Is there a simple description for $D$ modules over $\text{Spec}\left(k\left[x,y \right] / \left(xy\right) \right)$ (say k is algebraically closed of characteristic 0)? Is there a description of the $D$-module pushforward from its normalization?
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3$\begingroup$ The ($\infty$-categorical) theory of D-modules satisfies descent along proper surjective maps (see Gaitsgory-Rozenblyum “Crystals and D-modules”). For a nodal curve this implies that a D-module is equivalently a D-module on the normalization with identifications on the nodes $\endgroup$– Exit pathCommented Feb 27, 2023 at 21:18
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If $C$ and $D$ are the two branches of the nodal curve, and $p\in C$ and $q\in D$ are the points that are identified to form $X=C\cup D$, then I think that a $\mathcal D$-module on $X$ consists of:
• A vector bundle $V$ with flat connection over $C$.
• A vector bundle $W$ with flat connection over $D$.
• A vector space $U$.
• A injective linear map $V\otimes \Omega^1_C|_{\{p\}}\to U$.
• A injective linear map $W\otimes \Omega^1_D|_{\{q\}}\to U$.
answered Feb 27, 2023 at 20:50
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2$\begingroup$ Why not general $\mathcal D$-modules on $C$ and $D$? $\endgroup$ Commented Feb 27, 2023 at 20:56
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2$\begingroup$ As @WillSawin points out, I think this is incorrect as stated. For example, how would one obtain a skyscraper D-module at a smooth point using this description? $\endgroup$ Commented Feb 27, 2023 at 21:29