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Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ pullbacks. If $f$ was smooth, then the two operations would agree for all sheaves up to a shift.

What if $f$ isn’t smooth? Are there still conditions we can put on $A$ to guarantee that the two pullbacks agree (up to a shift)? I am particularly interested in the case when $f$ is a closed immersion.

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    $\begingroup$ Maybe it's true when $f$ is transverse to the characteristic cycle of $A$. $\endgroup$
    – Will Sawin
    Commented Jul 15, 2021 at 13:30
  • $\begingroup$ @WillSawin could you say anything more about the intuition behind your guess? $\endgroup$
    – Exit path
    Commented Jul 15, 2021 at 15:21
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    $\begingroup$ If $Y = X \times Z$ and the inclusion $X\to Y$ is by a point in $Z$, then this is true if $A$ is pulled back from $Z$, and also in that case the transversality holds. Also I know some similar properties generalize from smooth morphisms to the characteristic-cycle-transversal condition. $\endgroup$
    – Will Sawin
    Commented Jul 15, 2021 at 17:02

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