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Moduli of objects in dg-categories. (English) Zbl 1140.18005

The paper begins a research about moduli spaces (or stacks) of objects in a triangulated category of geometric or algebraic origin. It proves the existence of an algebraic moduli classifying objects in a given triangulated category satisfying some finiteness conditions and having a dg-enhancement.
To any dg-category \({\mathcal T}\) (i.e., a set of objects together with complexes of morphisms between two objects, the composition preserving the linear and differential structures) a stack \(M\) is associated. This one classifies compact objects in the triangulated category associated to \({\mathcal T}\) when \({\mathcal T}\) is saturated. Under some finiteness conditions on \({\mathcal T}\), \(M\) is locally geometric.
From that the algebraicity of the group of auto-equivalences of saturated dg-categories and the existence of reasonable moduli for perfect complexes on a smooth and proper scheme are proved.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D20 Enriched categories (over closed or monoidal categories)
18E30 Derived categories, triangulated categories (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18D99 Categorical structures
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