TWO-PRIMARY ALGEBRAIC K -THEORY

OF CURVES OVER FINITE FIELDS

John Rognes

February 19th 1998

Abstract. We apply the mod 2^{∞} Bloch–Lichtenbaum spectral sequence to describe the two-completed algebraic K -theory of a curve X over a finite field k of odd characteristic. When the curve is smooth and affine or projective, the even K -groups are given in terms of invariants of the Frobenius action on the Iwasawa module of the curve.

Introduction

Let X be any curve over a finite field k of odd characteristic. In Proposition 1.8 of this note we compute the 2-adic algebraic K -theory of X in terms of its étale cohomology, up to an extension question.

Suppose further that X is connected, smooth and either projective or affine. In Propositions 2.7 and 2.8 we rewrite the étale cohomology of X in terms of the Frobenius action on groups of units and Picard groups related to X. In Theorems 3.5 and 3.6 we express the 2-adic algebraic K -theory of X in terms of the Frobenius action on the Iwasawa module of X.

These results are applications of Voevodsky’s proof of the Milnor conjecture, and the Bloch–Lichtenbaum spectral sequence, together with extensions to finite coefficients and positive characteristics from [RW] and [Wb]. Related results for varieties are expected in a paper [Ka] by B. Kahn, that compares algebraic K - theory to étale K -theory.

1. Algebraic curves

Let k = 𝔽_{q} be a finite field, let X be an algebraic curve defined over k, and let K = k(X) be the function field on X. So K has transcendence degree 1 over k. As our focus is on X rather than on k, we may and will assume that k is algebraically closed in K.

Throughout this paper, let ℓ = 2 be the even prime, and suppose that k has odd characteristic p = char(k) ≠ ℓ. There is a natural mod ℓ^{∞} Bloch–Lichtenbaum spectral sequence converging to the mod ℓ^{∞} algebraic K -theory of K:

(1.1)  E_{2}^{s,t} = H_{ét}^{s−t}(K; ℚ_{ℓ}/ℤ_{ℓ}(−t)) ⟹ K_{−s−t}(K; ℚ_{ℓ}/ℤ_{ℓ})

Here t ≤ s ≤ 0. See [RW, §1]. The identification of the E_{2} -term is obtained by functorially replacing K by a field L of characteristic zero, with naturally

Typeset by AMS-TEX

isomorphic ℓ-adic étale cohomology groups and ℓ-adic algebraic K -groups as K, and considering the mod ℓ^{∞} Bloch–Lichtenbaum spectral sequence for L. See [Wb].

In some more detail, this is done as follows. First pass to the perfect closure K^{p} of K = K_{0} , by adjoining the p-th roots of all elements of K_{i} to obtain K_{i+1} , and iterate countably often. Then form the Witt ring R = W (K^{p}). This is a complete discrete valuation ring of characteristic zero, with residue field K^{p}. Let L_{0} be the quotient field of R = R_{0} . Adjoin the ℓ-th roots of a uniformizing parameter π in R_{i} to obtain R_{i+1} , with quotient field L_{i+1} , and iterate countably often. Let R_{∞} be the union of the R_{i}, with quotient field L_{∞}. The Galois group of K^{p} over K is a pro-p group, and acts on R_{∞} and L_{∞}. Let L be the invariant field for the latter Galois action. Then L has the asserted properties.

The ℓ-adic cohomological dimension of k is 1, so the ℓ-adic cohomological dimension of K is at most 2. Hence H_{ét}^{n}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 for all n ≥ 3. Thus the spectral sequence above collapses at the E_{2} -term.

