# Betti numbers of a Cohen-Macaulay Module in small projective dimension

I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. I have the following information about the minimal free resolution of $R_\Delta$:

1. The resolution has two twists at first level and then it is pure, i.e. $\beta_{1, j}\neq 0$ iff $j=j_1$ and $j=j_2$ for some $j_1\neq j_2$ and $\beta_{i,j}\neq 0$ for unique values of $j$ if $i\geq 2$.
2. I know all shiftings, i.e. values of $j$ when $\beta_{i, j}\neq 0$.

3. I also know the values of $\beta_{1, j_1}$ and $\beta_{1, j_2}$.

Is it possible to compute all other Betti numbers then? I am interested in something similar to Herzog-Kuhl equation.

Yes, using Boij-Söderberg theory. Your Betti table is a convex combination of precisely two pure tables, each determined by Herzog-Kuhl, and the values of $\beta_{1,j}$ let you find the coefficients of the convex combination.