Reduced scheme

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A scheme whose local ring at any point does not contain non-zero nilpotent elements. For any scheme there is a largest closed reduced subscheme , characterized by the relations

where is the ideal of all nilpotent elements of the ring . A group scheme over a field of characteristic 0 is reduced [3].


[1] M. Artin, "Algebraic approximation of structures over complete local rings" Publ. Math. IHES , 36 (1969) pp. 23–58
[2] A. Grothendieck, J. Dieudonné, "Eléments de géométrie algebrique I. Le langage des schémas" Publ. Math. IHES , 4 (1960)
[3] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)


That a group scheme over a field of characteristic is reduced is called Cartier's theorem, cf. also [a1].

It may happen that a scheme over a base scheme is reduced but that is not reduced (even with and reduced). The classical objects of study in algebraic geometry are the algebraic schemes which are reduced and which stay reduced after extending the base field.


[a1] F. Oort, "Algebraic group schemes in characteristic zero are reduced" Invent. Math. , 2 (1969) pp. 79–80
[a2] R. Hartshorne, "Algebraic geometry" , Springer (1977)
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Reduced scheme. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article