triangular algebra finite global dimension Let $A$ be a finite dimensional $k$-algebra, and $\{S_1,...,S_n\}$ be set of all simple $A$-modules up to isomorphism. Construct a quiver $Q_A$ with vertex set $\{1,...,n\}$, and $i \to j$ is an arrow in $Q_A$ if $\text{Ext}_A^1(S_i,S_j) \neq 0$. $A$ is a triangular algebra if $Q_A$ has no oriented cycle.
I saw a statement that if $A$ is a triangular algebra, then  $gl.dim(A) < \infty$, see https://webusers.imj-prg.fr/~bernhard.keller/ictp2006/lecturenotes/delapena-all.pdf page 5 line 13.
And I don't know how to proof it any reference would be great.
 A: Let $S_1, \cdots, S_n$ be an ordering of the simple $A$-modules such that $\mathrm{Ext}^1(S_i, S_j) = 0$ for $1 \leq i \leq j \leq n$.
Suppose $M$ is a module with $\mathrm{top} M \in \mathsf{add}(S_1 \oplus \cdots \oplus S_k)$.
We claim $\mathrm{top}\,\mathrm{rad} M \in \mathsf{add}(S_1 \oplus \cdots \oplus S_{k-1})$: Indeed let $\pi \colon \mathrm{rad} M \to S $ be an epi to a simple $A$-module and considering the pushout diagram
$\require{AMScd}$
\begin{CD}
0 @>>> \mathrm{rad}M @>\iota>> M @>>> \mathrm{top} M @>>> 0 \\
@. @V \pi VV @V \pi' VV @| @. \\
0 @>>> S @>\iota'>> X @>>> \mathrm{top M} @>>> 0.
\end{CD}
Notice that $\iota$ has radical image and so does $\pi' \circ \iota = \iota' \circ \pi$.
Because $\pi$ is epic $\iota'$ has radical image and the lower sequence does not split.
Hence $\mathrm{top} M$ has a non-trivial extension by $S$ and therefore $S \in \mathsf{add} (S_1 \oplus \cdots \oplus S_{k-1})$ which shows the claim.
It follows that for any module $M$ with $\mathrm{top} M \in \mathsf{add}(S_1 \oplus \cdots \oplus S_k)$ any submodule $M' \subset \mathrm{rad} M$ has a composition series with composition factors in $\mathsf{add} (S_1 \oplus \cdots \oplus S_{k-1})$.
Now let $P_n \to P_{n-1} \to \cdots \to P_1 \to P_0 \to M$ be the start of a minimal projective resolution of $M$. Using the above it is easy to show that $P_i$ has a composition series with composition factors in $\mathsf{add}(S_1 \oplus \cdots \oplus S_{n-i})$ using that $P_i \to P_{i-1}$ is radical. In particular $P_n = 0$, i.e. $\mathrm{gldim} A \leq n-1$.
