\(K_2\) of finite abelian group algebras
Let \(F\) be a finite field of odd characteristic \(p\), and \(G\) a finite abelian group. A covering of \(K_2(F[G])\) in terms of \(G \otimes A[G]\), where \(A\) is the unramified \(p\)-ring such that \(A/pA = F\), was given by \textit{R. Oliver} [J. Algebra 94, 425--487 (1985; Zbl 0595.18004)]. The purpose of the current paper is to give a more precise description of \(K_2(F[G])\) in various cases. The authors first note that \(K_2(F[G])\) is a finite \(p\)-group, annihilated by the exponent of the \(p\)-Sylow subgroup of \(G\). They prove that \(K_2[G \times \mathbb{Z}/p\mathbb{Z}]\) is a direct sum of \(K_2[G]\) and an elementary abelian \(p\)-group of rank \([F : \mathbb{F}_p] \cdot \left|G\right| \cdot (1-p^{-r} + (p-1)r)\), where \(r\) is the dimension of \(G/G^p\). By induction it follows that when \(G\) has \(p^2\)-rank at most \(1\), \(K_2[G]\) is the elementary abelian \(p\)-group of rank \([F : \mathbb{F}_p] \cdot \left|G\right| \cdot (t-1) (1-p^{-r})\), where \(t\) is the \(p\)-rank of \(G\), similarly to the case \(p = 2\) which was computed by \textit{B. A. Magurn} [J. Pure Appl. Algebra 209, No.~3, 801--811 (2007; Zbl 1120.19002)]. More generally, one defines the groups \(\Phi_i(F[G])\) as the kernels of the projection \[ K_2(F[G][t]/\left<t^i\right>) \rightarrow K_2(F[G][t]/\left<t^{i-1}\right>) \] then the order of \(K_2(F[G \times \mathbb{Z}/p^s\mathbb{Z}])\) is the product \(\left|K_2(F[G])\right| \cdot \prod_{i=2}^{p^s} \left|\Phi_i(F[G])\right|\). For \(i \not \equiv 1 \pmod{p}\), the groups \(\Phi_i(F[G])\) can be described in terms of the Kahler differentials \(\Omega_{F[G]/\mathbb{Z}}^1\). However the structure of \(\Phi_i(F[G])\) for \(i \equiv 1 \pmod{p}\) is far more complicated, and is related to the de Rham complex of \(F[G]\).
scientific article; zbMATH DE number 5544374