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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.

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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.

In case you are not familiar with Boij-Söderberg theory, some introductory treatments are available:

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    $\begingroup$ Thanks @ Zach. I did not know much about Boij-Söderberg but I see now. It is useful to me. $\endgroup$
    – Singh
    Commented Aug 27, 2018 at 8:59
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    $\begingroup$ @Singh I'm glad that the information was useful! Do you have any follow-up questions? If not, would you please consider accepting the answer, so that the question is marked as having been answered? $\endgroup$
    – Zach Teitler
    Commented Aug 21, 2019 at 15:21
  • $\begingroup$ @ Zach I do not have any follow up questions. You can make it as an answered question. Thanks again $\endgroup$
    – Singh
    Commented Sep 9, 2019 at 14:22
  • $\begingroup$ @Singh You have to do it because it's your question. meta.mathoverflow.net/a/3735/88133 $\endgroup$
    – Zach Teitler
    Commented Sep 10, 2019 at 0:56

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