2 Iwasawa algebras and modules

Throughout the article, p will always denote a fixed prime. Let G be a compact p-adic Lie group. The Iwasawa algebra of G over $\mathbb {Z}_p$ is defined by

where U runs over the open normal subgroups of G and the inverse limit is taken with respect to the canonical projection maps.

For now, let us assume that the group G is pro-p and has no p-torsion. It is then well-known that is a Noetherian Auslander regular ring (cf. [Reference Venjakob36, Theorem 3.26]) with no zero divisors (cf. [Reference Neumann29]). As a consequence, it admits a skew field $Q(G)$ which is flat over (see [Reference Goodearl and Warfield10, Chapters 6 and 10] or [Reference Lam19, Chapter 4, Sections 9 and 10]). This in turn enables us to define the -rank of a finitely generated -module M, which is given by

The -module M is said to be a torsion -module if . We also recall a useful equivalent formulation of a torsion -module: (cf. [Reference Venjakob36, Remark 3.7]). In the event that the torsion -module M satisfies , we say that M is a pseudo-null -module. For a finitely generated -module M, denote by $M[p^\infty ]$ the -submodule of M consisting of elements of M which are annihilated by some power of p. Howson [Reference Howson13, Proposition 1.11], and independently Venjakob [Reference Venjakob36, Theorem 3.40], showed that there is a -homomorphism

whose kernel and cokernel are pseudo-null -modules, and where the integers s and $\alpha _i$ are uniquely determined. The $\mu _G$ -invariant of M is then defined to be $\mu _G(M) = \displaystyle \sum _{i=1}^s\alpha _i$ .

We now extend the notion of torsion modules and pseudo-null modules to the case when G is a general compact p-adic Lie group. A well-known theorem of Lazard asserts that G contains an open normal subgroup $G_0$ which is pro-p with no p-torsion (cf. [Reference Dixon, Du Sautoy, Mann and Segal5, Theorem 8.32]). By [Reference Neukirch, Schmidt and Wingberg28, Proposition 5.4.17], we have

for every finitely generated -module M. In view of this, we shall say that M is a torsion -module (resp., psuedo-null -module) if (resp., for $i=0,1$ ). Equivalently, this is saying that M is a torsion -module (resp., pseudo-null -module), whenever M is a torsion -module (resp., pseudo-null -module) as in the preceding paragraph. (Also, compare with [Reference Venjakob36, Discussion after Definition 2.6]).

3 K-groups and Galois cohomology

We now describe a relation between the algebraic K-groups and Galois cohomology. To begin with, F will always denote a global function field of characteristic $\ell $ . We shall, once and for all, fix a finite nonempty set S of primes of F. Let $F_S$ be the maximal algebraic and separable extension of F unramified outside S. For every extension L of F contained in $F_S$ , write $G_S(L)$ for the Galois group $\operatorname {\mathrm {Gal}}(F_S/L)$ . Finally, we denote by $S(L)$ the set of primes of L above S, and for brevity, we shall write $\mathcal {O}_{L,S}$ for the ring of $S(L)$ -integers of L.

For a ring R with identity, write $K_n(R)$ for the algebraic K-groups of R in the sense of Quillen [Reference Quillen31] (also see [Reference Kolster17, Reference Weibel39]). In this article, we are interested in the group $K_n(\mathcal {O}_{F,S})$ . A theorem of Harder [Reference Harder11, Korollar 3.2.3] asserts that the group $K_n(\mathcal {O}_{F,S})$ is finite for every $n\geq 2$ . On the other hand, the group $K_n(F)$ is p-divisible by a result of Geisser–Levine [Reference Geisser and Levine9, Theorem 8.1]. By appealing to this and the localization sequence, one can then show that the group $K_n(\mathcal {O}_{F,S})$ has order coprime to $\ell $ (we refer readers to [Reference Weibel39, Chapter VI, Theorem 6.1 and Corollary 6.2] for details). Note that when $n=2$ , this was earlier established by Bass–Tate [Reference Bass and Tate1, Chapter II, Theorem 2.1]).

