Grothendieck's connectedness theorem

In mathematics, Grothendieck's connectedness theorem,[1][2] 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.

[3] It is a local analogue of Bertini's theorem.

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