Partial tilting complexes and beyond

Some “proper” non classical partial tilting modules T have the following property: even though their projective resolution T ° is not a tilting complex, for every non - zero module M there is a morphism from T ° to the projective resolution of M which is not homotopic to zero. On the other hand, a characterization of tilting complexes given by Y. Miyachi [M] guarantees the existence of non - zero right bounded complexes C ° (with projective components) with the property that any morphism from T ° to any shift of C ° is homotopic to zero. Confining ourselves to indecomposable complexes C° with a “combinatorial” structure, suggested by Shaps and Zachay - Illouz [S- ZI] (i.e. with the property that every non - zero component of C ° is an indecomposable projective module), we will show that there is no restriction on the cardinality of a representative system (up to shift)of the indecomposable complexes C ° as above. We will also see that complexes with injective components play a big role. [M] J. Miyachi , “Extensions of rings and tilting complexes” , J. Pure Appl. Algebra 105, (1995) 183 - 194 . [S - ZI] M. Shaps and E. Zachay Illouz, Combinatorial partial tilting complexes for the Brauer star algebras, M. Dekker (2002), 187 - 207 .