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High Quality Content by WIKIPEDIA articles! 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.