The Number Of Complements Of Almost Complete Tilting Module Over M-replicated Algebra

Let A be a hereditary algebra over an algebraically closed field k and A(m) be the m-replicated algebra of A.In this paper,first.we introduce the notion of tilting module and some tilting module theories that has proved;And the case that a faithful basic almost complete tilting A(1)-module T with pdA(1)T≤1has4non-isomorphic complements;second,we prove that a faithful basic almost complete tilting A(m)-module T with pdA(m)T≤m has2m+2non-isomorphic complements if and only if the complement X to T with pdA(m)X=2m is such that E(X) of X is projective. And prove that if a faithful basic almost complete tilting A(m)-module T has a complement X with pdA(m)X=1such that P(X) is injective, T has2m+1non-isomorphic complements;at last,we also prove that the case after we enlarge the projective dimension of tiling module.