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Formalising the Proj Construction in Lean

Author Jujian Zhang



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Jujian Zhang
  • Department of Mathematics, Imperial College London, UK

Acknowledgements

I want to thank Eric Wieser for his contribution and suggestion in formalising homogeneous ideals and homogeneous localisation; Andrew Yang and Junyan Xu for their review and comments on my code; Kevin Buzzard for suggesting this project and all the contributors to mathlib for otherwise this project would not have been possible.

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Jujian Zhang. Formalising the Proj Construction in Lean. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 35:1-35:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITP.2023.35

Abstract

Many objects of interest in mathematics can be studied both analytically and algebraically, while at the same time, it is known that analytic geometry and algebraic geometry generally do not behave the same. However, the famous GAGA theorem asserts that for projective varieties, analytic and algebraic geometries are closely related; the proof of Fermat’s last theorem, for example, uses this technique to transport between the two worlds [Serre, 1955]. A crucial step of proving GAGA is to calculate cohomology of projective space [Neeman, 2007; Godement, 1958], thus I formalise the Proj construction in the Lean theorem prover for any ℕ-graded R-algebra A and construct projective n-space as Proj A[X₀,… , X_n]. This is the first family of non-affine schemes formalised in any theorem prover.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Mathematics of computing → Topology
Keywords
  • Lean
  • formalisation
  • algebraic geometry
  • scheme
  • Proj construction
  • projective geometry

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References

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