CALCULATIONS OF 2-PRIMARY ALGEBRAIC K-THEORY OF SOME GROUP RINGS

PAUL ARNE ØSTVÆR

Abstract. In this paper we will use an excision result in algebraic K-theory supplied with new 2-primary calculations of algebraic K-theory of number rings to obtain results on the 2-primary algebraic K-theory of some group rings.

1. Introduction and summary of results

This section will serve as a motivation for our work and a summary of the results in this paper. Recently J. Rognes and C. Weibel have in [12] obtained strong 2-primary information on the algebraic K-theory of number rings. Inspired by their results we will in this paper focus on the 2-primary algebraic K-theory of a large class of group rings. The link between number rings and group rings is provided by the Cartesian square of rings

\xymatrix{ℤ[C_{p}] \ar[d] \ar[r] & ℤ[ζ_{p}] \ar[d] \\ ℤ \ar[r] & 𝔽_{p}}

where C_{p} denotes the cyclic group of order p, ζ_{p} a primitive p-th root of unity, and 𝔽_{p} the field of p elements. A result of Charney in [4] tells us that the K-theory diagram

\xymatrix{K(ℤ[C_{p}]; ℤ[\frac{1}{p}]) \ar[d] \ar[r] & K(ℤ[ζ_{p}]; ℤ[\frac{1}{p}]) \ar[d] \\ K(ℤ; ℤ[\frac{1}{p}]) \ar[r] & K(𝔽_{p}; ℤ[\frac{1}{p}])}

is homotopy Cartesian. Hence the iterated homotopy fibre, say X, in the diagram has trivial ℤ[\frac{1}{p}]-homotopy. So for p odd we get that π_{n}(X)_{2} = 0 for all n. Here _{2} indicates that the group is completed at 2. This tells us that the K-theory diagram

\xymatrix{K(ℤ[C_{p}])_{2} \ar[d] \ar[r] & K(ℤ[ζ_{p}])_{2} \ar[d] \\ K(ℤ)_{2} \ar[r] & K(𝔽_{p})_{2}}

1991 Mathematics Subject Classification. 19D50.

Key words and phrases. K-theory of group rings.

I thank John Rognes to whom I am very much indebted.

is homotopy Cartesian, i.e., the vertical maps have the same homotopy fibres up to homotopy. All the homotopy groups of the spaces in this square are finitely generated abelian groups. Hence there is no difference in completing at 2 topologically or algebraically, and the notation above should not lead to any misunderstandings. Proofs for the finitely generatedness for the group ring, the finite field and the number ring can be found in [7], [9] and [10], respectively.

We introduce the notation

K(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2}

for the vertical homotopy fibre in the 2-completed diagram above.

We call a prime number p “simple” if the 2-rank of the Picard group of the 2-integers in ℚ(ζ_{p}) equals zero and 2 is a primitive root modulo p. The last condition excludes the cases p ≡ 1, 7 mod 8. For an abelian group A we let A{2} denote the 2-torsion subgroup of A. Further, let ord_{2}(n) denote the exact 2-power of the integer n.

Theorem 1. Let p be a “simple” prime number congruent to 3 or 5 modulo 8. The induced map on homotopy

π_{2n−1} : K_{2n−1}(ℤ[ζ_{p}])_{2} ⟶ K_{2n−1}(𝔽_{p})_{2}

is surjective. In fact π_{2n−1} is an isomorphism on the torsion part unless p ≡ 5 mod 8 and n is odd. In that case π_{2n−1} is injective on the torsion part. Hence the connecting map

δ_{n} : K_{n}(𝔽_{p})_{2} ⟶ K_{n−1}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2}

is trivial, and we have the exact sequence

0 ⟶ K_{n}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} ⟶ K_{n}(ℤ[ζ_{p}])_{2} ⟶ K_{n}(𝔽_{p})_{2} ⟶ 0.

The sequence is split unless p ≡ 5 mod 8 and n ≡ 1, 5 mod 8.

Remark 1. Calculations of K_{n}(ℤ[ζ_{p}])_{2} in [13] gives us a calculation of K_{n}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} that we will expose in the next result. There are oodles of examples of primes as in the theorem. One can for example take p a Sophie Germain prime, see [13].

Theorem 2. We have the following calculation of the 2-primary algebraic K-theory for the group ring ℤ[C_{p}] where p ≡ 3, 5 mod 8 is a “simple” prime and n > 1.

\begin{matrix}n mod 8 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ K_{n}(ℤ[C_{p}])\{2\} & ℤ/2 & ℤ/2 & ℤ/16 & 0 & 0 & 0 & ℤ/2^{1+ord_{2}(n+1)} & 0 \end{matrix}

More precisely the map from K_{n}(ℤ)_{2} to K_{n}(ℤ[C_{p}])_{2} is an isomorphism on the 2-torsion part.

