Reduced scheme

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].

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

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.