Indecomposables live in all smaller lengths. (Q2836482)

The author shows that there are no gaps in the lengths of indecomposable objects in \(k\)-abelian categories over a field \(k\) provided all simple objects are absolutely simple.NEWLINENEWLINE One of the main results of the paper is the following theorem. Theorem. Let \(A\) be an associative algebra of finite dimension over an algebraically closed field \(k\). If there is an indecomposable \(A\)-module of length \(n\geq 2\), there is also one of length \(n-1\).NEWLINENEWLINE The presented proof of this theorem splits into two cases: \(A\) is not distributive or \(A\) is distributive. In the first case an indecomposable module of countable dimension is constructed. This module has an indecomposable subquotient of dimension \(m\), for any \(m\). In the distributive case covering techniques are applied.NEWLINENEWLINE The following corollaries of this theorem are also presented in the paper.NEWLINENEWLINE Corollary. Let \(C\) be an abelian category over an algebraically closed field \(k\). Suppose that all simple objects in \(C\) have endomorphism algebra \(k\). If there is an indecomposable object in \(C\) of length \(n\geq 2\), there is also one of length \(n-1\).NEWLINENEWLINE The next corollary the author calls ``The naive criterion for finite representation type''.NEWLINENEWLINE Corollary. Let \(A\) be an algebra of finite dimension over an algebraically closed field \(k\). The following conditions are equivalent. (a) \(A\) is representation finite. (b) There is a natural number \(n\) such that there is no indecomposable \(A\)-module of that length.