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Grothendieck’s Unifying Vision of Geometry

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Foundations of Mathematics and Physics One Century After Hilbert

Abstract

Grothendieck’s “vast unifying vision” provided new working and conceptual foundations for geometry, and even led him to logical foundations. While many pictures here illustrate the geometry, Grothendieck himself favored apt words and commutative diagrams over pictures and did not think of geometry pictorially.

If one thing has fascinated me in mathematics since childhood, it is this power to identify in words, and perfectly express, the essence of certain mathematical things which on first approach present themselves as elusive or mysterious beyond words.      

— A. Grothendieck Récoltes et Semailles p. 14

These notes attempted to show something that was still very controversial at that time: that schemes really were the most natural language for algebraic geometry and that you did not need to sacrifice geometric intuition when you spoke “scheme.”      

— David Mumford The Red Book (1988, p. VIII)

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Notes

  1. 1.

    While this holds of many geometers, [14] puts toposes among Grothendieck’s best ideas. Only Deligne emphasizes what Grothendieck also knew: you can think with topos intuitions while officially using only small sites in proofs. Cf. [40, p. 263].

  2. 2.

    This information from Pierre Cartier, discussion February 2015.

  3. 3.

    The following is based on de Rham cohomology. Local contour maps represent closed 1-forms, while actual contour maps represent exact 1-forms.

  4. 4.

    The geometric point is that, like level lines on a topographical map, contour lines never cross or merge with each other; and never start or stop except at the edges of the chart.

  5. 5.

    Periods are often defined as line integrals, \(i_{\gamma }=\oint \gamma \). This agrees with our definition when the line of the integral goes once around the annulus and \(\alpha \) has \(\oint \alpha =1\).

  6. 6.

    [32, pp. 252ff] has a gentle introduction, for more see Tennison [49].

  7. 7.

    He plays with the fact that the French soleil means both sun and sunflower.

  8. 8.

    Marble sculptures are made by hammer and chisel.

  9. 9.

    MacLane [33, Ch. VII] is a masterful account of Abelian categories. For homological algebra through derived functors and spectral sequences see Lang [27, Ch. XX].

  10. 10.

    In e-mails of June and July 2004 Serre argues that Grothendieck mis-remembered the events. Certainly Grothendieck was wrong to say the 1948–49 seminar discussed spectral sequences (ReS [21, p. 19]) as Serre did not know of them then. Serre suggests Grothendieck did not often attend the seminar, whose contents would not have interested him at the time: “Grothendieck spent the year 48–49 in Paris (straight from his “licence" at Montpellier) and he stated in print several times that he attended the Cartan seminar of that year. I don’t doubt this, but I have no memory of him then.... He probably got the written texts; I am not even sure he looked at them before 53 or even 54. They only influenced him in retrospect—just as a book one reads and finds interesting” (26 June 2004).

  11. 11.

    Grothendieck [18, p. 141], Tennison [49, p. 128], Hartshorne [25, p. 206], et alia.

  12. 12.

    Grothendieck cites [45] which loosely presages unramified covers. The truly germane [46] was not yet in print.

  13. 13.

    Sometimes Grothendieck distinguishes a petit topos of sheaves on a generalized space, from a gros topos which is a category of generalized spaces. Lawvere [28, 29] has developed this idea further. This distinction has nothing to do with set theoretic size. Gros and petit Grothendieck toposes are both proper classes in naive set theory.

  14. 14.

    See the work in progress [44].

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  1. Department of Philosophy, Case Western Reserve University, Cleveland, OH, USA

    Colin McLarty

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  1. Nice and Sophia Antipolis University, Nice, France

    Joseph Kouneiher

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McLarty, C. (2018). Grothendieck’s Unifying Vision of Geometry. In: Kouneiher, J. (eds) Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham. https://doi.org/10.1007/978-3-319-64813-2_4

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