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About: Grothendieck's connectedness theorem     Goto   Sponge   NotDistinct   Permalink

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Type: anatomical structureyago:WikicatTheoremsInAlgebraicGeometryyago:Abstraction100002137yago:Communication100033020yago:Message106598915yago:Proposition106750804yago:Statement106722453yago:Theorem106752293 New Facet based on Instances of this Class

In mathematics, Grothendieck's connectedness theorem , 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. It is a local analogue of Bertini's theorem.