Remark 12.8.8 in Arinkin--Gaitsgory

Question

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?

Answer 1

The answers to your questions can be found in this article: https://arxiv.org/abs/1108.5351. I highly recommend reading it before trying to understand Arinkin-Gaitsgory.

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.)