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 $\phi\_A:= max \{ d\_{i+1}-d\_i | 1 \leq i \leq r-1 \}$. 



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> Question: Is there a class of examples where $\phi\_A$ gets arbitrary large for a fixed $n$? 
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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](https://mathoverflow.net/questions/340068/projective-dimensions-of-simple-modules-in-acyclic-quiver-algebras/340070#340070) , 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 $2n-2$).



Gaps in the projective dimensions of simple modules
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 $\mathbb{N}$.


