Donovan's conjecture, crossed products and algebraic group actions
Donovan's conjecture is that for a given finite \(p\)-group \(D\), there are only finitely many Morita classes of \(p\)-blocks of group algebras of finite groups with defect group \(D\). Here, the group algebras are to be taken over an algebraically closed field of characteristic \(p\). The author reduces this conjecture to the case where the ambient finite group is generated by conjugates of \(D\). The proof uses Dade's theory of block extensions. Some of the steps of the proof are of independent interest. To wit, it is shown that if \(G\) is a finite group and \(C\) is a commutative \(G\)-algebra over an algebraically closed field \(F\) whose characteristic does not divide \(|G|\), then the cohomology groups \(H^i(G,U(C))\) are finite, where \(i\geq 1\) and \(U(C)\) is the group of units of \(C\). Also, if \(H\) is any linear algebraic group over \(F\), then there are only finitely many homomorphisms \(G\to H\), up to \(H\)-conjugacy.
scientific article; zbMATH DE number 836169
scientific article