The edge map K_{2i}(K; ℚ_{ℓ}/ℤ_{ℓ}) ⟶ H_{ét}^{0}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) of the mod ℓ^{∞} Bloch– Lichtenbaum spectral sequence, and naturality with respect to the composite map Spec(K) → X → Spec(k), gives us the following commutative square:

(1.2)  \xymatrix{K_{2i}(k; ℚ_{ℓ}/ℤ_{ℓ}) \ar[r] \ar[d]^{edge}_{≅} & K_{2i}(X; ℚ_{ℓ}/ℤ_{ℓ}) \ar[r] & K_{2i}(K; ℚ_{ℓ}/ℤ_{ℓ}) \ar[d]^{edge} \\ H_{ét}^{0}(k; ℚ_{ℓ}/ℤ_{ℓ}(i)) \ar[r]^{≅} & H_{ét}^{0}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) \ar[r]^{≅} & H_{ét}^{0}(K; ℚ_{ℓ}/ℤ_{ℓ}(i))}

The left hand map is an isomorphism since k has cohomological dimension 1, and the lower maps are isomorphisms because k is algebraically closed in K. Hence the edge map from K_{2i}(K; ℚ_{ℓ}/ℤ_{ℓ}) is split surjective. This proves the following:

Proposition 1.3. Let ℓ = 2. For m ≥ 1 we have

K_{m}(K; ℚ_{ℓ}/ℤ_{ℓ}) ≅ \{ \begin{matrix}H_{ét}^{1}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) & for m = 2i − 1, \\ H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⊕ H_{ét}^{0}(K; ℚ_{ℓ}/ℤ_{ℓ}(i − 1)) & for m = 2i − 2. \end{matrix}

Definition 1.4. Let K̃_{2i}(K; ℚ_{ℓ}/ℤ_{ℓ}) ≅ H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i + 1)) be the kernel of the split surjection

K_{2i}(K; ℚ_{ℓ}/ℤ_{ℓ}) ⟶^{edge} H_{ét}^{0}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)),

and let K̃_{2i}(X; ℚ_{ℓ}/ℤ_{ℓ}) be the kernel of the composite split surjection

K_{2i}(X; ℚ_{ℓ}/ℤ_{ℓ}) ⟶ K_{2i}(K; ℚ_{ℓ}/ℤ_{ℓ}) ⟶^{edge} H_{ét}^{0}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ H_{ét}^{0}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) .

The localization sequence in étale cohomology for the curve X contains the following exact sequence:

(1.5) 0 → H_{ét}^{1}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶ H_{ét}^{1}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶^{∂^{e}} ⨁_{x∈X_{1}} H_{ét}^{0}(k(x); ℚ_{ℓ}/ℤ_{ℓ}(i − 1))

⟶ H_{ét}^{2}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶ H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i))

TWO-PRIMARY K -THEORY OF CURVES 3

Here X_{1} is the set of closed points x of X, and k(x) is the residue field at such a point x. The last map is surjective for i ≠ 1. We will map the exact sequence above to the corresponding exact localization sequence in algebraic K -theory for X:

(1.6) 0 → K_{2i−1}(X; ℚ_{ℓ}/ℤ_{ℓ}) ⟶ K_{2i−1}(K; ℚ_{ℓ}/ℤ_{ℓ}) ⟶^{∂} ⨁_{x∈X_{1}} K_{2i−2}(k(x); ℚ_{ℓ}/ℤ_{ℓ})

⟶ K_{2i−2}(X; ℚ_{ℓ}/ℤ_{ℓ}) ⟶ K_{2i−2}(K; ℚ_{ℓ}/ℤ_{ℓ}) → 0

Let us first show that we can choose isomorphisms ϕ_{x}: H_{ét}^{0}(k(x); ℚ_{ℓ}/ℤ_{ℓ}(i − 1)) ≅ K_{2i−2}(k(x); ℚ_{ℓ}/ℤ_{ℓ}) for all x ∈ X_{1} that are compatible with the isomorphism K_{2i−1}(K; ℚ_{ℓ}/ℤ_{ℓ}) ≅ H_{ét}^{1}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) from (1.3) under the boundary maps ∂ and ∂^{e}.