We end the introduction by giving an outline of the proof. For Theorem 1 we need a description of the involved groups. The 2-completed copies of K_{n}(𝔽_{p}) is easily found from Quillen’s calculation of algebraic K-theory of finite fields. Results in [15] and [13] take care of K_{n}(ℤ)_{2}, and K_{n}(ℤ[ζ_{p}])_{2} for p a “simple” prime. An étale cohomology computation proves the stated property of the map. Theorem 2 is a direct consequence from the calculation of K(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2}. In Section 2 we will verify the theorems for p ≡ 3 mod 8, while the theorems for p ≡ 5 mod 8 will be taken care of in Section 3. The reason is that the argument will differ in some parts. Special attention is needed to establish the surjectivity of π_{2n−1} when p ≡ 5 mod 8 and n is odd, otherwise the proofs will be the same.

2. Proof of Theorem 1 and Theorem 2 for p ≡ 3 mod 8

In this section we work with the assumption that p ≡ 3 mod 8. We start with some recollections. The choice of p will affect the 2-completed copies, K_{n}(𝔽_{p})_{2}, of K_{n}(𝔽_{p}). However, these groups are easy to control. First we recall Quillen’s famous calculation, see [9].

Theorem 3. (Quillen ) The algebraic K-groups of 𝔽_{p} are given by

K_{n}(𝔽_{p}) = \begin{cases}ℤ & for n = 0, \\ ℤ/(p^{i} − 1) & for n = 2i − 1, \\ 0 & otherwise. \end{cases}

The condition p ≡ 3 mod 8 implies that

ord_{2}(p^{i} − 1) = \begin{cases}1 & for i odd, \\ 2 + ord_{2}(i) & for i even. \end{cases}

Corollary 1. For p ≡ 3 mod 8 the 2-completed algebraic K-groups K_{n}(𝔽_{p})_{2} are given by the following table.

K_{n}(𝔽_{p})_{2} = \begin{cases}ℤ_{2} & for n = 0, \\ ℤ/2 & for i odd, n = 2i − 1, \\ ℤ/2^{2+ord_{2}(i)} & for i even, n = 2i − 1, \\ 0 & otherwise. \end{cases}

Before we start to work with the maps, we recall the calculation of the 2-primary algebraic K-theory of ℤ[ζ_{p}] and ℤ from [13] and [15], respectively.

Theorem 4. We have the following calculation of the 2-primary algebraic K-theory of ℤ and ℤ[ζ_{p}] where p is a “simple” prime and n > 1.

\begin{matrix}n mod 8 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ K_{n}(ℤ)\{2\} & ℤ/2 & ℤ/2 & ℤ/16 & 0 & 0 & 0 & ℤ/2^{1+ord_{2}(n+1)} & 0 \\ K_{n}(ℤ[ζ_{p}])\{2\} & ℤ/2 & 0 & ℤ/8 & 0 & ℤ/2 & 0 & ℤ/2^{1+ord_{2}(n+1)} & 0 \end{matrix}

We can now begin with the proof of Theorem 1.

Proof. The ring of integers in ℚ_{p}(ζ_{p}) equals ℤ_{p}(ζ_{p}), the primitive element is (1 − ζ_{p}) and the residue field is 𝔽_{p}. Let l denote a prime. We let W (i) denote the colimit over the coefficient injections ℤ/l^{n}(i) ⟶ ℤ/l^{n+1}(i) where the integer i indicates the ith Tate twist. For us l = 2.

Consider the diagram:

[figure: K_{2i}(𝔽_{p}; ℚ_{2}/ℤ_{2}); K_{2i}(ℤ[ζ_{p}, \frac{1}{2}]; ℚ_{2}/ℤ_{2}); δ; ≅; π_{2i}; K_{2i}(ℤ_{p}[ζ_{p}]; ℚ_{2}/ℤ_{2}); δ; ≅; ≅; H_{ét}^{0}(ℤ[ζ_{p}, \frac{1}{2}]; W (i)); H_{ét}^{0}(ℤ_{p}[ζ_{p}]; W (i))]

