\(NK_{0}\) and \(NK_{1}\) of the groups \(C_{4}\) and \(D_{4}\)
Let \(R\) be a ring and \(K_i(R)\) denote the \(i\)-th \(K\)-theory group of \(R\), (\(i\in\mathbb{Z}\)). Recall that \(NK_i(R)\) is defined as the quotient \(K_{i}(R[X])/K_{i}(R)\). The author explicitly computes these groups in the following cases: \(R=\mathbb{Z}[G]\) (the integral group ring), \(G=D_{2}, C_{4}\) or \(D_4\) (dihedral of order \(2\), cyclic of order \(4\) and dihedral of order \(8\), respectively) and \(i=0,1\). The computations keep track of the module structure of these groups given by Verschiebung and Frobenius operators, as well as the continuous module structure over the ring of big Witt vectors \(W(\mathbb{Z})\). Let \(V\) denote the continuous \(W(\mathbb{F}_2)\)-module \(x\mathbb{F}_2[x]\): it is a countable direct sum of copies of \(\mathbb{F}_2\) on generator \(x^{i}\), \(i>0\) and the module structure is determined by \(V_{m}(x^{n})=x^{mn}\), \([a]x^{n}=a^{n}x^{n}; F_m(x^{n})=0\) if \( (m,n)=1\) \((m>1)\) and \( F_d(x^n)=dx^{n/d}\) wif \(d|n\). The results are as follows. Theorem 1.3: For \(D_2=C_2\times C_2\) we have \(NK_0(\mathbb{Z}[D_2])\cong V, NK_1(\mathbb{Z}[D_2])\cong \Omega_{\mathbb{F}_2[x]}\), Theorem 1.4: \(NK_1(\mathbb{Z}[C_4])\cong \Omega_{\mathbb{F}_2[x]}\) and \(NK_0(\mathbb{Z}[C_4])\cong V\), Theorem 2.5: \(NK_0(\mathbb{Z}[D_4])\) is isomorphic to the cyclic Cartier module \(NK_1(\mathbb{Z}[C_2],1-\tau)\) (described in the paper). As a group, this is the direct sum of a countable infinite free \(\mathbb{Z}/4\)-module and a countable infinite free \(\mathbb{Z}/2\)-module. The author also proves that \(NK_0(\mathbb{Z}[D_4])\) surjects onto \(NK_1(\mathbb{Z}[D_4])\).
scientific article; zbMATH DE number 5545773