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Let $X$ be a smooth Noetherian scheme over a field $K$. It is known that every coherent module with integrable connection on $X$ is locally free.

Is the same true for coherent modules with log-integrable connection with respect to a normal crossing divisor on $X$?

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    $\begingroup$ No, e.g. the structure sheaf of a component of the divisor is a quotient of $\mathcal{O}_X$ in the category of log connections, because its ideal is preserved by log derivations. $\endgroup$
    – Piotr Achinger
    Commented Feb 26 at 22:06

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