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

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?

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.