Foundations of Algebraic Geometry

Short description: 1946 book by André Weil

For the book by Grothendieck, see Fondements de la Géometrie Algébrique.

Foundations of Algebraic Geometry Author| André Weil

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Subject| Mathematics
Genre| Non-fiction

Publication date

| 1946

Foundations of Algebraic Geometry is a book by André Weil (1946, 1962) that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two subvarieties.

Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over finite fields.

Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.

In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by Grothendieck's development of schemes.

## See also

* Weil cohomology theory

## References

* Raynaud, Michel (1999), "André Weil and the foundations of algebraic geometry", Notices of the American Mathematical Society 46 (8): 864–867, https://www.ams.org/notices/199908/fea-raynaud.pdf
* van der Waerden, Bartel Leendert (1971), "The foundation of algebraic geometry from Severi to André Weil", Archive for History of Exact Sciences 7 (3): 171–180, doi:10.1007/BF00357215
* Weil, André (1947), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29, Providence, R.I.: American Mathematical Society, https://books.google.com/books?id=ML7u26rkEkIC ISBN 9780821874622
* Weil, André (1962), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29 (2 ed.), Providence, R.I.: American Mathematical Society, https://books.google.com/books?id=ML7u26rkEkIC ISBN 978-0-8218-1029-3
* Zariski, Oscar (1948), "Book Review: Foundations of algebraic geometry", Bulletin of the American Mathematical Society 54 (7): 671–675, doi:10.1090/S0002-9904-1948-09040-1, Bibcode: 1948Sci...107...75W

## External links

* Extracts from the preface of Foundations of Algebraic Geometry

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