How to prove a Proposition of Rouquier?

Question

Proposition 7.15 in Rouquier's paper (see Publication paper) "Dimension of triangulated categories J. K-theory, 1 (2008) 193-256" as follows, for details please see Rouquier's paper (or arXiv):

``Let $A$ be a noetherian finite dimensional $k$-algebra of global dimension $d\in \mathbf{N}\cup \{\infty\}$. Assume $k$ is perfect. Then, $d$ is the minimal integer $i$ such that $A\mathrm{-perf}=\langle A\rangle_{i+1}$.''

I am trying to give a proof as follows:

1. If the global dimension is $\infty$, then Proposition 7.15 is hold.

2. Assume that the global dimension is finite. Denote by $min$ the minimal integer $i$ such that $A\mathrm{-perf}=\langle A\rangle_{i+1}$.

   (a) Proposition 7.4 and Proposition 7.25 implies that $A\mathrm{-perf}=\langle A\rangle_{d+1}$, and this implies that $min\leq d$.

   (b) Suppose that there is an $A$-module $W$ such that $\mathrm{pdim}\,W\geq min+1$. Then Lemma 7.13 implies that $W\notin\langle\mathrm{ads}(A)\rangle_{min+1}$. Then there should be a contradiction ! Hence, for any $A$-modules $W$, $\mathrm{pdim}\,W\leq min$, therefore $d\leq min$.

Note that Lemma 7.13, Proposition 7.4 and Proposition 7.25 are in Rouquier's publication paper, for details please see Rouquier's paper.

Q. What I don't understand is that why there is a contradiction?

Answer 1

Since $W$ has finite projective dimension, it is isomorphic in the derived category to a bounded complex of projective $A$-modules. Thus it belongs to $A$-perf = $\langle A \rangle_{min + 1}$. That contradicts Lemma 7.13 as you stated in (b).