View source for Reduced scheme

A [[Scheme|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|group scheme]] over a field of characteristic 0 is reduced [[#References|[3]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> M. Artin, "Algebraic approximation of structures over complete local rings" ''Publ. Math. IHES'' , '''36''' (1969) pp. 23–58 {{MR|0268188}} {{ZBL|0181.48802}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique I. Le langage des schémas" ''Publ. Math. IHES'' , '''4''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0118.36206}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR>
</table>

====Comments====
That a group scheme over a field of characteristic 0 is reduced is called Cartier's theorem, cf. also [[#References|[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|algebraic geometry]] are the algebraic schemes which are reduced and which stay reduced after extending the base field.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Oort, "Algebraic group schemes in characteristic zero are reduced" ''Invent. Math.'' , '''2''' (1969) pp. 79–80 {{MR|0206005}} {{ZBL|0173.49002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

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[[Category:Algebraic geometry]]