The idea now is to establish an isomorphism in the latter part of the diagram. The indicated isomorphism between K_{2i}(𝔽_{p}; ℚ_{2}/ℤ_{2}) and K_{2i}(ℤ_{p}[ζ_{p}]; ℚ_{2}/ℤ_{2}) in the diagram is known as the rigidity theorem of Gabber, see [5]. Here K_{2i}(𝔽_{p}; ℚ_{2}/ℤ_{2}) is calculated in Corollary 1. The map δ comes from the edge map in the Bloch-Lichtenbaum spectral sequence, and in our case δ is an isomorphism. What we mean is the following. The edge map δ_{BL} in the Bloch-Lichtenbaum spectral sequence induces the diagram

[figure: K_{2i}(ℚ(ζ_{p}); ℚ_{2}/ℤ_{2}); δ_{BL}; ≅; H_{ét}^{0}(ℚ(ζ_{p}); W (i)); K_{2i}(ℤ[ζ_{p}]; ℚ_{2}/ℤ_{2}); δ; ≅; ≅; H_{ét}^{0}(ℤ[ζ_{p}, \frac{1}{2}]; W (i)); ≅]

and similarly in the local case. A reference for the fact that δ_{BL} is an isomorphism is Theorem 3.7 and Theorem 5.3 in [12]. Commutativity of the diagram follows from the naturality of the Bloch-Lichtenbaum spectral sequence. The calculation of the étale cohomology groups is given in [12]. Let F be a field. Define the integer w_{i}(F) by

w_{i}(F) = max {2^{ν} | Gal(F (ζ_{2^{ν}})/F) has exponent dividing i}

for each integer i. If no maximum can be achieved, we put w_{i}(F) = 2^{∞}. Proposition 2.9 in [12] tells us that H_{ét}^{0}(F; W (i)) ≅ ℤ/w_{i}(F) for any field F. Again from [12] a special case of Proposition 2.15 tells us that H_{ét}^{0}(ℚ(ζ_{p}); W (i)) ≅ H_{ét}^{0}(ℤ[ζ_{p}, \frac{1}{2}]; W (i)). Hence the two étale cohomology groups in the latter part of the diagram are isomorphic. This follows from Corollary 1, the isomorphisms on the right side of the diagram and the calculation

w_{i}(ℚ(ζ_{p})) = \begin{cases}2 & for i odd, \\ 2^{2+ord_{2}(i)} & for i even. \end{cases}

It is easy to verify this calculation, see also [13]. This implies in turn that the map between them is an isomorphism since the map is compatible with the inclusion of the two groups into H_{ét}^{0}(ℂ; W (i)) ≅ ℤ/2^{∞}.

We deduce that the map from K_{2i}(ℤ[ζ_{p}, \frac{1}{2}]; ℚ_{2}/ℤ_{2})) to K_{2i}(ℤ_{p}[ζ_{p}]; ℚ_{2}/ℤ_{2}) in the diagram is an isomorphism. Hence π_{2i} is an isomorphism as well. From the universal coefficient theorem in algebraic K-theory with ℚ_{2}/ℤ_{2} ≅ ℤ/2^{∞}-coefficients we have the commutative diagram:

\xymatrix{0 \ar[r] & K_{2i}(ℤ[ζ_{p}, \frac{1}{2}])/2^{∞} \ar[r] \ar[d] & K_{2i}(ℤ[ζ_{p}, \frac{1}{2}]; ℚ_{2}/ℤ_{2}) \ar[r] \ar[d] & K_{2i−1}(ℤ[ζ_{p}, \frac{1}{2}])\{2\} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K_{2i}(𝔽_{p})/2^{∞} \ar[r] & K_{2i}(𝔽_{p}; ℚ_{2}/ℤ_{2}) \ar[r] & K_{2i−1}(𝔽_{p})\{2\} \ar[r] & 0}

The group K_{2i}(𝔽_{p}) is trivial from Theorem 3, while K_{2i}(ℤ[ζ_{p}, \frac{1}{2}])/2^{∞} is trivial from Theorem 4. Now it follows immediately that the map between K_{2i−1}(ℤ[ζ_{p}, \frac{1}{2}])\{2\} and K_{2i−1}(𝔽_{p})\{2\} is an isomorphism, and we are done.

Now for Theorem 2.

