A limit on the length of the indecomposable modules over a hereditary algebra (Q1090403)

Let k be an algebraically closed field and A be a hereditary algebra of finite representation type, that is, there are only finitely many nonisomorphic indecomposable finitely generated A-modules. (It is known that then A is Morita equivalent to a finite product of path algebras of quivers whose underlying unoriented graphs are Dynkin diagrams.) For any A-module N denote by P(N) (resp. I(N), Soc(N), \(\ell (N))\) projective cover (resp. injective hull, socle, length) of N. The author proves that under the above assumptions for any indecomposable finitely generated A-module K holds the inequality \[ \ell (P(Soc(K))+\ell (I(K))-\ell (K)\leq \ell (M) \] where M is an indecomposable module of maximal length. - Let now R denote the trivial extension of A by its injective cogenerator \(Hom_ k(A,k)\), and p(R) the number of nonempty preprojective classes in the preprojective partition for R. It is also proved in the paper that if \(X_ 0\) is an indecomposable R-module of maximal length, then \[ p(R)-1\leq \ell (X_ 0)\leq p(R) \] and \(\ell (X_ 0)=p(R)\) iff the indecomposable A-module of maximal length is projective or injective.