Reduced scheme

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A scheme whose local ring at any point does not contain non-zero nilpotent elements. For any scheme $\left({ X,\mathcal{O}_X }\right)$ there is a largest closed reduced subscheme $\left({ X_{\mathrm{red}},\mathcal{O}_{X_{\mathrm{red}}} }\right)$, characterized by the relations $$ \mathcal{O}_{X_{\mathrm{red}},x} = \mathcal{O}_{X,x}/r_x $$ where $r_x$ is the ideal of all nilpotent elements of the ring $\mathcal{O}_{X,x}$. 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 MR0268188 Zbl 0181.48802
[2] A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique I. Le langage des schémas" Publ. Math. IHES , 4 (1960) MR0217083 MR0163908 Zbl 0118.36206
[3] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701


That a group scheme over a field of characteristic 0 is reduced is called Cartier's theorem, cf. also [a1].

It may happen that a scheme $X \rightarrow S$ over a base scheme $S$ is reduced but that $X \times_S T$ is not reduced (even with $S$ and $T$ 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 MR0206005 Zbl 0173.49002
[a2] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
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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