Proof. To proceed with the proof of Theorem 2 we should have in mind the rational rank of the abelian groups we are working with. The rank of the involved algebraic K-groups are found from the following theorems, found in [2] and [6], respectively.

rank K_{n}(O_{F}) = \begin{cases}1 & for n = 0, \\ r_{1} + r_{2} − 1 & for n = 1, \\ r_{1} + r_{2} & for n ≡ 1 mod 4, n ≠ 1, \\ r_{2} & for n ≡ 3 mod 4, \\ 0 & for n > 0 even. \end{cases}

For n > 1

rank K_{n}(ℤ[G]) = \begin{cases}r & for n ≡ 1 mod 4, \\ c & for n ≡ 3 mod 4, \\ 0 & otherwise. \end{cases}

Here O_{F} denotes the number ring of the number field F, and G a finite group. The number r denotes the number of irreducible real representations, c of them are of complex type. For F = ℚ(ζ_{p}) with p odd there are no real embeddings, r_{1} = 0, while we have r_{2} = (p − 1)/2 pairs of complex embeddings. For G cyclic of order p we have that r = (p + 1)/2, and c = (p − 1)/2 of them are of complex type. One finds c from neglecting the trivial representation, and the rest of them come in pairs corresponding to complex conjugation.

Next the homotopy fibre K(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} is easily found since the map

π_{2n−1} : K_{2n−1}(ℤ[ζ_{p}])_{2} ⟶ K_{2n−1}(𝔽_{p})_{2}

is an isomorphism on the 2-torsion part from Theorem 1.

Corollary 2. Let n > 1. The homotopy fibre K(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} has no 2-torsion. Its rank equals (p − 1)/2 in odd degrees, and 0 in even degrees.

Proof. Combine the exact sequence

0 ⟶ K(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} ⟶ K_{n}(ℤ[ζ_{p}])_{2} ⟶ K_{n}(𝔽_{p})_{2} ⟶ 0

with Theorem 4 and the remarks above.

The map ℤ[C_{p}] ⟶ ℤ has a canonical splitting, and we get the description

K_{n}(ℤ[C_{p}])_{2} ≅ K_{n}(ℤ)_{2} ⊕ K_{n}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2}

of the algebraic K-theory of ℤ[C_{p}]. We know the two direct summands from Theorem 4 and Corollary 2, and we end up with the claimed table.

3. Proof of Theorem 1 and Theorem 2 for p ≡ 5 mod 8

In this section we work with a “simple” prime p congruent 5 modulo 8. We begin at once with the proof of Theorem 1.

Proof. Theorem 4 calculated K_{n}(ℤ[ζ_{p}, \frac{1}{2}])_{2}. Further an easy application of Theorem 3 gives us the calculation

K_{n}(𝔽_{p})_{2} = \begin{cases}ℤ_{2} & for n = 0, \\ ℤ/4 & for i odd, n = 2i − 1, \\ ℤ/2^{2+ord_{2}(i)} & for i even, n = 2i − 1, \\ 0 & otherwise, \end{cases}

for every prime number p congruent 5 modulo 8.

A copy of the proof of Theorem 1 for p ≡ 3 mod 8 combined with tables for K_{n}(ℤ[ζ_{p}, \frac{1}{2}])_{2} and K_{n}(𝔽_{p})_{2} establish Theorem 1 except the surjectivity of the map

K_{n}(ℤ[ζ_{p}])_{2} ⟶ K_{n}(𝔽_{p})_{2}

for n ≡ 1, 5 mod 8.

For K_{1} this is a classical result of Rim, see pp. 29-32 in [8]. We will use the K_{1} result and then multiply up with the Adams A^{n} element in order to prove that the same holds for all degrees congruent 1 modulo 8. We have the commutative diagram:

[figure: K_{1}(ℤ[ζ_{p}, \frac{1}{2}]; ℤ/2); A^{n}; K_{1}(𝔽_{p}; ℤ/2) = ℤ/2; ≅; A^{n}; K_{8n+1}(ℤ[ζ_{p}, \frac{1}{2}]; ℤ/2); K_{8n+1}(𝔽_{p}; ℤ/2) = ℤ/2; K_{8n+1}(ℤ[ζ_{p}, \frac{1}{2}])/2; ≅; K_{8n+1}(𝔽_{p})/2; ≅]

The element A^{n} constructed in [1] acts on mod 2 homotopy shifting the degree with eight. A^{n} acts isomorphically on the mod 2 homotopy of BU from Bott periodicity. Further Quillen in [9] proved that

[figure: K(𝔽_{p}); ℤ × BU; ψ^{p}−1; BU]

is a fibration. The 5-lemma implies that A^{n} acts isomorphically on the mod 2 homotopy of K(𝔽_{p}) as well. The isomorphisms in the lower square come from the calculations of K_{8n}(ℤ[ζ_{p}, \frac{1}{2}]) and K_{8n}(𝔽_{p}). None of these groups have any 2-torsion, so an easy application of the universal coefficient theorem in algebraic K-theory gives us the isomorphisms. The diagram shows that

K_{8n+1}(ℤ[ζ_{p}])_{2} ⟶ K_{8n+1}(𝔽_{p})_{2}

is surjective.

