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Consider an holonomic D-module on a smooth algebraic variety $X$ over a field $k$ of caracteristic 0.

Let $i: Y \hookrightarrow X$ be a regular embedding.

$Li^* M = \mathcal{D}_{Y\to X} \otimes^L_{i^{-1}\mathcal{D}_X} i^{-1}M$

As an $\mathcal{O}_Y$-module we have $Li^* M = (\mathcal{O}_Y\otimes_{i^{-1}\mathcal{O}_X} i^{-1}\mathcal{D}_X) \otimes^L_{i^{-1}\mathcal{D}_X} i^{-1}M = \mathcal{O}_Y \otimes^L_{i^{-1}\mathcal{O}_X} i^{-1}M$ so we can compute $i_*L i^*M = i_*\mathcal{O}_Y \otimes_{\mathcal{O}_X}^L M\in D^bQCoh(X)$ using the Koszul resolution of $i_* \mathcal{O}_Y$ as a $\mathcal{O}_X$-module.

But this is not a resolution by $i_*\mathcal{D}_Y$-modules so how can you compute the action of $\mathcal{D}_Y$ on $i_*Li^* M$?

Thanks

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  • $\begingroup$ It might help that in this case the answer in the end has to be the algebraic local cohomology of $M$ w.r.t. $Y$ which always has a canonical action of $\mathcal{D}_X$. $\endgroup$
    – Saal Hardali
    Commented Jul 29, 2017 at 22:52

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