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 formal GAGA.

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, 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.
• Lurie, Jacob (2011), Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem (PDF).