Cambridge Journal of Mathematics

Volume 3 (2015)

Number 1–2

Compact generation of the category of $\mathrm{D}$-modules on the stack of $G$-bundles on a curve

Pages: 19 – 125



V. Drinfeld (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

D. Gaitsgory (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


Let $G$ be a reductive group. Let $\textrm{Bun}_G$ denote the stack of $G$-bundles on a smooth complete curve over a field of characteristic $0$, and let $\textrm{D-mod} (\textrm{Bun}_G)$ denote the $\textrm{DG}$ category of $\textrm{D}$-modules on $\textrm{Bun}_G$. The main goal of the paper is to show that $\textrm{D-mod} (\textrm{Bun}_G)$ is compactly generated (this is not automatic because $\textrm{Bun}_G$ is not quasi-compact). The proof is based on the following observation: $\textrm{Bun}_G$ can be written as a union of quasi-compact open substacks $j : U \hookrightarrow \textrm{Bun}_G$, which are “co-truncative”, i.e., the functor $j_!$ is defined on the entire category $\textrm{D-mod}(U)$.

Published 5 June 2015