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The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological spaces is representable by a spectral Deligne--Mumford stack iff

  • There exists a functor $Y:CAlg^{cn}\rightarrow S$ representable by Deligne--Mumford stack such that the restrictions to the full subcategory $CAlg^{\bullet}\subset Calg^{cn}$ spanned by discrete $E_{\infty}$-algebras are equivalent: $X|_{CAlg^{\bullet}}\simeq Y|_{CAlg^{\bullet}}$
  • $X$ admits a cotangent complex
  • $X$ is nilcomplete
  • $X$ is infinitesimally cohesive.

Some applications of this theorem include

  1. A proof of existence of the moduli stack of oriented elliptic curves (and thus of topological modular forms)
  2. Representability of relative Picard functor for families of spectral algebraic spaces
  3. Representability of Weil restriction functor
  4. Relationship between dilatations and formal modifications of spectral DM stacks

My question is: what other applications of spectral Artin representability in homotopy theory/algebraic geometry have been found?

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    $\begingroup$ A simpler application than (1) is the existence of a sheaf of E_oo-rings on the moduli stack of nonsingular quadratic curves, whose global sections is the 8-periodic E_oo-ring KO (and, of course, there is a more complicated application, to PEL Shimura varieties à la Behrens-Lawson). I think (but I have not seen a proof) another, more algebraic, application is the representability of the Hilbert scheme functor by an algebraic space. (After all, Artin used his criterion to prove the representability of the classical Hilbert functor.) $\endgroup$ – skd Jun 19 '18 at 4:32

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