Let A be an algebra. We denote by by A-proj the full subcategory of A-mod consisting of projective modules. An A-module T is called a tilting module if $proj.dim(_{A}T)=n < \infty$, $Ext_{A} ^{j} (T,T) =0$ for all $j > 0$, and there is an exact sequence $ 0 \rightarrow _A A \rightarrow X_0 \rightarrow X_1 \rightarrow \cdots \rightarrow X_n \rightarrow 0$ in A-mod with all $X_j \in add(T)$.

Let T be a tilting A-module of projective dimension $n \geq 1$, then fix a minimal projective resolution of T as follows: $ 0 \rightarrow P_n \rightarrow P_{n-1} \rightarrow \cdots \rightarrow P_1 \rightarrow P_0 \rightarrow T \rightarrow 0$.

Then can anyone tell me why A-proj =$ add(\oplus _{i=0} ^n P_i)$? Thank you.