Abstract: 

After a brief introduction, I will present some details of the proof of the result we have obtained with O. Leon Sanchez, which says that if D is a local algebra, then the theory of fields with D-operators satisfying a general compatibility condition is always companionable in characteristic zero, and is companionable in positive characteristic if the maximal ideal of D coincides with the kernel of the Frobenius homomorphism on D.

Examples falling into this framework include the theory of fields with commuting D-operators, the theory of fields with Lie-commuting derivations, the theory of fields with iterative Hasse-Schmidt derivations, and, more generally, the theory of fields with finite group scheme actions.

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