Formal moduli problems and formal derived stacks

Damien Calaque (1, 2) , Julien Grivaux (3, 4, 5)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IUF - Institut universitaire de France
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 I2M - Institut de Mathématiques de Marseille
5 IHES - Institut des Hautes Études Scientifiques

This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy, which gives a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasising the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of Gaitsgory and Rozenblyum.

Domaines: Géométrie algébrique [math.AG]; Topologie algébrique [math.AT]

Damien Calaque, Julien Grivaux. Formal moduli problems and formal derived stacks. Société Mathématique de France. Derived algebraic geometry, 55, , pp.85-145, 2021, Panoramas et Synthèses, 978-2-85629-938-8. ⟨hal-01721991⟩