Indecomposables live in all smaller lengths

Let Λ be a finite‐dimensional k ‐algebra with k algebraically closed. Bongartz has recently shown that the existence of an indecomposable Λ‐module of length n > 1 implies that also indecomposable Λ‐modules of length n − 1 exist. Using a slight modification of his arguments, we strengthen the assertion as follows: If there is an indecomposable module of length n , then there is also an accessible one. Here, the accessible modules are defined inductively, as follows. First, the simple modules are accessible. Second, a module of length n ⩾ 2 is accessible provided it is indecomposable and that there is a submodule or a factor module of length n − 1 which is accessible.