nLab Artin-Lurie representability theorem

Idea

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

Statement

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

Theorem

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

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

over some $Spec R$, 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 $R$:

1. For every discrete commutative ring, $\mathcal{F}(A)$ 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 $Spec R$ 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.

Related concepts

• Grothendieck existence

• Artin representability theorem

References

• Jacob Lurie, Representability theorems