Examples of non-hypercomplete sheaves on affine schemes

Let A be a commutative ring and let O be a sheaf of E-ring spectra on SpecA such that π0O=OSpecA. Lurie provides a criterion when (SpecA,O) coincides with SpecO(SpecA), namely if the homotopy groups πnO are quasi-coherent sheaves on SpecA and 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 O that is not hypercomplete, but satisfies the other conditions?

By Spectral Algebraic Geometry, Corollary 1.1.3.6, this cannot happen if SpecA is a noetherian space of finite Krull dimension. So a counterexample has to be something big like A=Z[x1,x2,x3,].

Answer

Attribution
Source : Link , Question Author : Lennart Meier , Answer Author : Community

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