We provide two examples of D-modules in prime characteristic p that answer two open questions in [G. Lyubeznik, 'A characteristic-free proof of a basic result on D-modules', J. Pure Appl. Alg. 215 (2011) 2019-2023] in the negative
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The second author gratefully acknowledges support through NSF grant DMS 0701127. The first author wishes to acknowledge support through EPSRC grant EP/H040684/1. The third author is partially supported by NSF grant DMS 1068946.