# Examples of non-hypercomplete sheaves on affine schemes

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 1.6.1.1). 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 1.1.3.6, this cannot happen if $$\mathrm{Spec} A$$ is a noetherian space of finite Krull dimension. So a counterexample has to be something big like $$A = \mathbb{Z}[x_1, x_2, x_3, \dots]$$.