For the remainder of the article, p will always denote a prime $\neq \ell $ . Denoting by $\mu _{p^n}$ the cyclic group generated by a primitive $p^n$ -root of unity, we write $\mu _{p^\infty }$ for the direct limit of the groups $\mu _{p^n}$ . The natural action of $G_S(F)$ on $\mu _{p^\infty }$ induces the cyclotomic character

$$\begin{align*}\chi: G_S(F) \longrightarrow \mathrm{Aut}(\mu_{p^\infty}) \cong \mathbb{Z}_p^{\times}.\end{align*}$$

For a discrete or compact $G_S(F)$ -module X, we denote by $X(i)$ the i-fold Tate twist of X. More precisely, $X(i)$ is the $G_S(F)$ -module which is X as a $\mathbb {Z}_p$ -module but with a $G_S(F)$ -action given by

$$\begin{align*}\sigma\cdot x = \chi(\sigma)^i\sigma x, \end{align*}$$

where the action on the right is the original action of $G_S(F)$ on X. Plainly, we have $X(0)=X$ and $\mu _{p^{\infty }} \cong \mathbb {Q}_p/\mathbb {Z}_p(1)$ . In [Reference Soulé34], Soulé connected the higher K-groups with (continuous) Galois cohomology groups via the p-adic Chern class maps

$$\begin{align*}\mathrm{ch}_{i,k}^{(p)}: K_{2i-k}(\mathcal{O}_{F,S})\otimes \mathbb{Z}_p \longrightarrow H^k(G_{S}(F), \mathbb{Z}_p(i))\end{align*}$$

for $i\geq 2$ and $k =1,2$ . (For the precise definition of these maps, we refer readers to loc. cit.) Soulé also proved that these maps are surjective (see [Reference Soulé34, Théorème 6(iii)]; also see the work of Dwyer–Friedlander [Reference Dwyer and Friedlander6]). It is folklore (see, for instance, [Reference Kolster17, Theorem 2.7]) that the bijectivity of the p-adic Chern class maps follows as a consequence of the norm residue isomorphism theorem (previously also known as the Bloch–Kato(-Milnor) conjecture; see [Reference Bloch and Kato2, Reference Milnor26]), and the latter was proven by Rost and Voevodsky [Reference Voevodsky37] with the aid of a patch from Weibel [Reference Weibel38]. As a consequence of these developments, we have the following identification between the Sylow p-subgroup of K-groups and Galois cohomology.

Proposition 3.1 For $i\geq 2$ , one has the following isomorphisms of finite abelian groups:

$$\begin{align*}K_{2i-1}(\mathcal{O}_{F,S})[p^\infty] \cong H^1\big(G_{S}(F), \mathbb{Z}_p(i)\big),\end{align*}$$

$$\begin{align*}K_{2i-2}(\mathcal{O}_{F,S})[p^\infty] \cong H^2\big(G_{S}(F), \mathbb{Z}_p(i)\big).\end{align*}$$

4 Iwasawa cohomology groups

We now introduce the Iwasawa cohomology groups which will be an important object for our study of K-groups. As before, F will denote a global function field of characteristic $\neq p$ , and we let $i\geq 2$ . For every extension $\mathcal {L}$ of F contained in $F_S$ , the Iwasawa cohomology groups are defined by

$$\begin{align*}H^k_{\mathrm{Iw},S}\big(\mathcal{L}/F, \mathbb{Z}_p(i)\big):= \displaystyle \mathop{\varprojlim}\limits_L H^k\big(G_S(L), \mathbb{Z}_p(i)\big), \end{align*}$$

where L runs through all finite extensions of F contained in $\mathcal {L}$ and the transition maps are given by the corestriction maps. In the event that $\mathcal {L}/F$ is a finite extension, we plainly have $H^k_{\mathrm {Iw},S}\big (\mathcal {L}/F, \mathbb {Z}_p(i)\big )= H^k\big (G_S(\mathcal {L}), \mathbb {Z}_p(i)\big )$ .

Now suppose that $F_\infty $ is a p-adic Lie extension of F contained in $F_S$ . In other words, $F_\infty $ is a Galois extension of F, whose Galois group $G=\operatorname {\mathrm {Gal}}(F_\infty /F)$ is a compact p-adic Lie group. We can now state and prove the first theorem of the article.

The remainder of the section is devoted to the proof of this theorem. As a start, we record the following version of Tate’s descent spectral sequence for Iwasawa cohomology.

