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DOI: 10.4230/LIPIcs.ITP.2024.38

The Functor of Points Approach to Schemes in Cubical Agda

Authors: Max Zeuner and Matthias Hutzler

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

We present a formalization of quasi-compact and quasi-separated schemes (qcqs-schemes) in the Cubical Agda proof assistant. We follow Grothendieck’s functor of points approach, which defines schemes, the quintessential notion of modern algebraic geometry, as certain well-behaved functors from commutative rings to sets. This approach is often regarded as conceptually simpler than the standard approach of defining schemes as locally ringed spaces, but to our knowledge it has not yet been adopted in formalizations of algebraic geometry. We build upon a previous formalization of the so-called Zariski lattice associated to a commutative ring in order to define the notion of compact open subfunctor. This allows for a concise definition of qcqs-schemes, streamlining the usual presentation as e.g. given in the standard textbook of Demazure and Gabriel. It also lets us obtain a fully constructive proof that compact open subfunctors of affine schemes are qcqs-schemes.

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Max Zeuner and Matthias Hutzler. The Functor of Points Approach to Schemes in Cubical Agda. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{zeuner_et_al:LIPIcs.ITP.2024.38,
  author =	{Zeuner, Max and Hutzler, Matthias},
  title =	{{The Functor of Points Approach to Schemes in Cubical Agda}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{38:1--38:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.38},
  URN =		{urn:nbn:de:0030-drops-207667},
  doi =		{10.4230/LIPIcs.ITP.2024.38},
  annote =	{Keywords: Schemes, Algebraic Geometry, Category Theory, Cubical Agda, Homotopy Type Theory and Univalent Foundations, Constructive Mathematics}
}

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DOI: 10.4230/LIPIcs.TYPES.2022.14

A Univalent Formalization of Constructive Affine Schemes

Authors: Max Zeuner and Anders Mörtberg

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

Abstract

We present a formalization of constructive affine schemes in the Cubical Agda proof assistant. This development is not only fully constructive and predicative, it also makes crucial use of univalence. By now schemes have been formalized in various proof assistants. However, most existing formalizations follow the inherently non-constructive approach of Hartshorne’s classic "Algebraic Geometry" textbook, for which the construction of the so-called structure sheaf is rather straightforwardly formalizable and works the same with or without univalence. We follow an alternative approach that uses a point-free description of the constructive counterpart of the Zariski spectrum called the Zariski lattice and proceeds by defining the structure sheaf on formal basic opens and then lift it to the whole lattice. This general strategy is used in a plethora of textbooks, but formalizing it has proved tricky. The main result of this paper is that with the help of the univalence principle we can make this "lift from basis" strategy formal and obtain a fully formalized account of constructive affine schemes.

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Max Zeuner and Anders Mörtberg. A Univalent Formalization of Constructive Affine Schemes. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{zeuner_et_al:LIPIcs.TYPES.2022.14,
  author =	{Zeuner, Max and M\"{o}rtberg, Anders},
  title =	{{A Univalent Formalization of Constructive Affine Schemes}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{14:1--14:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.14},
  URN =		{urn:nbn:de:0030-drops-184574},
  doi =		{10.4230/LIPIcs.TYPES.2022.14},
  annote =	{Keywords: Affine Schemes, Homotopy Type Theory and Univalent Foundations, Cubical Agda, Constructive Mathematics}
}

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The Functor of Points Approach to Schemes in Cubical Agda

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A Univalent Formalization of Constructive Affine Schemes

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