Structure theorem of Kummer étale \(K\)-group. II. (Q504295)

The author deals with a \(K\)-theory peculiar to logarithmic geometry, the so called Kummer étale \(K\)-Theory. Let \(k\) be a field of characteristic \(p\), \(X\) a scheme smooth, separated and of finite type over \(k\) and let \(D\) be a strictly normal crossing divisor, with \(\{D_i\}_{i\in I}\) its irreducible components. Let's denote by \(X\) the log scheme associate with \((X,D)\). In a previous paper [\(K\)-Theory 29, No. 2, 75--99 (2003; Zbl 1038.19002)] the same author proved a comparison theorem between \(K_q(X_{K\text{ét}})\) and the usual \(K\)-groups of \(X\). In this paper that result has been improved, by constructing, for a logarithmic variety satisfying some good conditions, an isomorphism between its Kummer étale \(K\)-groups and the usual \(K\)-groups associated with its stratification, preserving their \(\lambda\)-structures up to torsion. The main result is the following Theorem 0.1. There is an isomorphism of rings \[ K_0(X_{K\text{ét}})\otimes_{\mathbb Z}\simeq\varprojlim K_0(D_{J_2})\otimes_{\mathbb Z}\mathbb Q[(\mathbb Z_{(p)}/\mathbb Z)^{\oplus J_1}] \] which is compatible with the actions of Adam's operations. Here, for \(J=\{i_1,\dots,i_r\}\subset I\) we put \(D_J=D_{i_1}\cap\dots\cap D_{i_r}\) and \[ C_I=\{(J_1,J_2)|J_1\subset J_2\subset I\}. \] The limit is taken over \(C_I\), regarded as an ordered set by defining \((J_1,J_2)\geq (J'_1,J'_2)\) if \(J'_1\subset J_1\subset J_2\subset J'_2\). The transition morphism in the limit is induced by the natural closed inclusion \(D_{J_2}\supset D_{J'_2}\) and a projection \((\mathbb Z_{(p)} /\mathbb Z)^{\oplus J_1}\to (\mathbb Z_{(p)} /\mathbb Z)^{\oplus J'_1}\). The Adam operations \(\{\Psi^m\}_{m>0}\) on \(\mathbb Q[(\mathbb Z_{(p)} /\mathbb Z)^{\oplus J}]\) are defined by \(\Psi^m ([\alpha]=[m\alpha]\), for \(m\in\mathbb N\) and \(\alpha\in (\mathbb Z_{(p)} /\mathbb Z)^{\oplus J}\). Then the Adams operations on the right hand side are naturally induced from the usual ones on \(K_0(D_{J_2})_{\mathbb Q}\) and on \(\mathbb Q[(\mathbb Z_{(p)}/\mathbb Z)^{\oplus J_1}\).