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I have encountered what feels like a very basic confusion relating to the Beilinson Bersnstein localisation theorem.

This theorem in particular states that if $F$ is a $D_{\mathbb{P}^{1}}$ module, assumed quasi-coherent as an $\mathcal{O}$-module, then $H^{1}(\mathbb{P^{1}},F)=0$.

Now let $F$ be an arbitrary quasi-coherent sheaf on $\mathbb{P}^{1}$ and let $JF\rightarrow F$ denote the sheaf of jets of $F$. This can be defined as $\pi_{1*}\pi_{2}^{*}F$, where $\pi_{i}$ are the projections from the formal neighbourhood of the diagonal inside $\mathbb{P}^{1}\times\mathbb{P}^{1}$. This has a flat connection and so defines a $D$-module on $\mathbb{P}^{1}$, this is explained in the introduction to Bezrukavnikov and Kaledin's Fedosov Quantization paper. On the other hand, it appears to surject naturally onto $F$, and so induces a surjection on $H^{1}$, as there is no $H^{2}$ obstructing this.

Now this clearly contradicts my understanding of the BB localisation theorem. As far as I can tell, there are no finiteness assumptions in the localisation theorem and so this doesn't seem to resolve the issue.

The only thing I can think that might explain it is a confusion whereby one is really considering the jet sheaf as an object of the pro- category of coherent sheaves, and not as a quasi-coherent one.

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  • $\begingroup$ Isn't $JF=D_{\mathbb{P}^1} \otimes_{\mathcal{O}_{\mathbb{P}^1}} F$? What is the map $JF \to F$? $\endgroup$
    – Exit path
    Commented Feb 27, 2023 at 23:37
  • $\begingroup$ No, it is Hom(D,F) (a sort of coinduced D module), and the map to F is gotten from O ---> D. With respect to the description in post this comes down to fact that D pairs perfectly with functions on formal nhd of diagonal, as explained eg in Lurie's notes on crystals $\endgroup$
    – user108998
    Commented Feb 28, 2023 at 2:27

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