# How to prove a Proposition of Rouquier?

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?

• how can it be so that some statements in the text indicate that it's not clear how to prove some other statement in the same text?! Commented Aug 18, 2014 at 17:44
• The paper is in arXiv. Commented Aug 18, 2014 at 17:45
• It might help if you gave a statement of the Proposition and explained what you do and don't understand about the proof. Commented Aug 18, 2014 at 17:49
• If an edit such as Jeremy Rickard suggests is made I will vote to reopen. I think questions like this should be allowed in general, though obviously one has to give some background and point to where the difficulty lies Commented Aug 18, 2014 at 23:07
• I am sorry for the first time edit! Commented Aug 19, 2014 at 3:38

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).
• We know that objects in $A-perf$ are bounded complexes of finite generated projective A-modules, why $W \in A-perf$? Commented Aug 20, 2014 at 2:54
• @ParksJonehan There is always a finitely generated (in fact, cyclic) module whose projective dimension is equal to the global dimension of the ring. So $W$ can be chosen to be finitely generated, in which case, if the global dimension is finite and the ring is noetherian, its minimal projective resolution is a bounded complex of finitely generated projectives, so $W$ is perfect. Commented Aug 20, 2014 at 18:45