Grothendieck's connectedness theorem

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In mathematics, Grothendieck's connectedness theorem (Grothendieck 2005, XIII.2.1,Lazarsfeld 2004, 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.

[edit] References

• Grothendieck, Alexandre & Raynaud, Michèle (2005), Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2) (Updated edition ed.), Documents Mathématiques 4, Société Mathématique de France, pp. x+208, ISBN 2-85629-169-4 
• Lazarsfeld, Robert (2004), Positivity in Algebraic Geometry, Springer, ISBN 3540225331