# Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully someone knowledgeable could help me.

Authors claim that for any (connected) algebraic stack $$Y$$ and any compact object $$M$$ of the DG category of D-modules on $$Y$$ the map $$H_{dR}(Y)\otimes M\rightarrow M \qquad (*)$$ vanishes on a sufficiently high power of the augmentation ideal of $$H_{dR}(Y)$$.

The first difficulty is that $$H_{dR}$$, I believe, was only defined for the stack $$pt/G$$ where $$G$$ is a connected reductive group over an algebraically closed field of characteristic 0. So I am not completely sure what is that supposed to mean for other stacks.

The second difficulty is what would the map (*) be.

The third difficulty is why do we talk about augmentation ideal. Is it true that the image of the map $$H_{dR}(pt/G)\rightarrow H_{dR}(Bun_G)$$ lies in the augmentation ideal?

The fourth difficulty is how do we actually show that the above claim is true?

• A title related to the mathematical contents would be better.
– YCor
Dec 12, 2018 at 21:55

Let me try to resolve your difficulties. I will take for granted the existence of a dg (or equivalent, stable $$k$$-linear infinity-) category of $$\mathcal{D}$$-modules on $$Y$$, satisfying standard functorialities ($$!$$-pullback and $$*$$-pushforward). Denote this category by $$\mathcal{D}(Y).$$

Taking $$!$$-pullback along the map $$Y\rightarrow \operatorname{Spec}k$$, we get a dualizing object $$\omega_Y$$ in the category of $$\mathcal{D}$$-modules. This is the unit for a natural monoidal $$\otimes^!$$ structure on $$\mathcal{D}(Y)$$, defined via the formula $$\mathcal{F}\otimes^!\mathcal{G}\cong\Delta^!(\mathcal{F}\boxtimes\mathcal{G}),$$ where $$\Delta:Y\rightarrow Y\times Y$$ is the diagonal map. A warning: For stacks, even one as simple as $$Y\cong \operatorname{pt}/\mathbb{G}_m$$, $$\omega_Y$$ often fails to be compact.

Regarding your first difficulty: You can define $$H^{\bullet}_{\operatorname{dR}}(Y)$$ to be $$\operatorname{Hom}^{\bullet}(\omega_Y,\omega_Y).$$ To understand why this is a reasonable definition, check that it agrees with the usual one for $$Y$$ a smooth variety.

Regarding your second difficulty: Because $$\omega_Y$$ is the unit for the $$\otimes^!$$ monoidal structure, there is a map $$\operatorname{Hom}^{\bullet}(\omega_Y,\omega_Y)\rightarrow\operatorname{Hom}^{\bullet}(\omega_Y\otimes^! M,\omega_Y\otimes^! M)\cong\operatorname{Hom}^{\bullet}(M,M),$$ which leads to your map $$(*).$$

Unfortunately, I don't understand your third difficulty.

Regarding your fourth difficulty: Look at the Drinfeld-Gaitsgory paper I linked above. One of their main theorems is that $$\mathcal{D}(Y)$$ is compactly generated by induced $$\mathcal{D}$$-modules. This implies that it suffices to check your claim for $$M$$ induced, which I believe is a straightforward computation, possibly using their Lemma 2.1.4. (Unfortunately I have to run right now - I'll try and write the details later.)

• regarding the third difficulty, I think I understand now. The augmentation Arinkin-Gaitsgory have in mind is the projection from $Hom(\omega_Y, \omega_Y)$ to its 0-th graded component. So for any positive $n$, $H^{2n}_{dR}(Y)$ lies in the augmentation ideal. Do you think this is correct? Anyway, thank you very much for helping me. Dec 13, 2018 at 6:26
• @geometer: Yes, that is correct.
– dhy
Dec 13, 2018 at 7:25