On a deformation theory of finite dimensional modules over repetitive algebras

Abstract

Let Λ be a basic finite dimensional algebra over an algebraically closed field k, and let Λ be the repetitive algebra of Λ. In this article, we prove that if V is a left Λ-module with finite dimension over k, then V has a well-defined versal deformation ring R( Λ, V ), which is a local complete Noetherian commutative k-algebra whose residue field is also isomorphic to k. We also prove that in this situation R( Λ, V ) is stable after taking syzygies and that R( Λ, V ) is universal provided that End Λ ( V ) = k. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over P 1 k .2010 Mathematics Subject Classification. 16G10 and 16G20 and 16G70. Key words and phrases. Repetitive algebras and (Uni)versal deformation rings and stable endomorphism rings and Frobenius categories.