Reduced scheme

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].

References

[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 algebrique 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

Comments

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.

References

[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