On solvability of the automorphism group of a finite-dimensional algebra
Alexander Perepechko
TL;DR
This work completes Halperin's conjecture by extending Schulze's solvability criterion from local to global finite-dimensional algebras and showing that the unity component of the automorphism group is solvable for complete intersections. It analyzes extremal algebras (where $\dim(I/\frak m I)=l+n-1$) to understand when ${\rm Der} S$ can be non-solvable, showing such algebras decompose as a tensor product $S_1\otimes S_2$ with a specific $S_1$; this structure underpins a self-contained path to Yau's theorem, especially in the quasi-homogeneous case. The paper also develops a global criterion that reduces to local cases, provides lower bounds and constructions for automorphism groups (including a large unipotent subgroup) and presents an explicit unipotent example, thereby enriching the toolkit for understanding automorphism solvability in complete intersections and related moduli algebras.
Abstract
Consider an automorphism group of a finite-dimensional algebra. S. Halperin conjectured that the unity component of this group is solvable if the algebra is a complete intersection. The solvability criterion recently obtained by M. Schulze provides a proof to a local case of this conjecture as well as gives an alternative proof of S.S.--T. Yau's theorem based on a powerful result due to G. Kempf. In this note we finish the proof of Halperin's conjecture and study the extremal cases in Schulze's criterion, where the algebra of derivations is non-solvable. This allows us to reduce a direct, self-contained proof of Yau's theorem.