Indecomposable Modules and AR-Components of Domestic Finite Group Schemes

Kirchhoff, Dirk

In representation theory one studies modules to get an insight of the linear structures in a given algebraic object. Thanks to the theorem of Krull-Remak-Schmidt, any finite-dimensional module over a finite-dimensional algebra can be decomposed in a unique way into indecomposable modules. In this way, one reduces this problem to the study of indecomposable modules. In this work we are interested in representation theory of the group algebra of a finite group scheme. Examples of these algebras are given by group algebras of ordinary groups or by universal enveloping algebras of Lie algebras. Our main interest lies in the finite group schemes of domestic representation type. One of the main results in this work provides a full classification of the indecomposable modules for a certain subclass of the domestic finite group schemes. Based on this classification we will make some observations regarding the Auslander-Reiten quiver and geometric invariants which lead us to more general results. The shape of these quivers is well understood for the algebras we are investigating in this work. We will give a concrete description of the Euclidean components with respect to the McKay quiver of a certain binary polyhedral group scheme. An important fact for this work is that the module category of a group scheme is closed under taking tensor products. Friedlander and Suslin proved that the even cohomology ring of a finite group scheme is a finitely generated commutative algebra. The variety defined by this algebra is the cohomological support variety of the group scheme. These varieties contain many interesting information about the representation theory of the group schemes they are assigned to. In this work we will study the ramification index of a morphism between two support varieties. As we will see, this number has a connection to the ranks of the tubes in the Auslander-Reiten quivers.

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Kirchhoff, Dirk: Indecomposable Modules and AR-Components of Domestic Finite Group Schemes. 2016.

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