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Grothendieck's connectedness theorem
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.
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Grothendieck's connectedness theorem
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.