For each closed point x ∈ X_{1} we may form the Henselization K_{x}^{h} of K, which admits a natural map K → K_{x}^{h}. This is a complete discrete valuation ring with residue field k(x). There are corresponding localization sequences in étale cohomology and algebraic K -theory for K_{x}^{h}, in which the connecting maps

∂_{x}^{e}: H_{ét}^{1}(K_{x}^{h}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶^{≅} H_{ét}^{0}(k(x); ℚ_{ℓ}/ℤ_{ℓ}(i − 1))

and

∂_{x}: K_{2i−1}(K_{x}^{h}; ℚ_{ℓ}/ℤ_{ℓ}) ⟶^{≅} K_{2i−2}(k(x); ℚ_{ℓ}/ℤ_{ℓ})

are isomorphisms. This follows by rigidity for étale cohomology and algebraic K - theory. The following lemma is then a consequence of naturality of the localizations sequences.

Lemma 1.7. The boundary map ∂^{e} in the étale cohomology localization sequence factors as the composite

H_{ét}^{1}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶ ⨁_{x∈X_{1}} H_{ét}^{1}(K_{x}^{h}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶_{≅}^{⊕_{x}∂_{x}^{e}} ⨁_{x∈X_{1}} H_{ét}^{0}(k(x); ℚ_{ℓ}/ℤ_{ℓ}(i − 1)) .

Likewise the boundary map ∂ in the algebraic K -theory localization sequence factors as the composite

K_{2i−1}(K; ℚ_{ℓ}/ℤ_{ℓ}) ⟶ ⨁_{x∈X_{1}} K_{2i−1}(K_{x}^{h}; ℚ_{ℓ}/ℤ_{ℓ}) ⟶_{≅}^{⊕_{x}∂_{x}} ⨁_{x∈X_{1}} K_{2i−2}(k(x); ℚ_{ℓ}/ℤ_{ℓ}) .

The left hand maps are induced by the embeddings K → K_{x}^{h}.

For each x ∈ X_{1} the isomorphism

ϕ_{x}: H_{ét}^{0}(k(x); ℚ_{ℓ}/ℤ_{ℓ}(i − 1)) ⟶ K_{2i−2}(k(x); ℚ_{ℓ}/ℤ_{ℓ})

is now defined to correspond to the isomorphism

H_{ét}^{1}(K_{x}^{h}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ K_{2i−1}(K_{x}^{h}; ℚ_{ℓ}/ℤ_{ℓ})

of (1.3) via the isomorphisms ∂_{x}^{e} and ∂_{x}. Then by naturality of the Bloch– Lichtenbaum spectral sequence (1.1) with respect to the field embeddings K → K_{x}^{h} we obtain an isomorphism

⊕_{x}ϕ_{x}: ⨁_{x∈X_{1}} H_{ét}^{0}(k(x); ℚ_{ℓ}/ℤ_{ℓ}(i − 1)) ⟶^{≅} ⨁_{x∈X_{1}} K_{2i−2}(k(x); ℚ_{ℓ}/ℤ_{ℓ})

which is compatible with the boundary maps ∂^{e} and ∂.

Hence the induced map of kernels ker(∂^{e}) → ker(∂) is an isomorphism

H_{ét}^{1}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ K_{2i−1}(X; ℚ_{ℓ}/ℤ_{ℓ}) .

Likewise the induced map of cokernels is an isomorphism cok(∂^{e}) ≅ cok(∂). The group K̃_{2i−2}(X; ℚ_{ℓ}/ℤ_{ℓ}) is an extension of K̃_{2i−2}(K; ℚ_{ℓ}/ℤ_{ℓ}) ≅ H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) by cok(∂) ≅ cok(∂^{e}), due to (1.2), (1.4) and (1.6). By (1.5) also H_{ét}^{2}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) is an extension of the same groups when i ≠ 1. Without asserting that the two extensions of H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) agree, we have proven:

Proposition 1.8. Let ℓ = 2 and let X be a curve over a finite field k of characteristic p ≠ ℓ. For m ≥ 1 there are isomorphisms

K_{m}(X; ℚ_{ℓ}/ℤ_{ℓ}) ≅ \{ \begin{matrix}H_{ét}^{1}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) & for m = 2i − 1, \\ H_{ét}^{2}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⊕ H_{ét}^{0}(X; ℚ_{ℓ}/ℤ_{ℓ}(i − 1)) & for m = 2i − 2, \end{matrix}

up to extensions within H_{ét}^{2}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)).

