On local categories of finite groups.
Let \(\mathcal P\) be a partially ordered set and let \(G\) be a group. Then \(\mathcal P\) is a \(G\)-poset if there is a group homomorphism \(G\to\Aut(\mathcal P)\) giving an action of \(G\) on \(\mathcal P\). The transporter category \(G\propto\mathcal P\) has the same objects as \(\mathcal P\) and morphisms from \(x\) to \(y\) are couples \((g,gx\leq y)\) where \(g\in G\) and where \(\leq\) is the order relation of \(\mathcal P\). For a small category \(\mathcal C\) denote the category algebra \(k\mathcal C\) as the \(k\) vector space with basis given by the morphisms in \(\mathcal C\) and composition of morphisms induces the multiplication in \(k\mathcal C\). The composition of two basis elements is put to \(0\) if they do not compose in \(\mathcal C\). The author recalls the main properties and basic examples of these constructions. In particular, the author recalls that for a category algebra one can always construct a trivial module, such as for group algebras, and Ext-groups including a notion of a cup product. The author shows as main result that for two modules \(\mathcal M\) and \(\mathcal N\) over \(k(G\propto\mathcal P)\), then \(\mathrm{Ext}^*_{k(G\propto\mathcal P)}(\mathcal{M,N})\) is finitely generated as \(\mathrm{Ext}^*_{k(G\propto\mathcal P)}(k,k)\)-modules. The author gives applications to fusion systems.
scientific article; zbMATH DE number 6116864
scientific article