Moduli of objects in dgcategories
Abstract
To any dgcategory $T$ (over some base ring $k$), we define a $D^{}$stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$dgmodules. 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 substacks). As a consequence we prove the algebraicity of the group of autoequivalences of a saturated dgcategory. 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2005
 DOI:
 10.48550/arXiv.math/0503269
 arXiv:
 arXiv:math/0503269
 Bibcode:
 2005math......3269T
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 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