Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.

Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category of $\mathcal{D}_X$-modules (unbounded complexes of $\mathcal{D}_X$-modules localized at quasi-isomorphisms). Assuming this is the correct category, i've seen in many places the statement that this category is compactly generated. I haven't been able to find a proof of this however. So...

Is $\mathcal{C}$ compactly generated? If so, who are the compact objects in $\mathcal{C}$? Where can I find a good reference for this?


1 Answer 1


One place where this is discussed is

Drinfeld, Vladimir; Gaitsgory, Dennis, On some finiteness questions for algebraic stacks, Geom. Funct. Anal. 23, No. 1, 149-294 (2013). ZBL1272.14005.


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