Minimal Attached Primes of Local Cohomology Modules of Binomial Edge Ideals of Block Graphs

Abstract

We calculate the minimal attached primes of the local cohomology modules of the binomial edge ideals of block graphs. In particular, we obtain a combinatorial characterisation of which of these modules are non-vanishing. We also show that the main result of this paper follows from a recent result of Lax, Rinaldo, and Romeo (arXiv:2405.08671, Theorem 3.2), which was published independently during the writing of this paper. This provides a short alternative proof of our result.

17 pages, v3 replaces the erroneous propositions A.5 and A.6 and fixes typos; v2 replaces what was Section 5 in v1 to take into account recent work of Lax, Rinaldo, and Romeo (arXiv:2405.08671), in which they had in fact already proved a result we stated as a conjecture. Many thanks to Ernesto Lax for bringing this to our attention