  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
$J^{(j)}$.
