×

Formal moduli problems and formal derived stacks. (English. French summary) Zbl 1472.14004

Ben-Zvi, David et al., Derived algebraic geometry. Paris: Société Mathématique de France (SMF). Panor. Synth. 55, 85-145 (2021).
Summary: 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 J. P. Pridham [Adv. Math. 224, No. 3, 772–826 (2010; Zbl 1195.14012)] and J. Lurie [“Derived algebraic geometry X: formal moduli problems”, (2011)]) which gives a precise mathematical formulation for Drinfeld’s derived deformation theory philosophy. This theorem provides 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, emphasizing 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 J. Nuiten [Adv. Math. 354, Article ID 106750, 63 p. (2019; Zbl 1433.14007)], 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 D. Gaitsgory and N. Rozenblyum [A study in derived algebraic geometry. Volume I: Correspondences and duality. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1408.14001)].
For the entire collection see [Zbl 1471.14005].

MSC:

14A30 Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.)
13D10 Deformations and infinitesimal methods in commutative ring theory
14A20 Generalizations (algebraic spaces, stacks)
14B10 Infinitesimal methods in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
18N40 Homotopical algebra, Quillen model categories, derivators
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
PDFBibTeX XMLCite
Full Text: arXiv