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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.

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A more general argument is that if you delete T from the minimal projetive resolution, you get a tilting complex. (see for example the book by Happel) But by definition, for a tilting complex M, add(M) generates the bounded homotopy category of finitely generated projective modules, and this is here only possible if the A-proj =$add(\bigoplus P_i)$ condition is fullfilled.

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  • $\begingroup$ Do you mean this book "Triangulated Categories in the Representation Theory of Finite Dimensional Algebras"? Can you tell me which page talked about tilting complex? Thank you. $\endgroup$ – Xiaosong Peng Sep 20 '16 at 0:57
  • $\begingroup$ gdz.sub.uni-goettingen.de/pdfcache/PPN358147735_0062/… page 344 shows that a tilting module induces a derived equivalence and to this derived equivalence corresponds a tilting complex which is given by the minimal projective resolution where T is deleted. The rest of the information can be found in chapter 6 of the recent book by Zimmermann. $\endgroup$ – Mare Sep 20 '16 at 5:08

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