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Grothendieck existence theorem

From Wikipedia, the free encyclopedia

In mathematics, the Grothendieck existence theorem, introduced by Grothendieck (1961, section 5), gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme S to schemes over S.

The theorem can be viewed as an instance of (Grothendieck's) formal GAGA.

See also

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References

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  • Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11: 5–167. doi:10.1007/bf02684274. MR 0217085.
  • Illusie, Luc (2005), "Grothendieck's existence theorem in formal geometry with a letter from Jean-Pierre Serre", Fundamental Algebraic Geometry: Grothendieck's FGA Explained, Mathematical surveys and monographs, vol. 123, American Mathematical Society, pp. 179–234, doi:10.1090/SURV/123, ISBN 9780821842454.
  • Kosarew, Siegmund (1987), Grothendieck's existence theorem in analytic geometry and related results, Regensburger mathematische Schriften, vol. 14, Fakultät für Mathematik der Universität Regensburg, ISBN 9783882461206.
  • Kosarew, Siegmund (1991). "Grothendieck's existence theorem in analytic geometry and related results". Transactions of the American Mathematical Society. 328 (1): 259–306. doi:10.1090/S0002-9947-1991-1014252-X. JSTOR 2001883.
  • Lurie, Jacob (2011), Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem (PDF).
  • Olsson, Martin C. (2005). "On proper coverings of Artin stacks". Advances in Mathematics. 198 (1): 93–106. doi:10.1016/j.aim.2004.08.017.

formal GAGA

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formal GAGA

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