In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
of dimension less than ''k'' is connected.
It is a local analogue of
Bertini's theorem In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broades ...
.
See also
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Zariski connectedness theorem In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varie ...
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Fulton–Hansen connectedness theorem In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimensi ...
References
Bibliography
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Theorems in algebraic geometry
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