Grothendieck's connectedness theorem
In mathematics, Grothendieck's connectedness theorem,{{harvnb|Grothendieck|Raynaud|2005|loc=XIII.2.1}}{{harvnb|Lazarsfeld|2004|loc=theorem 3.3.16}} states that if A is a complete Noetherian 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.{{harvnb|Grothendieck|Raynaud|2005|loc=XIII.2.1}}
It is a local analogue of Bertini's theorem.
See also
References
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Bibliography
- {{citation
| last1 = Grothendieck
| first1 = Alexander
| authorlink1 = Alexander Grothendieck
| first2 = Michel |last2=Raynaud | authorlink2=Michel 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
| 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 | authorlink=Robert Lazarsfeld
|year=2004
|publisher=Springer
|isbn =3-540-22533-1
}}
Category:Theorems in algebraic geometry
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