Contents

Idea

A scheme is reduced if it has no “purely infinitesimal directions”.

Definition

An algebraic scheme is reduced if all stalks of the structure sheaf are local rings without nonzero nilpotent elements.

This means that it is a reduced object in the sense of synthetic differential geometry.

Equivalently a scheme $X$ is reduced iff it can be covered by a family $\{U_\alpha\}_\alpha$ of open sets such that $\mathcal{O}_X(U)$ has no nonzero nilpotent elements. Equivalently, every open $U\subset X$ has the property that $\mathcal{O}_X(U)$ has no nonzero nilpotent elements.

Reducedness can also be characterized by the internal language of the sheaf topos $\mathrm{Sh}(X)$: $X$ is reduced iff $\mathcal{O}_X$ is a reduced ring in $\mathrm{Sh}(X)$ iff $\mathcal{O}_X$ is a residue field in $\mathrm{Sh}(X)$ (i.e. every non-unit is zero). This can be used to give an internal proof of the fact that $\mathcal{O}_X$-modules of finite type over a reduced scheme are locally free iff their rank is constant (by internalizing the constructive proof of the linear algebra fact that finitely generated modules over residue fields are free, provided the minimal number of elements needed to generate the module exists as a natural number).

Examples

For instance the dual $Spec k[\epsilon](\epsilon^2)$ of the ring of dual numbers is not reduced, since it is the point with an infinitesimal extension added to it. Its reduction is the point itself. Generally, formal schemes are not reduced.

Related concepts

• formal scheme

• reduced object

• de Rham space

• formally smooth scheme