nLab
Artin-Lurie representability theorem

Contents

Context

Higher geometry

Higher algebra

Contents

Idea

The generalization of the Artin representability theorem from algebraic geometry to spectral algebraic geometry.

Statement

Write CAlg cnCAlg^{cn} for the (∞,1)-category of connective E-∞ rings, and ∞Grpd for that of ∞-groupoids.

Theorem

Necessary and sufficient conditions for an (∞,1)-presheaf

:CAlg cnGrpd \mathcal{F} \;\colon\; CAlg^{cn}\longrightarrow \infty Grpd

over some SpecRSpec R, on the opposite (∞,1)-category of connective E-∞ rings to be represented by a spectral Deligne-Mumford n-stack locally of almost finite presentation over RR:

  1. For every discrete commutative ring, (A)\mathcal{F}(A) is n-truncated.

  2. \mathcal{F} is an ∞-stack for the étale (∞,1)-site.

  3. \mathcal{F} is nilcomplete, integrable, and an infinitesimally cohesive (∞,1)-presheaf on E-∞ rings.

  4. \mathcal{F} admits a connective cotangent complex.

  5. the natural transformation to SpecRSpec R is locally almost of finite presentation.

(Lurie Rep, theorem 2)

Remark

The condition that \mathcal{F} be infinitesimally cohesive implies that the Lie differentiation around any point, given by restriction to local Artin rings (formal duals of infinitesimally thickened points), is a formal moduli problem, hence equivalently an L-∞ algebra.

Applications

The motivating example of the Artin-Lurie representability theorem is the re-proof of the Goerss-Hopkins-Miller theorem. See there for more.

References

Last revised on July 29, 2016 at 14:20:03. See the history of this page for a list of all contributions to it.