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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 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 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).

Mathematics Encyclopedia