4
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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:

$\endgroup$
4
  • 1
    $\begingroup$ Thanks @ Zach. I did not know much about Boij-Söderberg but I see now. It is useful to me. $\endgroup$
    – Singh
    Aug 27 '18 at 8:59
  • 1
    $\begingroup$ @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? $\endgroup$ Aug 21 '19 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
    Sep 9 '19 at 14:22
  • $\begingroup$ @Singh You have to do it because it's your question. meta.mathoverflow.net/a/3735/88133 $\endgroup$ Sep 10 '19 at 0:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.