2. Smooth curves

Now suppose the curve X/k is connected, smooth and projective, of genus g. We adopt some notations from [DM]: Let S be the finite set of points at infinity of X. Let X′ = X \S be the corresponding smooth affine curve, and write X′ = Spec(O). Let k̄ be an algebraic closure of k, let X̄ = X ×_{Spec(k)} Spec(k̄), and let X̄′ = X̄ \S̄ where S̄ is the set of points in X̄ above S. Also write X̄′ = Spec(Ō).

We refer to the summary in [DM, §2] and [Mi, Ch.III] for the following facts about étale cohomology of curves.

The Picard group Pic(X) = H_{ét}^{1}(X; 𝔾_{m}) admits a split surjective degree map to ℤ, hence sits in the split extension

(2.1)  0 → Pic^{0}(X) ⟶ Pic(X) ⟶^{deg} ℤ → 0

with Pic^{0}(X) finite. Passing to the algebraic closure, Pic^{0}(X̄) is ℓ-divisible with ℓ-torsion subgroup (ℚ_{ℓ}/ℤ_{ℓ})^{2g}. In the affine case, Pic(X′) is finite and Pic(X̄′) is ℓ-divisible with ℓ-torsion subgroup (ℚ_{ℓ}/ℤ_{ℓ})^{2g}. We write A{ℓ} for the ℓ-torsion subgroup of an Abelian group A. As in [DM, 3.7] we obtain

(2.2)  H_{ét}^{n}(X̄; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℚ_{ℓ}/ℤ_{ℓ}(i) & for n = 0, \\ Pic^{0}(X̄)\{ℓ\}(i − 1) & for n = 1, \\ ℚ_{ℓ}/ℤ_{ℓ}(i − 1) & for n = 2, \\ 0 & for n ≥ 3, \end{matrix}

in the projective case, and

(2.3)  H_{ét}^{n}(X̄′; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℚ_{ℓ}/ℤ_{ℓ}(i) & for n = 0, \\ E(i − 1) & for n = 1, \\ 0 & for n ≥ 2, \end{matrix}

in the affine case. Here E is an extension

(2.4)  0 → Ō^{×} ⊗ ℚ_{ℓ}/ℤ_{ℓ} ⟶ E ⟶ Pic(X̄′){ℓ} → 0

which is a finite sum of copies of ℚ_{ℓ}/ℤ_{ℓ}. (We will retain the notation E for this extension.)

In particular each cohomology group H_{ét}^{n}(X̄; ℤ/ℓ^{ν}(i)) is finite, so H_{ét}^{n}(X̄; ℤ_{ℓ}(i)) is a finitely generated ℤ_{ℓ} -module. The same assertions hold for each cohomology group of X̄′, as well as for X and X′ by Galois descent.

Since each group H_{ét}^{n}(X̄; ℚ_{ℓ}/ℤ_{ℓ}(i)) is a direct sum of copies of ℚ_{ℓ}/ℤ_{ℓ}, the sequence

(2.5)  0 → H_{ét}^{n}(X̄; ℤ_{ℓ}(i)) ⟶ H_{ét}^{n}(X̄; ℚ_{ℓ}(i)) ⟶ H_{ét}^{n}(X̄; ℚ_{ℓ}/ℤ_{ℓ}(i)) → 0

is short exact for all n and i. This also holds for the cohomology of X̄′.

