Almost complete tilting modules

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite-dimensional hereditary algebra over an algebraically closed field. All modules will be finite-dimensional left <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules. We are concerned with partial tilting modules which can be completed to a tilting module by one indecomposable module which will be called a complement. As a main result we show that such a partial tilting module allows (up to isomorphism) at most two complements and there are two such complements if and only if the partial tilting module is sincere.