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 is -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.
