A Tate cohomology sequence for generalized Burnside rings
The fundamental theorem for the Burnside ring \(B(G)\) of a finite group \(G\) is an exact sequence of abelian groups \[ 0 \to B(G) \to C(G) \to \bigoplus_K \mathbb{Z}/|N_G(K):K|\,\mathbb{Z} \to 0; \] here \(C(G)\) denotes the ghost ring of \(G\). The authors show that this result follows from an exact sequence of Mackey functors \[ 0 \to \widehat{\text{Ext}}^{-1}_\gamma(\rho,D) \to D_+ @>{\varphi}>> D^+ \to \widehat{\text{Ext}}^{0}_\gamma(\rho,D) \to 0. \] Here \(D\) is a restriction functor for \(G\) over a noetherian commutative ring \(R\), as defined by \textit{R. Boltje} [J. Algebra 206, No. 1, 293--343 (1998; Zbl 0913.20001)], and the definitions of \(D_+, D^+\) and \(\varphi\) go also back to this paper. Moreover, \(\rho\) and \(\gamma\) are the restriction and conjugation algebra, respectively, of \(G\), defined in a way similar to the construction of the Mackey algebra. The Ext-groups appearing above can be described in terms of Tate cohomology for groups.
scientific article; zbMATH DE number 5544384