Grothendieck's connectedness theorem
- Grothendieck's connectedness theorem
In mathematics, Grothendieck's connectedness theorem (harvnb|Grothendieck|2005|loc=XIII.2.1,harvnb|Lazarsfeld|2004|loc=theorem 3.3.16) states that if "A" is a complete local ring whose spectrum is "k"-connected and "f" is in the maximal ideal, then Spec("A"/"fA") is ("k" − 1)-connected. Here a Noetherian scheme is called "k"-connected if its dimension is greater than "k" and the complement of every closed subset of dimension less than "k" is connected. Grothendieck XIII.2.1
It is a local analogue of Bertini's theorem.
References
*citation
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
first2 = Michèle |last2=Raynaud
title = Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2)
series=Documents Mathématiques 4
origyear = 1968
edition = Updated edition
year = 2005
publisher = Société Mathématique de France
language = French
pages = x+208
ISBN =2-85629-169-4
*citation|title=Positivity in Algebraic Geometry
first=Robert |last=Lazarsfeld
year=2004
publisher=Springer
ISBN =3540225331
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