The tilting module correpsonding to a tilting object of cluster category $\mathcal{C}$

Question

I am reading the paper "representation dimension of cluster-concealed algebras", the link is here: https://arxiv.org/pdf/1102.1048v1.pdf

Let $H$ be a finite dimensional hereditary algebra. $\mathcal{C}$ is the cluster category associated to $H$.

In section 2.2 of this paper, there is a theorem: each basic tilting module over $H$ induces a basic tilting object for $\mathcal{C}$ and each basic tilting object in $\mathcal{C}$ is induced by a basic tilting module over a hereditary algebra $H'$, derived equivalent to $H$

At the start of section 3, there are the following words: Let $\widetilde{T}$ be a tilting object in a cluster category $\mathcal{C}$ and let $B=End_{\mathcal{C}}(\widetilde{T})$ be the associated cluster-tilted algebra. To simplify some proofs, we choose without lose of generality $T$ and $\tau T$ without projective summands.

I want to know that why we could choose $T$ and $\tau T$ without projective summands without lose of generality? Could the theorem in section 2.2 make sure we choose $T$ such that $T$ and $\tau T$ without projective summands?

Answer 1

The authors of the paper are only interested in the endomorphism algebra $B$ of $\tilde{T}$. If one applies any automorphism $\Phi$ to $\tilde{T}$, one gets an isomorphism between $End_{\mathcal{C}}(\tilde{T})$ and $End_{\mathcal{C}}(\Phi(\tilde{T}))$.

In the last sentence of the first paragraph of Section 3, the authors also assume that $H$ is of infinite representation type. Thus there exists an integer $n$ such that $\tau^n \tilde{T}$ and its Auslander-Reiten translation do not have any projective direct summand. Then $B \cong End_{\mathcal{C}}(\tau^n\tilde{T})$.

Thus the authors can assume, without loss of generality, that $\tilde{T}$ and $\tau\tilde{T}$ do not have any projective direct summands.