In
mathematics, Grothendieck's connectedness theorem , states that if ''A'' is a complete
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
whose spectrum is ''k''-connected and ''f'' is in the
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
, then Spec(''A''/''fA'') is (''k'' − 1)-connected. Here a
Noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thu ...
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 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 broad ...
.
See also
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Zariski connectedness theorem
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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 dim ...
References
Bibliography
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Theorems in algebraic geometry
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