Gaps in the projective dimensions of simple modules

Question

Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $n$ simple modules. Let $d_1<d_2<...<d_r$ be the sequence of projective dimension of simple $A$-modules in increasing order. Define $\phi_A:= max \{ d_{i+1}-d_i | 1 \leq i \leq r-1 \}$.

Question: Is there a class of examples where $\phi_A$ gets arbitrary large for a fixed $n$?

I did not even find an example with $\phi_A >2$ (feel free to post an example if you know one).

Here are two examples for $\phi_A$:

In Projective dimensions of simple modules in acyclic quiver algebras , Jeremy Rickard proved that $\phi_A=1$ for acyclic quiver algebras.

Dag Madsen showed me an argument, which shows that $\phi_A \leq 2$ for any Nakayama algebra (this gives also a very nice proof that the global dimension of a Nakayama algebra with $n$ simple modules is at most $2n-2$).

Answer 1

For $n>0$, let $A_n=KQ_n/I_n$ as follows: $Q_n$ is the quiver with vertex set $\{1,2\}$ and arrows $\alpha_i\colon 1\to 2$ and $\beta_i\colon 2\to 1$ for $1\leq i\leq n$, and $I_n$ is generated by $\beta_j\alpha_i$ for $i\leq j$ and $\alpha_j\beta_i$ for $i\leq j-1$ (reading composition from right-to-left). That is, to write down a path that is non-zero in $A_n$, an $\alpha$ may only be followed by a $\beta$ with a strictly lower index, and a $\beta$ may only be followed by an $\alpha$ with an equal or lower index.

These algebras were introduced by Green in Example 2.1 of Remarks on projective resolutions, who shows that the projective dimensions of $S(1)$ and $S(2)$ are $2n$ and $2n-1$ respectively. (Here I am taking only the even terms of Green's sequence since they are marginally easier to describe and still answer the question.)

Now let $A_n'=KQ_n'/I_n'$ as follows: $Q'_n$ is the quiver obtained from $Q_n$ by adding a vertex $3$ and a single arrow $2\to 3$ , and $I'$ is the ideal of $KQ_n'$ with the same generators as those given above for $I$. Then the simple module $S(3)$ for $A_n'$ is projective, i.e. has projective dimension zero, but one can check combinatorially that the simple modules $S(1)$ and $S(2)$ have the same projective dimension as the simple modules for $A_n$ of the same name. Thus $\Phi_{A'_n}=2n-1$ gets arbitrarily large.

The same construction will also work for the odd terms of Green's sequence, showing that $\Phi_A$ can take any value in $\mathbb{N}$.