Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\Delta$ in $S$. It is well-known that the Betti numbers (graded or otherwise) appearing in the free resolution of $I$ may depend on the ground field $\mathbb{k}$. A classic example of this is the minimal triangulation of the real projective plane with Stanley-Reisner ideal
$$I = \langle abc, abe, acf, ade, adf, bcd, bdf, bef, cde, cef \rangle.$$
When the characteristic of $\mathbb{k}$ is $2$ then the projective dimension of $I$ (as an $S$-module) is 3 while in all other cases it is 2. This example and others can be found in Dalili and Kummini's paper [Dependence Of Betti Numbers On Characteristic](https://arxiv.org/pdf/1009.4243.pdf) and the references there.


In all of the examples I've seen of complexes whose Betti numbers are characteristic-dependent, the offending characteristic has always been 2, and only 2. That is, the Betti numbers are stable in all cases except when the characteristic of $\mathbb{k}$ is 2. My intuition tells me there must be ideals with other offending characteristics. As I have no idea how to construct one and my computer experiments (using [RandomIdeals](http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.12/share/doc/Macaulay2/RandomIdeals/html/index.html)) have come up empty my questions are the following.


**Question** 1: Are there known examples of square-free monomial ideals with characteristic-dependent Betti numbers for characteristics other than two?


**Question** 2: Is there one whose only offending characteristic is 3?



Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Consider the presentation complex of $\mathbb{Z}/3\mathbb{Z}=\langle x|x^3=1 \rangle$, consisting of a full triangle with all three edges identified in the same orientation. It has a triangulation with 9 vertices and facets $abd,acd,abf,acf,abe,ace,def,bcg,ceg,deg,bdg,bch,cfh,efh,beh,bci,cdi,dfi,bfi$.


The projective dimension of the corresponding Stanley-Reisner ideal is 6 in characteristic 3 and 5 in all others characteristics, which answers question 2 positively.


