Hecke actions on Picard groups (Q1062135)

The Hecke category \({\mathfrak H}_ G\) of a group \(G\) is defined as the category of the \({\mathbb{Z}}G\)-permutation modules \({\mathbb{Z}}G/H\) for all subgroups \(H\) of \(G\). For any given \({\mathbb{Z}}G\)-module \(M\) one defines in a natural way a contravariant additive functor \(\Phi_ M: {\mathfrak H}_ G\to\) Abelian groups, with \(\Phi_ M({\mathbb{Z}}G/H)=the\) group of fixed points \(M^ H\). One can also extend \({\mathfrak H}_ G\) with the finite direct sums of its objects. In case \(G\) is a group of automorphisms of a commutative ring \(A\), one obtains such a functor \(\Phi\) with \(\Phi ({\mathbb{Z}}G/H)=Pic A^ H\). If \(G\) is the Galois group of a finite Galois extension \(K/{\mathbb{Q}}\), one obtains a functor \(\Phi\) with \(\Phi ({\mathbb{Z}}G/H)=class\) group of \(K^ H\), thus extending a construction of \textit{R. Perlis} [J. Number Theory 10, 489-509 (1978; Zbl 0393.12009)]. Some applications to reduction theorems for the computation of class groups generalize results of \textit{H. Nehrkorn} [Abh. Math. Semin. Univ. Hamb. 9, 318-334 (1933; Zbl 0007.10303)] and of \textit{C. D. Walter} [Acta Arith. 35, 33-40 (1979; Zbl 0339.12007)].