5.5 The Grothendieck group of D <r (G, K)
5.5.2 Injectivity
As mentioned in the introduction, the motivation of Chapter 5 is to be able to compute the Grothendieck group of D(G, K). In this section, we take a step to-wards it. We prove that the group homomorphismK0(K[[G]])→K0(D<r(G, K)) induced by the natural injection of rings K[[G]] → D<r(G, K) is injective. This has a nice consequence, namely that there is a natural injective homomorphism Zc ↪→ K0(D<r(G, K)) and more importantly, it implies that there is an injective group homomorphism Zc↪→K0(D(G, K)). We will see what this homomorphism exactly is. We very much suspect that it is in fact an isomorphism. Let us denote byd the dimension of H.
Theorem 5.5.3. There is an injective mapZc ↪→K0(D<r(G, K)), where cis the number of p-regular conjugacy classes of G/H.
Proof. Let us denote by I0 the kernel of the ring homomorphism ϕ0 : K[[G]] → K[G/H] induced by the surjective group homomorphism G → G/H. Then the kernel I0 is generated by the elements bi for i = 1, . . . , d. One may look
at the algebra K[G/H] as the distribution algebra of G/H, which is a com-pact p-adic analytic group of dimension 0. Obviously, in this case all the al-gebras K[[G/H]], D(G/H, K), Dr(G/H, K), D<r(G/H, K) are the same, the group algebra K[G/H]. In the previous section we denoted by IK the kernel of the surjection ϕK :D<r(G, K)→ K[G/H], induced by the group homomorph-ism G → G/H. As before, IK is generated by bi for all i = 1, . . . , d. It is easy to see that IK is the scalar extension of I0 via the canonical at ring map K[[G]]↪→D<r(G, K). Hence we have the following commutative diagram:
K[[G]]
ϕ0
↓↓ →→D<r(G, K)
ϕr
↓↓
K[G/H] = →→K[G/H]
(41)
Lemma 5.5.4. K0(ϕ0) :K0(K[[G]])↪→K0(K[G/H])is injective.
Proof. It is easy to see that the diagram OK[[G]]
↓↓π →→K[[G]]
↓↓π
OK[G/H] →→K[G/H]
is commutative where the horizontal maps are the natural inclusions the the ver-tical maps are the natural surjections. Hence it induces a diagram
K0(OK[[G]])
↓↓ →→K0(K[[G]])
↓↓
K0(OK[G/H]) →→K0(K[G/H]).
By Theorem 5.2.4 the upper horizontal map is an isomorphism.By Proposition 3.3 (b)in [8], OK[[G]]is complete with respect to the augmentation ideal I(H): It is the kernel of the map OK[[G]]→ OK[G/H]). ActuallyI(H) is in the Jacobson radical of OK[[G]], which is contained in the radical, the intersection of all open maximal left ideals of OK[[G]]. The Iwasawa algebra is complete with respect to the ltration induced by the radical, by Corollary 5.2.19 in [30]. Hence by Proposition 2.5.6, the vertical map on the left hand side is also an isomorphism.
By Corollary 2.9.9, the lower horizontal map is injective. Hence the vertical map on the right hand side must be injective.
Diagram (41) induces a commutative diagram after applying K0( ): K0(K[[G]])
K0(ϕ0)
↓↓ →→K0(D<r(G, K))
K0(ϕr)
↓↓
K0(K[G/H]) = →→K0(K[G/H]) .
By Lemma 5.5.4, K0(ϕ0) is injective. Hence the upper horizontal map must also be injective, by commutativity.
Remark 5.5.5. Note that for injectivity, we did not need assumption (E).
Corollary 5.5.6. The map K0(K[[G]]) → K0(D(G, K)) induced by the natural inclusion K[[G]] → D(G, K) is injective. Hence we have an injective map Zc → K0(D(G, K))
Proof. It is an easy consequence of the fact that the natural inclusion K[[G]] ↪→ D<r(G, K)factorizes throughD(G, K)since know that D(G, K)⊂D<r(G, K)for all rQ such that 1/p < r <1. Then we can use that K0( ) is a functor to get the desired injective map.
Now the proof of Theorem 5.5.1: Hence the theorem follows from the well-known structure theorem for nitely generated modules over PID's since by Theorem 5.5.3 and by Theorem 5.5.2, we have an injective map Zc ↪→K0(D<r(G, K)) and a surjective map Zc→K0(D<r(G, K)). Hence Zc∼=K0(D<r(G, K)).
Corollary 5.5.7. Let G be a compactp-adic analytic group. Let r ∈pQ, 1/p <
r < 1 and assume that K satisies (E). Then there is an injective map Zc → Dr(G, K).
Proof. The mapK0(K[[G]])↪→K0(D<r(G, K))factorizes throughK0(Dr(G, K)). It follows thatZc ↪→K0(Dr(G, K)). Hence using that K0 is a functor, we get the injective map.
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Selbstständigkeitserklärung
Ich erkläre, dass ich die Dissertation selbständig und nur unter Verwendung der von mir gemäÿ 7 Abs. 3 der Promotionsordnung der Mathematisch-Naturwissenschaftlichen Fakultät, veröentlicht im Amtlichen Mitteilungsblatt der Humboldt-Universität zu Berlin Nr. 126/2014 am 18.11.2014 angegebenen Hilfsmittel angefertigt habe.
Berlin, den 04.10.2016 Tamás Csige