$\mathcal{D}$-quasi-isomorphisms and coherent $\Omega$-modules

2012-04-29T15:16:44Z

Let $X$ be a smooth $\mathbf{C}$-scheme of finite type. In section 7.2 of their preprint "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld define an adjunction \[\mathcal{D}:Z^0(\mathbf{dgMod}_{\mathrm{qcoh}}(\Omega^{\cdot}_{X/\mathbf{C}}))\leftrightarrows \mathbf{Cplx}(\mathbf{Mod}^{\mathrm{r}}_{\mathrm{qcoh}}(\mathcal{D}_X))):\Omega\] between the category of (closed morphisms of) quasi-coherent dg-modules over the de Rham algebra of $X/\mathbf{C}$ and that of complexes of quasi-coherent right $\mathcal{D}_X$-modules. Their aim is to obtain a derived equivalence between a certain localization of this category of dg-modules over $\Omega^{\cdot}_{X/\mathbf{C}}$ and the usual derived category of the abelian category $\mathbf{Mod}^{\mathrm{r}}_{\mathrm{qcoh}}(\mathcal{D}_X)$ . The functor $\mathcal{D}$ does not send all quasi-isomorphisms to quasi-isomorphisms and Beilinson and Drinfeld's solution is to invert the morphisms of dg-modules that are sent to quasi-isomorphisms by $\mathcal{D}$, which they call $\mathcal{D}$-quasi-isomorphisms. In 7.2.6(iii) (and also in 2.1.10 of their "Chiral algebras"), Beilinson and Drinfeld assert that any quasi-isomorphism of bounded $\mathcal{O}_X$-coherent dg- $\Omega^{\cdot}_{X/\mathbf{C}}$ -modules is a $\mathcal{D}$-quasi-isomorphism (the converse is true without the finiteness hypothesis). In light of Kapranov's paper ("On dg-modules over the de Rham complex and the vanishing cycles functor"), this sounds reasonable.

On the other hand, as suggested by 6.23 in these lecture notes, if we take $X=\operatorname{Spec}(\mathbf{C}[t])$ and consider the left $\mathcal{D}_X$-module $\mathcal{O}_X\operatorname{e}^t$ given by the free $\mathcal{O}_X$-module of rank $1$ generated by $\operatorname{e}^t$, i.e. the integrable connection $f\mapsto (f+f')\mathrm{d}t:\mathbf{C}[t]\to\mathbf{C}[t]\mathrm{d}t$, then its de Rham complex appears to be acyclic, $\mathcal{O}_X$-coherent and bounded. I think this means that the image of the corresponding right $\mathcal{D}_X$-module under the functor $\Omega$ is acyclic, bounded and $\mathcal{O}_X$-coherent, hence $\mathcal{D}$-acyclic. As the adjuction morphism $\mathcal{D}\Omega\to\operatorname{id}$ is a quasi-isomorphism by [BD, 7.2.4], the right $\mathcal{D}_X$-module corresponding to $\mathcal{O}_X\operatorname{e}^t$ is zero, which sounds absurd.

Question: Have I made a stupid error somewhere and, if not, how does one reconcile this example with Beilinson-Drinfeld's description of $\mathcal{D}$-quasi-isomorphisms for $\mathcal{O}_X$-coherent $\Omega^{\cdot}_{X/\mathbf{C}}$ -modules?

Comment by M4tth3w

2012-04-30T22:02:48Z

Thank you! Great answer!

User m4tth3w - MathOverflow

$\mathcal{D}$-quasi-isomorphisms and coherent $\Omega$-modules

Comment by M4tth3w