Let $A$ be a commutative ring and let $\mathcal{O}$ be a sheaf of $E_{\infty}$-ring spectra on $\mathrm{Spec} A$ such that $\pi_0\mathcal{O} = \mathcal{O}_{\mathrm{Spec} A}$. Lurie provides a criterion when $(\mathrm{Spec} A, \mathcal{O})$ coincides with $\mathrm{Spec}\, \mathcal{O}(\mathrm{Spec} A)$, namely if the homotopy groups $\pi_n\mathcal{O}$ are quasi-coherent sheaves on $\mathrm{Spec} A$ and $\mathcal{O}$ is hypercomplete (Spectral Algebraic Geometry, Proposition To get a better understanding why this last condition is really necessary, I would like to know the answer to the following question:

Is there an example of sheaf $\mathcal{O}$ that is not hypercomplete, but satisfies the other conditions?

By Spectral Algebraic Geometry, Corollary, this cannot happen if $\mathrm{Spec} A$ is a noetherian space of finite Krull dimension. So a counterexample has to be something big like $A = \mathbb{Z}[x_1, x_2, x_3, \dots]$.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy