Ind-coherent sheaf

Definition

For a suitable derived scheme or stack $Y$, the category $\operatorname{IndCoh}(Y)$ is the ind-completion of the category $\operatorname{Coh}(Y)$ of coherent complexes. An ind-coherent sheaf on $Y$ is an object of $\operatorname{IndCoh}(Y)$.

Unlike quasi-coherent sheaves, ind-coherent sheaves carry a functorial notion of singular support when $Y$ is quasi-smooth. On a smooth $Y$, the distinction is largely a renormalization; on a singular derived stack it records additional directions.