Proposition 4.2 Suppose that $i\geq 2$ . Let U be a closed normal subgroup of $G=\operatorname {\mathrm {Gal}}(F_\infty /F)$ and write $\mathcal {L}$ for the fixed field of U. Then we have a homological spectral sequence

$$ \begin{align*}H_r\big(U, H^{-s}_{\mathrm{Iw},S}(F_\infty/F, \mathbb{Z}_p(i))\big)\Longrightarrow H^{-r-s}_{\mathrm{Iw},S}\big(L_\infty/F, \mathbb{Z}_p(i)\big)\end{align*} $$

and an isomorphism

$$\begin{align*}H^2_{\mathrm{Iw},S}\big(F_\infty/F, \mathbb{Z}_p(i)\big)_U \cong H^2_{\mathrm{Iw},S}\big(\mathcal{L}/F, \mathbb{Z}_p(i)\big).\end{align*}$$

Furthermore, if U is an open subgroup of G, then $\mathcal {L}$ is a finite extension of F and we have

$$\begin{align*}H^2_{\mathrm{Iw},S}\big(F_\infty/F, \mathbb{Z}_p(i)\big)_U \cong H^2\big(G_S(\mathcal{L}), \mathbb{Z}_p(i)\big).\end{align*}$$

We record another observation pertaining to the p-cohomological dimension of the group $G_S(F_{\mathrm {cyc}})$ which will be an important component to our subsequent discussion.

We are in position to prove Theorem 4.1.

Proof (Proof of Theorem 4.1) We first prove statement (a). The torsionness can be proved via a similar argument to that in [Reference Lim23, Proposition 4.1.1]. We will focus on the finite generation assertion. By base changing F, we may assume that G is pro-p with no p-torsion. Since

$$\begin{align*}H^2_{\mathrm{Iw},S}\big(F_\infty/F, \mathbb{Z}_p(i)\big)_H \cong H^2_{\mathrm{Iw},S}\big(F_{\mathrm{cyc}}/F, \mathbb{Z}_p(i)\big)\end{align*}$$

by Proposition 4.2, a Nakayama’s lemma argument reduces us to showing that $H^2_{\mathrm {Iw},S}\big (F_{\mathrm {cyc}}/F, \mathbb {Z}_p(i)\big )$ is finitely generated over $\mathbb {Z}_p$ . But since we also have

$$\begin{align*}H^2_{\mathrm{Iw},S}\big(F_{\mathrm{cyc}}/F, \mathbb{Z}_p(i)\big)/p \cong H^2_{\mathrm{Iw},S}\big(F_{\mathrm{cyc}}/F, \mathbb{Z}/p(i)\big), \end{align*}$$

this in turn reduces us to showing that $H^2_{\mathrm {Iw},S}\big (F_{\mathrm {cyc}}/F, \mathbb {Z}/p(i)\big )$ is finite, or equivalently, $H^2_{\mathrm {Iw},S}\big (F_{\mathrm {cyc}}/F, \mathbb {Z}/p(i)\big )$ is torsion over , where $\Gamma = \operatorname {\mathrm {Gal}}(F_{\mathrm {cyc}}/F)$ . By considering the initial term of the spectral sequence

(see [Reference Fukaya and Kato8, Subsection 1.6.12], [Reference Lim and Sharifi25, Theorem 4.5.1] or [Reference Nekovář27, Theorem 5.4.5]), we obtain an isomorphism

where we note that the latter vanishes by virtue of Lemma 4.3. Hence it follows that $H^2_{\mathrm {Iw},S}\big (F_{\mathrm {cyc}}/F, \mathbb {Z}_p(i)\big )$ is torsion over as required. The proof of statement (a) is therefore complete.

We now turn to the verification of statement (b). As before, we may assume that G is pro-p with no p-torsion. Since the first assertion of statement (a) is telling us that

we therefore have

By a formula of Howson [Reference Howson12, Theorem 1.1], the latter sum can be expressed as

$$\begin{align*}\sum_{r,s\geq 0}(-1)^{r+s} \operatorname{\mathrm{rank}}_{\mathbb{Z}_p}\Big(H_r\big(G, H^s_{\mathrm{Iw},S}\big(F_\infty/F, \mathbb{Z}_p(i)\big)\big)\Big). \end{align*}$$

Taking the spectral sequence of Proposition 4.2 into account, this sum is equal to

$$\begin{align*}\sum_{m\geq 0}(-1)^{m} \operatorname{\mathrm{rank}}_{\mathbb{Z}_p}\Big(H^m\big(G_S(F), \mathbb{Z}_p(i)\big)\Big) =0, \end{align*}$$

where the final zero is a consequence of the cohomology groups appearing in the sum being finite. We have therefore established the first assertion of statement (b).

