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We extend a classical theorem of Hartshorne concerning the connectedness of the punctured spectrum of a local ring by analyzing the homology groups of a simplicial complex associated with the minimal primes of a local ring., Comment: Results on Stanley-Reisner rings have been added. Comments welcome!

We provide two examples of $\mathcal{D}$-modules in prime characteristic $p$ which answer two open problems in \cite{Lyubeznik} in the negative.

We prove a result relating the Jacobian ideal and the generalized test ideal associated to a principal ideal in $R=k[x_1,...,x_n]$ with $[k:k^p]0$. As a consequence of this result, we establish an upper bound on the number of $F$-jumping coefficients of a principal ideal with an isolated singularity.