Tilting theory

Remarks

Example: $T = A$ is a tilting module, since it is projective and its indecomposable summands are the indecomposable projectives, one for each simple module. Example: let $A$ be the path algebra of $1 \to 2$, with $S_2 = P_2$, $P_1$ and $S_1$ as indecomposables, and let $T = P_1 \oplus S_1$. The projective resolution $0 \to P_2 \to P_1 \to S_1 \to 0$ gives clause 1. Clause 2 holds since $P_1$ is projective and injective and $\operatorname{Ext}^1_A(S_1, S_1) = 0$, and clause 3 holds with $2$ summands. Non-example: $T = S_1 \oplus S_2$ fails clause 2, since the non-split sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ gives $\operatorname{Ext}^1_A(S_1, S_2) \ne 0$. Happel proved that for a tilting module $T$ the bounded derived categories of $\operatorname{mod} A$ and $\operatorname{mod} B$ are equivalent as triangulated categories.