The Galois group Gal(X̄/X) ≅ Gal(k̄/k) is topologically generated by the (geometric) Frobenius automorphism ϕ = ϕ_{q}: z ↦ z^{q}. Let ϕ^{∗} be the (algebraic) Frobenius map acting on H_{ét}^{n}(X̄; ℚ_{ℓ}(0)) induced by ϕ^{−1} . The action of ϕ^{∗} on H_{ét}^{n}(X̄; ℚ_{ℓ}(i)) is scaled by a factor q^{−i} in comparison to that on H_{ét}^{n}(X̄; ℚ_{ℓ}(0)).

By a theorem of A. Weil [Wl], the eigenvalues of ϕ^{∗} acting on H_{ét}^{n}(X̄; ℚ_{ℓ}(0)) have absolute value q^{n/2} , since X is smooth and projective. Thus 1 is only an eigenvalue of ϕ^{∗} acting on H_{ét}^{n}(X̄; ℚ_{ℓ}(i)) when i = n/2, and therefore ϕ^{∗} −1 maps H_{ét}^{n}(X̄; ℚ_{ℓ}(i)) isomorphically to itself as long as i ≠ n/2. Since X is a curve the only exceptions are (n, i) = (0, 0) and (2, 1), which only affect K_{0}(X).

By (2.5) it follows that ϕ^{∗} − 1 maps H_{ét}^{n}(X̄; ℚ_{ℓ}/ℤ_{ℓ}(i)) surjectively to itself, whenever i ≠ n/2. As in [DM, 3.12] we can apply descent to go from X̄ to X. We find that when i ≠ n/2, the coinvariants of ϕ^{∗} acting on H_{ét}^{n}(X̄; ℚ_{ℓ}/ℤ_{ℓ}(i)) are zero, while the invariants of ϕ^{∗} acting on H_{ét}^{n}(X̄; ℚ_{ℓ}/ℤ_{ℓ}(i)) are finite, and are identified with H_{ét}^{n}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)).

Definition 2.6. For an Abelian group A with a self-map ϕ^{∗}: A → A let A^{ϕ} = ker(ϕ^{∗} − 1) denote the subgroup of invariants under the action by ϕ^{∗}, and let A_{ϕ} = cok(ϕ^{∗} − 1) denote the quotient group of coinvariants under the action.

Let w_{i}(k) denote the maximal power ℓ^{ν} of ℓ such that the Galois group of k with all ℓ^{ν} -th roots of unity adjoined, over k, has exponent dividing i. Then H_{ét}^{0}(k; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ ℤ/w_{i}(k). (We allow w_{i}(k) = ℓ^{∞}, when ℤ/ℓ^{∞} ≅ ℚ_{ℓ}/ℤ_{ℓ}.)

Proposition 2.7. Let X/k be a connected smooth projective curve, as above. For (n, i) ≠ (0, 0) or (2, 1) we have

H_{ét}^{n}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℤ/w_{i}(k) & for n = 0, \\ Pic^{0}(X̄)\{ℓ\}(i − 1)^{ϕ} & for n = 1, \\ ℤ/w_{i−1}(k) & for n = 2, \\ 0 & for n ≥ 3. \end{matrix}

These groups are finite.

We now turn to the affine case.

Proposition 2.8. Let X′/k be a connected smooth affine curve, as above. For (n, i) ≠ (0, 0) or (2, 1) we have H_{ét}^{0}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ ℤ/w_{i}(k), H_{ét}^{n}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 for n ≥ 2, and there is an exact sequence

0 → (Ō^{×} ⊗ ℚ_{ℓ}/ℤ_{ℓ})(i − 1)^{ϕ} ⟶ H_{ét}^{1}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i))

⟶ Pic(X̄′){ℓ}(i − 1)^{ϕ} ⟶ (Ō^{×} ⊗ ℚ_{ℓ}/ℤ_{ℓ})(i − 1)_{ϕ} → 0 .

These cohomology groups are finite, except for (n, i) = (1, 1).

