# Examples of non-hypercomplete sheaves on affine schemes

Let $$AA$$ be a commutative ring and let $$O\mathcal{O}$$ be a sheaf of $$E∞E_{\infty}$$-ring spectra on $$SpecA\mathrm{Spec} A$$ such that $$π0O=OSpecA\pi_0\mathcal{O} = \mathcal{O}_{\mathrm{Spec} A}$$. Lurie provides a criterion when $$(SpecA,O)(\mathrm{Spec} A, \mathcal{O})$$ coincides with $$SpecO(SpecA)\mathrm{Spec}\, \mathcal{O}(\mathrm{Spec} A)$$, namely if the homotopy groups $$πnO\pi_n\mathcal{O}$$ are quasi-coherent sheaves on $$SpecA\mathrm{Spec} A$$ and $$O\mathcal{O}$$ 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 $$O\mathcal{O}$$ that is not hypercomplete, but satisfies the other conditions?

By Spectral Algebraic Geometry, Corollary 1.1.3.6, this cannot happen if $$SpecA\mathrm{Spec} A$$ is a noetherian space of finite Krull dimension. So a counterexample has to be something big like $$A=Z[x1,x2,x3,…]A = \mathbb{Z}[x_1, x_2, x_3, \dots]$$.

## Answer

Attribution
Source : Link , Question Author : Lennart Meier , Answer Author : Community
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.820296585559845, "perplexity": 308.8077564196632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500671.13/warc/CC-MAIN-20230208024856-20230208054856-00566.warc.gz"}