On \(K_ 3\) of Witt vectors of length two over finite fields (Q1372606)

Let \(W_2(\mathbb{F}_q)\) denote the local ring of Witt vectors of length two over \(\mathbb{F}_q\). For \(p\geq5\) the author proves \(K_3(W_2(\mathbb{F}_{p^f}))=(\mathbb{Z}/p^2)^f\oplus \mathbb{Z}/(p^{2f}-1)\) and if \((3,f)=1\) he shows \(K_3(W_2(\mathbb{F}_{3^f}))=(\mathbb{Z}/9)^{f-1}\oplus (\mathbb{Z}/3)^2\oplus \mathbb{Z}/(3^{2f}-1) \). There is a conflict with some results of Aisbett. The author attributes the problem to Prop. II 4.5 of \textit{J. Aisbett} [``On \(K_3(\mathbb{Z}/p^n)\) and \(K_4(\mathbb{Z}/p^n)\)'', in: Mem. Am. Math. Soc. 329, 1-90 (1985; Zbl 0576.18006)]. In view of the very technical nature of this type of work it comes as no surprise that a mistake has gone undetected. As in the earlier work of Aisbett, Lluis-Puebla, Snaith, Evens, Friedlander, Parshall, the theorem is based on computations in a Hochschild-Serre spectral sequence. A new ingredient is to exploit the action of the outer automorphisms of SL on this spectral sequence. Other input comes from recent work on the higher \(K\)-theory of local rings, in particular the ring of dual numbers over a finite field. For \(p=3\) the author needs to work especially hard to find a differential in the spectral sequence explicitly.