Moduli of objects in dg-categories
Abstract
Description
To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the triangulated category $[T]$ associated to $T$. The main result of this work states that under certain finiteness conditions on $T$ (e.g. if it is saturated) the $D^{-}$-stack $\mathcal{M}_{T}$ is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of a saturated dg-category. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as well as complexes of representations of a finite quiver.
64 pages. Minor corrections. Section 3.4 including some corollaries has been added. Sections 1 and 2.5 added, as well as some remarks. To appear in Annales de l'ENS
64 pages. Minor corrections. Section 3.4 including some corollaries has been added. Sections 1 and 2.5 added, as well as some remarks. To appear in Annales de l'ENS