Moduli problems for ring spectra. (English) Zbl 1244.55007

Lurie, Jacob

Bhatia, Rajendra (ed.) et al., Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency (ISBN 978-981-4324-32-8/hbk; 978-81-85931-08-3/hbk; 978-981-4324-30-4/set; 978-981-4324-35-9/ebook). 1099-1125 (2011).

Let \(X\) be a moduli space for some moduli functor defined in characteristic zero. It is well known that a formal neighbourhood of any point of \(X\) is controlled by some differential graded Lie algebra. There are many examples of this principle (see, e.g., [S. Kosarew, “Local moduli spaces and Kuranishi maps”, Manuscr. Math. 110, No. 2, 237–249 (2003; Zbl 1102.32300)] for some geometric examples).
The main aim of the paper under review is to give a precise meaning to the above principle in terms of higher category theory (see [J. Lurie, Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)]). The author traces the above principle to the paper of D. Quillen in [“Rational homotopy theory”, Ann. Math. (2) 90, 205–295 (1969; Zbl 0191.53702)] and some unpublished work of Deligne, Drinfeld and Feigin.
The problem of formalizing the above principle is considered in several cases. First, the author considers classical moduli problems: global (functor from the category of commutative rings) and formal (corresponding to local Artin algebras). However, the main part of the paper is devoted to derived moduli problems: global (\(E_{\infty}\)-ring spectra) and formal (\(E_n\)-rings). The main theorems are stated as equivalences of categories between the category of moduli problems over a field of characteristic zero and \(\infty\)-categories of differential graded Lie algebras (or their analogues).
For the entire collection see [Zbl 1220.00032].

Reviewer: Adrian Langer (Warszawa)

MSC:

55P43	Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
14D15	Formal methods and deformations in algebraic geometry
18D05	Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)

Keywords:

moduli space; ring spectra; deformation theory; derived algebraic geometry

Citations:

Zbl 1102.32300; Zbl 1175.18001; Zbl 0191.53702