Tilting modules and a theorem of Hoshino

Let k be an algebraically closed field, and A a finite dimensional k-algebra, which we shall assume, without loss of generality, to be basic and connected. By module is meant throughout a finitely generated right A-module. Following Happel and Ringel [10], we shall say that a module Tλ is a tilting (respectively, cotilting) module if it satisfies the following three conditions: (1) (2) (3) the number of non-isomorphic indecomposable summands of T equals the rank of the Grothendieck group K0(A) of A.

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