Comparing direct limit and inverse limit of even \(K\)-groups in multiple \(\mathbb{Z}_p\)-extensions (Q6642860)

Let \(p\) be a prime. There is a well-known duality between the direct and inverse limit of \(p\)-class groups along a \(\mathbb Z_p^d\)-extension of a number field \(F\). In this paper the author proves an analogue for higher even \(K\)-groups.\N\NMore precisely, let us denote the \(p\)-class group of a number field \(L\) be \(A_L\). Let \(F_{\infty}\) be a \(\mathbb Z_p^d\)-extension of a number field \(F\) and consider the inverse and direct limits\N\[\N\varprojlim_L A_L, \quad \varinjlim_L A_L,\N\]\Nwhere \(L\) runs over all finite extensions of \(F\) in \(F_{\infty}\). Set \(\Gamma := \mathrm{Gal}(F_{\infty}/F) \simeq \mathbb Z_p^d\). When \(d=1\), \textit{K. Iwasawa} [Ann. Math., II. Ser. 98, 246--326 (1973; Zbl 0285.12008)] proved that there is a pseudo-isomorphism\N\[\N(\varprojlim_L A_L)^{\iota} \sim (\varinjlim_L A_L)^{\vee},\N\]\Nwhere \((-)^{\vee} = \mathrm{Hom}(-, \mathbb Q_p / \mathbb Z_p)\) denotes the Pontryagin dual and the upper \(\iota\) indicates that \(\gamma \in \Gamma\) now acts by \(\gamma^{-1}\). This result has been generalized to arbitrary \(d \geq 1\) by \textit{J. Nekovář} [Selmer complexes. Astérisque 310. Paris: Société Mathématique de France (2007; Zbl 1211.11120)], \textit{D. Vauclair} [Ann. Inst. Fourier 59, No. 2, 691--767 (2009; Zbl 1254.11098)] and more recently by \textit{K.F. Lai} and \textit{K.-S. Tan} [Res. Math. Sci. 8, No. 2, Paper No. 20, 18 p. (2021; Zbl 1472.11283)].\N\NNote that the \(p\)-class group \(A_L\) naturally identifies with the torsion subgroup of \(K_0(\mathcal{O}_L)\). For \(n \geq 1\) the higher even \(K\)-groups \(K_{2n}(\mathcal{O}_L)\) are known to be finite. The main result of the paper now states that there is a pseudo-isomorphism\N\[\N(\varprojlim_L K_{2n}(\mathcal{O}_L)[p^{\infty}])^{\iota} \sim (\varinjlim_L K_{2n}(\mathcal{O}_L)[p^{\infty}])^{\vee}\N\]\Nin complete analogy with the \(n=0\) case. \par The main input of the proof is a consequence of an exact sequence due to \textit{U. Jannsen} [Adv. Stud. Pure Math. 17, 171--207 (1989; Zbl 0732.11061)]. If \(M\) is a finitely generated \(\mathbb Z_p[[\Gamma]]\)-module, then this sequence (use Jannsen's Theorem 2.1 with \(r=1\)) specializes to\N\[\N0 \rightarrow (\varinjlim_U M_U[p^{\infty}])^{\vee} \rightarrow \mathrm{Ext}^1_{\mathbb Z_p[[\Gamma]]}(M, \mathbb Z_p[[\Gamma]]) \rightarrow (\varinjlim_U H_1(U,M) \otimes \mathbb Q_p / \mathbb Z_p)^{\vee} \rightarrow 0,\N\]\Nwhere \(U\) runs over all open subgroups of \(\Gamma\). If the coinvariant modules \(M_U\) are all finite, then \(H_1(U,M)\) is also finite and hence \(H_1(U,M) \otimes \mathbb Q_p / \mathbb Z_p\) vanishes. Moreover, such a module is necessarily torsion so that \(\mathrm{Ext}^1_{\mathbb Z_p[[\Gamma]]}(M, \mathbb Z_p[[\Gamma]])\) is pseudo-isomorphic to \(M^{\iota}\). Now take \(M = \varprojlim_L K_{2n}(\mathcal{O}_L)[p^{\infty}]\). \par Jannsen's proof uses a spectral sequence argument. In an appendix, the author provides a more direct proof of the required isomorphism\N\[\N(\varinjlim_U M_U)^{\vee} \simeq \mathrm{Ext}^1_{\mathbb Z_p[[\Gamma]]}(M, \mathbb Z_p[[\Gamma]])\N\]\Nwhenever all \(M_U\) are finite (the latter property is called `systematically coinvariant-finite' in the paper).