In this thesis, we apply local cohomology to the study of binomial edge ideals in order to establish three main results:
1. A combinatorial characterisation of graphs whose binomial edge ideals are of König type.
2. The calculation of the local cohomology modules of the binomial edge ideals of the complements of connected graphs of girth at least 5.
3. The calculation of the minimal attached primes of the local cohomology modules of the binomial edge ideals of block graphs.
These first appeared in https://arxiv.org/abs/2310.14410, https://arxiv.org/abs/2312.01613, and https://arxiv.org/abs/2406.14368 respectively.
We employ several prime characteristic tools, such as the Frobenius functor and Lyubeznik’s ℋ-functor, in the proofs of some of these results, and prove several results of independent interest concerning these tools.
Specifically, we use Lyubeznik’s ℋ-functor to establish a correspondence between F-stable secondary representations and primary decompositions in the category of F-finite F-modules, and construct a concrete example to demonstrate that there are rings for which this category is not closed under taking primary decompositions.
We also include a chapter compiling many graded analogues of local results, including some which do not seem to have appeared previously in the literature, which we hope will be of value to other researchers.