For the same map, but now in degrees 5 modulo 8 we use the diagram

\xymatrix{ & K_{n}(ℤ)_{2} \ar[dl] \ar[dr] \\ K_{n}(ℤ[ζ_{p}])_{2} \ar[rr] &  & K_{n}(𝔽_{p})_{2}}

induced by the triangle

\xymatrix{ & ℤ \ar[dl] \ar[dr] \\ ℤ[ζ_{p}] \ar[rr] &  & 𝔽_{p}}

to conclude that it will be sufficient to prove that the map

K_{8n+5}(ℤ)_{2} ⟶ K_{8n+5}(𝔽_{p})_{2}

is surjective. From [14] p. 45 the image of K_{3}(ℤ) in K_{3}(𝔽_{p}) is of order 24 for p ≥ 5. So we know that

K_{3}(ℤ)_{2} ⟶ K_{3}(𝔽_{p})_{2}

is surjective. Consider the commutative diagram:

[figure: K_{3}(ℤ)_{2}; ∘\overline{η}_{2}; K_{3}(𝔽_{p})_{2}; ∘\overline{η}_{2}; K_{5}(ℤ; ℤ/2); K_{5}(𝔽_{p}; ℤ/2)]

Here ∘\overline{η}_{2} indicates pre-composition with the map \overline{η}_{2}. This map is defined by the diagram

[figure: S^{4}; 2; 2Ση; S^{4}; Ση; i; S^{4}/2; _{\overline{η}_{2}}; S^{3}]

where η denotes the Hopf map and S^{4}/2 = S^{4} ∪_{2} D^{5} the mod 2 Moore space. Recall that 2Ση is null homotopic making it possible to define \overline{η}_{2}.

Recall the fibre sequence

[figure: ΩBU; δ; K(𝔽_{3}); i; ℤ × BU; Ψ^{3}−1; BU]

from [9]. The map δ combined with \overline{η}_{2} gives the commutative diagram:

[figure: π_{4}(BU); ∘\overline{η}_{2}; δ; K_{3}(𝔽_{3}); ∘\overline{η}_{2}; π_{6}(BU; ℤ/2); δ; K_{5}(𝔽_{3}; ℤ/2)]

The horizontal maps are surjective according to [9]. Further the map on the left hand side is surjective by Bott periodicity and the fact that the complex e-invariant of η equals 1/2, see [11]. We conclude that the map from K_{3}(𝔽_{p}) to K_{5}(𝔽_{p}; ℤ/2) is surjective. Hence the map from K_{5}(ℤ; ℤ/2) to K_{5}(𝔽_{p}; ℤ/2) is surjective as well. Since K_{4}(ℤ) = 0 we finally deduce from the universal coefficient theorem in algebraic K-theory that the map from K_{5}(ℤ)_{2} to K_{5}(𝔽_{p})_{2} is onto. The next step is to use the Adams element A^{n} to derive the same conclusion in all degrees 5 modulo 8. We have already seen that trick for the K-groups in degree 1 modulo 8, and the proof is complete.

Next we turn to the algebraic K-theory of the group ring ℤ[C_{p}], i.e., Theorem 2 for p ≡ 5 mod 8.

Proof. The only “problem” now is to identify the fibre K_{n}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} with the kernel of the map from K_{n}(ℤ[ζ_{p}])_{2} to K_{n}(𝔽_{p})_{2} for n ≡ 1, 5 mod 8. In all other degrees we will have an isomorphism between the torsion in K_{n}(ℤ[C_{p}])_{2} and the torsion in K_{n}(ℤ)_{2}. Consider the sequence:

[figure: 0; K_{8n+1}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2}; K_{8n+1}(ℤ[ζ_{p}])_{2}; K_{8n+1}(𝔽_{p})_{2}; 0; 0; ℤ2^{\frac{p−1}{2}} ⊕ (?); ℤ2^{\frac{p−1}{2}} ⊕ ℤ/2; ℤ/4; 0]

We deduce that the torsion part (?) of K_{8n+1}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} must be trivial as the copy of ℤ/2 injects into the copy of ℤ/4. Exactly the same diagram for the K-groups in degrees 5 modulo 8 shows that K_{8n+5}(ℤ[ζ_{p}] ⟶ 𝔽_{p})_{2} is torsion free as well. Hence all the torsion in K_{n}(ℤ[C_{p}])_{2} is detected by the torsion in K_{n}(ℤ)_{2}. Again the rest of the calculation follows from the direct sum decomposition of the algebraic K-theory of ℤ[C_{p}] described in Section 2.

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Department of Mathematics, University of Oslo, Norway.

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