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Grothendieck existence theorem

In mathematics, the Grothendieck existence theorem, introduced by {{harvs|txt|last=Grothendieck|authorlink=Alexander Grothendieck|year=1961|loc=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

References

{{refbegin}}

  • {{EGA|book=3-1| pages = 5–167}}
  • {{citation|first=Luc|last=Illusie|contribution=Grothendieck's existence theorem in formal geometry with a letter from Jean-Pierre Serre|pages=179–234|title=Fundamental Algebraic Geometry: Grothendieck's FGA Explained|volume=123|series=Mathematical surveys and monographs|publisher=American Mathematical Society|year=2005|isbn=9780821842454|doi=10.1090/SURV/123|doi-access=free}}.
  • {{citation|title=Grothendieck's existence theorem in analytic geometry and related results|volume=14|series=Regensburger mathematische Schriften|first=Siegmund|last=Kosarew|publisher=Fakultät für Mathematik der Universität Regensburg|year=1987|isbn=9783882461206}}.
  • {{cite journal |doi=10.1090/S0002-9947-1991-1014252-X|title=Grothendieck's existence theorem in analytic geometry and related results |year=1991 |last1=Kosarew |first1=Siegmund |journal=Transactions of the American Mathematical Society |volume=328 |issue=1 |pages=259–306 |doi-access=free|jstor=2001883 }}
  • {{citation|first=Jacob|last=Lurie|authorlink=Jacob Lurie|title=Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem|url=http://www.math.harvard.edu/~lurie/papers/DAG-XII.pdf|year=2011}}.
  • {{cite journal |doi=10.1016/j.aim.2004.08.017 |doi-access=free |title=On proper coverings of Artin stacks |year=2005 |last1=Olsson |first1=Martin C. |journal=Advances in Mathematics |volume=198 |issue=1 |pages=93–106 }}

{{refend}}

= formal GAGA =

  • {{cite book |doi=10.1090/CONM/388|title=Snowbird Lectures in Algebraic Geometry |series=Contemporary Mathematics |year=2005 |volume=388 |isbn=9780821837191|chapter=Rigid-analytic geometry and the uniformization of abelian varieties |chapter-url={{Google books|t5wbCAAAQBAJ|page=158|plainurl=yes}}}}

External links

{{refbegin}}

  • {{cite web |url=https://stacks.math.columbia.edu/tag/087V |author=The Stacks Project authors |title=30.24 Grothendieck's existence theorem, I}}
  • {{cite web |url=https://stacks.math.columbia.edu/tag/089N |author=The Stacks Project authors |title=75.42 Grothendieck's existence theorem}}
  • {{cite web

| url = https://indico.ictp.it/event/a0255/session/14/contribution/9/material/0/0.pdf

| title =Grothendieck's existence theorem in formal geometry (Advanced School in Basic Algebraic Geometry | (smr 1487))

| first =L.

| last = ILLUSIE

| date =7–18 July 2003

| website = The Abdus Salam International Centre for Theoretical Physics

| series =

| publisher =

| agency =

| location = Trieste - Italy

}}

{{refend}}

=formal GAGA=

  • {{cite web |last1=Mathew |first1=Akhil |title=Étale π1 OF A SMOOTH CURVE|url=http://math.uchicago.edu/~amathew/pi1.pdf}}
  • {{cite web |url=http://math.stanford.edu/~conrad/papers/formalgaga.pdf|s2cid=296658 |title=Formal Gaga on Artin Stacks |year=2005 }}

Category:Theorems in algebraic geometry

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