Proof. We know that H_{ét}^{3}(X′; ℤ_{ℓ}(i)) vanishes since each H_{ét}^{2}(X′; ℤ/ℓ^{ν}(i)) is finite. Hence there is a surjection H_{ét}^{2}(X′; ℚ_{ℓ}(i)) → H_{ét}^{2}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) and we deduce that H_{ét}^{2}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) must be divisible. But it is also finite for i ≠ 1, due to the surjection

H_{ét}^{2}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶ H_{ét}^{2}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i))

from the localization sequence for X′ → X. Hence H_{ét}^{2}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 for i ≠ 1.

By descent, the invariants and coinvariants of the action of ϕ^{∗} on

H_{ét}^{1}(X̄′; ℚ_{ℓ}/ℤ_{ℓ}(i)) = E(i − 1)

are identified with H_{ét}^{1}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) and H_{ét}^{2}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)), respectively. Here E is the extension from (2.4). We have just seen that the coinvariants vanish for i ≠ 1, hence the invariants are finite in these cases. The kernel-cokernel sequence for ϕ^{∗} − 1 acting on the extension E(i − 1) now simplifies to the asserted exact sequence. □

3. Iwasawa modules

Let k_{∞} ⊂ k̄ be obtained by adjoining all ℓ^{ν} -th roots of unity in k̄ to k, for all ν ≥ 1. Let X_{∞} = X ×_{Spec(k)} Spec(k_{∞}) and similarly for X′_{∞} = Spec(O_{∞}).

Let X̂ be the maximal unramified pro–Galois pro-ℓ Abelian cover of X_{∞}. The Iwasawa module M_{X} is the Galois group of X̂ over X_{∞}. Hence the Pontryagin dual M_{X}^{#} = Hom(M_{X}, ℚ/ℤ) is identified with H_{ét}^{1}(X_{∞}; ℚ_{ℓ}/ℤ_{ℓ}(0)). By descent over Gal(X̄/X_{∞}) ≅ Gal(k̄/k_{∞}) we have

(3.1)  H_{ét}^{n}(X_{∞}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℚ_{ℓ}/ℤ_{ℓ}(i) & for n = 0, \\ M_{X}^{#}(i) & for n = 1, \\ ℚ_{ℓ}/ℤ_{ℓ}(i − 1) & for n = 2, \\ 0 & for n ≥ 3. \end{matrix}

Here M_{X}^{#} is a retract of Pic^{0}(X̄){ℓ}. Thus it has the form (ℚ_{ℓ}/ℤ_{ℓ})^{r} for some 0 ≤ r ≤ 2g. Dually M_{X} is a free ℤ_{ℓ} -module of rank r, and the action of ϕ^{∗} on M_{X}^{#}(i) has zero coinvariants and finite invariants. Thus

(3.2)  H_{ét}^{n}(X; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℤ/w_{i}(k) & for n = 0, \\ M_{X}^{#}(i)^{ϕ} & for n = 1, \\ ℤ/w_{i−1}(k) & for n = 2, \\ 0 & for n ≥ 3. \end{matrix}

Likewise let X̂′ be the maximal unramified pro–Galois pro-ℓ Abelian cover of X′_{∞}. The Iwasawa module M_{X′} is the Galois group of X̂′ over X′_{∞}. Hence the Pontryagin dual M_{X′}^{#} = Hom(M_{X′}, ℚ/ℤ) is identified with H_{ét}^{1}(X′_{∞}; ℚ_{ℓ}/ℤ_{ℓ}(0)). By descent over Gal(X̄′/X′_{∞}) ≅ Gal(k̄/k_{∞}) we have

(3.3)  H_{ét}^{n}(X′_{∞}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℚ_{ℓ}/ℤ_{ℓ}(i) & for n = 0, \\ M_{X′}^{#}(i) & for n = 1, \\ 0 & for n ≥ 2. \end{matrix}

