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On graded \({\mathbb {E}}_{\infty }\)-rings and projective schemes in spectral algebraic geometry

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Abstract

We introduce graded \({\mathbb {E}}_{\infty }\)-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective \({\mathbb {N}}\)-graded \({\mathbb {E}}_{\infty }\)-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the \(\infty \)-category of almost perfect quasi-coherent sheaves over a spectral projective scheme \(\text { {Proj}}\,(A)\) associated to a connective \({\mathbb {N}}\)-graded \({\mathbb {E}}_{\infty }\)-ring A can be described in terms of \({{\mathbb {Z}}}\)-graded A-modules.

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Acknowledgements

The first author would like to express her thanks to Professor David Gepner for his valuable advice on grading. She also would like to express her thanks to Professor Gonçalo Tabuada for a lot of advice. She had an opportunity to talk with them during a program “K-theory and related fields” at Hausdorff Institute in Bonn. The authors would like to thank the anonymous referee for useful suggestion of including Remarks 5.24. 5.26, and 5.27.

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Authors and Affiliations

  1. General Education, National Institute of Technology, Oshima College, 1091-1, Komatsu, Suo-oshima town, Oshima district, 742-2193, Japan

    Mariko Ohara

  2. Department of Mathematics, Okayama University, Okayama, 700-8530, Japan

    Takeshi Torii

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  1. Mariko Ohara
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  2. Takeshi Torii
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Correspondence to Takeshi Torii.

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Communicated by Tyler Lawson.

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Takeshi Torii was partially supported by JSPS KAKENHI Grant Number JP17K05253.

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Ohara, M., Torii, T. On graded \({\mathbb {E}}_{\infty }\)-rings and projective schemes in spectral algebraic geometry. J. Homotopy Relat. Struct. 17, 105–144 (2022). https://doi.org/10.1007/s40062-021-00298-0

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  • DOI: https://doi.org/10.1007/s40062-021-00298-0

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