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On extensions for gentle algebras. (English) Zbl 1465.16010

This paper concerns the representation theory of gentle algebras. An algebra is gentle if it is Morita equivalent to an algebra defined by quiver and relations with some restrictive properties on the quiver and relations. Although these properties are quite strong, gentle algebras nevertheless have a variety of applications. Gentle algebras are tame, which means that their indecomposable modules can be classified, and this has been done in general; the indecomposable modules are the so-called string and band modules. The aim of the paper under review is to compute the Ext\(^1\) space between indecomposable modules. The method of proof involves direct calculations with cohomology, but made much more accessible via a diagrammatic approach; in fact, the paper is surprisingly pleasant to read.
At the start of the paper the authors give a good account of the literature on gentle algebras, so this paper is a very good starting point for readers wanting to get into this subject.

MSC:

16G10 Representations of associative Artinian rings
16E35 Derived categories and associative algebras
05E10 Combinatorial aspects of representation theory
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