Syst\`emes de points dans les dg-cat\'egories satur\'ees

TL;DR

This paper introduces the concept of 'system of points' in saturated dg-categories and demonstrates how such systems can be used to construct algebraic spaces that realize these categories, linking dg-categories to algebraic geometry.

Contribution

It defines 'system of points' in saturated dg-categories and shows how to construct algebraic spaces from them, establishing criteria for equivalence with dg-categories.

Findings

Construction of algebraic space M from a system of points

Equivalence of dg-category T with sheaves on M when M is proper

Study of t-structures on algebraic families of objects in T

Abstract

In this work we consider the question of realizing triangulated dg-categories by derived categories of algebraic varieties. For this, we introduce the notion of "system of points" in saturated dg-categories. We show that given such a system on a dg-category T, we can construct an algebraic space M, of finite type, smooth and separated, together with a dg-functor from T to a certain twisted dg-category of sheaves on M. We prove that this functor is furthermore an equivalence if and only if M is proper. All along this work we study t-strutcures on algebraic families of objects in T, which might be of independant interest.