To continue, we shall work under the assumption that G has dimension $\geq 2$ . We now call upon another spectral sequence of Nekovář

(cf. [Reference Fukaya and Kato8, Subsection 1.6.12], [Reference Lim and Sharifi25, Theorem 4.5.1] or [Reference Nekovář27, Theorem 5.4.5]), whose low degree terms fit into the following exact sequence

As the group G is of dimension $\geq 2$ , it follows from a classical observation of Jannsen (cf. [Reference Jannsen15, Corollary 2.6(b)(c)]) that whenever M is finitely generated over $\mathbb {Z}_p$ . Since this is the case for the module $H^{0}\big (G_S(F_\infty ), \mathbb {Q}_p/\mathbb {Z}_p(i)\big )^\vee $ , we see that the leftmost term of the above exact sequence is zero. Consequently, the module $H^{1}_{\mathrm {Iw},S}\big (F_\infty /F,\mathbb {Z}_p(i)\big )$ , which is torsion over as seen in the previous paragraph, injects into which is a reflexive -module (cf. [Reference Venjakob36, Proposition 3.11(i)]). Thus, we must have $H^{1}_{\mathrm {Iw},S}\big (F_\infty /F,\mathbb {Z}_p(i)\big )=0$ . This therefore completes the proof of the theorem.

We end with a byproduct of the discussion in this section. For the remainder of the section, we assume that G is a compact p-adic Lie group with no p-torsion. The G-Euler characteristics of a -module M is said to be well-defined if $H_j(G,M)$ is finite for every $j\geq 0$ . In the event of such, the G-Euler characteristics of M is then defined to be

$$\begin{align*}\chi(G,M) = \prod_{j\geq 0} \big(\# H_j(G,M)\big)^{(-1)^j}.\end{align*}$$

Note that since the group G is assumed to have no p-torsion, it has finite p-cohomological dimension (see [Reference Serre32, Corollaire 1]), and therefore, the alternating product is a finite product. We now evaluate the G-Euler characteristics of the second Iwasawa cohomology group for p-adic Lie extension with dimension at least $2$ .

Theorem 4.5 Notation as before. Suppose further that G has dimension $\geq 2$ and has no p-torsion. Then one has

$$\begin{align*}\chi\big(G,H^2_{\mathrm{Iw}}\big(F_\infty/F, \mathbb{Z}_p(i)\big)\big) = \frac{\#K_{2i-2}(\mathcal{O}_{F,S})[p^\infty]}{\#K_{2i-1}(\mathcal{O}_{F,S})[p^\infty]}.\end{align*}$$

Proof Indeed, from Theorem 4.1(b), we have $H^1_{\mathrm {Iw}}\big (F_\infty /F, \mathbb {Z}_p(i)\big )=0$ . Taking this latter observation into account, the spectral sequence

$$\begin{align*}H_r\big(G, H^{-s}_{\mathrm{Iw}}\big(F_\infty/F, \mathbb{Z}_p(i)\big)\big)\Longrightarrow H^{-r-s}\big(G_S(F),\mathbb{Z}_p(i)\big) \end{align*}$$

degenerates yielding

$$\begin{align*}H_0\big(G,H^2_{\mathrm{Iw}}\big(F_\infty/F, \mathbb{Z}_p(i)\big)\big) \cong K_{2i-2}(\mathcal{O}_{F,S})[p^\infty], \end{align*}$$

$$\begin{align*}H_1\big(G,H^2_{\mathrm{Iw}}\big(F_\infty/F, \mathbb{Z}_p(i)\big)\big) \cong K_{2i-1}(\mathcal{O}_{F,S})[p^\infty] \end{align*}$$

and $H_j\big (G,H^2_{\mathrm {Iw}}\big (F_\infty /F, \mathbb {Z}_p(i)\big )\big ) =0$ for $j\geq 2$ . The conclusion is now immediate.

