Short chains and short cycles of modules

Abstract

We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not the middle of a short chain if and only if there is no short cycle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M right-arrow upper N right-arrow upper M"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M \to N \to M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module that is not the middle of a short chain, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a tilted algebra.