Local Cohomology and Degree Complexes of Monomial Ideals

This paper examines the dimension of the graded local cohomology $H_\mathfrak{m}^p(S/K^s)_\gamma$ and $H_\mathfrak{m}^p(S/K^{(s)})$ for a monomial ideal $K$. This information is encoded in the reduced homology of a simplicial complex called the degree complex. We explicitly compute the degree complexes of ordinary and symbolic powers of sums and fiber products of ideals, as well as the degree complex of the mixed product, in terms of the degree complexes of their components. We then use homological techniques to discuss the cohomology of their quotient rings. In particular, this technique allows for the explicit computation of $\text{reg} ((I + J + \mathfrak{m}\mathfrak{n})^{(s)})$ in terms of the regularities of $I^{(i)}$ and $J^{(j)}$.