The source algebras of nilpotent blocks over arbitrary ground-fields
Let \(\mathcal O\) be a complete discrete valuation ring with residue field \(k\) of prime characteristic \(p\) that is not necessarily algebraically closed, let \(G\) be a finite group, set \(A = {\mathcal O} G\) and let \(b\) be a block idempotent of \(A\). Clearly \(A\) is an interior \(G\)-algebra. For any subgroup \(H\) of \(G\), a conjugacy class \(\beta\) of primitive idempotents of \(A^ H\) is said to be a point of \(H\) on \(A\) and the pair \(H_ \beta\) is called a pointed group on \(A\). For any \(i \in \beta\), \(iAi\) is a subalgebra of \(A\) and is unique up to an \(H\)-algebra embedding so set \(A_ \beta \equiv iAi\). For \(\alpha = \{b\}\), a point of \(G\) on \(A = {\mathcal O}G\), \(G_ \alpha\) is a pointed group on \(A\) and \(A_ \alpha = {\mathcal O}Gb\). If \(H_ \beta\), \(K_ \gamma\) are two pointed groups on \(A\), then \(K_ \gamma\) is contained in \(H_ \beta\) \((K_ \gamma \subset H_ \beta)\) if \(K \leq H\) and there are \(j \in \gamma\), \(i \in \beta\) such that \(ji = j = ij\). A pointed group \(Q_ \delta\) is said to be local if \(\text{Br}_ Q(\delta) \neq 0\) (hence \(Q\) is a \(p\)-subgroup). The maximal local pointed groups \(P_ \gamma\) contained in \(G_ \alpha\) (where \(\alpha = \{b\}\)) are called the defect pointed groups of \(G_ \alpha\) and are unique up to \(G\)-conjugation and the \(P\)-algebra \(A_ \gamma\)-is defined to be the source algebra of \(\alpha = \{b\}\). The local category of \(A_ \alpha\) as a \(G\)-algebra is defined by: objects are the local pointed groups on \(A_ \alpha\) and the set of morphisms \(E_ G(Q_ \delta,R_ \varepsilon)\) from \(Q_ \delta\) to \(R_ \varepsilon\) is the set \(\{g \in G \mid (Q_ \delta)^ g \subset R_ \varepsilon\) modulo \(\text{Inn}(R)\}\). A local pointed group \(Q_ \delta\) on \(A_ \gamma\) is also a local pointed group on \(A_ \alpha\) and \(E_ P(Q_ \delta,R_ \varepsilon) \subseteq E_ G(Q_ \delta,R_ \varepsilon)\). Thus there is an inclusion functor from the local category of \(A_ \gamma\) to the local category of \(A_ \alpha\). When \(k\) is algebraically closed, then Puig has shown that \(b\) is a nilpotent block if and only if the inclusion functor of the local category of \(A_ \gamma\) to the local category of \(A_ \alpha\) is part of a categorial equivalence (i.e., \(E_ P(Q_ \delta,R_ \varepsilon) = E_ G(Q_ \delta,R_ \varepsilon)\) for any two local pointed groups \(Q_ \delta\) and \(R_ \varepsilon\) on \(A_ \gamma\)). This paper is devoted to generalizing the fundamental results of Puig on the source algebras of nilpotent blocks to the case that \(k\) is not necessarily algebraically closed. Definition (1.1). The \({\mathcal O}\)-block \(b\) of \(G\) is \(\mathcal O\)-nilpotent if \(G_ \alpha\) is locally controlled by \(P_ \gamma\) (i.e., \(E_ G(Q_ \delta,R_ \varepsilon) = E_ P(Q_ \delta,R_ \varepsilon)\) for any two local pointed groups \(Q_ \delta\), \(R_ \varepsilon\) contained in \(P_ \gamma\)). Let \(\widehat{\mathcal O}\) be the integral extension of \(\mathcal O\) by adjoining all \(\varphi(x')\) where \(\varphi\) is an irreducible Brauer character in \(b\) and \(x'\) is a \(p'\)-element of \(G\). Main Theorem (1.3). The following two statements are equivalent: (1.3.1) \(b\) is an \(\mathcal O\)-nilpotent block of \(G\); (1.3.2) there is an interior \(P\)-algebra \(S\) over \(\mathcal O\) which is isomorphic to the matrix algebra \(M_ n(\widehat{\mathcal O})\) as an \(\mathcal O\)-algebra and such that \(A_ \gamma \cong S \otimes_{\mathcal O} {\mathcal O}P\) as interior \(P\)-algebras over \(\mathcal O\). When these conditions hold, \((p,n) = 1\) and as a \(P\)-algebra, \(S\) has a \(P\)-stable \(\widehat{\mathcal O}\)-basis \(B\) such that \(C_ B(P) = \{1_ S\}\) and \(S\) is unique up to isomorphism. This paper also contains an interesting example (1.6) and results on the Brauer categories in this context.
scientific article; zbMATH DE number 589963