5 Growth of even K-groups

In this section, we present growth formulas for even K-groups over certain p-adic Lie extensions of global function fields. For a positive integer m, we write $\mathrm {ord}_p(m)$ for the highest power of p dividing m, i.e., $m = p^{\mathrm {ord}_{p}(m)}m'$ for some $p\nmid m'$ . If N is a finite group, we then write $\mathrm {ord}_p(N)$ for $\mathrm {ord}_p(|N|)$ . In particular, if N is a finite p-group, one has $|N| = p^{\mathrm {ord}_p(N)}$ .

We begin with the cyclotomic $\mathbb {Z}_p$ -extension.

Theorem 5.1 Let $i\geq 2$ be given. Denote by $F_n$ the intermediate subfields of $F_{\mathrm {cyc}}/F$ with $|F_n:F|=p^n$ . Then we have

$$\begin{align*}\mathrm{ord}_p\Big(K_{2i-2}(\mathcal{O}_{F_n,S})[p^\infty]\Big) = \operatorname{\mathrm{rank}}_{\mathbb{Z}_p}\Big(H^{2}_{\mathrm{Iw},S}\big(F_\infty/F,\mathbb{Z}_p(i)\big)\Big) n + O(1).\end{align*}$$

Proof Write $\Gamma _n=\operatorname {\mathrm {Gal}}(F_\infty /F_n)$ . By a combination of Propositions 3.1 and 4.2, we have

$$\begin{align*}H^2_{\mathrm{Iw},S}\big(F_\infty/F, \mathbb{Z}_p(i)\big)_{\Gamma_n} \cong H^2\big(G_{S}(F_n), \mathbb{Z}_p(i)\big)\cong K_{2i-2}(\mathcal{O}_{F_n,S})[p^\infty].\end{align*}$$

The conclusion of the theorem now follows from an application of [Reference Neukirch, Schmidt and Wingberg28, Proposition 5.3.17] and taking Theorem 4.1(a) into account.

Recall that Lazard’s theorem (cf. [Reference Dixon, Du Sautoy, Mann and Segal5, Corollary 8.34]) tells us that every compact p-adic Lie group contains an open normal subgroup which is a uniform pro-p group in the sense of [Reference Dixon, Du Sautoy, Mann and Segal5, Definition 4.1]. Therefore, one can always reduce consideration for a general compact p-adic Lie group to the case of a uniform pro-p group which we will do for the remainder of this section. For a uniform pro-p group, we shall write $G_n$ for the lower p-series $P_{n+1}(G)$ which is defined recursively by $P_{1}(G) = G$ , and

$$\begin{align*}P_{n+1}(G) = \overline{P_{n}(G)^{p}[P_{n}(G),G]}, ~\mbox{for}~ n\geq 1. \end{align*}$$

For every n, we have $|G:G_n| = |G:P_{n+1}(G)| = p^{dn}$ (see [Reference Dixon, Du Sautoy, Mann and Segal5, Sections 3 and 4]), where d is the dimension of the uniform pro-p group G. We now record two results which we require.

Proposition 5.3 Let G be a uniform pro-p group which contains a closed subgroup $H\cong \mathbb {Z}_p^{d-1}$ such that $G/H\cong \mathbb {Z}_p$ . Suppose that M is a -module which is finitely generated over and that $M_{G_n}$ is finite for every n. Then we have

Proposition 5.4 Let G be a uniform pro-p group which contains a closed subgroup H such that $G/H\cong \mathbb {Z}_p$ . Suppose that M is a -module which is finitely generated over . Then we have

Building on our Theorem 4.1, we can prove the following asymptotic formula for the growth of even K-groups for a special class of p-adic Lie extensions.

Theorem 5.5 Let $i\geq 2$ be given. Suppose that $F_\infty $ is a p-adic Lie extension of F containing $F_{\mathrm {cyc}}$ with $\operatorname {\mathrm {Gal}}(F_\infty /F_{\mathrm {cyc}})\cong \mathbb {Z}_p^{d-1}$ . Let $F_n$ be the intermediate subfield of $F_\infty /F$ with $\operatorname {\mathrm {Gal}}(F_\infty /F_n) = G_n$ . Then we have

For a general p-adic Lie extension which contains $F_{\mathrm {cyc}}$ , we at least have an upper bound.