Here M_{X′}^{#} is a retract of the extension E of (2.4), hence is of the form (ℚ_{ℓ}/ℤ_{ℓ})^{r′} for some finite r′, and dually M_{X} is a free ℤ_{ℓ} -module of rank r′. As noted in the proof of (2.8) the action of ϕ^{∗} on E(i − 1) has zero coinvariants, so the action of ϕ^{∗} on M_{X′}^{#}(i) has zero coinvariants and finite invariants. Thus

(3.4)  H_{ét}^{n}(X′; ℚ_{ℓ}/ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℤ/w_{i}(k) & for n = 0, \\ M_{X′}^{#}(i)^{ϕ} & for n = 1, \\ 0 & for n ≥ 2. \end{matrix}

Recall that K_{2i−1}(k) ≅ ℤ/(q^{i}−1) when k has q elements [Qu]. Let the notation A ≅_{(ℓ)} B mean that A and B are isomorphic modulo odd finite groups.

Theorem 3.5. Let X be a connected smooth projective curve over a finite field k of characteristic ≠ ℓ = 2, with Iwasawa module M_{X}. When m ≥ 1 we have

K_{m}(X) ≅_{(ℓ)} \{ \begin{matrix}K_{m}(k) ⊕ K_{m}(k) & for m = 2i − 1 odd, \\ H_{ét}^{2}(X; ℤ_{ℓ}(i)) & for m = 2i − 2 even. \end{matrix}

There are isomorphisms

H_{ét}^{2}(X; ℤ_{ℓ}(i)) ≅ Pic^{0}(X̄){ℓ}(i − 1)^{ϕ} ≅ M_{X}^{#}(i)^{ϕ} .

Proof. From 2.8 and the localization sequence we deduce that H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 for i ≠ 1. Hence there is no ambiguity about the extension in 1.8 for X smooth. The theorem follows from (1.8), (2.7), (3.2), the universal coefficient theorem, and finiteness of the groups K_{m}(X) [Gr]. □

Theorem 3.6. Let X′ = Spec(O) be a connected smooth affine curve over a finite field k of characteristic ≠ ℓ = 2. When m ≥ 2 we have

K_{m}(X′) ≅_{(ℓ)} \{ \begin{matrix}K_{m}(k) & for m = 2i − 1 odd, \\ H_{ét}^{2}(X′; ℤ_{ℓ}(i)) & for m = 2i − 2 even. \end{matrix}

There is an exact sequence

0 → (Ō^{×} ⊗ ℚ_{ℓ}/ℤ_{ℓ})(i − 1)^{ϕ} ⟶ H_{ét}^{2}(X′; ℤ_{ℓ}(i))

⟶ Pic(X̄′){ℓ}(i − 1)^{ϕ} ⟶ (Ō^{×} ⊗ ℚ_{ℓ}/ℤ_{ℓ})(i − 1)_{ϕ} → 0

and an isomorphism H_{ét}^{2}(X′; ℤ_{ℓ}(i)) ≅ M_{X′}^{#}(i)^{ϕ}.

Proof. Again H_{ét}^{2}(K; ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 for i ≠ 1, so we can use (1.8), (2.8) and (3.4), as in the previous proof. We omit the case m = 1, and in fact K_{1}(X′) ≅ O^{×} is typically not finite. □

References

[DM] W.G. Dwyer and S.A. Mitchell, On the K -theory spectrum of a smooth curve over a finite field, Topology 36 (1997), 899–929.

[Gr] D.R. Grayson, Finite generation of K -groups of a curve over a finite field. (After Daniel Quillen), Algebraic K-theory, Proc. Conf., Oberwolfach 1980, Part I, Lect. Notes Math., vol. 966, Springer, 1982, pp. 69–90.

[Ka] B. Kahn, K -theory of semi-local rings with finite coefficients and étale cohomology, UIUC K-theory preprint server, no. 177 (1997).

[Mi] J.S. Milne, Etale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, 1980.

[Qu] D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math., II. Ser. 96 (1972), 552–586.

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Dept. of Math., University of Oslo, Norway

   E-mail address: rognes@math.uio.no