Moduli of objects in dg-categories

By Bertrand Toën, M. Vaquié. 2005

378 citations recorded by the source corpus.

The purpose of this work is to prove the existence of an algebraic moduli classifying objects in a given triangulated category. To any dg-category T (over some base ring k), we define a D -stack MT in the sense of [HAGII], classifying certain T op -dg-modules. When T is saturated, MT 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 MT 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 saturated dg-categories. 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d10ba5ef50e0706fb751bf1151b298b82c1f091

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Citations in the corpus

Higher Segal Spaces (2012, background): “Here is the main result of [TV07] (stated in a lesser generality, sufficient for our purposes).”
Derived categories of coherent sheaves (2006, background): “perverse point sheaves. The trick we used in dimension three will not work in higher dimensions. Instead one must construct the moduli space directly. Recent results of Inaba [22] and Toen and Vaquie [49] construct spaces parameterising very general classes of objects of derived categories. For appl”
Moduli of complexes on a proper morphism (2005, background): “In fact, Toën and Vaquié have recently posted a preprint [20] which (among other things) carries out this derived program for perfect complexes on a proper smooth scheme over a field.”
Cluster varieties from Legendrian knots (2015, methodology): “ objects in Sh (X;˙). It is a locally geometric derived stack. We write RM n( ; ;˙) for the substack parameterizing sheaves of microlocal rank n. Proof. The existence of these spaces is guaranteed by [TVa], which constructs derived moduli stacks of pseudoperfect modules of ﬁnite-type dg categories (”
Hirzebruch–Riemann–Roch‐type formula for DG algebras (2007, background): “ototype is the pull-back functor associated with a morphism of schemes). The real deﬁnition is more subtle and we won’t discuss it here referring the reader to more thorough treatments of the subject [20, 35, 62, 63, 65, 66]. Here is an interesting question: Is it possible to tell whether a noncommu”