the splitting principle (e.g., see section 2.5 in [Ko1]), one easily shows that, for any locally free O_{X}-module F, the equality

λ_{−1}([F_{X}^{∗}(F)]) · θ^{p}(F)^{−1} = λ_{−1}([F]) in K_{0}(X_{Y})[p^{−1}]

holds. (Note that F_{X}^{∗}(F) ≅ F^{⊗p} if rank(F) = 1.) Thus we have:

λ_{−1}([F_{Y}^{∗}(Ω)]) · F_{∗}(θ^{p}(Ω)^{−1})
= F_{∗} (λ_{−1}([F_{X}^{∗}(Ω)]) · θ^{p}(Ω)^{−1})
= F_{∗}(λ_{−1}([Ω]))
= λ_{−1}([F_{Y}^{∗}(Ω)]) in K_{0}(X_{Y})[p^{−1}].

Here, the first equality follows from the projection formula and the last equality follows from the Cartier isomorphism (see Theorem 7.2 on p. 200 in [Ka]). This proves Proposition 5.3.

In contrast to the element λ_{−1} ([⊕^{n}(M ⊗ H_{X^{l},l})]), the element λ_{−1}([F_{Y}^{∗}(Ω)]) is not invertible, it is even nilpotent. Thus, Proposition 5.3 does not imply the equality F_{∗}(θ^{p}(Ω)^{−1}) = 1.

In addition to the previous assumptions, we now suppose that each coherent O_{Y} - module on Y is a quotient of a locally free O_{Y} -module and that f is projective. We set

Num_{0}(f) := {y ∈ K_{0}(X_{Y})[p^{−1}] : (f_{Y})_{∗}(y · F_{Y}^{∗}(x)) = 0 for all x ∈ K_{0}(X)[p^{−1}]}.

The notation Num_{0}(f) should remind the reader of the group of cycles numerically equivalent to zero which is defined in a similar way.

Example 5.4.

(a) Let f : ℙ(E) → Y be the canonical projection of the projective space bundle associated with a locally free O_{Y} -module E. Then we have Num_{0}(f) = 0. This follows from the projective space bundle theorem (see Theorem 2.3 on p. 115 in [FL]).

(b) Let f : X → Spec(𝔽_{p}) be a smooth projective curve. Then we have Num_{0}(f)⊗ℚ = 0. This follows from the Riemann-Roch theorem for curves (e.g., see Theorem 1.3 on p. 295 in [H]) because K_{0}(X) ≅ Pic^{0}(X) ⊕ ℤ ⊕ ℤ.

The following proposition may be considered as an analogue of the fact that Theorem 3.1 implies the Adams-Riemann-Roch formula for ℂ-schemes (see section 4). This proposition also clarifies to what extent, conversely, the Adams-Riemann-Roch formula implies the equality F_{∗}(θ^{p}(Ω)^{−1}) = 1.

Proposition 5.5. The difference F_{∗}(θ^{p}(Ω)^{−1})−1 is contained in Num_{0}(f) if and only if, for all x ∈ K_{0}(X), the Adams-Riemann-Roch formula

ψ^{p}(f_{∗}(x)) = f_{∗}(θ^{p}(Ω)^{−1} · ψ^{p}(x)) in K_{0}(Y)[p^{−1}]

holds.

Proof. For all x ∈ K_{0}(X), we have

ψ^{p}(f_{∗}(x)) = F_{Y}^{∗}(f_{∗}(x)) = (f_{Y})_{∗}(F_{Y}^{∗}(x)) in K_{0}(Y)

by Proposition 5.1. Furthermore, using the projection formula, we obtain:

f_{∗}(θ^{p}(Ω)^{−1} · ψ^{p}(x))
= (f_{Y})_{∗} (F_{∗} (θ^{p}(Ω)^{−1} · F^{∗}(F_{Y}^{∗}(x))))
= (f_{Y})_{∗} (F_{∗}(θ^{p}(Ω)^{−1}) · F_{Y}^{∗}(x)) in K_{0}(Y)[p^{−1}].

This immediately implies Proposition 5.5.

The following remark may be considered as an analogue of the equality Δ_{∗}(1) = 1 established in the proof of Theorem 4.9.

Remark 5.6. Let f : X → Y be an étale morphism. Then we have Ω = 0, and F : X → X_{Y} is an isomorphism (see Lemma 3.7 on p. 163 in [FK]). Hence, we have:

F_{∗}(θ^{p}(Ω)^{−1}) = F_{∗}(1) = 1 in K_{0}(X_{Y}).

As in Proposition 5.5, a version without denominators of the Adams-Riemann-Roch formula ψ^{p}f_{∗} = f_{∗}ψ^{p} can be deduced from this.

In the following example, we describe what the equality F_{∗}(θ^{p}(Ω)^{−1}) = 1 means in case of a smooth curve.

Example 5.7. Let f : X → Spec(𝔽_{p^{n}}) = Y be a smooth connected curve (for some n ≥ 0). Then we have

θ^{p}(Ω)^{−1} = \frac{2}{p} − \frac{1}{p^{2}} θ^{p}(Ω) in K_{0}(X)[p^{−1}]

since the square of the augmentation ideal ker(rank : K_{0}(X) → ℤ) vanishes by Corollary 3.10 on p. 125 in [FL]. For instance, we have in the case p = 2:

θ^{2}(Ω)^{−1} = \frac{1}{4}(3 − Ω).

Now, let X = Spec(R) be affine. Using the Cartier isomorphism (see the proof of Proposition 5.3), one easily shows that, in the case p = 2, Question 5.2 is equivalent to the following question: Is there an r ≥ 0 such that

⊕^{2^{r}} (F_{∗}(R) ⊕ F_{∗}(R)) ≅ ⊕^{2^{r}}(R ⊕ R ⊕ R ⊕ F_{Y}^{∗}(Ω))?

Furthermore, if X is an elliptic curve, this is equivalent to the following question: Is there an r ≥ 0 such that the direct sum of 2^{r} copies of F_{∗}(R) is R-free?

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Mathematisches Institut II der Universität Karlsruhe, D-76128 Karlsruhe, Germany; e-mail: Bernhard.Koeck@math.uni-karlsruhe.de.