Proposition 5.6 Let $i\geq 2$ be given. Suppose that $F_\infty $ is a p-adic Lie extension of F containing $F_{\mathrm {cyc}}$ with G being a uniform pro-p group of dimension d. Let $F_n$ be the intermediate subfield of $F_\infty /F$ with $\operatorname {\mathrm {Gal}}(F_\infty /F_n) = G_n$ . Then we have

6 Comparing limits

We come to the final section of the article, whose goal is to prove a relation between the direct limit and the inverse limit of even K-groups. For two finite subextensions $L\subseteq L'$ of F contained in $F_\infty $ , the inclusion $\mathcal {O}_{L,S}\longrightarrow \mathcal {O}_{L',S}$ induces a map $\jmath _{L/L'}: K_{2i-2}(\mathcal {O}_{L,S})[p^\infty ]\longrightarrow K_{2i-2}(\mathcal {O}_{L',S})[p^\infty ]$ by functoriality. (Note that the induced map may not be injective in general.) In the other direction, there is the norm map (also called the transfer map) $\mathrm {Tr}_{L'/L}: K_{2i-2}(\mathcal {O}_{L',S})[p^\infty ]\longrightarrow K_{2i-2}(\mathcal {O}_{L,S})[p^\infty ]$ . Consider the following direct limit and inverse limit

$$\begin{align*}\displaystyle \mathop{\varinjlim}\limits_L K_{2i-2}(\mathcal{O}_{L,S})[p^\infty] \quad \mbox{and}\quad \displaystyle \mathop{\varprojlim}\limits_L K_{2i-2}(\mathcal{O}_{L,S})[p^\infty],\end{align*}$$

whose transition maps are given by the maps $\jmath _{L/L'}$ and $\mathrm {Tr}_{L'/L}$ respectively. The main result of this section is as follows.

Theorem 6.1 Let $i\geq 2$ and let $F_\infty $ be a p-adic Lie extension of F which is contained in $F_S$ . Then there is an isomorphism

of -modules.

The following is an immediate corollary of the theorem.

We now give the proof of Theorem 6.1.

Proof of Theorem 6.1

Now, if the dimension of G is $1$ , then by base-changing, we may assume that $F_\infty /F$ is a $\mathbb {Z}_p$ -extension, which is a commutative p-adic Lie extension. In this situation, the theorem follows from a direct similar argument to that in [Reference Lim24, Theorem 3.1]. Therefore, we may assume that G has dimension $\geq 2$ . From the discussion in [Reference Soulé34, Chapter III], there is a commutative diagram

which in turn yields

$$\begin{align*}H^2_{\mathrm{Iw}}\big(F_\infty/F, \mathbb{Z}_p(i)\big) \cong \displaystyle \mathop{\varprojlim}\limits_{L} K_{2i-2}(\mathcal{O}_{L,S})[p^\infty]. \end{align*}$$

For brevity, write $M=H^2_{\mathrm {Iw}}\big (F_\infty /F, \mathbb {Z}_p(i)\big )$ . By Jannsen’s result (cf. [Reference Jannsen15, Theorem 2.1] or [Reference Neukirch, Schmidt and Wingberg28, Theorem 5.4.13]), there is a short exact sequence

(6.1)

where the transition map in the limit is given by norm map given as follow: For two open normal subgroups $V\subseteq U$ of G, the norm map $N_{U/V}$ on $M_V$ factors through $M_U=(M_V)_{U/V}$ to yield a map $M_U\longrightarrow M_V$ , which by abuse of notation is also denoted by $N_{U/V}$ .

By a similar argument to that in [Reference Lim24, Lemma 3.6], we obtain the following commutative diagram

which yields an identification of the leftmost term in the short exact sequence (6.1) with

$$\begin{align*}\displaystyle \mathop{\varinjlim}\limits_{L} K_{2i-2}(\mathcal{O}_{L,S})[p^\infty].\end{align*}$$

It therefore remains to show the rightmost term in the short exact sequence (6.1) vanishes. For this, it suffices to show that $H_1(U,M)$ is finite for every open subgroup U of G. But as already seen in the proof of Theorem 4.5, if we denote by L the fixed field of U, then one has

$$\begin{align*}H_1\Big(U,H^2_{\mathrm{Iw}}\big(F_\infty/F, \mathbb{Z}_p(i)\big)\Big) \cong K_{2i-1}(\mathcal{O}_{L,S})[p^\infty], \end{align*}$$

where the latter K-group is finite by Harder’s theorem. The proof of our theorem is now complete.