Grothendieck existence theorem

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

• Chow's lemma

References

• 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

• "Rigid-analytic geometry and the uniformization of abelian varieties". Snowbird Lectures in Algebraic Geometry. Contemporary Mathematics. Vol. 388. 2005. doi:10.1090/CONM/388. ISBN 9780821837191.

External links

• The Stacks Project authors. "30.24 Grothendieck's existence theorem, I".
• The Stacks Project authors. "75.42 Grothendieck's existence theorem".
• ILLUSIE, L. (7–18 July 2003). "Grothendieck's existence theorem in formal geometry (Advanced School in Basic Algebraic Geometry | (smr 1487))" (PDF). The Abdus Salam International Centre for Theoretical Physics. Trieste - Italy.

formal GAGA

• Mathew, Akhil. "Étale π₁ OF A SMOOTH CURVE" (PDF).
• "Formal Gaga on Artin Stacks" (PDF). 2005. S2CID 296658.